In 2010, I 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.

(See an excursion at the end for some discussion of “social construct” in general.)

“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.

Addendum:Here there is, admittedly, some ambiguity in terminology—and one that is common and commonly problematic in the sciences. Contrast the proper division supplied by “history” and “historiography” and note how the same division is missing for e.g. math, physics, chemistry. At the same time, we would have to postulate an inconsistent choice of meanings between “mathematics” and e.g. “physics” by the author of the criticized text if any benefit of doubt were to be extended.

The author confuses the ideas of “being true” and “applying to the physical world”: A mathematical result is true, and might or might not be apply to the physical world. A physics result is (to some approximation) always applying to the physical world. An XY-plane does not have a physical existence, but certainly exists as a mathematical abstraction, while the keyboard I am typing on is very physical.

Side-note:More generally, a common error among the mathematically unversed is to only measure math according to whether it describes (known aspects of) physical reality, which makes as much sense as dismissing Monet because his paintings were not sufficiently realistic looking.

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. This, of course, unless the author confuses the mathematical conventions that form a mathematical language with math as such. (In this language, we can have “2 + 2 = 4” represent a mathematical claim, but it is the claim it self that is math and/or a statement about math, even be it trivial math, while the representation is just characters on a screen/piece of paper/whatnot.)

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

(Moved to excursion.)

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

(For more on logic, an illogical bedmate for physics and chemistry, see below.)

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.

Then there is the paradox of seeing

*absence*of something in a human society as proof that something would be a social construct... If, however and arguendo, we were to take the absence of a concept of mathematics in one group of humans to imply that mathematics were a social construct, a consistent application of that idea would turn almost anything else into a social construct, making the idea overly broad and pointless.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.

There is also an internal contradiction, in that “two-and-two” clearly is intended to have the same implication as “four” does to a Brit, which is not compatible with the sheep–rice example. (Or, as case may have it, with his interpretation: if the example is truthful, chances are that the shepherd was just extremely bad at arithmetic.)

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 arithmetic, a small and trivial sub-set of mathematics, with mathematics as a whole. 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 world of the ancient Greeks, 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 that the angles so formed would be unequal, in violation of Euclid’s fourth postulate.

Side-note:

The restriction to the “world of the ancient Greeks” above is important: We do have a non-Euclidean universe, but the differences to a Euclidean one are so small that the ancient Greeks would not realistically have noticed. (I dropped the ball on this formulation in an earlier version of this page.)

On the outside, there might have been some case of e.g. a spherical surface (with a locally spherical two-dimensional geometry applicable) embedded in a (virtually) Euclidean three-dimensional space.

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}.

At best, 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 affected by the culture and society. 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.

(This will likely be expanded and moved to a separate page at some later date.)

As time has gone by, I have grown ever more sceptical to the idea of a “social construct”—something poorly defined, used more often as a propaganda trick by rabid Leftists (“X is just a social construct!!!”) than as a legitimate scientific concept, and often so broad as to be virtually pointless (if applied consistently). Consider e.g. race: race is by no means an unproblematic concept, but the type of pseudo-argument used against race in order to claim that “Race is just a social construct!!!’ (or, e.g., “Race does not exist!!!”) would also apply to e.g. sand heaps and hot/cold. Sand heaps would then be social constructs, and the claim that “the water is hot” would just express something about water relative a social construct. Sometimes, I have the impression that someone has re-discovered issues, e.g. relating to the difference between perception and underlying reality, that philosophers have had in discussion since there was philosophy, ignored all previous thought on the matter, and grabbed at the worst resolution possible.

Two particular problems around alleged social constructs is (a) a common attitude that what is or (here and elsewhere) is claimed to be a social construct can be ignored or has no practical effects (both non sequiturs), (b) the seeming assumption that those who deal with such social constructs without calling them “social constructs” would be unaware of the complications that are often legitimately involved, e.g. subjectivity, differences in definitions, the need for a nuanced view of an issue. In reality, these often have far better insights into the matter than the “social constructionists”. For instance, a typical mathematician will be well aware of the aforementioned complications around Euclidean and non-Euclidean geometries, ditto how different past mathematicians have approached the same problems in different manners, ditto how this-and-that standard notation might have looked very differently in just a slightly different alternate reality, ditto how equivalent algebraic structures can arise in very different manners, ditto of how the idea of an algebraic structure, it self, is a somewhat arbitrary thing. Even something as basic as the true nature of various types of numbers, even the reality of certain types of numbers, is something that mathematicians have pondered at length.

As to what might be a social construct in a narrower and more legitimate sense, this must investigated with a scientific mind, we must not try to call everything a social construct in a blanket manner (or those things that we disagree with for political/ideological reasons), 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. (Below, I will assume an at least somewhat narrower sense, as a laxer approach renders the concept pointless.)

Even the claim that language is a social construct (cf. the criticized text), plausible as it might seem at first glance, is at best 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. (I leave the issue of non-human communication and proto-languages aside, notwithstanding that it could further weaken the idea of language as a social construct.) 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. Then there is the issue of languages like Esperanto: it was constructed by a human and certainly drew on already existing languages, but can it be justified to refer to such an explicit construction as “social construct”?

From here we have a natural segue to
criticism against the phrase “*social*
construct”. As with languages above,
much of what is referred to as a “social construct” has strong biological
underpinnings, and biological factors will directly or indirectly
affect all aspects of human lives, how memes develop, etc.—including
effects on how humans view and explore math. From another point of view,
much of what is called a “social” construct might be better referred to
as a “cultural” or “societal” construct. Many others are not actual
constructs, but (like with math) simply reflect how we interpret the shadows on
the cave wall.

Further, the word “social” is itself highly dubious, with no obvious and clear meaning in this context and of limited compatibility with other uses (e.g. in “social event”, “social animal”, or “Social Democrat”), raising some doubts as to whether the phrase actually means what its coiners intended.

Possibly, it would be best to speak of a “human construct” (in some special cases, “human influenced construct” or “human interpretation”), bringing across the meaning that something has arisen (or been influenced or interpreted) through some subjective mechanism involving humans, be it through biology, cultural conventions, or some variation of “in the eye of the beholder” rather than in the thing itself.

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