Polytropy

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 R² 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.

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 write blithely as

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,

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

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.