I continue with the recent posts about mathematics, which so far have been as follows.

“What Mathematics Is”: As distinct from the natural sciences, mathematics is the science whose findings are proved by deduction. I say this myself, and I find it at least implicit in an address by Euphemia Lofton Haynes.

“More of What It Is”: Some mathematicians do not distinguish mathematics from physics.

“Knottedness”: Topologically speaking, there is a sphere whose outside is not that of a sphere. The example is Alexander’s Horned Sphere, but it cannot actually be physically constructed.

“Why It Works”: Why there can be such a thing as the horned sphere.
When I first drafted the first post above, I said a lot more than I eventually posted. I saved it for later, and later is starting to come now.
I still said a lot, even in that first post. I want to review it some more, before moving on.
I spent some time to explain that I had discovered Euphemia Lofton Haynes, because I had ignored the call of a mathematician on Twitter to unfollow a certain account. Other mathematicians supported the call, but were reluctant to explain their support.
I have had other encounters with mathematicians who take positions on a social issue, but aren’t prepared to explain why.
They may see it differently, and indeed I have recognized in a couple of posts that people like Steve Bannon may not debate in good faith. Who knows whether a rando on Twitter is going to be such a person?
Still, anybody may question you in good faith, and mathematics ought to train you to be prepared to answer. Practically speaking, you could have your answer ready on a blog.
I say that mathematics is one science, namely the deductive science, and that deduction is “universally valid reasoning.” From this I draw the conclusion, given in such posts as “Salvation,” that in the case of mathematical disagreement, at least, we have to allow the possibility that we ourselves are wrong.
As I suggested in a post written during the 2019 Logic Colloquium in Prague, we cannot say “My way or the highway”; but I was not even talking about mathematics. There are in fact differences and disagreements among us. The unity of the universe, the possibility of universal truth, is not an observation, but a conviction. A key component of this conviction is that resolution of disputes is possible.
Such resolution does not happen if we dismiss our opponents. This is a lesson that I hope students learn from mathematics, although we mathematicians ourselves have not always learned it.
In “What Mathematics Is,” I took up the theorem that the sum of the first n counting numbers is half the product of n and n + 1. This is the theorem that Gauss is said to have recognized at the age of seven, when tasked by a teacher to add up the first hundred numbers. In the blog post, I gave three proofs.

One proof was by formal mathematical induction.

Another was a diagram of a single case.

A third was an algebraic expression of what the diagram was intended to show.
It could be said that the latter two proofs are not real proofs. I would say that of the first as well, if it is considered as a string of symbols. We call the string or the picture a proof, if we can use it to understand that there really is a proof; but the real proof is seen with the mind’s eye.
The proof is universal, in the sense that calling it a proof expresses our conviction that it is true for everybody, everywhere, for all time.
Not every field has proofs like that. Through my window right now, I seem to see a fine rain falling. I can prove that it is falling by stepping onto the balcony and holding out my hand; but this tells me nothing about how long the rain will continue.
Suppose however I prove the Pythagorean Theorem, and a student seems to accept the proof, but then asks, “Will it still be true tomorrow?” I don’t think I can give an additional proof that it will be. This is just something that you have to understand, in order to able to do mathematics. You have to have something like the object permanence that children somehow pick up, without being explicitly taught.
We do not prove this universal validity of mathematics, any more than experiments prove the universal validity of Newton’s laws of motion. We have to believe that such universal laws are possible in the first place; then experiment may prove that we have found a particular law. Playing with a socalled Newton’s cradle here on Earth, I somehow know it will behave the same on Neptune; but the play alone does not show this.
Mathematics is similar, except that the standard of proof is different: deductive rather than inductive.
Some persons take issue with the putative universality of mathematics. They may cite Alan J. Bishop, “Western mathematics: the secret weapon of cultural imperialism,” Race & Class 32(2), 1990. I learned about this paper from a tweet, though not one of those that you can find by searching on “math is not universal” or “mathematics is not universal.” The people saying this may be correct according to their meaning; those denying it, correct according to their meaning. Working out that meaning needs more than a tweet or even a blog article. I’m trying to do what I can.
There may well be kinds of mathematics different from socalled Western mathematics. Even within Western mathematics, there are differences unseen by those unprepared to look for them. Examples of such differences have been created by

the discovery of nonEuclidean geometry;

the abstraction of numbers from numbers of things, and the resulting treatment of ratios as numbers.
People may overlook the differences; they may also learn to see them.
Bishop acknowledges a kind of universality of mathematics in his fourth paragraph:
There is no doubt that mathematical truths like those [namely “two twos are four, a negative number times a negative number gives a positive number, and all triangles have angles which add up to 180 degrees”] are universal. They are valid everywhere, because of their intentionally abstract and general nature. So, it doesn’t matter where you are, if you draw a flat triangle, measure all the angles with a protractor, and add the degrees together, the total will always be approximately 180 degrees. (The ‘approximate’ nature is only due to the imperfections of drawing and measuring—if you were able to draw the ideal and perfect triangle, then the total would be exactly 180 degrees!) Because mathematical truths like these are abstractions from the real world, they are necessarily contextfree and universal.
An additional qualification should be made: the total is 180 degrees for the measures of the angles of “the ideal and perfect triangle” in the Euclidean plane, as distinct from, say, the “Lobachevskian” or hyperbolic plane.
Bishop is aware of nonEuclidean geometry, but only in a different sense:
The conception of space which underlies Euclidean geometry is also only one conception—it relies particularly on the ‘atomistic’ and objectoriented ideas of points, lines, planes and solids. Other conceptions exist, such as that of the Navajos where space is neither subdivided nor objectified, and where everything is in motion.
Perhaps hyperbolic geometry is still to be considered “Euclidean” according to Bishop’s conception, since it comes out of the Western, Euclidean tradition. Still, Bishop ought to know that, within this tradition, people tried for two thousand years to prove that Euclid’s was the only geometry, before Lobachevsky and Bolyai showed that it didn’t seem to be, and then Beltrami confirmed it.
Bishop says,
Recognising symbolisations of alternative arithmetics, geometries and logics implies that we should, therefore, raise the question of whether alternative mathematical systems exist.
They do exist, even within “Western” mathematics. Even on the subject of the counting numbers, Euclid’s way of thinking is not clear to mathematicians today. Today we have a notion of “fraction” that Euclid didn’t, because he made a distinction between numbers and their ratios that we do not. I wrote about my understanding of Euclid in a 2017 post called “The geometry of numbers in Euclid” (reposted at the De Morgan Forum), and more recently, with references to all of the relevant literature that I know of, on pages 15–17 of the notes (in pdf format) to a talk I gave at the University of Maryland in June, 2019. As I wrote in “Salvation” in February of this year (2020), Euclid’s numbers are magnitudes, or at least multitudes of magnitudes, but ratios are dimensionless.
I find a paper called “Navajo spatial representation and Navajo geometry” by Rik Pinxten, a researcher whom Bishop refers to. Greek geometry comes from Egypt, by the account of Herodotus that I referred to in “Thales of Miletus”; the Egyptians needed geometry to measure their fields for taxation after the annual Nile floods. Herodotus’s term γεωμετρίη should be understood in the literal sense of earthmeasurement, that is, surveying.
Euclid’s geometry is not that, but a system for deducing theorems and problems (that is, constructions) from simple assumptions. If Navajo geometry has a Euclid, let us learn from him or her. If somebody said that Navajo people could not learn from the Greek Euclid, that would be a remarkable claim.
There’s a remarkable claim that the Pirahã language spoken in Amazonas does not exhibit recursion. However, writing in Harper’s (August 2016), Tom Wolfe quotes Chomsky as saying about this,
The speakers of this language, Pirahã speakers, easily learn Portuguese, which has all the properties of normal languages, and they learn it just as easily as any other child does, which means they have the same language capacity as anyone else does.
I would likewise expect any child to learn mathematics—any particular kind of mathematics—as easily as any other child does.
This is like saying Newton’s cradle behaves on Neptune as on Earth.
It also says nothing about the mathematics that children ought to learn, any more than it says children ought to be taught in Portuguese in Brazil, or Turkish in Turkey, or English in the United States.
Not everybody does learn mathematics with the same ease. In “Why It Works,” I quoted Douglas Hofstadter on not being able to handle graduate mathematics. However, Hofstadter goes on to say:
If, at that crucial moment in my life, someone had suggested that before abandoning mathematics, I take a look at geometry, I might have discovered the works of Donald Coxeter and followed a very different pathway in life.
If students do perform differently in school, that doesn’t mean they have to. With human beings, there is no telling in advance what is not possible.
Pinxten recommends teaching Navajo children according to their experience:
In the case of Navajo children we have the particularity that we should not only start from the world of experience of a child, but from the world of experience of a child in another culture. Therefore, we aim at developing a geometry course by means of terms and expressions which are Navajo, and in a cultural context of representations and choices which are, again, Navajo.
Pinxten proposes having children construct and analyze a model rodeo ground, because “Rodeo has grown to be a major point of interest for Navajo children.” By children he means boys though:
Boys especially are eager to assist rodeo events and not seldom the dream of a boy or a young adult is to become a rodeo rider himself … Moreover, it often happens now that upon the birth of a boy one of the parents buries a string of horsetail under the hooghan “so that he may become a good rodeoman” …
Pinxten’s aim is that students learn geometry:
Starting from there, geometric abstractions will be gradually introduced. Finally, children and teacher alike will search for strong relationships between the terms, leading eventually to a genuine formalized (or even axiomatized) geometry.
Pinxten does refer to a geometry, as if it were one among many. Each of us has a geometry; I think this is our individual understanding of the one thing called geometry.
The image near the top of this post is of an icosahedron inscribed in an octahedron. The vertices of the former divide the edges of the latter in the golden ratio. It must take some training to see the image as being of a threedimensional object. In “June [2014] in the New World,” I recalled not having been able to see the bull in a painting called “Tijuana” by Elaine de Kooning.
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