I recently stumbled upon a page discussing
math as a social construct^{e},
and doing so in a manner that demonstrated a
considerable ignorance and lack of understanding of what math actually is.

In particular, I suspect that the author has been influenced by one of the pseudo-scientific politically-correct attempts to re-classify everything as a social construct, irrespective of facts. (I stress, however, that I have not investigated this further.) Seeing that this is one of the most dangerous tendencies in modern society, resulting e.g. in some absurd claims by gender theorists, I decided to write a refutation.

Side-note:

Naturally, many things *are* social constructs; however, they must be
investigated with a scientific mind, we must not try to call everything a
social construct in a blanket manner, and a clear differentiation between
things that are social constructs and those that are merely viewed through
a subjective lens (while themselves being independent of social constructs)
must be made.

“Mathematics describes truths.” This does not mean that mathematics is the truth. That’s what physics and logic and chemistry and all the other sciences are about. Mathematics is merely the language used to explore these sciences, and language is most definitely a social construct.

The fundamental error of the author seems to be confusing mathematics with the human exploration and experience of mathematics.

The author confuses truth and factuality: A mathematical result is true, and may or may not be factual. A physics result is (to some approximation) factual. An XY-plane is not factual; but the keyboard I am typing on is (with some very advanced reservations, cf. the end of the page).

To say that mathematics is a language is, at the very outside, a half-truth. It would be better to claim that exploration of mathematics has provided a language usable in other fields, as well as a set of tools and methods.

Certainly, mathematics is far more than a language: An entire scientific field in its own right—and one that many consider the highest science.

Even the claim that language is a social construct is not entirely true (although certainly far more than a half-truth), considering both that there is a biological influence on human languages and that languages that are abstracted from social influence are, to some approximation and in theory, conceivable. Further, it does not automatically follow that everything called a “language” underlies the same restrictions—just like natural laws and legislative laws are very different. (I grant, however, that the language of math is far closer to natural language than natural laws are to legislative laws.)

The quoted (possibly straw-man) claim “Mathematics describes truths.” is incorrect in a nonsensical way, somewhat similar to “Music describes symphonies.”.

Some time ago, as I began reading a history of mathematics, I was surprised and amused to find that there are African cultures alive and well today, in the modern age, that simply have no concept of mathematics. I don’t mean that they don’t recognize the formula for the area of a circle. I mean that they have no concepts for numbers greater than two. After two they literally begin using “two-and-one”, “two-and-two”, “two-and-two-and-one”, and so forth. If you bought one sheep for two bags of rice, the shepherd would smile and be happy, but if you tried to buy two sheep for four bags of rice, he’d look confused and wonder if he was being cheated. The concepts just aren’t there.

This has nothing to do with mathematics, per se. It merely shows that these groups have not explored certain concepts related to a small sub-field of mathematics. In fact, this is a phenomenon of anthropology and some neighbouring fields—not mathematics.

It does not follow from the text that they have no concept of numbers larger than two—just as we cannot claim that e.g. Brits have no concept of numbers larger than ten (or twelve, depending on opinion). All we can conclude is that the language of, at least, the former is not well adapted for stating numbers beyond a certain limit.

It is theoretically possible (if practically extremely unlikely) for a certain culture to have developed mathematical knowledge (say of geometry) without knowing basic arithmetic—in this case, arguably, not even arithmetic, but counting.

Side-note:

One of the tell-tale signs of someone truly ignorant of math is the tendency to confound arithmetics with mathematics. This is roughly comparable to equating “3 barrels of wheat”, inscribed in cuneiform on a clay tablet, with literature.

Even when you get into mathematics as we study it today, you discover that concepts can be relative. Remember geometry? Circles, rectangles, parallel lines and so forth? Many centuries ago a fellow named Euclid tried to prove the parallel postulate, and found that he couldn’t. It was one of those things that simply had to be assumed to be true in order for Euclidean geometry to be consistent. But as the centuries went on, non-Euclidean geometries were unveiled. Here, if you defined a plane as the surface of a sphere or a hyperboloid, redefined lines appropriately, and changed the parallel postulate, then you would still have a consistent geometry.

Eventually it was proved that if one geometry was consistent, then all the other ones must be as well. And while no culture in history ever began exploring geometry from a non-Euclidean point of view, it’s intriguing to think of what would happen if one did. (The Pythagorean Theorem would look vastly different, for a start.)

This portion reads largely like a non-sequitur—and, to the degree that it is not, it displays the same flaws of thinking as above. Apart from that, it contains several oddities. Consider the hypothetical development of non-Euclidean geometry (in a form not deviating in trivial manner from the Euclidean) by e.g. the ancient Greeks: This geometry would not apply to the real world, it would not make sense, the results obtained would often have no practical application, ... In short, if it had happened, it would have been a short-lived and fruitless experiment. Alternatively, take the claim “[...] if one geometry was consistent, then all the other ones must be as well.”: This is simply not true. Firstly, there are many non-Euclidean geometries that are not consistent (typically, immediately rejected); secondly, the proof that certain geometries were consistent would have been made on an individual basis or, on the outside, on a group basis.

Side-note:

A trivial example of an inconsistent geometry would be a variation of the parallel postulate claiming that two non-parallel lines would intersect in two points, forming a right angle at both, and the angles so formed would be unequal, in violation of Euclid’s fourth postulate.

Mathematics is not “real”. It’s not something that was ever discovered, it was invented and reinvented throughout history as a way of explaining the physical world using abstract symbols. Logic, you could argue, is immutable and culturally objective. Same goes for physics. But these are not mathematics, and should never be confused with it.

Mathematics is not something concrete and tangible, but, in many ways, it is very real. It certainly something that has been discovered: Piece by piece, highly incompletely, and with varying interpretations—true. Nevertheless, to discover (to remove a cover, to find something hidden) is a very good description of what mathematical research is all about (the same applies to most research, in general).

Logic, depending on POV, could be considered a sub- or super-set of mathematics, it could be considered an abstraction or something culturally dependent; however, the claim that logic would be immutable and culturally objective is actually harder to defend than the same claim for math.

Physics is even trickier. Depending on exactly what is understood by “physics”, it can be anything from objective to highly subjective.

Even the above glosses over some major complications in areas like epistemology and perception. Consider e.g. Gödel’s incompleteness theorem, or how we can know that the well-accepted first four postulates of Euclid match reality. (They are intuitively obvious, but, ultimately, unprovable and unknowable with regard to the real world—if there is a real world, at all, ...) For that matter, how do we know that the logical system used to complete a certain proof is actually valid? Factor these in, and the claims made about logic and physics fall like a house of cards in hurricane.

Side-note:

For a good layman’s introduction to some of these themes, I recommend
Gödel, Escher, Bach^{w}.

Even in a kind evaluation, the text mixes odd (but ultimately subjective) terminology and unsound thinking. At worst, it borders on nonsense and grave misconceptions.

How we humans *view* mathematics, what is considered true and not, which
parts of this infinite topic have been explored, ..., these are all things
that have been effected by the culture and society we live in. This, however,
does not make math, in and by itself, a social construct. Notably, any
mathematician worth his salt will be aware that his experience of mathematics
is akin to watching the shadows in Plato’s cave—he will not mistake the
shadows for the objects casting the shadows, as the author of the analyzed
text appears to do.

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