The last post, “Knottedness,” constructed Alexander’s Horned Sphere and proved, or sketched the proof, that
the horned sphere itself is topologically a sphere, and in particular is simply connected, meaning
it’s path-connected: there’s a path from every point to every other point;
loops contract to points—are null-homotopic;
the space outside of the horned sphere is not simply connected.
This is paradoxical. You would think that if any loop sitting on the horned sphere can be drawn to a point, and any loop outside the horned sphere can be made to sit on the sphere and then drawn to a point, then we ought to be able to get the loop really close to the horned sphere, and let it contract it to a point, just the way it could, if it were actually on the horned sphere.
You would think that, but you would be wrong. Mathematics is pathological that way. Some of us may note the pathologies, then pass to areas where intuition works better. Other persons may become obsessed with the pathologies.
If not by just reading this blog, you might find such persons by doing an image search on the Alexander Horned Sphere. I made such a search, but could not find images like the ones I am going to give here. I am trying to clarify why the horned sphere has the properties it does.
The idea of the proof of those properties is that the horned sphere is the intersection of a descending chain,
X_{0} ⊃ X_{1} ⊃ X_{2} ⊃ …,
of sets X_{n} that are not simply connected, but are compact.
I did not try to define compactness in the last post. Perhaps one does not normally learn about this concept until one has spent some time with calculus. Calculus is the practical side of what has the theoretical side called analysis. From high school I have Apostol’s Mathematical Analysis (second edition, Reading, Mass.: Addison-Wesley, 1974); we used selections from it with Mr Brown to learn about uniform convergence, a concept needed also for the proof of Dirichlet’s Theorem that I discussed in “LaTeX to HTML.”
Apostol defines compactness for ℝ^{n} on page 59; for an arbitrary metric space, page 63. Thus I must have learned about compactness when I started to work my way through Apostol before graduate school. I did some of this work at the farm.
Alexander’s Horned Sphere sits in ℝ^{3}, which is an example of ℝ^{n}, which is an example of a metric space, which is an example of a topological space. Such a space was apparently not defined till the 20th century. It is a set of points, and certain subsets are called closed, and these must satisfy certain axioms, namely:
the union of two closed sets is closed;
the intersection of any collection of closed sets is closed (here the intersection of the empty collection is understood to be the whole space);
the empty set is closed.
That’s all.
The friend who gave me the book about Helaman Ferguson that inspired my last post (and now this one) gave me also Siobhan Roberts, King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry (Toronto: Anansi, 2006). There’s a foreward by Douglas R. Hofstadter, whom I discussed in “Writing, Typography, and Nature.” In Roberts’s book, Hofstadter recalls discovering, in adolescence, Kelley’s General Topology. This “did not, in its hundreds of pages, contain a single diagram.” At first, Hofstadter was not dismayed.
Within a few years, however, I discovered that I personally could not survive in such an arid atmosphere. Diagrams … were the oxygen of mathematics to me, and without them I would simply die. And thus, when the air of abstraction for abstraction’s sake became too thin for me to breathe, I wound up with no choice by to bail out of graduate school in mathematics. It was a terrible trauma.
Had he known about Coxeter’s geometry, he might have stayed in mathematics.
There are many kinds of mathematics. Some can be visualized in a two-dimensional diagram; some not. The ellipsis in the quotation of Hofstadter stands for a parenthesis:
Diagrams (or at least mental imagery that could be thought of as personal, inner diagrams) were the oxygen of mathematics to me …
Reading anything gives you mental imagery, unless you are not really reading, but only looking at the pages. However, topology as such is so abstract that drawings on paper may mislead. I was thrilled to learn in graduate school that you could prove a lot of interesting theorems, just from the axioms that I wrote above. I was so thrilled that I wrote about the axioms to a friend in New York who had got a job with a publisher.
The friend was not amused, but he did send me the book of mathematical short stories that his firm published: Rudy Rucker (editor), Mathenauts: Tales of Mathematical Wonder (Arbor House, 1987). A theme of one of the stories (“The New Golden Age,” by the editor) is that, if you figure out a way for people to enjoy mathematics without actually understanding it, then the work they enjoy may be that of cranks.
The friend himself later became a chiropractor.
The notion of a topological space is general enough that the structures of a logic can serve as the points of a topological space. The closed sets are the sets of models of theories. For example, the union of the sets of models of formulas φ and ψ respectively is the set of models of the formula (φ ∨ ψ). The connection between topology and logic was a reason why I decided to specialize in model theory.
In “Boolean Arithmetic,” I worked out the proof that the space of models of propositional logic is compact. The topology of that space is the topology of a Cantor set, discussed in “The Tree of Life.”
As a quotation in the last post noted,
there is a Cantor set of ‘bad’ points on the horned sphere.
We constructed that sphere by, for each finite binary sequence σ, attaching the horns H_{σ0} and H_{σ1} to H_{σ}. Here if σ is the empty sequence, we can understand H_{σ} to be the ball B_{0}. Each infinite binary sequence τ yields the sequence of horns H_{τ|n}, where τ|n is the sequence (τ(k): k < n) of the first n entries of τ. The union (over n) of the horns H_{τ|n} has a unique “tip,” a unique accumulation point that doesn’t belong to it. Such points are the “bad” points mentioned in the quotation.
The same sequence τ determines the model of propositional logic in which the true propositions P_{n} are those for which τ(n) = 1.
In our present situation, working inside Euclidean space, I suggested in the last post that being compact is equivalent to being closed and bounded.
Being bounded means being part of a ball. A ball is the inside of a sphere, this being understood in the usual sense of the set of points sharing a common distance from a single point, which is then the center of the sphere and the ball.
A set F is closed if every point not belonging to F is the center of a ball with no points in common with F. I believe the letter F is commonly used for closed sets because it stands for the French fermé(e).
A subset K of Euclidean space is closed and bounded if and only if, whenever K is covered by a collection of balls, meaning the union of the balls includes K, a finite number of those balls are enough to cover K. This theorem, or the difficult part, the “only if” part, is called the Heine–Borel Theorem. The condition involving coverings is now a standard definition of a compact set (in German, kompakt), and I quoted the definition in “Poetry and Mathematics” as being “so poetic.”
The complement of a ball is a closed set. Then being compact is equivalent to the condition whereby, if the intersection of a family of closed subsets is empty, then some finite number of those closed subsets have empty intersection. This yields that the outside of the horned sphere fails to be simply connected, as in the last paragraph of the previous post.
But why does the proof work? Where does the horned sphere come from?
I used the idea of contorting a lump of clay to capture a loop of string. Freeing the string would mean contracting it to a point. During this contraction, the loop would trace out a surface S. Being closed and bounded, this surface would be compact. Its points would therefore have a minimum distance from the horned sphere. Proof: The horned sphere B being closed, each point of the surface S would be center of a ball that contained no points of B; but finitely many of those balls would cover S, by its compactness. The minimum radius of those finitely numerous balls would be the minimum distance from S to B.
Thus, for any distance, no matter how small, there should be arms of the lump of clay
that are that close together, but
that the loop would have to slip through to be free.
An obvious first attempt is to have two arms that grow indefinitely close, as in the figure below. This attempt fails, because the compactness of figure ensures that the two arms will actually touch.
Still, the figure can have, attached to it somehow, infinitely many pairs of arms, with no lower bound on the gap between them, as below.
But how can such arms trap the string? The string cannot initially be made to pass through all of the hoops that are almost formed by the arms; for some of those hoops will be too narrow. (If they are not, then we are back at the first failed attempt.)
However, two hoops, if linked, can form another hoop, which the string would have to break out of to be free.
Thus we let the gap between two arms be almost closed by two more pairs of arms, and so on. This leads to Alexander’s construction, which I have tried to depict below. The original pdf file has size A1, if you want to see more detail.
]]>If you roll out a lump of clay into a snake, then tie a string loosely around it, can you contort the ends of the snake, without actually pressing them together, so that you cannot get the string off?
You can stretch the clay into a Medusa’s head of snakes, and tangle them as you like, again without letting them touch. If you are allowed to rest the string on the surface of the clay, then you can get it off: you just slide it around and over what was an end of the original snake.
If however no point of the string may touch the snake, then you can trap the string so that it cannot be removed. You let the clay start as an amphisbaena, with heads at both ends; you let each head be a hydra’s head, sprouting two new ones from the stump when cut off; and you cut off all heads as they appear, ad infinitum. The result is a solid binary tree, as discussed in “The Tree of Life.” With the branches of the tree intertwined just so, you get what is called Alexander’s Horned Sphere.
You may object that you cannot actually do something ad infinitum. A person called James R. Meyer has this objection. His website is devoted to showing what’s wrong with mathematics and philosophy. I too sometimes take issue with contemporary professional philosophy in my blog. I like to think that I am saved from being a crank by knowing, as a mathematician, that the truth cannot be given uniquely to me. Meyer himself has a page about cranks, taking issue with how others define them. I found first his page about the horned sphere. The page shows no sign of understanding what the horned sphere is for.
The horned sphere is a topological sphere. It arises as a counterexample to a formerly conjectured three-dimensional version of the Jordan–Schönflies Theorem.
I read about that theorem as a child, or at least about the simpler form, the Jordan Curve Theorem: a simple closed curve divides the plane into an inside and an outside. I did not understand what there was to be excited about. I suppose now the point is the following. A simple closed curve is a continuous function f from the unit interval [0, 1] into ℝ^{2} that repeats a value precisely at 0 and 1, so that, f being the function,
f(t) = f(u) ⇔ t = u ∨ {t,u} = {0, 1}.
A simple example is g, given by
g(t) = (sin (2πt), cos (2πt)),
tracing out the unit circle; but f might not be given by a formula. How are you going to define a function h on ℝ^{2} so that h(x,y) = 1, if (x,y) is “inside” the curve given by f, and otherwise h(x,y) = 0? I just did define h, but only by begging the question of what inside means.
If g is as above, and f is some simple closed curve, then f∘g^{−1} is a homeomorphism between the two curves, just for being continuous in both directions. According to the Jordan–Schönflies Theorem, the homeomorphism extends to a homeomorphism from the whole plane ℝ^{2} to itself.
Passing to three dimensions, one may think that if there is a homeomorphism from a sphere, considered as a surface in ℝ^{3}, to some other surface in space, then that homeomorphism should extend to a homeomorphism from the whole space ℝ^{3} to itself.
The horned sphere shows that one would be wrong. Here I want to work out some details of the proof. It may serve as another example of my recent theme, that mathematics is the science whose findings are proved by deduction. Topology in particular can seem to be a counterexample, since it seems to rely on physical intuition, albeit an intuition that tolerates supposed absurdities like completed infinite processes.
The horned sphere has been the inspiration of some sculptures pictured in Claire Ferguson, Helaman Ferguson: Mathematics in Stone and Bronze (Erie, Penn.: Meridian Creative Group, 1994). A friend recently gave me the book, and the book is a reason for this post.
J. W. Alexander described his construction in “An example of a simply connected surface bounding a region which is not simply connected” (Proc. N. A. S. 10, 1924). A simply connected space is one in which
you can carry a string from any point to any other like Ariadne, and
if you carry the string back to where you started from, then you can hold the two ends and draw the whole string to yourself.
In more technical language, the loop of string must be null-homotopic, meaning there is a continuous function f from the square [0, 1] × [0, 1] into the space in question such that the function t ↦ f(0, t) is the original loop, and t ↦ f(1, t) is constant.
Alexander describes his construction with words and a drawing:
The surface Σ is the limiting surface approached by the sequence Σ_{1}, Σ_{2}, Σ_{3}, .. It will be seen without difficulty that the interior of the limiting surface Σ is simply connected, and that the surface itself is of genus zero and without singularities, though a hasty glance at the surface might lead one to doubt this last statement. The exterior R of Σ is not simply connected, however, for a simple closed curve in R differing but little from the boundary of one of the cells γ_{i} cannot be deformed to a point within R.
The surface Σ is the surface of our clay. To say that it is of genus zero means it has zero holes; a torus has genus one. I myself do see without difficulty that Σ will have no holes. I am not sure what Alexander means by singularities. That the exterior of Σ is not simply connected is not clear without a proof. Alexander himself confesses, at the end of his short article,
This example shows that a proof of the generalized Schönfliess theorem announced by me two years ago, but never published, is erroneous.
If he was wrong then, why is he not wrong now?
The relevant Wikipedia article, “Alexander Horned Sphere,” lists a reference on my shelves, Spivak’s Comprehensive Introduction to Differential Geometry, Volume One (2nd ed., Houston: Publish or Perish, 1979). Defining Alexander’s Σ using a drawing like his, an exercise asks the reader to show what I said was clear, that Σ has no holes. (Spivak calls the surface S and asks, “Show that S is homeomorphic to S^{2},” the latter being the surface, which is two-dimensional, of a sphere.) The exercise then asserts, without explicitly asking for a proof, that the S together with the the region outside it fails to be a manifold-with-boundary, though the student has shown that it would have to be one, if S were differentiable.
I pass to another of the Wikipedia references, Hocking and Young, Topology (Reading, Mass.: Addison-Wesley, 1961), which says on page 175 (where the circumflex on ĥ is a tilde in the original),
Let S be a simple closed surface in [Euclidean space] E^{3}, that is, S is a homeomorph of S^{2}, and let h be a homeomorphism of S onto the unit sphere S^{2} in E^{3}. Is there an extension ĥ of h such that ĥ is a homeomorphism of E^{3} onto itself? … Alexander … gave a famous example, the Alexander horned sphere, showing that the answer must be “no” in the general case. This example is pictured [below]. We can see from the picture alone that it is quite obvious that the complement of the horned sphere is not simply connected. Since the complement of S^{2} in E^{3} is simply connected, it follows that no homeomorphism of E^{3} onto itself will throw the horned sphere onto S^{2}. Note that there is a Cantor set of “bad” points on the horned sphere.
My sense is that in topology nothing is obvious. In Real Analysis II in graduate school, the professor made an “obvious” topological assertion that I questioned and later disproved with the help of a book I found: Steen and Seebach, Counterexamples in Topology. I am going to have to work out a proof of what Hocking and Young think is obvious. I am helped by yet another reference in the Wikipedia article, though a reference to which the link was dead, till I revived it: Allen Hatcher, Algebraic Topology.
We started with a lump of clay and rolled it out into a snake. Alternatively, we pulled two horns from it. Now continuing, from each horn we pull two more horns, ad infinitum.
More precisely, letting the initial lump of clay be B_{0}, we pull two horns out of it to get B_{1}. In general, B_{n} will have 2^{n} horns, and when we make each horn into two, we get B_{n+1}. Each B_{n} is closed and bounded, hence compact.
Perhaps ordinary clay is incompressible, so that our original lump retains its volume throughout the reshaping. We prefer the volume to grow, as if we add horns at each step. Let us say that B_{n} has 2^{n} horns, one for each binary sequence σ of length n. Let the corresponding horn itself be called H_{σ}. To obtain B_{n+1}, to each H_{σ} we attach the horns H_{σ0} and H_{σ1}. Now we have a strictly increasing chain:
B_{0} ⊂ B_{1} ⊂ B_{2} ⊂ …
We may think of B_{n} as a balloon filled with air; when we lower the pressure outside, the balloon expands from B_{n} into B_{n+1}.
Since it depends on physical intuition, this description is perhaps not “universally valid.” We could write down equations in three variables, defining the surfaces of the B_{n}. Then inequalities would define the solids themselves.
There is a homeomorphism h_{n} from B_{n} to B_{n+1}. We “could” define it precisely, but we don’t want to bother. Still there are conditions it must satisfy. For each n, we have that the composite function
h_{n}∘ … ∘h_{0}
is a homeomorphism from B_{0} to B_{n+1}. We want the sequence
(h_{0}, h_{1}∘h_{0}, h_{2}∘h_{1}∘h_{0}, …)
of these functions to converge uniformly to an injective function h. This is not automatic; we have to choose the h_{n} right. The uniform convergence will imply that h itself is continuous. Every closed subset of B_{0} is bounded and therefore compact, so its image under h is compact and therefore closed. Thus the inverse of h is also continuous, so h is a homeomorphism onto its image.
We call that image B. Now this is homeomorphic to the original B_{0}. Why then have we bothered to create B? We want the complement of B in space not to be homeomorphic to the complement of B_{0}. We achieve this by attaching in two steps to H_{σ} the horns H_{σ0} and H_{σ1}, for each binary sequence σ of length n, for each n:
Attach a “handle” to H_{σ}.
Cut out part of the handle, leaving the horns H_{σ0} and H_{σ1}.
The cut-out part will include the handles that will be attached to the horns H_{σ0} and H_{σ1} in the next step. Thus B_{n} gets 2^{n} handles attached, resulting in X_{n}. These form a decreasing chain of compact sets. In particular, we now have
B_{0} ⊂ B_{1} ⊂ B_{2} ⊂ … ⊂ B ⊂ … ⊂ X_{2} ⊂ X_{1} ⊂ X_{0}.
It is not automatic that B is the intersection of the X_{n}, but we can have ensured that this will be so.
The key move really needs three dimensions, and the pictures just above don’t handle this. For each n, for each σ of length n, the handles attached respectively to the horns H_{σ0} and H_{σ1} should be interlocking. This is to ensure that
the complement of X_{n+1} is not simply connected,
every loop in the complement of X_{n} that is null-homotopic in the complement of X_{n+1} was already null-homotopic in the complement of X_{n}.
If one accepts this, then the complement of B also fails to be simply connected; for, by the compactness of X_{0}, any loop in the complement of B that is null-homotopic must already have been null-homotopic in the complement of one of the X_{n}, and we know that that complement has loops that are not null-homotopic. Indeed, suppose we are given a homotopy of loops in the complement of B, namely a continuous function f from the square [0, 1] × [0, 1] into the complement of B such that f(0,0) = f(0,1) and f(1,0) = f(1,1). Let K be the image of the square under f. Then K ∩ B is empty, but is the intersection of the descending chain of closed subsets K ∩ X_{n} of the compact set X_{0}. Therefore, by compactness, some set in the chain must be empty. Thus f is a homotopy in the complement of some X_{n+1}.
]]>I say that mathematics is the deductive science; and yet there would seem to be mathematicians who disagree. I take up two cases here.
In November of 2017 I presumed, as an amateur, to write a post called “What Philosophy Is.” From his Essay on Philosophical Method (Oxford, 1933), I quoted R. G. Collingwood on what is commonly understood as Plato’s “doctrine of recollection.” Bertrand Russell was dismissive of this doctrine, in A History of Western Philosophy (New York: Simon and Schuster, 1945), as I had recalled in a post on the first chapter of Collingwood’s New Leviathan (Oxford, 1942); but in the Essay, Collingwood avers sensibly,
in a philosophical inquiry what we are trying to do is … to know better something which in some sense we knew already.
Thus I didn’t think philosophy should be defined as the taking up of certain specific questions, no matter how “general” or “fundamental” in form, such as “What is consciousness?” What matters is what you are trying to do with your questions. “There is only one philosophical question,” I said: “‘What are we going to do now?’”
What I am going to do now is continue the activity of my most recent post, the activity of asking “What Mathematics Is.”
I say in that post that mathematics is the science whose findings are proved by deduction. Deduction is reasoning that is valid universally. At least the reasoning is intended to be so. I don’t tell you that my mathematical proof is correct for everybody, for all time, everywhere; I aim to make it so. I may fail. If you think I have, you tell me, and I reconsider, or I try to tell you what you have missed.
As evidence about mathematics, I used a talk of unknown date by Euphemia Lofton Haynes. There is always more evidence. Now I want to consider two mathematicians who think our subject is not purely deductive, or else do not distinguish deduction as something special.
I shall take up the words of Doron Zeilberger at the end; meanwhile, according to Vladimir Arnold in a 1997 address,
Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.
The Jacobi identity (which forces the heights of a triangle to cross at one point) is an experimental fact in the same way as that the Earth is round (that is, homeomorphic to a ball). But it can be discovered with less expense.
Delivered in Paris, the address is called “On teaching mathematics”; the full citation is, Arnolʹd, V. I., “On the teaching of mathematics” (Russian), Uspekhi Mat. Nauk 53 (1998), no. 1(319), 229–234; translation in Russian Math. Surveys 53 (1998), no. 1, 229–236.
We shall look presently at Arnold’s mathematical example. Meanwhile, writing in Mathematical Reviews, V. Ya. Kreinovich seems right to point out that the “impact” of Arnold’s article
may be somewhat lessened by arguable statements such as “mathematics is a part of physics” (this is actually the very first sentence of the paper). Surely, the author is right that geometry and physics often clarify and simplify the understanding of mathematical notions, but he may be overemphasizing physics, as other mathematical notions may be better understood in the context of applications to biology, computer science, cryptography, etc.
I would go further, saying that the mathematics illuminated by a science that applies it is not part of that science, be it physics, biology, or any other.
Why should it matter to say this? Arnold’s next paragraph reads:
In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy’s warning that ugly mathematics has no permanent place under the Sun).
This may be correct as history, and beneficially pointed out, as Kreinovich allows in his review:
The author gives a lot of (mainly anecdotal) evidence that there exists a tendency to teach mathematics in an unnecessarily abstract form, without giving students motivation and intuitive examples, and that this tendency often decreases the quality of mathematical education.
Nonetheless, though they may have been indistinguishable for much of their history, physics and mathematics are different. An example of the difference is seen in
the discovery of Lobachevskian or hyperbolic geometry, and
the discovery (attributed to Beltrami) that each of Lobachevskian and Euclidean geometry proves the consistency of the other, regardless of whether any one of them fits the physical world.
Arnold knows these discoveries. In referring to the Jacobi identity and the concurrence of the altitudes of a triangle, he alludes to his own work in hyperbolic geometry. In a paper dedicated to Arnold’s memory, Nikolai V. Ivanov writes that Arnold
showed that the Jacobi identity
[[ A, B], C] + [[B, C], A] + [[C, A], B] = 0
lies at the heart of the theory of altitudes in hyperbolic geometry. In his approach, Arnolʹd used the Jacobi identity for the Poisson bracket of quadratic forms on R^{2} endowed with its canonical symplectic structure.
I quote these words, from “Arnolʹd, the Jacobi identity, and orthocenters,” Amer. Math. Monthly 118 (2011), no. 1, 41–65, mainly to illustrate the sentence that follows them:
Unfortunately, the use of these advanced notions renders Arnolʹd’s approach nonelementary.
That the altitudes of a triangle in the Euclidean plane have a common point, which we now call the orthocenter, was supposed to be known to the readers of the Book of Lemmas, which survives only in Arabic, but is attributed to Archimedes. The proof of Proposition 5, about the arbelos, calls on the existence of the orthocenter. I discussed all of this in “Thales and the Nine-point Conic,” § 2.1.
The same section of that paper reviews a Euclidean proof that the medians of a triangle also concur. Archimedes proved it by physical considerations in On the Equilibrium of Planes.
In an exercise that could be done in school, though I don’t remember doing it myself, if you cut a triangle out of heavy cardboard, and you hang from a vertex both the triangle itself and a plumb-line, then that line will lie along a median of the triangle.
For the same reason, if you hold the line taut horizontally, and you let a median of the triangle lie along the line, then the triangle will balance, albeit unstably. That is true by symmetry, if the median is perpendicular to the side that it meets, so that the triangle itself is isosceles; but in any case, if you cut the triangle into thin strips, parallel to the side met by the median, then the median crosses the center of each strip, which therefore balances at the median; so the whole triangle balances. That is the idea of the proof of Archimedes.
Moreover, the triangle must balance at some point along the median; and then all three medians concur at that point, which is the center of gravity or centroid of the triangle.
Strictly, the strips of the triangle are sure to balance along the median, if you cut triangles from their ends with cuts that are parallel to the median, so that the remainder is a parallelogram. However, you can make the area of the sum of all of those cut-off triangles as small as you like. This would yield a contradiction, if the center of gravity were not on the median.
ΑΔ is a median of triangle ΑΒΓ.
If the centroid of the triangle lies elsewhere, at Θ,
then we can halve ΔΓ, and halve the halves, and so on,
till one division point is closer to ΑΔ than Θ is.
Then the parallelograms ΜΝ, ΚΞ, and ΖΟ together
have their centroid at some Ρ on ΑΔ,
and the remainder of ΑΒΓ must have its centroid out at Χ.
But then
ΡΘ : ΡΧ :: ΣΜ : ΔΓ,
so Χ is beyond Φ, which is absurd.
Archimedes thus uses the technique that we know today as calculus. Euclid before him used the technique to prove that circles are in the ratio of the squares on their diameters: the theorem that today we blithely write blithely as
A = πr².
I wrote in “Salvation” about a misunderstanding of Euclid’s theorem and ratios in general in a recent popular book about calculus. The book or its promoters praised calculus for the technology it made possible. I praised mathematical deduction, for being able to save the world the way Jane Austen can.
Possibly Archimedes discovered by trial the concurrence of the medians of a triangle. Then he proved it, using simple postulates about balancing, in addition to Euclidean geometry. His proof is deductive, like (presumably) Arnold’s proofs about the orthocenter.
By Collingwood’s account in the Prologue of Speculum Mentis (Oxford, 1924), art and philosophy won their independence from religion in the Renaissance. You may judge this independence a catastrophe, as people may do who are nostalgic for the middle ages; but the three pursuits are in fact distinct from one another.
Just so are mathematics and physics distinct. Proof in mathematics is deductive; physics, inductive. There may be a deductive element of a physical proof, but the ultimate test is agreement with experience, and such agreement is always provisional. As Arnold says, one may produce a mathematical model, as for weather forecasting, and draw conclusions from this; and yet,
It is obvious that in any real-life activity it is impossible to wholly rely on such deductions. The reason is at least that the parameters of the studied phenomena are never known absolutely exactly and a small change in parameters (for example, the initial conditions of a process) can totally change the result.
That’s fine. Arnold goes on to say,
In exactly the same way a small change in axioms (of which we cannot be completely sure) is capable, generally speaking, of leading to completely different conclusions than those that are obtained from theorems which have been deduced from the accepted axioms. The longer and fancier is the chain of deductions (“proofs”), the less reliable is the final result.
I don’t know what kind of “small change” there can be in axioms. I think normally the researcher who is curious to see what will happen will change an axiom to its negation. This is what Lobachevsky does.
Proposition 17 of Book I of Euclid’s Elements is that any two angles of a triangle are together less than two right angles. The converse is Euclid’s fifth and last postulate—or axiom, if you like— , the parallel postulate: only in a triangle can two facing angles sharing a side be together less than two right angles.
In “Geometric Researches on the Theory of Parallels” (Geometrische Untersuchungen zur Theorie der Parallellinien, 1840), Lobachevsky assumes the negation of the parallel postulate, leaving the rest of Euclidean geometry intact. In this case, for any particular distance a, there is an acute angle θ such that, if a perpendicular is erected at a point B of a line BC, and a point D on the perpendicular is at distance a from the point B, then lines through the point D making
less than angle θ with the line DB will cut BC;
angle θ itself or greater with the line DB will not cut BC.
Denoting θ by Π(a), Lobachevsky shows that, for some distance u, for every a,
cot (Π(a)/2) = exp (a/u).
In Euclidean geometry, Π(a) is always a right angle, and the cotangent of half of this is 1, which is exp (0/u). Thus Lobachevskian geometry is the more Euclidean, so to speak, the smaller the distances being considered. As Lobachevsky says,
the imaginary geometry passes over into the ordinary, when we suppose that the sides of a rectilineal triangle are very small.
One might also say that “imaginary” or Lobachevskian geometry approaches “ordinary” or Euclidean geometry as u increases without bound; also, a small change in u creates big changes at large-enough distances. This may be the kind of thing that Arnold is referring to, when he says, “a small change in axioms … is capable, generally speaking, of leading to completely different conclusions …” However, Lobachevsky has no such notation as my u; he effectively lets it be the unit of measurement.
When Arnold remarks parenthetically of axioms that “we cannot be completely sure” of them, I have to say that I thought we had learned from Lobachevsky not to expect to be “sure” of axioms anymore. So I don’t know what Arnold is talking about here. Of course we can fail in our mathematical deductions, as a model can fail to predict tomorrow’s thunderstorm, or a pipeline can fail to deliver water. These are not all “exactly” the same kind of failure. (I argued this in “Anthropology of Mathematics.”)
To be able to declare a vaccine safe, you test it on what is assumed to be a representative sample of people, while knowing that your assumption is not 100% correct. To be able to declare that there are no odd perfect numbers at all, there is no representative sample to be tested, unless you can prove it by deductive reasoning.
Mathematical theorems are not sought with such reasoning. Reason is a tool, not for discovery, but for confirmation. How we discover theorems is mysterious. It may involve induction, in the sense of generalization from special cases. Arnold suggests that it always does, but that’s a stretch:
The scheme of construction of a mathematical theory is exactly the same as that in any other natural science. First we consider some objects and make some observations in special cases. Then we try and find the limits of application of our observations, look for counter-examples which would prevent unjustified extension of our observations onto a too wide range of events (example: the number of partitions of consecutive odd numbers 1, 3, 5, 7, 9 into an odd number of natural summands gives the sequence 1, 2, 4, 8, 16, but then comes 29).
Here it will be valuable to look at what Victor V. Pambuccian says in Mathematical Reviews about Ivanov’s article on Arnold’s work.
Pambuccian has reviewed also an article of mine, “Abscissas and Ordinates.” I am glad that Mathematical Reviews was willing to ask somebody to do this, and that Pambuccian was willing to take on the job. He was succinct:
The paper is a walk through the Greek origin of the words “abscissa”, “ordinate”, “upright side” and “latus rectum” of a parabola, “transverse side” of a hyperbola—interspersed with lexicographic and literary (Blake, Orwell) asides—focusing on their meaning in Apollonius’ Conics.
That’s the whole review. I might have wished to hear more about those literary asides, which concerned the treatment of children. Sometimes Mathematical Reviews just prints an author’s own summary of their paper, and my summary was,
In the manner of Apollonius of Perga, but hardly any modern book, we investigate conic sections as such. We thus discover why Apollonius calls a conic section a parabola, an hyperbola, or an ellipse; and we discover the meanings of the terms abscissa and ordinate. In an education that is liberating and not simply indoctrinating, the student of mathematics will learn these things.
Pambuccian writes more than two pages (size A4) on Ivanov’s article, mostly explaining what Ivanov and Arnold leave out. In a way to suggest what is missing from Arnold’s account of the “scheme of construction of a mathematical theory,” here is how Pambuccian sets out:
There are two distinct traditions in mathematics. The first one believes in the unity of all existing mathematics and values very highly unexpected uses of results and techniques of one area of mathematics in another, preferably distant, one. Practitioners of this tradition who look at geometry tend to believe that the continuity of space is one of its essential attributes, and thus believe in an intimate connection between geometry and the real numbers (understood as a unique structure, insensitive to potential disturbances of its uniqueness coming from set theory), take advantage of the manifold benefits to be had from its topological or differential structure, or else assume that the geometry in question is built over special classes of fields, to unleash the powerful results of algebraic geometry.
The second, and oldest one, looks at geometry (and by extension, at any area of mathematics) as consisting of statements on the words of a language, that are not there to be verified in an algebraic or a continuous realm, but to be deduced from other such statements, the main aim of this undertaking being the deduction of statements from the weakest possible set of some special statements that are endowed with a certain hard-to-define quality of simplicity.
Arnold sets himself in the first tradition here by saying,
These discoveries of connections between heterogeneous mathematical objects can be compared with the discovery of the connection between electricity and magnetism in physics or with the discovery of the similarity in the geology of the east coast of America and the west coast of Africa.
The emotional significance of such discoveries for teaching is difficult to overestimate. It is they who teach us to search and find such wonderful phenomena of harmony in the universe.
It is inductive, the initial speculation that “heterogeneous” objects may nonetheless be instances or consequences of one law. One may then be able to proof a theorem about such a law, deductively.
Does this scheme account for one of the first mathematical theorems? According to Proclus in his Commentary on the First Book of Euclid’s Elements (translated by Glenn R. Morrow; Princeton University Press, 1970),
The famous Thales is said to have been the first to demonstrate that the circle is bisected by the diameter.
This theorem does not seem like a generalization from the observation of special cases. As Proclus goes on to say,
If you wish to demonstrate this mathematically, imagine the diameter drawn and one part of the circle fitted upon the other. If it is not equal to the other, it will fall either inside or outside it, and in either case it will follow that a shorter line is equal to a longer. For all the lines from the center to the circumference are equal, and hence the line that extends beyond will be equal to the line that falls short, which is impossible. The one part, then, fits the other, so that they are equal. Consequently the diameter bisects the circle.
Fold a circle along a diameter, and the two halves match up. If you remark on a single occurrence of of this, would you need to try again, in order to be sure that it will always happen?
Perhaps not; but you still have generalized from some number of special cases, at least if you allow one to be a number. Collingwood observes the possibility of such generalization, albeit in a different context:
There is nothing in recurrence that is not already present in the single instance. Indeed some determinists have argued that because a certain man once did a certain action, therefore he was bound to do it. This seems a reductio ad absurdum; and yet if we can argue from frequency to necessity, the question “How often must a thing happen before you know it was bound to happen?” can have only one answer:—“Once is enough.”
That is from the last chapter of Religion and Philosophy (London: Macmillan, 1916). Collingwood is proving that there is no miracle in the the sense of a divine interference with the ordinary course of nature; everything is miraculous that we can see as such, because miracles are those events that “compel us to face reality as it is, free, infinite, self-creative in unpredicted ways.”
I took up Collingwood’s treatment of miracles in “Effectiveness,” which was mainly about Eugene Wigner’s 1960 article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” an article that Arnold also brings up.
When Thales realized that the two halves a circle must match up, perhaps this was like noticing that the coastlines of Africa and the Americas match up. A difference is that the matching of the two halves of a circle is a theorem, θεώρημα, literally a spectacle, something to be seen, as in a theater, θέατρον. The matching of coastlines is a random occurrence, until explained by a theory such as plate tectonics.
Religion and Philosophy was Collingwood’s first book, published when he was 27. The first chapter argues
Theology = philosophy = religion.
That is actually how the Index summarizes several pages. Later Collingwood disavowed such identifications and developed the doctrine of the “overlap of classes,” expounded in An Essay on Philosophical Method. You can find things, such as art and craft, bound up together in every observable instance; it doesn’t make them the same. They need not agree just in “extension,” while differing in “intension”; they may overlap in intension. Collingwood’s example is from ethics:
Duty rejects expediency in the sense of refusing to accept it as even a legitimate kind of goodness, and regarding it rather as the inveterate enemy of morality, but reaffirms it in the sense of accepting it, when modified by subordination to its own principles, as a constituent element in itself. Thus duty and expediency overlap … the overlap is essentially not … an overlap of extension between classes, but an overlap of intension between concepts, each in its degree a specification of their generic essence, but each embodying it more adequately than the one below.
Collingwood alludes here to a progression that he will take up in the “Reason” chapter of The New Leviathan. Historically, we have justified our behavior, first as expedient, then as right, and finally as dutiful.
We might suggest then that the recognition of several divisions within what was formerly called natural philosophy is a progression. The several sciences grew up together, and they are still all studied at single universities; but this is done in different departments.
Members of different departments will publish their research in different ways. Our mathematical findings as such will have no physical component; thus we must justify them deductively, as Archimedes justified his theorem locating the centroid of a triangle.
There is however a mathematician who objects to this requirement. On his webpage he publishes a lot of his opinions, and one of these opinions is that the Euclidean deductive approach ruined mathematics and justified absolutism in politics. See for yourself.
Zeilberger is evidently one of those persons who enjoy being controversial. Here’s an article of his, “What is Mathematics and What Should it Be?” saying,
Today’s Mathematics Is a Religion
Its central dogma is thou should[st] prove everything rigorously.
Call it a dogma, or call it, following Collingwood, an absolute presupposition: the requirement of rigorous deductive proof is not a restriction, but a platform for discovery.
According to Zeilberger, “Scientists, by definition, are trying to discover the truth about the outside world.”
We are trying to find the truth, I would say. That there is an “outside world” at all is a dogma, or absolute presupposition, of many sciences, but not of mathematics.
One may say that what we study in mathematics is outside us, but in a different way from the physical world.
Says Zeilberger,
Mathematicians do not care about discovering the truth about the mathematical world. All they care about is playing their artificial game, called [rigorous] proving, and observing their strict dogmas.
I almost said that was true, until I noticed that Zeilberger said not “outside world,” but “mathematical world.” The second sentence is true, though again Zeilberger introduces a needless qualification. Every game is artificial. Zeilberger is welcome to play another.
]]>Mathematics “has no generally accepted definition,” according to Wikipedia today. Two references are given for the assertion. I suggest that what has no generally accepted definition is the subject of mathematics: the object of study, what mathematics is about. Mathematics itself can be defined by its method. As Wikipedia currently says also,
it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.
I would put it more simply. Mathematics is the science whose findings are proved by deduction.
That is my definition. It has only science in common with a dictionary definition:
măthėmă′tĭcs n. pl. (also treated as sing.) (Pure) ⁓, abstract science of space, number, and quantity; (applied) ⁓, this applied to branches of physics, astronomy, etc.; (as pl.) use of mathematics in calculation etc.
That’s the Concise Oxford Dictionary (6th ed., 1976). The Grolier International Dictionary (1981) does not even refer to science:
math⋅e⋅mat⋅ics (măth′ə-măt′ĭks) n. Abbr. math. Plural in form, used with a singular verb. The study of number, form, arrangement, and associated relationships, using rigorously defined literal, numerical, and operational symbols.
Presumably the dictionaries reflect the common view. People think mathematics is numbers, because that is all they learn about it in school. The teaching of more than this is urged by Euphemia Lofton Haynes, who says,
Mathematics is no more the art of reckoning and computation than architecture is the art of making bricks, no more than painting is the art of mixing colors.
This is in an address to high-school teachers, probably in Washington, D.C. The typescript, which I have digitized, is undated, but Haynes’s dates are 1890–1980.
The ideal example of an explanation of a pursuit is Collingwood’s 1938 book, The Principles of Art (1938). I talked about Collingwood’s procedure in the pdf booklet that is linked to from the post about art and mathematics called “Discrete Logarithms.” In the middle of his book, Collingwood offers the definition,
By creating for ourselves an imaginary experience or activity, we express our emotions; and this is what we call art.
He immediately points out:
What this formula means, we do not yet know. We can annotate it word by word; but only to forestall misunderstandings …
In my formula for mathematics (“the science whose findings are proved by deduction”):
“Science” is used in the general sense of a system of knowledge. The Latin root scient- means knowing, more precisely discerning or separating out. The same root is found in our scission, though not scissors, whose initial ess was added by analogy, because the French original is the plural cisoires of cisoir, from the Latin cisorium, from the verb caedo, meaning cut and giving us words like decide and homicide.
Mathematics is not a deductive science, but the deductive science. Other sciences may use deductive arguments, but not exclusively. For example, a friend pointed out to me an article, Ada Marinescu, “Axiomatical examination of the neoclassical economic model. Logical assessment of the assumptions of neoclassical economic model,” Theoretical and Applied Economics, Volume XXIII (2016), No. 2(607), Summer, pp. 47–64:
We analyze in this paper the main assumptions of the neoclassical theory, considered as axioms, like the rationality of the economic actor, equilibrium of the markets, perfect information or methodological instrumentalism … Correspondence with reality matters less in this strictly abstract approach compared to the possibility to build a coherent and convincing system.
Correspondence with reality may matter less; I think it still matters, unless one is going to say that neoclassical economics is simply mathematics.
The “findings” of any science may be found by any means. The means of finding them may not, and usually will not, be the same as the method of proving them.
“Deduction” is universally valid reasoning. That mathematical proofs are universally valid is a “metaphysical proposition,” in the sense of Collingwood’s Essay on Metaphysics (1940). This means the proposition carries implicitly the “metaphysical rubric,” namely, “Certain persons believe.” As mathematicians, we believe our proofs to be universally valid. In the same way, Anselm proves not that God exists, but that we believe it:
Whatever may have been in Anselm’s mind when he wrote the Proslogion, his exchange of correspondence with Gaunilo shows beyond a doubt that on reflection he regarded the fool who ‘hath said in his heart, There is no God’ as a fool not because he was blind to the actual existence of un nommé Dieu, but because he did not know that the presupposition ‘God exists’ was a presupposition he himself made.
In mathematics, we believe not that our proofs are universally valid, but that they can be and ought to be, and if they are not, then they are not proofs.
“Deduction” can be called “deductive logic.” The address by Euphemia Lofton Haynes is called “Mathematics—Symbolic Logic.” The speaker discusses two “scientific movements,” originating in the 19th century:
in mathematics, to criticize the foundations of calculus and geometry;
in logic, to overcome the inadequacy of Aristotelian logic.
Says Haynes,
Thus originating in apparently distinct domains and following separate parallel paths these two scientific movements led to the same conclusion, viz. that the basis of mathematics is the basis of logic also. Symbolic logic is Mathematics and Mathematics is Symbolic Logic.
On the page with the transcript, I mention that:
Haynes is said to have been the first African American woman to receive a doctorate in mathematics, in 1943 at Catholic University of America;
I learned of her address to teachers through a certain Twitter account.
That account is called Great Women of Mathematics, and it was denounced, earlier this year, apparently for not affirming the slogan “Trans women are women.” Perhaps there were additional failures to toe a line. Twitter users were called on to unfollow the account. Had I heeded the call, I might not have learned of Haynes’s address, or even of herself. Let me note:
I try to follow Twitter accounts that I disagree with, at least if they offer actual arguments (either directly or through links).
I haven’t got great expectations for dialogue on Twitter with people I disagree with; however, I have hoped mathematicians would be different, because, as I say, we aim for universality in our arguments. This means we may be wrong, if somebody disagrees with us.
I had an argument with two other mathematicians over the proposed cancellation of Great Women of Mathematics. I shall not try to review the argument. I do have two related blog posts, called “Sex and Gender” and “Be Sex Binary, We Are Not,” composed respectively before and after the argument. I agree with Brian Earp in his 2016 essay, “In praise of ambivalence—‘young’ feminism, gender identity, and free speech”:
… “no-platforming” can be a bad idea … even when the person you want to exclude is a dyed-in-the-wool ideological opponent …
The political theorist Rebecca Reilly-Cooper, herself a controversial figure in this debate, argues that there is “a creeping trend among social justice activists of an identitarian persuasion” towards what she calls ideological totalism.
This is “the attempt to determine not only what policies and actions are acceptable, but what thoughts and beliefs are, too” …
My worry is that such thought-policing, to the extent that it exists, is unlikely to achieve its aims in the long run.
My aim here is to define mathematics. I say that its findings are proved by deduction. We usually refer to a deductive proof as a proof, simply. However, in her address, Haynes uses the word “proof” only in the context of physical science:
Let us consider for a moment a teacher of physics or chemistry. In order to examine the validity of an hypothesis, he does not rush through the proof, i.e. the experiment, concentrating on the findings, and spending the major portion of his time trying to discover whether or not his pupil has retained the results or findings. No …
I have put “and spending” where the typescript as “but spends.” I think the kind of proof that Haynes refers to is inductive, in one of the senses that I discussed in the post “On Gödel’s Incompleteness Theorem.” Haynes uses the term “incomplete induction” for such proofs, as when she says of the mathematician,
His processes and methods are similar to those of his colleagues in natural science. The importance of observation, experimentation, testing of hypotheses by the laboratory method, and incomplete induction cannot be too strongly emphasized.
I assume the qualifier “incomplete” is meant to distinguish this kind of induction from specifically mathematical induction, which in spite of its name is a method of deductive proof. In mathematics, we may be given a set of counting numbers, such as the set of all n such that twice the sum 1 + … + n is the product of n with its successor n + 1. If that set contains
the successor of its every element, and
1 itself,
then the set contains every counting number. Since indeed
2 ⋅ 1 = 1(1 + 1) and,
if, for some k,
2(1 + … + k) = k(k + 1),
then
2(1 + … + (k + 1)) = 2(1 + … + k) + 2(k + 1)
= k(k + 1) + 2(k + 1) = (k + 1)(k + 2),
we conclude that, for all counting numbers n,
2(1 + … + n) = n(n + 1).
That’s a proof. We call it a proof by induction, but it’s a deductive proof. There’s nothing missing, nothing left to prove; the proof is “complete.” The method might then be called “complete induction,” as I find that it is in one source:
The method of mathematical (or complete) induction is a very strong tool in mathematical proofs. Unfortunately, in secondary school it does not receive the attention which it deserves. Most students have a rather hazy idea concerning this important method.
That’s from page 21 of Dorofeev, Potapov, and Rozov, Elementary Mathematics: Selected Topics and Problem Solving (Moscow: Mir Publishers, 1973). My teacher Donald J. Brown had us use the text for precalculus at St Albans School in Washington, D.C.
I assume George Yankovsky, translator of the Soviet book, knew what he was doing in using the word “complete” to describe mathematical induction. Nonetheless, looking around, I see that some people use the term “complete induction” as an alternative to “strong induction.” This is the method of proof whereby a set of counting numbers contains all of them if it contains
the successor of every number for which the set contains all numbers up to and including that number, and
1 itself.
The two conditions collapse to one, if you follow the grammar: a set of counting numbers contains all of them if it contains
every number than which it contains all numbers less.
The condition of containing 1 then follows, since every set contains every counting number that is less than 1, there being no such numbers.
I find a webpage actually called “Complete Induction,” although it is only about the method that I just described as strong induction. The page is part of an online textbook on the foundations of mathematics, and the previous page in the text is called “Mathematical Induction.” Unfortunately the text teaches the same error that Donald Brown taught us from Spivak’s Calculus: the error that ordinary mathematical induction and “complete” or strong induction are equivalent conditions on the natural numbers. They are equivalent, in the sense that they are both true of the natural numbers; but what is meant is that you can prove either of the conditions after assuming the other. The proofs make tacit assumptions that ought to be explicit. Ordinary induction involves only 1 and the operation of succession, while strong induction involves a linear ordering. The explicit assumptions for the putative proofs of equivalence are that
different numbers have different successors;
1 is the successor of no number.
If we make a third assumption, namely
then we can indeed prove that the counting numbers are linearly ordered; however, the takes a lot of work, which is usually forgotten. One approach is that of Landau, whose Elementary Number Theory I used for my last post, “LaTeX to HTML.” In The Foundations of Analysis (1929/1966), given the three axioms above, Landau proves that there is a unique operation of addition with the property
n + (k + 1) = (n + k) + 1.
Then he can define the relation “<” by the rule
k < n ⇔ ∃x k + x = n.
Alternatively, one can prove the “Recursion Theorem,” or at least the special case that every natural number has a well-defined set of predecessors according to the rules,
1 has no predecessor,
the predecessors of n + 1 are precisely n and its predecessors.
Now one can define k < n to mean precisely that k is a predecessor of n.
Either way, one goes on to prove that “<” is a linear ordering and even a “well-ordering,” meaning every nonempty set of numbers has a least element. This is equivalent to strong induction, at least in the compact form with the single condition that I stated.
Conversely, given a nonempty well-ordered set with no greatest element, we can call
its least element 1,
the least of the elements greater than n the successor (namely n + 1) of n
Every limit ordinal in the sense of my post “Ordinals” is a nonempty well-ordered set with no greatest element, but only ω is isomorphic to the set of counting numbers.
There is more discussion of the logic here in Example 1.2.3 (pages 37–8) of Model Theory and the Philosophy of Mathematical Practice (Cambridge University Press, 2018), by John Baldwin, who calls the general problem my paradox (that is, “Pierce’s paradox”). I mentioned the so-called paradox in “Anthropology of Mathematics,” suggesting that we may learn some things when we are too young to question them, then teach them when we are older without going back to question them.
According to Euphemia Lofton Haynes, as in a quotation I made earlier, we mathematicians also use incomplete induction. Her example is the Pythagorean Theorem:
It was observation of the fact that the squares of certain numbers are each the sum of two other squares; the collection of these sets of numbers by the method of trial; the observation that apparently these and only these triplets are the measures of the sides of a right triangle—that is, observation, experimentation, incomplete induction—processes common to the experimental sciences—that led to the discovery of the Pythagorean Theorem.
That is how I would edit the typescript, which actually reads as follows:
It wasby observation of the fact that the squares of certain numbers are each the sum of two other squares; the collection of these sets of numbers by the method of trial; the observation that apparently these and only these triplets are the measures of the side of a triangle. That is by obser- vation, experimentation, incomplete induction processes, common to the experimental sciences, led to the discovery of the Pythagorean Theorem.
The account is plausible. I’m not sure I didn’t offer a similar account of the Pythagorean Theorem, when an English teacher at St Albans tried to explain the distinction between deductive and inductive logic. I may have been in the eighth grade; at any rate, my classmates and I were too young to have seen official proofs in mathematics. We must have been taught the Pythagorean Theorem somehow, so that our English teacher could give it as an example of a general rule from which specific cases could be deduced. He may even have said that the rule itself was established deductively. In my memory at least, I responded that the rule must have been discovered inductively.
Perhaps before hypothesizing the Pythagorean Theorem itself, in all of its generality, somebody did observe that 3^{2} + 4^{2} = 5^{2}, and 5^{2} + 12^{2} = 13^{2}, and 8^{2} + 15^{2} = 17^{2}, and that the bases in each equation were sides of a right triangle. The observation about the sides of a right triangle may have come not from measuring with a ruler, but from pictures. Pictures make it clear that the triangles are indeed right-angled. But then a single picture can also serve as a proof of the general theorem, and it is conceivable that one may thus discover the theorem, without having considered particular triangles whose sides one knows the measures of.
Likewise may a picture replace our inductive proof that for all counting numbers n,
2(1 + … + n) = n(n + 1).
Here’s the picture:
An array of n rows, each row consisting of n + 1 dots, contains n(n + 1) dots in all. It can also be broken into two triangles as indicated, each triangle consisting of 1 + … + n dots.
You may say that’s not a proof. Dorofeev, Potapov, and Rozov say it’s not. The picture establishes the assertion, only for a special case.
The incompleteness of [such a] proof is obvious. We establish the formula for a few values of n and then draw the conclusion that it is true for any [counting number] n. With that approach, it is possible to “prove” the following assertion: for an arbitrary integer n, n^{2} + n + 41 is prime. Indeed, for n = 1, 2, 3, 4 we have 43, 47, 53, 61—all primes. “Consequently”, the assertion is proved, though it is clear that, for example, when n = 41 the number n^{2} + n + 41 is divisible by 41.
Our picture proof that 2(1 + … + n) = n(n + 1) is based on the case when n = 5; but it should be obvious that there is nothing special about 5 here. This makes the proof different from the false proof of the primality of all numbers n^{2} + n + 41. We could also write our picture proof in algebraic form:
2(1 + … + n)
= (1 + … + n) + (1 + … + n)
= (n + … + 1) + (1 + … + n)
= (n + 1) + … + (n + 1)
= n(n + 1).
However, such a proof may be too obscure, needing too much intuition. In that case, one can always fall back on the proof by induction.
If I understand correctly, in the US there’s an attempt to teach arithmetic in a more intuitive way than by just applying the traditional algorithms. Thus for example to add 79 and 18, instead of first adding 9 and 8 to get 17, then adding the tens digit here to the sum of 7 and 1 to get 9, so that 79 + 18 = 97, you may do better to think
79 = 80 − 1,
18 = 20 − 2,
79 + 18 = 80 + 20 − (1 + 2) = 100 − 3 = 97.
That’s fine, but it seems to me one should have the fall-back algorithm of performing the addition digit by digit, right to left, as I described first. This provides a mechanism for resolving disputes about sums, as well as for not having to think and be creative, which nobody wants to do all the time.
We have suggested three proofs that 2(1 + … + n) = n(n + 1), not all of which may be accepted as proofs by everybody. I have suggested that some standard proofs found in textbooks are bogus. What then of my assertion that mathematical proofs are universally valid?
A proof is not a picture or an arrangement of typographical characters, any more than a work of art is pigment in oil on canvas. The physical things are just the means we use to understand the real thing.
How we come to see the real thing may itself be obscure. The Soviet textbook has some good comments here, though they be about seeing the assertion rather than its proof:
It must be stressed that the induction method is a method of proof of specified assertions and does not serve as a derivation of these assertions. For instance, this method cannot be used to obtain the formula of the general term [of an arithmetic progression or—let us add—of the sum of the first n counting numbers]; however, if we have found the formula in some way, say by trial and error, then the proof of it can be carried out by the induction method … In this process, of course, the method of trial and error, the mode of obtaining a formula or an assertion is not a necessary element of the proof. On the basis of some kind of reasoning or guessing we conjecture an assertion, then we can proceed to proof by induction.
This distinction between finding an assertion and proving it is one that we have seen Euphemia Lofton Haynes discuss for the natural sciences. How then is mathematics to be distinguished from, say, physics? According to Haynes,
Although a mathematical system is syllogistic and postulational in style and form, the assumption that syllogistic reasoning is the very foundation of all mathematical activity is another inherited fallacy which is the result of the reign of methodology.
I think another way to say “syllogistic and postulational” is deductive and axiomatic. Perhaps physics can be this, but the conclusions of the deductions still have to be checked, to see whether they fit the experimental data. By contrast,
the observation of the mathematician transcends that of the natural scientist in that it is not confined to observations of the physical eye …
The mathematician builds … worlds that are possible logically. Whether they are possible in our world of sense is of no concern to him.
That worlds are “possible logically” means they can be deduced from postulates. This is what distinguishes mathematics among the sciences. Mathematics is the deductive science.
Pages 22–3 of Joseph Needham,
Mathematics and the Sciences of Heaven and Earth,
Volume 3 of Science and Civilization in China
(Cambridge University Press, 1959),
concerning the Chou Pei Suan Ching or Zhoubi Suanjing
This is a little about mathematics, and a little about writing for the web, but mostly about the nuts and bolts of putting mathematics on the web. I want to record how, mainly with the pandoc
program, I have converted some mathematics from a LaTeX file into html
. Like “Computer Recovery” then, this post is a laboratory notebook.
The mathematics is a proof of Dirichlet’s 1837 theorem on primes in arithmetic progressions. This is the theorem that, if to some number you keep adding a number that is prime to it, there will be no end to the primes that you encounter in this way.
For some reason, I wanted to learn the proof. Maybe this had to do with having given courses on the Prime Number Theorem of 1896 at the Nesin Mathematics Village, but not having been able to teach there this summer, owing to the Covid pandemic. Dirichlet’s theorem could be part of a course at the Village.
I read the proof in Landau’s Elementary Number Theory, originally published in 1927, ninety years after Dedekind’s theorem. I wrote out the argument, according to my understanding. If you want to read what I wrote, there are:
a pdf file (24 pages, size A5) based on the LaTeX file that I composed;
an html file derived from my LaTeX file by means of the pandoc
program.
Recently I encountered a page of links to over two hundred expository articles by a mathematician. I looked at one article, and I wondered what kind of audience would both
need to be told that an automorphism of a field is a bijective homomorphism from the field to itself,
already know the field of p-adic numbers.
I might have thought somebody who knew the p-adic numbers would also know something of Galois theory; but maybe not.
In my own article on Dirichlet’s theorem, I had already given accounts of
what the reader should know,
what I know as an amateur of number theory.
I become increasingly aware of how webpages can be visited from anywhere, although their composers seem unaware of this. If you are a newspaper, what city are you in? If a university, what state or province? If a business, what country? If you are a blog, what are you trying to do, and how can your visitors decide whether to spend time with it?
I have tried to make my own “About” page useful in this way.
I wrote my article on Dirichlet’s theorem, to satisfy my curiosity. Then I remembered posting on this blog my memoir of life on a farm. I had obtained the html
file from a LaTeX file using pandoc
, and I had been pleased with the results. Notably, pandoc
had kept my footnotes as such: they were at the end of the same html
file.
I decided to try to convert the Dirichlet article to html
.
WordPress allows the embedding of TeX and LaTeX code, but converts the code to images. The tex4ht
program does this as well. I wanted to avoid images.
The pandoc
program does not create images, but tries to express mathematics (along with everything else) as text. However, there are mathematical expressions that LaTeX accommodates, but html
does not, or not so well. The pandoc
program leaves those untouched.
I therefore edited my original LaTeX file, to turn all of the mathematics into something that pandoc
could interpret. I tried to make it as easy as possible to switch between a LaTeX file as such—a file to be compiled by the latex
program—and a file to be converted to html
by pandoc
.
Here is what I did.
Apparently pandoc
can handle the commands that you define, even with arguments; but not default arguments. For example, my
\newcommand{\Zmod}[1][k]{\mathbb Z/#1\mathbb Z}
didn’t work until I removed [k]
.
I like the compactitem
and compactenum
environments of the paralist
package, but apparently pandoc
does not recognize these, so I switched back to itemize
and enumerate
.
Fractions are the big challenge. I used
\renewcommand{\frac}[2]{#1/#2}
.
One will then need to use parentheses if a numerator or denominator is a sum; but I had this problem in only one case. The reader still has to understand a/bc as meaning a/(bc), although I have allowed (a/b)c to appear as a/b⋅c (this happens when c is a summation with ∑).
Summations themselves are a problem; I rewrote each \sum_{j=i}^{n}
as \sum_{i\leq j\leq n}
. (One could also define a new command with three arguments here.)
For one use, I defined:
\DeclareMathOperator{\nCk}{C}
\renewcommand{\binom}[2]{\nCk(#1,#2)}
None of the environments align
, gather
, and multline
for displaying more than one line of mathematics together gets interpreted by pandoc
; therefore I put each line to be displayed into its own equation
environment.
There is a similar problem with the cases
environment; I recast using itemize
.
Apparently pandoc
cannot handle a negated symbol such as \not\equiv
, so I used \nequiv
from txfonts
.
I rewrote \pmod{#1}
as \;(\text{mod }#1)
.
Doing that much gave a file that pandoc
would render as pure html
. However, there were remaining issues.
The program was not dealing properly with the bibliography I had created with BibTeX. Also pandoc
gave my section headings <h1>
tags. I took care of this by running the following command:
pandoc --base-header-level=2 --bibliography ../../../references.bib --filter pandoc-citeproc dirichlet-simple.tex -o dirichlet-simple.html
The online documentation says --base-header-level=
is deprecated, and one should use --shift-heading-level-by=
; but this didn’t work for me. (If pandoc
didn’t come with my Ubuntu Linux installation, I may have installed it when converting the docx
file of somebody else’s philosophy paper to LaTeX.)
At this point, the problem remained that pandoc
did not deal properly with
theorem environments,
labels of equations.
I found discussion of these on the Google group for pandoc
. I have not understood why they should be a problem, or at least why references should be a problem.
For example, pandoc
prints the label of a theorem as plain text. It could print the label of an equation in the same way, but apparently it doesn’t.
One can apparently customize pandoc
, but for now I don’t know how. Therefore I have done the following (probably not something to be done on a regular basis):
formatted and numbered by hand my theorems and lemmas;
formatted the proofs by hand, inserting $\Box$
at the end if this is text, and \qquad\Box
if it’s an equation (pandoc
could not deal with \qed
);
changed \label{#1}
(which I habitually place just after \begin{equation}
) to \mylabel{#1}
, defined as (#1)\qquad
;
redefined \eqref{#1}
as (#1)
;
changed all of my equation labels to the desired serial numbers;
changed every {equation}
to {equation*}
.
After all of this, I cleaned up the html
file created by pandoc
by:
putting blank lines between paragraphs (for ease in reading the html
file);
changing <span class = "math display">
to <div style= "text-align:center;">
;
putting <div style = "text-align:justify; margin-left:10%; margin-right:10%;">
at the head, and </div>
at the foot, as with all of my blog posts and pages.
Now I am using pandoc
to create the present html
file from the plain text file that I originally typed.
The bulk of this post is a summary of the chapter on art in Collingwood’s Speculum Mentis: or The Map of Knowledge (1924). The motto of the book is the first clause of I Corinthians 13:12:
The chapter “Art” has eight sections:
Art as Pure Imagination
The Work of Art
The Monadism of Art
Meaning in Art
Knowledge as Question and Answer
Art as a form of Error
The Dialectic of Art
Play
The summary below follows these.
Art is one instance of an activity that may succeed or fail. In the strictest sense, every activity has this property; for there is a distinction between what we are doing and what we are trying to do.
I begin by observing that, with reference to quoted words of Agnes Callard about Aristotle. Callard happens to be named in the acknowledgments of Lost in Thought by Zena Hitz, whose words about analytic philosophy I shall ultimately quote as well.
Editing this post from an email I drafted on July 26, I have removed (as not having been intended for publication) some words of other persons that I had quoted from a 2012 email discussion of art and Speculum Mentis.
On July 21, Agnes Callard had a good essay in the New York Times, “Should We Cancel Aristotle?” Sure, Aristotle liked slavery and stuff; but reading him – reading him literally – doesn’t mean we agree with him. Neither does it make us agree with him, unless perhaps he is right after all. In Callard’s words:
Yet I would defend Aristotle, and his place on philosophy syllabuses, by pointing to the benefits of engaging with him. He can help us identify the grounds of our own egalitarian commitments; and his ethical system may capture truths – for instance, about the importance of aiming for extraordinary excellence – that we have yet to incorporate into our own.
And I want to go a step further, and make an even stronger claim on behalf of Aristotle. It is not only that the benefits of reading Aristotle counteract the costs, but that there are no costs. In fact we have no reason at all to cancel Aristotle. Aristotle is simply not our enemy.
I, like Aristotle, am a philosopher, and we philosophers must countenance the possibility of radical disagreement on the most fundamental questions. Philosophers hold up as an ideal the aim of never treating our interlocutor as a hostile combatant. But if someone puts forward views that directly contradict your moral sensibilities, how can you avoid hostility? The answer is to take him literally – which is to say, read his words purely as vehicles for the contents of his beliefs.
Aristotle urges “the importance of aiming for extraordinary excellence.” As philosophers, he and Callard “hold up as an ideal the aim of never treating our interlocutor as a hostile combatant.” They have aims; we have aims; and yet we may not achieve them.
In particular, the words of Aristotle are not simply “vehicles for the contents of his beliefs”; not boxcars in which the Philosopher loaded his beliefs in order to deliver them to us. Before the loading, what could the beliefs have been? The words represent an attempt to work out the beliefs. The attempt may have been only partially successful.
The idea is in Collingwood, who says for example in The Principles of Art (1938):
The proper meaning of a word … is never something upon which the word sits perched like a gull on a stone; it is something over which the word hovers like a gull over a ship’s stern. Trying to fix the proper meaning in our minds is like coaxing the gull to settle in the rigging, with the rule that the gull must be alive when it settles: one must not shoot it and tie it there. The way to discover the proper meaning is to ask not, ‘What do we mean?’ but, ‘What are we trying to mean?’
The word under consideration, by Collingwood then and by me now, is art.
A beginning of such considerations is recorded in “Poetry and Mathematics,” a post beginning with talk about writings in a couple of Australian literary publications. One of the essays was about poetry and mathematics, by a poet who had studied mathematics.
I found some reason to observe how Michael Oakeshott had worked on the theme of Collingwood’s Speculum Mentis: or The Map of Knowledge of 1924.
The book can be classified, with Religion and Philosophy of 1916, as belonging to Collingwood’s juvenilia. This could have made it more accessible than a mature work; but it didn’t.
Hegel’s Phenomenology of Spirit was inaccessible. In my day at St John’s College, we read short excerpts of this book, and if memory serves, one of our two seminar leaders admitted from the start that he didn’t know what the book was about.
The other seminar leader had read the tome in graduate school, but perhaps didn’t feel much more confident about it. He did suggest that just reading the whole thing might be the way to go, if reading little bits at a time did not clear it up.
Nonetheless, reading Speculum Mentis section by section seems to clear some things up for me.
Here is my attempt to summarize the chapter called “Art.”
“Art” is Chapter III, the first two chapters being
“Prologue,” which begins, “All thought exists for the sake of action”;
“Speculum Mentis,” which begins, “Our task, then, is the construction of a map of knowledge” – spoiler alert, the map never gets constructed.
Chapter III has eight sections. Collingwood will not explain till § 4 that the first three sections give a one-sided account of art, based on Vico and Croce: an account that, since the Renaissance, has tended to replace the ancient account.
The mode of thinking in the chapter then would seem to be hypothetical, so to speak, as art itself is.
“Art is the simplest and most primitive, the least sophisticated, of all possible frames of mind,” not because only children and “savages” make art or are even best at it, but because art imposes no requirement of literal truth.
For example, it is irrelevant to Cymbeline the play, as a work of art, whether there really was a king called Cymbeline who acted as Shakespeare’s character does.
“A philosophical theory must be capable of being conceived as a whole, a historical narrative, of being narrated as a whole – narrated, that is, as true – a work of art, of being imagined as a whole.”
“Art, then, is pure imagination.”
Nonetheless, art is trying to be something: not literally true, but beautiful. Art can fail to be this. The possibility of failure “is the standing refutation of all emotional and sensationalistic theories of art.”
Nonetheless, beauty is not a concept. There are no laws or principles for achieving it. If you try to explain the beautiful in terms of its form, you fail.
This is a paradox, whose resolution is that beauty is not a concept, but “the guise under which concepts in general appear to the aesthetic consciousness.”
The paradox seems real to me. A way I try to understand the resolution is to think of Collingwood’s 1916 essay “The Devil,” in which evil is found to be neither the negation of good, nor the opposite of good, but the counterfeit of good. As with being beautiful, so with being good, there is no law for it. “It is a duty, indeed it is the spring of all moral advance, to criticise current standards of morality,” and yet the essence of evil is also to engage in this criticism.
Meanwhile, being an act of imagination, the work of art is not a physical object. The physical object may help the rest of us achieve the aesthetic activity that the artist has accomplished; but this is not really true, since “one never sees anything in anybody’s work but what one brings to it.”
I would object here that Collingwood (or rather Croce, as Collingwood seems to write in his guise) discounts the possibility that the physical work of art teaches us how to see. We shall come back to this in § 4.
“Every fresh aesthetic act creates a new work of art, though one such act may last for five years at a time.”
“Works of art always ignore one another and begin each from the beginning: they are windowless monads.”
“Art in its pure form is therefore unaware even that it is imagination; the monad does not know that it is windowless; the artist does not say ‘I am only
imagining’, for that would be to distinguish imagination from knowledge, and this he does not do. Hence the aesthetic life of children and uneducated people results in what an unintelligent critic calls lying and hallucination.”
I would suggest, as an example, that the current POTUS does not lie, because he has no conception of the truth. He has been allowed to continue living in the dream-world of a child (and this is a terrible indictment of the United States).
Now Collingwood points out what I said at the beginning, that the foregoing account of art is one-sided.
The “special problem of the philosophy of art to-day” is to reconcile that account with the older one, whereby art could teach moral, religious, or philosophical truths.
Imagination must go to work on something:
“Our dreams have a certain continuity with our waking life, and imagination never cuts itself wholly adrift from fact.”
The idea of art as a kind of dreaming comes back in § 7.
To imagine is to suppose, which is to question. To answer is to assert.
“Questioning is the cutting edge of knowledge; assertion is the dead weight behind the edge that gives it driving force.”
I note that Pirsig uses a similar metaphor in Zen and the Art of Motorcycle Maintenance. What he calls Romantic Knowledge is the leading edge of the train of knowledge; Classical Knowledge is the train itself, the engine and cars.
Art wants only to ask questions. In life you can’t do that. “The artist is an artist only for short times; he turns artist for a while, like a werewolf.”
Thus the life of art is unstable. For example, a school of art, once founded, declines. (Herbert Read opens his Concise History of Modern Painting—which I first read while working on a farm in 1988—by quoting Collingwood on this decline.)
Collingwood does not spell out the reason that I would give for the decline. Founding a school of art (such as Impressionism, or Abstract Expressionism) means figuring out how to do something (e.g. paint shadows with colors, or just splash paint on canvas). This means finding a technique. Once you have the technique, you can go on applying it without any fresh act of imagination.
I think Collingwood will suggest such an idea about technique in The Principles of Art. There he finds it important to distinguish art from craft. They are not really separable in fact, and I think Collingwood says this now (in § 7), though without referring to craft as such:
“The cleavage between means and end, technique and inspiration, talent and genius, materials and result, is inevitable in the life of art, and only those who idealize that life by looking at it from the outside deny the dualism. The artist knows that he can only get his work of art by passing through a non-aesthetic world which is that of facts, training, daily life, and so forth.”
In The Principles of Art, the presence of distinctions between (i) means and end and (ii) raw material and finished product will be two of the differentia of craft as distinct from art.
Collingwood considers dreams as works of art. A dream has a structure: this is why we can call it a dream, one dream, as opposed to some kind of un-unified mélange. The structure is not “unconscious,” since everything we can know about the dream comes from our being conscious of it. But the structure is implicit until made explicit by psycho-analysis.
Knowing the possibility of such analysis, if you proceed to construct your “dreams” consciously, you have become an artist.
I put “dreams” in quotes because I take the word here to refer to any work of the imagination. The artist “conceives [the work of art] in advance of imagining it, in the sense that at any given moment in the process of creation he has in his mind a criterion which enables him to distinguish between the right and the wrong way of continuing the process of imagination itself.”
The artist’s awareness of such a criterion is why Collingwood will later (in The Principles of Art) coin the term “criteriological” for such sciences as aesthetics (also logic, ethics, and economics; I add the example of grammar, which I used in my best attempt to explicate the notion of criteriological science).
In art there is a distinction between form and content, but if it is “developed,” the work of art ceases to be that: for example, poetry becomes prose.
I mentioned the Australian poet (her name is Anupama Pilbrow) who had studied and written about mathematics. There may be poetry in mathematics, but the way mathematics is communicated is entirely prose.
As art is the most rudimentary thought, play is the most rudimentary action.
We can find reasons for play, whether childish or adult; but “all such explanations of play are in part mythological and forced.”
“God may be pictured as an artist, or as playing, with far more verisimilitude than as a scientist or a business man.”
Still people will ask why we play. “What is the use?” they ask.
It is no excuse “that the overstrained spirit must be allowed some relief from the burden of its responsibilities; for these responsibilities, properly understood, are nothing but its own highest and freest life, and to face them is to find, not to sacrifice, our happiness.”
Life is always an adventure, and “play, which is identical with art, is the attitude which looks at the world as an infinite and indeterminate field for activity, a perpetual adventure.”
In her recent book Lost in Thought: The Hidden Pleasures of an Intellectual Life (Princeton University, 2020), St John’s College alumna and tutor Zena Hitz says that in grad school, one thing she learned was
the delightful mental gymnastics of analytic philosophy, in which any manner of thesis whatsoever is defended, explored, and almost always refuted.
That is perhaps the comic version of a tragedy that Collingwood tells of a senior colleague at Oxford, now counted among the analytic philosophers:
[H. A.] Prichard, developing his extraordinary gift for destructive criticism, by degrees destroyed not only the ‘idealism’ he at first set out to destroy but the ‘realism’ in whose interest he set out to destroy it, and described a path converging visibly, as years went by, with the zero-line of complete scepticism.
That’s from An Autobiography (1939). Speculum Mentis is itself autobiographical, in the sense of reviewing careers that Collingwood might have selected. The book covers in turn art, religion, science, history, and philosophy. In 1939 (the autobiography being with the publisher), Collingwood wrote his wife from Java that he wished he had been a novelist.
Zena Hitz starts her book with a Prologue (“How Washing Dishes Restored My Intellectual Life”), which reviews her life story: growing up in San Francisco in a house full of books; heading east to Annapolis for college; grad school in philosophy; a reconsideration after 2001/09/11; staying with philosophy and getting a job in it; joining the Roman Catholic Church; investigating some convents and moving to a sort of Catholic commune (Madonna House) in Ontario; finding her calling to be back at St John’s.
]]>In the fall of 2017, I created what I propose to consider as being both art and mathematics. Call the art conceptual; the mathematics, expository; here it is, as a booklet of 88 pages, size A5, in pdf format.
More precisely, the work to be considered as both art and mathematics is the middle of the three chapters that make up the booklet. The first chapter is an essay on art, ultimately considering some examples that inspire my own. The last chapter establishes the principle whereby the lists of numbers in Chapter 2 are created.
Each of the lists is of the same ordered pairs of integers. There are a thousand pair, less five, in five columns on each page. For each pair (n, k) on the lists,
1 < n < 997,
0 < k < 996, and
7^{k} ≡ n (mod 997).
The last expression is read as, “7 to the power k is congruent to n modulo 997”; it means 7^{k} − n is a multiple of 997. One can say then that k is the (discrete) logarithm, to the base 7, of n modulo 997; and one can write
k ≡ log_{7} n (mod 996).
The first list of the pairs (n, k) is according to n; the second, k. One can use the lists in tandem to turn multiplication problems into addition problems, because of one of the laws of exponents, also expressed as a law of logarithms:
b^{k} ⋅ b^{ℓ} = b^{k + ℓ},
log_{b} (n ⋅ m) = log_{b} n + log_{b} m.
The good old-fashioned slide rule exploits these laws. A basic slide rule is like two rulers, each measuring 10 inches; but the number printed k inches from the beginning is 10^{k/10}, so that k/10 is the base-10 logarithm of this number. One multiplies the printed numbers by adding their distances from the beginning, which is marked 1, not 0.
Before the advent of the pocket calculator, the slide rule had practical value. It may still be useful for learning about logarithms, and logarithmic scales can be useful for graphing such things as Covid-19 cases (as on my friend’s handy site).
To show my students, I actually used discrete logarithms to make slide rules (such as the circular slide rule depicted on the cover of my booklet). I don’t however think my tables of discrete logarithms have any practical value. Thus they may be Chindōgu.
My tables can exist because
997 is a prime number;
prime numbers have “primitive roots”;
7 is a primitive root of 997.
I spent much of the fall semester of 2017, proving the theorem that prime numbers have primitive roots. This was in my department’s first-year number-theory course. Thus Chapter 3 of my booklet concentrates a lot of work for the beginner into a few pages. The chapter also looks at the historical origins of some of our terminology, such as “modulo” (which is not actually in any of the print dictionaries in my personal library).
For students in my 2017 course, I created a summary, “Sayılar Kuramına Giriş Özeti” (5 pages, size A4, two columns, landscape orientation, in our language of instruction, which is Turkish).
Chapter 3 of the booklet begins by reviewing Euclid’s proof of the commutativity of multiplication of counting numbers; I had taken this up also, earlier in 2017, in a blog post, “The geometry of numbers in Euclid.” I had once considered trying to teach number theory as Euclid did; but I decided this would be too strange.
In 2013, I had learned of a conceptual work by Claude Closky, “The first thousand numbers classified in alphabetical order.” It is what it is called. I translated it into Turkish as “Alfabe sırasına göre sınıflanmış ilk bin sayı” (8 pages, size A5), and I created for this document a page of my departmental website. Now I am delighted to see that my page is linked to by what appears to be Closky’s own page.
Also linked to from that page is “The First M Numbers Classified in Alphabetical Order,” by Nick Montfort (2013); the work is a program to list the first thousand Roman numerals in alphabetical order.
Not knowing about that, I made such a list myself, later that year, up to MMMCMXCIX; the result, “The Roman numerals in alphabetical order,” is 165 pages of size A5. Like the discrete logarithms, the list of Roman numerals may also be Chindōgu, because in principle one could use it to interpret such numerals, but nobody ever would; at least nobody should; but then my friend once said she had given her secretary a chart as a reminder that 1/2 = 0.5 and 1/4 = 0.25.
I am pleased to have at hand all of the books I used in preparing my booklet. Some I have only as electronic files; I photographed the others for this post. Then I realized that I had left out the book on Duchamp. I noticed that its cover was complemented by that of Rabih Alameddine, An Unnecessary Woman (New York: Grove Press, 2013). The woman of the title is useless for making translations that nobody ever reads.
Apparently I read her story in 2015, since the receipt tucked inside the novel is dated to April of that year. Another piece of paper tucked inside is a cinema ticket, marking the page of a nice paragraph:
In a silly essay on Crime and Punishment, a critic suggests that Raskolnikov is the epitome of the Russian soul, that to understand him is to understand Russia. Tfeh! Not that the proposition isn’t true; it may or may not be. I’ve yet to meet Russia’s soul. What the reviewer is doing is distancing himself from the idea that he too is capable of killing a pawnbroker. We’re supposed to infer that only someone with a Russian soul would.
Some art is good, precisely for not distancing you, but giving you both the idea that you can do it for yourself, and the impetus to do it.
]]>This past spring, when my university in Istanbul was closed (like all others in Turkey) against the spread of the novel coronavirus, I created for my students an exercise, to serve at least as a distraction for those who could find distraction in learning.
The exercise uses no more mathematical tools than may be found in an algebra course in high school; yet it serves the purposes of university mathematics, as I understand them.
My course was in analytic geometry, and our first-year students take this course in their second semester, after a first-semester course of reading and presenting the propositions of Book I of Euclid’s Elements. (The students also have courses in calculus, number theory, and linear algebra in their first year.)
The writings and instructions that I gave to my students are gathered on a page that I called “Analitik Geometri Özeti” (Summary of Analytic Geometry). That is what at first I expected the page (and linked pages) to be: a précis, both of what I had already lectured on, the few times I had met the students in person, and of what I would have gone on to lecture on.
Then I conceived the exercise that shall describe here. I am also posting, as a pdf file, a worked example, in English. (As of August 25, 2020, the linked document has 31 pages of size A5, including 14 figures. This is a shortening of the document that I originally posted, which has 35 pages and 15 figures. The old document is not in error, but I have seen a way to simplify some computations.)
The exercise illustrates the correspondence between conic sections and second-degree equations in two variables.
This correspondence is worked out in some books, such as M. Vygodsky, Mathematical Handbook: Higher Mathematics (Moscow: Mir, Fifth printing 1987), where there is a thorough treatment. However, here and in other sources, the correspondence is worked out by means of changes of coordinates. I use other means, namely the tools of high-school algebra, as I said.
Moreover, I do not contrive examples that will work out neatly. I ask students to supply their own examples.
Briefly, the exercise is to
choose three points in the coordinate plane,
use these to define a central conic, namely an ellipse or hyperbola,
find the axes of that conic.
One can plot one’s results on graph paper, by way of confirming that one’s computations are correct.
There could be a similar exercise involving the parabola.
Perhaps every course of analytic geometry will teach the ellipse as being given by an equation of the form
(x/a)^{2} + (y/b)^{2} = 1;
the hyperbola,
(x/a)^{2} − (y/b)^{2} = 1.
These are central conics, their centers being the origin. The conics have axes, one with endpoints (±a, 0); the other, (0, ±b). The latter segment, since it does not actually meet the hyperbola, may not normally be considered an axis of the hyperbola. We shall nonetheless call it that, following Apollonius, and noting that the two axes are shared with the conjugate hyperbola, given by
(y/b)^{2} − (x/a)^{2} = 1.
Apollonius proves that every chord of a central conic that passes through the center is a diameter and has a conjugate diameter. Vygodsky states the result without proof; it means that chords of the conic that are parallel to one of the diameters are bisected by the other.
In another post, “Elliptical Affinity,” I tried to make the Apollonian proof visual. The exercise that I am reviewing now suggests a simple “analytic” or algebraic proof.
The conic sections of Apollonius are just that, sections of a cone by a plane. As such, they come with diameters; but the cone may be oblique, and then the conjugate diameters may fail to be at right angles to one another. If they are at right angles, they are axes. It is then a theorem of Apollonius that every central conic has axes in this sense. Our exercise uses the idea of the proof.
Given, or choosing, linearly independent points (a, b) and (c, d), defining
δ = ad − bc,
we obtain an equation
(dx − cy)^{2} ± (bx − ay)^{2} = δ^{2}.
We can understand this to define an ellipse or hyperbola (depending on choice of sign) with conjugate diameters whose endpoints are the original two points and their negatives. In the terminology of Apollonius (concerning which see my article “Abscissas and Ordinates”), those endpoints are vertices. If we translate the center to a third given or chosen point, (e, f), this means replacing x with x − e and y with y − f. We can then rewrite the equation in the general form
Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0.
We have now created the exercise of analyzing this equation. By completing squares, we obtain an equation
(αx + βy + κ)^{2} ± (γy + λ)^{2} = δ^{2}.
This defines the same conic as before, while displaying a new pair of conjugate diameters: these are segments of the horizontal line given by γy + λ = 0 and the line given by αx + βy + κ = 0. These lines will intersect at (e, f). Translating this back to the origin, we have the conic defined by
(αx + βy)^{2} ± (γy)^{2} = δ^{2}.
In fact δ is an invariant of the conic, and the endpoints of the diameters will be (±γ, 0) and (±β, ∓α). In any case, there is a circle defined by the equation
(αx)^{2} + (αy)^{2} = δ^{2}.
The circle shares its horizontal diameter with the conic and also intersects the conic in two more points. The four points of intersection are the vertices of a rectangle. One can now compute the slopes of the sides of that rectangle. The axes of the conic have these slopes, and we can now compute their endpoints.
That’s the exercise, in a nutshell. I was honor-bound to work out some examples for myself, before asking the same of students. The computations can be fiendishly difficult to get right. Though having chosen one’s parameters to be single-digit whole numbers, one can end up having to work with square roots of square-free four-digit numbers. That in itself may not be a problem; but one has to get the computations for the radicand right in the first place.
One can do the computations with a calculator, but one still must enter the parameters correctly.
One can write a computer program to do all of the computations, given the initial parameters. If a student can write such a program, that’s great. I did it, in order to check every students’ work. (Most students chose parameters unique to them; I did allow students to work in groups, but many did not take this option.)
I wrote my program in LaTeX, using the ifthen package in order to have every equation and every coordinate typeset in simplest form. I used the pstricks and pst-eucl packages to draw the graphs, and the pgf package to print decimal approximations of coordinates. Doing all of this—learning how to do it—was a lot of work in itself; however, this work does not seem to duplicate what is done anywhere else, in particular by WolframAlpha.
Thus my exercise is practically immune to cheating. To get it right, you probably have to learn some mathematics, even if you have friends who already know the mathematics.
If you have the money, maybe you can pay some unscrupulous person to learn the mathematics for you and apply it, the way you can pay somebody to complete your writing assignments, even your doctoral dissertation. People who want to be ignorant are able to succeed in being so.
In the exercises of his own textbooks, Michael Spivak is fond of asking for complex computations, as for example of the derivative of the function f given by
f (x) = sin (x/(x − sin (x/(x − sin x)))).
This is the last of a collection (in Calculus, 2nd ed., 1980) of eighteen differentiation exercises that are prefaced by the remark,
It took the author 20 minutes to compute the derivatives for the answer section, and it should not take you much longer. Although rapid calculation is not the goal of mathematics, if you hope to treat theoretical applications of the Chain Rule with aplomb, these concrete applications should be child’s play—mathematicians like to pretend that they can’t even add, but most of them can when the have to.
The author of the present blog post spent countless hours over a number of weeks, engaged in both the theory and practice of the exercise being discussed here. Students should expect to spend hours understanding the exercise and working out their own examples.
The payoff is twofold.
Students can see for themselves when they have got the right answer.
The right answer is aesthetically pleasing, if symmetry be a form of beauty, which it classically is (see my essay, “On Commensurability and Symmetry”).
In particular, students can plot the endpoints that they find for the axes and see if they fit the original conic; the conic will then be symmetric about the axes.
If you solve Spivak’s exercise, there’s not a good way to see that you have done it right; also, the answer itself is of no interest. You can compare your answer to Spivak’s or somebody else’s. You can also just review your own work carefully, step by step; or you can do the same work again, to see if you get the same results.
In fact I had to do this a number of times in working out my analytic geometry problem. Even when I knew from graphing that there must be an error in my work, I could have a hard time finding it. Eventually I did find it.
Our graduates may never again use the particular mathematics that we try to teach them. If they learn
to persist until they have solved a problem,
to verify their solution, according to standards that they themselves recognize,
this is an ability that will benefit them.
Nonetheless, I have a hedonistic theory of education. Learning is a pleasure, however much it may be accompanied by travail. One then has to learn to accept the travail, or turn it into something else.
If there is any pleasure in solving the differentiation exercise above, I think it is the pleasure of having the power to do what is asked.
If one can analyze an arbitrary second-degree polynomial equation in two variables, in order to obtain (in the non-degenerate cases) a conic section with its axis or axes, this is like knowing how to find flowers in a forest.
In fact my students did not seem to appreciate the visual beauty of the exercise that I assigned them. Of the students who chose to undertake the exercise in the first place, most of them could imitate my own computations, using their own parameters; but not many could reach the end without errors. Of those who could, few confirmed their results with a proper graph, even though I did this in my own examples. (I initially posted my examples as typeset by LaTeX; later I posted handwritten versions, with hand-drawn graphs.)
Some students did plot vertices that obviously did not fit their curves; I wish they could have said (and thus I wish I had encouraged them to say), “I see that I must have made a mistake, but I cannot find it.”
Perhaps I must conclude that my exercise is too hard, even though I see one of its key concepts—the intersection of a circle and a concentric ellipse with oblique axes—in Weeks and Adkins, Second Course in Algebra (1971), which I used in my freshman year of high school.
My own students might better do my exercise if we could all work together face to face. Even then, time spent on the exercise might be better spent learning, for example, that a linear equation in three variables defines not a line, but a plane. I was warned, in fact, that students have seemed not to learn this from their first-year courses.
However, if students have not learned this, they would seem not to have learned to visualize properly. Part of learning linear algebra is learning what not to try to visualize: for example, how there can be four or more mutually orthogonal coordinate axes. Still, I suppose one does visualize how any two of the axes are at right angles to one another.
One should acknowledge here that the visual may be the palpable, perceived with the hands instead of the eyes; the word visual itself, from Latin, is related to idea from Greek, along with wit and wise from Old English.
My own exercise tries to make the point of Descartes: that algebraic equations describe something geometrical, something real.
Perhaps only algebra and computations have been emphasized in our students’ educations and on the national multiple-choice university entrance exam. Students learn how to find answers, but not how to know that they are right answers, except insofar as a teacher tells them that their technique is correct.
The beauty of mathematics is that we can know for ourselves that our answers are correct. However, society may discourage such independence.
Revised August 25, 2020
Content warning: suicide.
The following sentence is bold in the last paragraph of an essay: “the science is clear and conclusive: sex is not binary, transgender people are real.” I don’t know what the writer means by this. As far as I can tell, as a biological concept used for explaining reproduction, sex has two kinds or parts or sides or aspects, and the essay tacitly affirms this; at the same time, obviously persons called transgender exist.
☾ ☿ ♃ ♄ ☉
The title of the essay is a command: “Stop Using Phony Science to Justify Transphobia.” I can support that. I don’t even need the qualifier “phony.” If transphobia is the kind of morbid fear suggested by the suffix “-phobia,” then science ought to help dispel this, not promote it.
One might also just say, Stop using phony science.
The writer of the essay in question (dated June 13, 2019) is called Simón(e) D Sun. I imagine the parenthetical letter is meant to suggest alternation between masculinity and femininity. I shall therefore use the epicene, formerly plural “they,” with a singular verb, when I want a pronoun for the writer. Their essay being only a blog post like this one, I might have passed it by; but it is on the website of Scientific American.
I have respected that magazine since childhood, when my godfather gave me a subscription, and I became a devoted reader of Martin Gardner’s “Mathematical Games” column.
I saw the blog post recommended on Twitter to somebody who was trying to understand the slogan, “Trans women are women.”
In an earlier post, “Sex and Gender,” I suggested (though not in these terms) that Scientific American had engaged in phony science by publishing “Beyond XX and XY: The Extraordinary Complexity of Sex Determination” (September 1, 2017). The explicit information about intersex conditions is presumably correct; however, as far as I can tell, the pictorial suggestion that those conditions lie on a spectrum is phony, in the sense of being given neither a justification nor even a meaning that could be justified. For example, there is no clear sense in which a person with Turner syndrome is closer to being a “typical biological female” than is a person with congenital adrenal hyperplasia.
Those are intersex conditions. Being trans is different. The essay under review has a concluding remark,
The trans experience provides essential insights into the science of sex and scientifically demonstrates that uncommon and atypical phenomena are vital for a successful living system.
The essay has hardly mentioned trans persons, nor said what the “trans experience” is (I can imagine several possibilities).
I have no idea how anything could show “that uncommon and atypical phenomena are vital for a successful living system.” Every living system that we can observe is successful; otherwise it would not be there in the first place. Be that living system an organism or a species, there must be something atypical, something not only uncommon, but unique, that distinguishes it from others. I don’t know how this would be understood as a scientific conclusion. I don’t know, and the author makes no appearance of trying to let the reader know.
“Transgender people are real,” says Simón(e) D Sun, in the passage already quoted. I didn’t think there could be any question of this. Some persons do in fact disagree with the sex they were assigned at birth; and as far as I know, this is what it means to be transgender, or simply trans.
It’s not actually so simple. Some trans persons don’t disagree with their assigned sex; they are made dysphoric by it. They affirm that they always will be of the sex they were assigned as birth; they still prefer, or are led by an inner compulsion, to live as if they were of the other sex. These persons may take hormones, and undergo surgery, to make their bodies mimic the opposite sex, without actually becoming it. Debbie Hayton seems to be a notorious example of such a trans person.
As far as I can tell, some trans persons only adopt clothing and mannerisms associated with the other sex. Some persons do the same, while not considering themselves trans. Such persons may be called trans, posthumously; Marsha P. Johnson and Stormé DeLarverie would seem to be examples.
This whole discussion assumes that sex is binary, and there are two sexes, male and female. These are only ideals or concepts, and how they are to be applied is not always clear. Simón(e) D Sun may mean something like that. Nonetheless, the way they brings in science, I have to ask: who gets to decide? Some activists seem to assert that the individual gets to decide their sex—or their gender, which is independent of sex, though referred to with the same or similar vocabulary.
If the individual decides their sex, then biology would seem to be irrelevant. Nonetheless, Sun says the following:
A newly fertilized embryo initially develops without any indication of its sex. At around five weeks, a group of cells clump together to form the bipotential primordium. These cells are neither male nor female but have the potential to turn into testes, ovaries or neither. After the primordium forms, SRY – a gene on the Y chromosome discovered in 1990, thanks to the participation of intersex XX males and XY females – might be activated.
Intersex XX males and XY females exist, although, as the writer has told us,
Nearly everyone in middle school biology learned that if you’ve got XX chromosomes, you’re a female; if you’ve got XY, you’re a male … The popular belief that your sex arises only from your chromosomal makeup is wrong.
Before college, I myself had no biology course; instead, those of us who were interested in science were advised to satisfy other requirements, and then take chemistry and physics as our sciences. (This was at St Albans School for Boys, in Washington, DC. The “other requirements” were courses in ancient Greek history and the Bible.)
As for the writer’s words above, apparently they originally referred to XX males and XY females as being transgender. This would have been phony science, and it was corrected. The correction was properly noted (on 6/18/19), and this is how I can know it happened.
Members of a group may not agree with the activists who campaign in their name. Not all African Americans will agree with everything that is done under a banner that reads, “Black lives matter.” Activists may also make a poor case for a worthy cause; examples may include some of the white people who preach against racism.
Simón(e) D Sun does say why their cause is important:
… “intellectual” assertions are used by nonscientists to claim a scientific basis for the dehumanization of trans people. The real world consequences are stacking up: [1] the trans military ban, [2] bathroom bills, and removal of [3] workplace and [4] medical discrimination protections, [5] a 41-51 percent suicide attempt rate and [6] targeted fatal violence. It’s not just internet trolling anymore.
The “‘intellectual’ assertions” referred to here are attributed vaguely to the Intellectual Dark Web, though the writer cites only Bari Weiss’s New York Times opinion piece of May, 2018, as a reference for this entity.
I am not sure what Sun means by dehumanization. The Louisiana diner scene in Easy Rider (1969) might be an example; here local white men talk about the three male freaks who have arrived on two motorcycles.
—Check that one with the long hair. —I checked him. Might have to bring him up to the Hilton before it’s over with. —I think she’s cute. —Isn’t she though? Guess we’ll put ’em in a woman’s cell, don’t you reckon? —I think we oughta put ’em in a cage and charge admission to see ’em.
—You know, I thought at first that bunch over there … their mothers may have been frightened by a bunch of gorillas … but now I think they’re a cult. —One of ’em is Alley Oop, I think, from the beads on him. —One of ’em, darn sure, is not Oola. —Look like a bunch of refugees from a gorilla love-in. —A gorilla couldn’t love that. —Nor could a mother. —I wouldn’t even mate him up with one of those black wenches out there. —Oh, now, I don’t know about that. —That’s about as low as they come, I’ll tell you.
—Man, they’re green. —No, they’re not green. They’re white. —White? You’re color-blind. I just got to say that. —I thought most jails were built for humanity … and that won’t quite qualify. —Wonder where they got those wigs from? —They probably grew ’em. Look like they’re standin’ in fertilizer. —Nothin’ else would grow on ’em. —I saw two of ’em one time. They were just kissin’ away. Two males! Just think of it.
The girls in the diner adore the visitors. The men’s discussion ends ominously:
—What you think we oughta do with ’em? —I don’t know, but I don’t think they’ll make the parish line.
About ten years after that movie, my grandmother told me that my long hair made me look like a girl. I didn’t cut it.
As for the consequences of dehumanizing trans people, I have no detailed knowledge. I would say that one of the consequences, in the numbering that I supplied, is not like the others. All but number 5 may be addressed with laws that, properly framed, ought to be unobjectionable. Suicide can be addressed with law too: it can be made illegal. However, I don’t think this is what the writer has in mind.
Suicide attempt rates are disputed, as for example in another blog post, called “The Theatre of the Body: A detransitioned epidemiologist examines suicidality, affirmation, and transgender identity.” Here is just one of the epidemiologist’s remarks:
… clinicians in the earlier days of proper gatekeeping often reported that their male trans patients commonly used manipulative suicide threats to get more rapid approval for hormone drugs and genital de-masculinization surgery.
Journal articles from 1965 and 1979 are cited. The post is not on the site of a well-known magazine, but of 4thWaveNow, “A community of people who question the medicalization of gender-atypical youth.” The post itself is by Hacsi Horváth, who says,
For about 13 years, I also masqueraded “as a woman,” taking medical measures which suggest, shall we say, that I was completely committed to that lifestyle. Most men would have recoiled from this, but in my estrogen-drug-soaked stupor it seemed like a good idea. In 2013 I stopped taking estrogen for health reasons and very rapidly came back to my senses. I ceased all effort to convey the impression that I was a woman and carried on with life.
Such detransitioners are examples of persons who cannot always be trusted to know their own sex or gender, or what they want in general. Maybe you can usually trust people, but you have to allow for exceptions. This is a basic point made in the first book of Plato’s Republic, after Polemarchus says, quoting Simonides, “it is just to render to each his due,” in Shorey’s translation, or in Jowett’s, “the repayment of a debt is just,” (ὀφειλόμενα ἑκάστῳ ἀποδιδόναι δίκαιόν ἐστι 331e).
Justice is not actually so simple. Only the letter of the rule of Simonides, but not the spirit, would be observed in returning weapons to the hands of somebody who has gone mad. Such, at least, is Socrates’s observation.
You cannot always trust Socrates to say what he believes. He is also only a character in a dialogue of Plato. In any case, somebody’s believing something doesn’t make it true. If one needs an argument, the dialogue of the Republic makes clear enough that there is no foolproof formula for justice.
Besides the one by Sun, another Scientific American blog post is an interview of a trans woman scientist who says she would commit suicide if she could not obtain the hormones she takes.
If suicide can be a side effect of anti-depressant medication, maybe it can be a side-effect of not taking certain medications. Such, at least, would seem to be the claim.
Nobody should use suicide as a threat, and nobody else should be manipulated by such a threat. Though your lover may say they will kill themself when you propose to break up, you should still go through with it, according to a WikiHow article (with a number of references), “How to Break Up With Someone Who Is Threatening Suicide.”
Perhaps the trans scientist in the Scientific American blog post is not making a threat, but simply stating the facts as she sees them. In that case, she needs help. I do not know how that help can best be given; perhaps indeed with hormones. However, the judgment of any suicidal person is ipso facto questionable. Survivors of suicide attempts from the Golden Gate Bridge say they regretted their decision the moment they jumped. (I first learned of this regret from a recorded lecture of Nicholas Christakis, “The Sociological Science Behind Social Networks and Social Influence.”)
I return to the science of sex, the nominal subject of the essay under review. As far as I understand, the binary distinction between female and male is useful for explaining what is called sexual reproduction, which is the kind of reproduction that human beings engage in. Among us, an egg from a woman (or call her what you will) combines with a sperm from a man (or call him what you will) to produce a zygote, which can be nourished in the woman’s womb (or suitable replacement, such as another woman’s womb) so as to develop into a new human being.
The zygote has a sex, or is going to have a sex, but the expression of this sex may be ambiguous. It still seems reasonable to say that nature tries to make a zygote into a girl or boy. This is the purpose of nature, although this purpose may not be realized in individual cases.
Whatever the purpose of nature may be, it imposes no moral obligation on us. For instance, we need not try to “help” nature by performing genital surgery on intersex infants. I addressed the general issue in “A New Kind of Science,” noting that while a fever may be nature’s way of fighting an infection, it may not be our best way in any particular case.
In a book that I happen to have on hand, Tinbergen makes some useful remarks about nature’s purposes (the bold emphasis is mine):
Whereas the physicist or the chemist is not intent on studying the purpose of the phenomena he studies, the biologist has to consider it. ‘Purpose’ of course is meant here in a restricted sense. I do not mean that the biologist is more concerned with the problem of why there should be life at all, than the physicist with the problem of why there should be matter and movement at all. But the very nature of living things, their unstable state, leads us to ask: how is it possible that living things do not succumb to the omnipresent destructive influences of the environment? How do living things manage to survive, to maintain and to reproduce themselves? The purpose, end, or goal of life processes in this restricted sense is maintenance, of the individual, of the group, and of the species. A community of individuals has to be kept going, has to be protected against disintegration just as much as an organism, which, as its name implies, is a community of parts—of organs, of parts of organs, of parts of parts of organs.
That is from page 2 of Social Behavior in Animals (1953/1964). A bit later (page 8), Tinbergen mentions a mated pair of herring gulls in which the male had no “brooding urge.” The female alone sat on the couple’s eggs for twenty days, then gave up. Her personal purpose was defeated, though not the species’s:
However disastrous this was for the young, it was a blessing for the species, for what if the offspring inherited this defect from the father and supplied the species with three instead of one of these degenerates?
Such eugenic considerations are irrelevant to our own moral concerns. More precisely, you may use them in making your own reproductive decisions, but not impose them on others.
Nature attempts to make each of us male or female, but is not always successful. In the same way, gasses try to obey the gas laws, combined ultimately as
PV = nRT.
The attempt cannot be entirely successful, if only because gasses comprise discrete individuals, namely molecules, while pressure, volume, and temperature vary continuously, as far as the given equation is concerned.
I make no judgment of what this all means, ontologically.
In his book, Mathematics Under the Microscope (2007), as well as in his blog, Alexandre Borovik remarks on the challenge of taking a finitistic course of calculus, while working out the subject independently in the usual way: “I learned to love actual infinity – it makes life so much easier.” There may be no actual infinite or infinitesimal quantities in nature; we still find it convenient to study nature as if there were.
Simón(e) D Sun refers to research whereby the brain of a trans woman (for example) is in some ways like a cis woman’s, in other ways like a cis man’s, and in yet other ways in between. Studies were made “both before and after transitioning”; indeed, the title of one cited article explains: “Grey and white matter volumes either in treatment-naïve or hormone-treated transgender women: a voxel-based morphometry study.”
Is Sun really hoping for a way to use physiology to decide whether a person is trans? If so, there is a long way to go. The cited studies seem not to involve the testing of hypotheses arising from a theory of the gendered brain.
One of Sun’s references mentions such a theory. The article is called “Regional Grey Matter Structure Differences between Transsexuals and Healthy Controls—A Voxel Based Morphometry Study,” and it says,
According to a recent review about the sexual differentiation of the human brain, transsexualism might be the result of the fact that the development of the sexual organs in the fetal life occurs well before the sexual differentiation of the brain. Thus, if something disturbs the sexual differentiation of the brain, the fetus already has sexual organs according to his/her assigned sex, while his/her brain might develop differently.
So there is a potential theory, but I don’t see a proposal for testing it. It’s somebody else’s theory, anyway. The authors themselves have an hypothesis of sorts:
Our hypothesis was that the regional structural parameters of the brain of transsexual subjects will be different from that of control subjects with the same biological gender.
The verification of such an hypothesis would seem to be practically inevitable, since the transsexual subjects are few (n = 17), and the “regional structural parameters of the brain” are (apparently) numerous.
As a mathematician, I cannot normally write a paper that says, “My hypothesis is that such-and-such is true, and I have attempted to prove it in this way, but the proof fails.” I may be able to salvage something out of my work; but in mathematics we haven’t the custom that Feynman urges for natural science, whereby we must publish whatever we come up with. As he says in “Cargo Cult Science,”
If you’ve made up your mind to test a theory, or you want to explain some idea, you should always decide to publish it whichever way it comes out. If we only publish results of a certain kind, we can make the argument look good. We must publish both kinds of result.
However, it seems the publishing should also be part of such a project as Feynman describes:
When you have put a lot of ideas together to make an elaborate theory, you want to make sure, when explaining what it fits, that those things it fits are not just the things that gave you the idea for the theory; but that the finished theory makes something else come out right, in addition.
Thus Mendeleev’s hypothesis of the periodic table had to be confirmed by discovery of the elements whose existence it predicted; and Ventris’s hypothetical interpretation of Linear B had to be confirmed by its successful application to newly discovered documents.
Above was a quoted suggestion of a possible theory of the gendered brain. In describing the theory, the authors continue (emphasis mine):
These authors suggest that the disturbance of the testosterone surge that masculinize[s] the fetal brain might be at the background of GID [Gender Identity Disorder] in certain cases. Furthermore, they emphasize that there is no compelling evidence that postnatal environmental factors play a crucial role in sexual orientation and gender identity.
This theory is contradicted by the references cited in another of the papers that Sun themself links to, namely “A Review of the Status of Brain Structure Research in Transsexualism” (emphasis mine again, and I have removed the references):
In regard to environmental variables, parental and family factors have been reviewed; parental influences seem to be a contributing factor to the development of GID and play a role in social gender transitioning.
Something is fishy here. What should the lay reader conclude? Here is Feynman again:
I would like to add something that’s not essential to the science, but something I kind of believe, which is that you should not fool the layman when you’re talking as a scientist … I’m talking about a specific, extra type of integrity that is not lying, but bending over backwards to show how you’re maybe wrong, that you ought to do when acting as a scientist. And this is our responsibility as scientists, certainly to other scientists, and I think to laymen.
“Thanks to the participation of trans people in research,” says Sun, “we have expanded our understanding of how brain structure, sex and gender interact.” And yet if we actually look up Sun’s references, they do not seem to share a common understanding.
According to the article that Sun cites called “Neuroimaging studies in people with gender incongruence,”
Causal mechanisms for feelings of gender incongruence are unknown, but biological factors are suggested to play a role. Men and women have been shown to differ in several characteristics, but the largest difference may be found in gender identity: most women feel that they are women, and most men feel that they are men.
Has there really been a scientific study, showing that most men and women feel that they are men and women?
I wonder how such a study would proceed. It might mean asking people on the street two questions:
What is your gender?
Do you feel that you are of that gender?
I don’t think answers could be of much use, if they were even forthcoming. Perhaps the first question would be replaced with a request for consent to examine the subject’s genitals.
The reference in question cites no study of gender identity, though it does give citations for such claims as, “Boys and girls show differences in the development of grey and white matter volume over the course of puberty.”
Do trans people, do any of us, want to leave the question of who we are to a physiologist of some kind?
Speaking for myself, I don’t feel that I am a man. I may then differ from my fellow man Charlie Rich, who sang in 1973,
And when we get behind closed doorsThen she lets her hair hang downAnd she makes me glad that I’m a manOh, no one knows what goes on behind closed doors.
I used to hear this on the AM radio in my father’s Pontiac. When behind closed doors as Rich describes, I am glad to be who I am, where I am, with whom I am with; being a man as such has nothing to do with it.
Before Charlie Rich, though I heard them only later on my friend’s cassette player, the Grateful Dead were singing the lyrics of Robert Hunter and Bob Weir:
She can dance a Cajun rhythm,Jump like a Willys in four wheel drive,She’s a summer love in the spring, fall, and winter,She can make happy any man alive.
“Sugar Magnolia” may be the Dead’s most joyful song, but I reject the idea of a generic way of making me happy as a man.
I acknowledge responsibility for being a man, in the sense that, when men are boorish, even by writing that last verse above, this somehow reflects on me. There may not be much I can do about it, beyond not being that way myself.
When men fight on the street in Istanbul, I witness from a distance, counting on the intervention of men who know the belligerence better; and the intervention always happens. I suspect that some men try to fight, knowing others will restrain them.
I have once or twice approached a quarrelling, (sexually) mixed couple, to see that no violence would be done.
After participating in the March for Women’s Lives in April, 1989, in Washington, I was a member of the National Organization for Women for some years, until I started thinking that I could afford to give money to such organizations, only because I lived the kind of frugal lifestyle that the leadership of those organizations did not adopt. One of my five roommates worked as an assistant to the NOW president, Patricia Ireland, whom I had met when she spoke at the University of Maryland. After her talk, another man shook her hand when I did, but he turned out to be an anti-abortion activist. He didn’t get to ask whatever clever question he had in mind; a handler took Ireland off to an interview.
The trans debate, and what it is even about, remain in question; but the abortion debate can perhaps be informed by science, and more precisely ethology. Here is Angela Saini, from Inferior: The True Power of Women and the Science that Shows It (2017), reporting from the San Diego Zoo.
I’m transfixed by a fluffy two-year-old bonobo. She’s cheerfully hanging on to her mother’s fur as the ape leaps from branch to floor, letting go of her to playfully roll on the ground for a few seconds before quickly returning. I have a two-year-old as well. And the bonobos’ behavior reminds me of my own close relationship with my son. In the little bonobo I see a similar mischievousness and even the hint in her of his cheeky smile. They watch each other the same way that we do. The similarities between us are uncanny.
At close quarters like this, I start to understand why humans are sometimes regarded as another great ape, alongside bonobos, chimpanzees, gorillas, and orangutans. But as much as we have in common, there’s one important contrast between me and the bonobo mother. In the entire time I’m looking into the glass enclosure, I never see her lose contact with her infant. At no point does the little one fall out of her mother’s protectively tight orbit. My son, on the other hand, is already at the other end of the enormous zoo with his father.
Many primate species are like bonobos. Humans and tamarins are different; our females do not raise their young exclusively, and mothers who have no help are known to abandon their offspring. Primatologist Sarah Hrdy is reported to use such observations to argue for the availability of abortion and childcare.
I think the argument can go as follows. A priori, a woman might be expected to regret an abortion. She may also regret keeping the child, while letting others look after it. However, there seems to no “biological” reason why this should be so, because:
in some primate species (Hrdy observed the langur; Dawn Starin, the red colobus), mothers will carry around the corpses of their dead infants;
in other primate species, mothers will abandon even living children they cannot care for;
in those species, children do not cling so tightly to their mothers as, say, bonobos do;
neither do human children so cling.
That is all fine; but still we have to ask whether we want to be like tamarin monkeys. We probably do not wish any infants to be exposed in the manner intended for Oedipus.
Ethological studies only broaden the range of moral examples that we can consider, as we decide what to do.
After the bolded sentence from their last paragraph that I quoted in the beginning, Simón(e) D Sun says,
Defining a person’s sex identity using decontextualized “facts” is unscientific and dehumanizing.
I think defining a person’s sex using biological facts is simply what is done, scientifically.
Defining a person’s identity by any kind of classification is, if not dehumanizing, depersonalizing. I have written something about this in “On Knowing Ourselves.” It may be more or less accurate to call me a cis het white male, but if that tells you all you need to know about me, then perhaps that tells me all I need to know about you.
We have seen Sun’s conclusion,
The trans experience provides essential insights into the science of sex and scientifically demonstrates that uncommon and atypical phenomena are vital for a successful living system.
We have seen Tinbergen’s account of the kind of success studied by biology: “maintenance, of the individual, of the group, and of the species.” This is reproductive success, in a broad sense. For success in human reproduction, much more is needed than the joining of male and female gametes. Evolutionarily speaking, infertile persons, such as women past menopause, who can help care for the young may be an essential component of successful human reproduction; but Sun makes no suggestion that trans persons as such are a component of this reproduction.
There would be no need for such a suggestion. Trans persons are members of our species and deserving of respect as such.
Again, Sun had originally tried to say that trans persons had led to the discovery of the SRY gene; but it was intersex persons. I have so far skipped the section of the essay about hormones; but it does not mention trans persons at all. It says,
The binary sex model not only insufficiently predicts the presence of hormones but is useless in describing factors that influence them.
It seems to me such a sentence ought to be followed by an account of a better model than the binary sex model. Instead there is a brief elaboration on how
Environmental, social and behavioral factors also influence hormones in both males and females.
This itself would seem to show tacit acceptance of the binary sex model.
There are other models. I heard of the five-sex model in the 1990s, but apparently Anne Fausto-Sterling was proposing this “with tongue firmly in cheek,” by the account she gave later in “The Five Sexes, Revisited” (2000; available from the Wayback Machine). The main focus here is on intersex persons, rather than on those whom Fausto-Sterling calls “Transsexuals, people who have an emotional gender at odds with their physical sex.”
Simón(e) D Sun alludes to no such work as Fausto-Sterling’s, nor to any other alternative to the binary model of sexual reproduction. Neither does they make any suggestion of what an alternative model would be modeling.
“Stop Using Phony Science to Justify Transphobia,” says the title. I think I have looked at all of the examples: the ideas that (1) males by definition differ from females by having a Y chromosome in place of an X; (2) males and females have different brains; (3) males and females have different hormones. Sex is assigned on a more complicated basis, because being intersex is possible; and there’s a lot more to us anyway than sex.
I have said what I have to say about “Stop Using Phony Science to Justify Transphobia.” Now I would say more about science and how I come to it.
When writing “Sex and Gender,” I bought Saini’s Inferior, because the bookshop had this, but not Superior. I cited Audra J. Wolfe for the observation that the latter book was not really science, but journalism; the same is true for the former book.
Tinbergen’s Social Behavior in Animals (1953) was read in the freshman laboratory at St John’s College in Annapolis.
I have written about St John’s for the De Morgan Gazette, as well as on this blog in “The Tradition of Western Philosophy.”
In my freshman year at St John’s, my language tutor, Chaninah Maschler, noted the paradox of the gas laws.
This was in a conference with me about a paper I had written, but I don’t recall the connection. Mrs Maschler must have been thinking independently of the freshman laboratory, where we studied the development of the gas laws and of the notion of the cell. In a recent essay called “When ‘Academic Solidarity’ Is Sophistry,” College alumna and current tutor Zena Hitz lauds the possibility of having such conferences between student and tutor:
I expected small liberal-arts college teaching to be more fulfilling, but I was not prepared for the magnitude of its superiority over the teaching I had done earlier. With only 50 students a semester, in three classes, I knew whom I was teaching. I could meet with each student one on one and calibrate feedback according to the needs and the character of each.
More importantly, the small scale of teaching meant I could give students the freedom to set their own educational agenda, rather than raining PowerPoint bullet points down over a large lecture hall – the regurgitation of which too often stood as a standard of achievement. In my new role, students set their own paper topics. Their questions drive the discussion, not my research program.
At St John’s, one may select one’s own paper topics; but the books that are read collectively are selected by the College. I read recently that American students do not choose colleges for academic reasons; they figure the academic aspects are pretty much the same everywhere, so what they look for is social life and reputation. Not me; I chose St John’s College precisely for the academics (though it probably helped that the students in the promotional literature looked serious as they read their books).
In the College music tutorial, which took the place of laboratory in sophomore year, we read (among other things) the Gradus ad Parnassum of Johann Joseph Fux. In the summer after my graduation, I met a piano teacher who thought Fux’s was an odd book to study. Maybe somebody would say that of the Tinbergen that I quoted above. Readings at the College may be chosen more for the discussion they can provoke than their continued significance. In Fads and Fallacies in the Name of Science (1957), Martin Gardner (he of “Mathematical Games”) wrote of the College,
so heavy is the emphasis on highlights in the past history of science, that little time is left for acquiring a solid grasp of current scientific opinion.
Indeed, the point is not to learn the opinions of others, but to develop one’s own knowledge. This happens, not by looking at mere “highlights,” but by spending months studying Ptolemy’s obsolete geocentric theory of astronomy. This could be one of the most distinctive and valuable features of the College.
I recently encountered a critical allusion to Ptolemy by one of my fellow mathematicians:
This is the problem with physics. Contrivances that fit the observed data but are fundamentally wrong. In the ancient world think of the theory that the sun orbited the earth in a circle. To explain why the solstices and equinoxes are not quite evenly spaced through the year they offset the centre of this circle from the earth. Aristotle had averred that all cyclic motion must be based on circles, so ellipses were not acceptable. This idea led to the absurd contrivance of circles whose centres moved on other circles (epicycles) to explain the motion of the planets around the earth.
Thus “Dark Matters” (The Critic, 22 June, 2020), by Mark Ronan, whom I recall meeting at Antalya Algebra Days in 2001. It seems to me that contrivances fitting the observed data are precisely what physics wants to find. The contrivance of epicycles fit the data well enough for some centuries.
If ancient astronomers had thought elliptical orbits for planets “acceptable,” they still would have had to determine the planets’ rate of motion along those ellipses. “Obvious” choices might have been constant linear speed, or constant angular velocity about the center of the ellipse, or constant angular velocity about a focus of the ellipse. The last might have been most natural, since we can draw an ellipse with a loop of string and pins at the two foci.
Nonetheless, it seems a law of planetary motion along ellipses would not have fit the data without a second law, whereby the radius drawn from one of the foci sweeps out equal areas in equal times. Placing the sun at the relevant focus, Kepler was able to work out these laws from observations taken from the earth. He needed the hypothesis of Copernicus, that the sun was not a planet, a “wanderer” (πλανήτης), one of the seven whose symbols I put near the head of this essay; but the earth was.
I don’t see a basis for ridiculing the generations of astronomers who did not hypothesize Kepler’s laws. Or if there is a basis, it might serve also for ridiculing Kepler’s earlier hypothesis, that the planets moved along spheres nested in Platonic solids. We may also note that Copernicus had still explained the heavenly motions in terms of circles.
We are all living in the same world, be we in the second century or the twenty-first, in Africa or Europe. There is no reason to think any one of us understands the world any better than anybody else. Obviously we are going to have our opinions; but there is no test for whose ideas are better, other than dialogue.
So-called IQ tests have been administered to people of different countries, and “national” IQ scores inferred. This was the subject of a good recent thread of tweets. Some countries get very low IQ scores. You can conclude that the people of those countries must be intellectually disabled, whether from their genes or from environmental conditions; or you can question the idea that your IQ test measures something in the first place. I would go with the latter. The test is certainly meaningless unless the subject cooperates; this makes it not objective in the way that a test for body temperature is.
I would propose similar considerations when people in the past turn out not to have seen things that are obvious to us. Like Newton, we may stand on their shoulders. If we cannot literally have a dialogue with them, we can read them, preferably in a college, a collegivm, that takes them seriously.
In the senior laboratory of St John’s College, after trying to learn something of quantization from original papers, and performing such demonstrations as the Millikan Oil Drop Experiment, we read, as I recall, a Scientific American article on the EPR paradox. Somehow I heard, perhaps not directly from him, that one of our tutors thought an aim of our laboratory was to enable us to understand articles in Scientific American. This then is another personal reason of mine to worry if the magazine publishes phony science. I would however give a more serious account of the St John’s laboratory. It is a study of science as a human endeavor.
]]>This reviews some reading and thinking of recent weeks, pertaining more or less to the title subjects, of which it may be worth noting that
poetry is from ποιέω “make”;
mathematics is from μανθάνω “learn.”
Summary added August 23, 2020: Mathematics may bring out such emotions as poetry does; but in the ideal, a work of mathematics is correct or not, in a sense that everybody will agree on. Here I review work of
Lisa Morrow, writing in Meanjin as an immigrant to Istanbul, like me;
Wendell Berry, in “The Peace of Wild Things,” which things “do not tax their lives with forethought / of grief,” and include the stars;
Randall Jarrell, in The Animal Family;
Mary Midgley, in Evolution as a Religion, on how we see animals;
James Beall, astronomer, poet of the stars, tutor at my college;
Edith Södergran, in “God,” as translated by Nicholas Lawrence in Cordite;
Lukas Moodysson, in Fucking Åmål, where Agnes’s father notices that his daughter is reading Edith Södergran;
Thomas J.J. Altizer, in The Gospel of Christian Atheism, a book that I kept from my father’s collection;
Özge Samancı, in Dare to Disappoint, where the character to be disappointed is the father of the artist, and where Özlem (the artist’s friend and mine) praises the poetry of mathematics;
Fiona Hile, writing, quâ editor of an issue of Cordite featuring poetry of mathematics, about the set theory of Maryanthe Malliaris and Saharon Shelah;
Anupama Pilbrow, a poet writing in Meanjin about studying mathematics;
Robert Pirsig, about students who ask their teacher, “Is this what you want?”
R. G. Collingwood, who in Speculum Mentis analyzes Art, Religion, Science, History, and Philosophy as modes of existence;
Michael Oakeshott, supposedly influenced by Collingwood, but also considered a forefather of “postmodern conservatism,” and analyzing existence into different modes from Collingwood’s, the latter according to the article in the Stanford Encyclopedia of Philosophy by Terry Nardin, who reports, “to insist on the primacy of any single mode is not only boorish but barbaric”;
Allan Bloom, who suggests, in The Closing of the American Mind, that for Ronald Reagan, for the Soviet Union to be “the evil empire” and to “have different values” from the United States is the same thing;
Galen Strawson, who seems to belie the possibility of different modes of being by saying, “we know exactly what consciousness is,” and also, “The nature of physical stuff is mysterious except insofar as consciousness is itself a form of physical stuff,” when (according to me) consciousness is simply not physical, not in the sense of being studied by physics.
A Twitter friend living here in Istanbul announced (on June 16) her pleasure in having a memoir published in Meanjin.
Since this Australian literary publication is not giving away its articles for free, I bought the relevant issue (Winter 2020). In “Unpacking Home: Thoughts of a Displaced Traveller,” Lisa Morrow writes of teaching students of English to distinguish house and home, even as Australia is no longer her home, particularly now that her parents are dead and she has no relations with her two siblings.
The furniture and belongings I carry with me from country to country are the only relics of my past that remain. The heaviest component, my library, fits into 29 boxes, each large enough to hold a dozen bottles of wine. I only drank some of them myself, but I have read every title in this collection, which spans more than 30 years of my life. Who I am is written in the pages. I just have to find the right one.
That sounds a bit like me, although I haven’t changed countries in twenty years. Since then, I have brought more and more of my books to Turkey from the US, not in wine crates, but in Rubbermaid storage bins. One of these, sitting on a table, serves me as the standing desk at which I am writing now.
I remembered somehow discovering that an Australian literary magazine had had an issue on the theme of mathematics. That magazine turned out to be the one called Cordite, which I had made notes about on April 5, after an email friend shared a poem by Wendell Berry called “The Peace of Wild Things”:
When despair for the world grows in meand I wake in the night at the least soundin fear of what my life and my children’s lives may be,I go and lie down where the wood drakerests in his beauty on the water, and the great heron feeds.I come into the peace of wild thingswho do not tax their lives with forethoughtof grief. I come into the presence of still water.And I feel above me the day-blind starswaiting with their light. For a timeI rest in the grace of the world, and am free.
I could best do as Berry does in the valley of the North River, a tributary of the Cacapon River, a tributary of the Potomac. We buried my mother’s ashes there, six years ago.
I spent time there alone after graduating from college. From the books in the house, I picked up Randall Jarrell, The Animal Family, and I recall from this the Mermaid’s account of how little fish are willing to get close to the predators when the former know that the latter are not hungry.
That nature is in an eternal struggle for survival: this is one way of looking at nature, but it is our way, and it is not our only one. Mary Midgley points this out in Evolution as a Religion (1985/2002).
Looking at seagulls eying each other on the next roof over, here in Istanbul, I can be reminded of the two male classmates who lived on my corridor in our senior year of college. When near one another, they would put their hands over their crotches for protection.
“Wild things,” say Berry, “do not tax their lives with forethought of grief.” I imagine that’s true. By living in a country without a Protestant work ethic, I also imagine I’ve learned how we may not tax ourselves with forethought of grief.
A problem with seeking refuge in nature is that there will always be somebody wanting to pave your paradise and put up a parking lot.
A difficulty I have with Berry’s poem is the unexpected passage from night to day. Berry wakes in the night, but then, when he goes and lies down “where the wood drake rests in his beauty on the water,” day seems to have come; for the stars are “day-blind,” and I think this means the sun’s light has made them invisible. Berry could mean the stars are hemeralopic (I just learned that word), in the sense that always, even at night, they carry the property of being invisible during the day; but the stars are also “waiting with their light,” and I suppose this means they are waiting for night.
Berry feels the stars above him, but I suppose you can do this at night, as I recall from a poem whose text I have not got at hand, but it was by James Beall, a tutor at St John’s College. The poet puts his hand to a windowpane and feels starlight hitting it. If I recall correctly, he is in bed with a mate, and he likens the stars’ photons to his own seed.
I happened to encounter a poem called “God” that begins in bed and passes to the heavens:
God is a daybed, on which we lie outstretched in the universepure as angels, with saint-blue eyes answering the salutation of the stars;god is a pillow on which we rest our head,god is a support for our feet;god is a store of strength and a virginal darkness;god is the immaculate soul of the unseen and the already decayed body of the unthought;god is the stagnant water of eternity;god is the fertile seed of nothingness and the handful of ash from burnt-down worlds;god is the myriad insects and the ecstasy of the rose;god is an empty swing between the nothing and the universe;god is a prison for all free souls;god is a harp for the mightiest hand of wrath;god is what longing can persuade to descend upon the earth!
That’s Edith Södergran, as translated by Nicholas Lawrence (2016).
A girl called Agnes reads Södergran in Lukas Moodysson’s wonderful Fucking Åmål (or Show Me Love for Anglophone audiences). We happened to watch it the other night, or rather two nights; we have been turning on the home entertainment system about 45 minutes a night, which means we spent four nights watching Malcolm X, and we went on to spend the same watching Blue Is the Warmest Color. Like Åmål, Blue features high-schoolers, at least to start with, but they are not so convincing as in Moodysson’s film. Maybe French children mature earlier than Swedish children, or at least are better trained to pretend.
I don’t know what to think of Södergran, as I don’t know what to think of a book I have picked up, Thomas J.J. Altizer, The Gospel of Christian Atheism (1966). I have my father’s copy, but I don’t recall his talking about it. I kept the book, maybe in part because the bio on the back says the author “attended St. John’s College, Annapolis, Maryland” (though he “received his degrees of A.B., A.M., and Ph.D. at the University of Chicago”).
If I understand Altizer’s basic point, it might be more simply expressed. Probably then I don’t understand it, but it seems to be that the Gospels might as well end with Good Friday.
The Australian poetry review called Cordite where I found Södergran turns out to have had an issue on the theme of mathematics. I am not sure what to make of this either. Mathematics is already poetic. As the graphic version of my friend Özlem says in two speech balloons, as she is trying to help Özge graduate in mathematics from Boğaziçi University, in Özge Samancı’s delightful graphic memoir, Dare to Disappoint (i.e. dare not to be what your parents want you to be):
A subset S of a topological space X is compact, if every open cover of S has a finite subcover.
Try to imagine it. It is so poetic.
I tried to convey the poetry of a particular compact set in the lavishly illustrated “Tree of Life”; I effectively proved its compactness in “Boolean Arithmetic.” However, when I proposed this compactness as a topic for a student who wanted to do her diploma project with me, and I recommended a book on the subject (in Turkish, by Özlem’s husband), the student worked for some months before leaving me for an easier topic with an easier teacher.
She wasn’t really a mathematician, maybe. When my wife Ayşe eulogized an old teacher of hers who had died, she recalled his saying that the mathematician was somebody who, when reaching a fork in the road, continued on the more difficult path.
Says Fiona Hile, editor of the Cordite issue on mathematics (November, 2017):
I still don’t know how to “do” mathematics but in reading through the twelve hundred or so poems submitted to this special issue of Cordite I was looking for traces of the various ways in which it can make its presence felt.
It intrigues me that Hile goes on to mention another mathematician friend of mine, Maryanthe (she once came to our flat for a beer during a conference here in Istanbul):
… of particular interest to me here is the way in which Malliaris and Shelah stumbled onto their discovery. In his account of Badiou’s philosophical edifice, Norris explains how a subject’s fidelity to a generic truth procedure “can make room, via these concepts of the generic and indiscernible, for the advent of truths that as yet lie beyond the compass of achieved (or achievable) knowledge.” What at first seems insoluble or paradoxical can be turned via Cohen’s technique of forcing “into a fully operative concept”. In proving that the two properties they were working on were both maximally complex, Malliaris and Shelah were also able to show that two infinities (p and t) that were thought to be of different sizes were in fact equal. They did this by “cutting a path between set theory and model theory” in a move that deployed Paul Cohen’s method of “forcing” to solve one of the remaining problems of the continuum hypothesis. The move is reminiscent (in terms of audacity if not scope) of Cantor’s realisation that “the scandal of the infinite – of a part that must somehow be conceived as equal to the whole – could in fact serve as its very definition or distinguishing mark.”
I’m afraid I have not been able to retain an understanding of forcing, though I worked through Cohen’s book on the subject with several students in Ankara.
Before I could recover what I wrote above about Cordite, I found a post on mathematics in the Meanjin blog.
Anupama Pilbrow knows our subject as a suffering that is also a pleasure. She seems to have majored in it. Four months after graduation, she returns to it with a friend, to study a book that one of my non-math friends once paged through with bewilderment, when I was in grad school in the 1990s:
After our two-person seminar, my brain physically hurts and I want to have a nap. I try to do other work but it feels like there is glue in my head. I can barely read. Later in the afternoon, I write to Nick, “Can we please do that again next week? That was so fun, I miss maths so much.”
She misses it, because what she concentrates on now is poetry:
As a poet, I’ve been asked a few times to comment on the intersections between mathematics and poetry. I tend to hesitate in my reply. I flip-flop. I say either “they are very alike!’ or “they are hardly alike!” Mathematics and poetry push the limits of language a lot – in terms of what words mean, how new words and concepts rise up out of imagery, how meaning shifts to accommodate advances in understanding. And although the points of likeness and difference between the two disciplines are ripe, I don’t consider these things in a conscious way when I’m doing maths, or poetry. Because, for me, to do one means to decide, if temporarily, against the other.
I addressed the author in a tweet:
One reason to like your article is your acknowledgment that there are different ways of thinking (and thus doing).
I think the real “mind-body problem” is this: how can one person do such different things as, say, poetry and mathematics?
I continue with this. There are two fledgling yellow-legged gulls being raised on the roof that our balcony overlooks; so far they flap their wings only in frustration, unable to get much lift. They are two gulls; but the mind and body are not two in this way. They are two ways (or classes of ways) of thinking about ourselves.
Apparently this is not a popular idea. I wonder if this has to do with the difficulty of recognizing oneself as even one kind of thinker, let alone two.
I see this difficulty in my students, whom I could hardly get to know before the pandemic closed the universities this spring. I was to teach them analytic geometry, but didn’t know how to do this online, and didn’t think I could demand much from the students anyway. I seem still to have ended up demanding a lot: that the students choose non-orthogonal conjugate diameters for an hyperbola or ellipse, then find the orthogonal conjugate diameters – the axes – of the same conic. (I went on to write about this in “An Exercise in Analytic Geometry.”)
I gave the students my own examples, in print and in my own hand. Some students were able to imitate my work, using their own chosen parameters, and achieving correct results.
Ideally students would understand the work, then do it in their own way; but it is still an achievement to be able to copy me line by line, except for the numbers, which the students supplied and worked with.
I told the students what was true, that I myself easily made mistakes. I pointed out that they could check their work by plotting points on a graph. This was how I had checked my own work. However, this also turned out to be an aspect of my examples that practically no student could imitate.
Neither could the students cheat, as far as I could tell; so that was good. WolframAlpha will apparently not analyze your equation as required for my exercise. I ended up writing my own program, in TeX, in order to compute answers, and draw graphs, for each student’s parameters. This was a lot of work, and I might have saved time, had I had knowledge of existing mathematical software; but I have never cared to acquire this knowledge. Neither did any student seem to have done this, or to have a confederate who had.
One student did use a program to draw her ellipse and to plot the endpoints of the axes that she had computed by hand. Those endpoints did not actually lie on the ellipse. The student should then have gone back to her computations, in order to find her error. At least she could have said, “I thus know my computations are in error, but I have no time to check them.”
She did not do this. I understand not wanting to go back over your computations. But to my mind, an attractive point of the exercise was that, done correctly, the computations would give you something nice to look at. This “something nice,” the graph, would also allow the student to know that she was correct, without having to ask the teacher.
To learn that one can know one is correct, without having to consult anybody else: I think this is the point of studying mathematics.
The like may be true in art. You can be satisfied with your poem or your painting, regardless of what anybody else thinks. But many students seem to want to avoid the responsibility of being satisfied, as much in English as in mathematics. Robert Pirsig wrote about them in a letter to another English professor:
The problem being fought is the old problem that is renewed each time a student brings in a rewritten paper saying, “Is this what you want?” The question seems ordinary enough to the student but every time one tries to answer it honestly it becomes a frustrating and subtly maddening question. An instructor often gets the feeling that he could spend the rest of his life telling the student what he wanted and never get anywhere precisely because the student is trying to produce what the instructor wants rather than what is good.
There seems to be a difference though. In art, you may care what others think; but you don’t have to, the way you do in mathematics. Here, if somebody disputes your computation or your proof, you must resolve the disagreement. The student should learn this too. You cannot debate, much less come to blows, but must work together for mutual understanding and agreement.
That’s the ideal. Maybe as the shaky foundations of mathematics were being exposed in the nineteenth century, or even more recently, conferences experienced such brawls as are described in this memoir of an Edgar Allen Poe conference:
so the place goes dead fucking silent as every giant ass poe stan in the room is immediately thrust into a series of war flashbacks: the orangutan argument, violently carried out over seminar tables, in literary journals, at graduate student house parties, the spittle flying, the wine and coffee spilled, the friendships torn – the red faces and bulging veins – curses thrown and teaching posts abandoned – panels just like this one fallen into chaos – distant sirens, skies falling, the dog-eared norton critical editions slicing through the air like sabres – the textual support! o, the quotes!
I may have been too naïve to recognize it then, but I don’t think there was such a contentious spirit at St John’s, where faculty were not pressed to publish original research. I say that attending the College was the best thing I ever did, besides getting together with my wife (at the Fields Institute for Research in Mathematical Sciences in Toronto). To these two important actions, I should probably add getting to know Collingwood, who was a polymath, but didn’t know much of mathematics.
I had read on Wikipedia that Collingwood had been an influence on Michael Oakeshott. I have been moved to investigate this influence. I read the article on Oakeshott in the Stanford Encyclopedia of Philosophy.
At least I read the sections that mentioned Collingwood. I am not a philosopher, because I am not interested in struggling with philosophical writing that is not somehow personally appealing.
I can do this with mathematical writing, if the mathematics behind the writing is appealing. In writing a review of an article recently, I’ve enjoyed working out examples to illustrate the main theorem in a way that the authors didn’t do.
In the SEP article on Oakeshott, I read a lot that sounded like what I had learned from Collingwood:
Philosophers have used the word “mode” to refer to an attribute that a thing can possess or the form a substance can take. For Oakeshott, this thing or substance is experience … experience involves thinking and therefore ideas … It is an “autonomous” kind of thinking … A puzzle, then, is how the modes can talk to one another, and the solution is that as modes they don’t …
… Because propositions in one mode of discourse have no standing in another, truth is coherence, however defined, within a given mode. To argue across a modal boundary is to commit the fallacy of ignoratio elenchi (irrelevance). If there is any relationship between the modes it is conversational, not argumentative … there is no extra-modal definition of reason or rationality. The illusion that there is arises from privileging what counts as reasonable within a given mode and denigrating what is considered reasonable in other modes. This illusion of superiority generates the narrowness, and at times hubris, characteristic of each mode connoted by the labels “historicism”, “scientism”, “pragmatism”, and “aestheticism”. A conversational as opposed to argumentative juxtaposition of modal voices is respectful of differences and for that reason inherently civilized, which means that to insist on the primacy of any single mode is not only boorish but barbaric.
Right on. We must know and respect that there are different ways of thinking.
And yet sometimes it is said that we must not respect this. I have recalled the words of the man who spoke (though I do not recall how he was chosen) at my college graduation ceremony:
When President Ronald Reagan called the Soviet Union “the evil empire,” right-thinking persons joined in an angry chorus of protest against such provocative rhetoric. At other times Mr. Reagan has said that the United States and the Soviet Union “have different values” (italics added), an assertion that those same persons greet at worst with silence and frequently with approval. I believe he thought he was saying the same thing in both instances …
Thus Allan Bloom in The Closing of the American Mind (1987). Reagan may have thought that way; and yet it’s not evil to be a poet, or a mathematician, or a programmer, or a physicist, even though they all value different things.
The Stanford Encyclopedia article on Oakeshott (written by Terry Nardin) mentions Collingwood’s Speculum Mentis (1924) as being concerned with five modes:
R.G. Collingwood … begins with Hegel’s triad of art, religion, and philosophy, identifying philosophy broadly defined with “knowledge” and distinguishing three kinds of knowledge – science, history, and philosophy narrowly defined – to generate a fivefold hierarchy of modes … Oakeshott, partly in response to Collingwood, folds art and religion into practice, denies that modes can be ordered hierarchically, and defines philosophy as the activity of interrogating presuppositions, including its own, and therefore not itself a mode.
Somehow, I don’t think there’s a dispute here that needs resolving. We are not doing mathematics. For Collingwood, the “hierarchy” of modes, if there really is one, is in the order of development. Paleolithic humans had art; neolithic, religion; the Greeks, science. The child develops first an artistic sense, then a religious, then a scientific; then in adolescence starts over again, except it’s not really “again,” since an adolescent’s art is not a baby’s.
Speculum Mentis is a kind of abstract autobiography. The author writes:
If the person to whom is committed the task of judging between art, religion, science, history, and philosophy can prove to us that he has lived these lives for himself, has graduated successively in each of these schools, then we shall listen with respect to his opinion.
Collingwood is implicitly such a person. His parents were artists; he considered being a professional violinist (as I reported in “Anthropology of Mathematics,” citing Ray Monk); later in life, he regretted not having been a writer. In Speculum Mentis he alludes to his scholarship on Roman Britain:
… periods of history thus individualized are necessarily beset by “loose ends” and fallacies arising from ignorance or error of their context. Now there is and can be no limit to the extent to which a “special history” may be falsified by these elements. The writer insists upon this difficulty not as a hostile and unsympathetic critic of historians, but as an historian himself, one who takes a special delight in historical research and inquiry; not only in the reading of history-books but in the attempt to solve problems which the writers of history-books do not attack. But as a specialist in one particular period he is acutely conscious that his ignorance of the antecedents of that period introduces a coefficient of error into his work of whose magnitude he can never be aware …
I haven’t understood why Oakeshott wrote Experience and its Modes (1933), whose Introduction begins:
An interest in philosophy is often first aroused by an irrelevant impulse to see the world and ourselves better than we find them. We seek in philosophy what wiser men would look for in a gospel, some guidance as to le prix des choses, some convincing proof that there is nothing degrading in one’s being alive …
Somehow it makes sense that Oakeshott would be considered a forefather of postmodern conservatism.
Thinking about what we do is apparently hard. According to Galen Strawson,
we know exactly what consciousness is – where by “consciousness” I mean what most people mean in this debate: experience of any kind whatever. It’s the most familiar thing there is, whether it’s experience of emotion, pain, understanding what someone is saying, seeing, hearing, touching, tasting or feeling. It is in fact the only thing in the universe whose ultimate intrinsic nature we can claim to know. It is utterly unmysterious.
It seems to me the familiar is not always unmysterious. I have an urge now to write some words in response to words that I have read lately. I satisfy the urge by writing; but I cannot know the writing is satisfactory until it is done; and perhaps it is never finally done. How is this possible? It is a mystery, but it is my experience.
Strawson’s “experience” seems passive, except maybe for the part about “understanding what someone is saying”; but if Strawson recognized this part of experience as being especially active, he might have said so.
Strawson continues:
The nature of physical stuff, by contrast, is deeply mysterious, and physics grows stranger by the hour. (Richard Feynman’s remark about quantum theory – “I think I can safely say that nobody understands quantum mechanics” – seems as true as ever.) Or rather, more carefully: The nature of physical stuff is mysterious except insofar as consciousness is itself a form of physical stuff. This point, which is at first extremely startling, was well put by Bertrand Russell in the 1950s in his essay “Mind and Matter”.
This would make sense, if the second and third instances of the adjective “physical” were left out: “The nature of stuff is mysterious except insofar as consciousness is itself a form of stuff.” But I don’t see the point of applying the adjective “physical” to stuff that is not an object of the sciences called physical; and consciousness is not such an object.
According to Strawson,
The German philosopher Gottfried Wilhelm Leibniz made the point vividly in 1714. Perception or consciousness, he wrote, is “inexplicable on mechanical principles …”
… Leibniz’s basic point remains untouched. His mistake is to go further, and conclude that physical goings-on can’t possibly be conscious goings-on. Many make the same mistake today – the Very Large Mistake …
A physical going-on is studied by physics, which is itself a conscious going-on. To say these goings-on are the same kind of thing is like saying poetry and mathematics are the same. They are not, even though the same person can do them. To see this, just do them.
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