In light of this, a few people have asked me to elaborate on if math is created or discovered. My answer to this is "both," and it's complicated. Mathematics, however, like science is not a "social construct" as critical constructivists mean, which is "political construct." https://twitter.com/ConceptualJames/status/1312412818881339393

My belief is that most of the most basic axioms of mathematics, and certainly those of number theory and geometry from which so many others were derived or devised, were more or less discovered and that in that sense, they are largely the results of empirical inquiry.

It is, for example, straightforward once one defines a consistent abstraction of a common unit, to do the experiment of adding various numbers of units together by the simple acts of putting two enumerated piles together and counting. The results are shockingly consistent.

It's also not difficult to realize that what's going on here is pretty basic and simple and thus to realize that the abstractions themselves aren't a huge step away from accurately describing combinatorial features of reality. (PS: my PhD is actually in enumerative combinatorics)

Those rather self-evident basic combinatorial axioms, together with logic (a kind of approach to thought/philosophy, if you will), produce arithmetic pretty easily. At this point in the adventure, there's not much reason to say mathematics is "created" except in a banal sense.

Of course, critical constructivists love to exploit that banal sense like a high sophomore philosophy major and point out that technically humans created the abstraction. This is neatly handled by all mature philosophies of realism, though, and is really just a dumb distraction.

Things get a little weird from here. If you have whole or counting numbers, the idea of forming ratios of those is not only intuitive but empirically reasonable. Fill a glass halfway, and the proportion becomes immediately apparent. Again, we're discovering math. Realism wins.

But the basic logic and fundamentally *discovered* axioms of enumeracy become a bit tricky now. We can find ratios, like circumference of a circle to diameter, that don't fit. We can imagine adding more units indefinitely, raising the question of what infinity is and means.

These two ideas, infinity and the real numbers that in some sense depend upon them, have a strange relationship in that sense, and they seem to follow quite naturally from our basic axioms, but they don't. They require new axioms that aren't anywhere discovered except in the math

And it's wrong to say they're discovered "in" the math. They're implied by the math. This isn't all that profound, though. It's just a statement that our models are limited and are divorced from reality by being abstractions about it. Again, realisms understand this.

Now there's a second "meta" layer to the question of "discovered or created?" Once the axioms are agreed upon, be those self-evident via realistic observation or inferred extensions of those or even otherwise, an abstract universe is created, an "axiomatic system."

An axiomatic system is a huge collection of propositions with truth values assigned to them: true, false, and indeterminable within the system. It takes very little work to lay out axioms and choose a logical approach, and then this defines an entire abstract universe.

When using mathematical axioms (mostly those somehow relevant to measuring things), we call that abstract system a "mathematics," and we tend to know surprisingly little about it. Mathematicians, in some sense, then discover features of this abstract universe. Not creating them.

The truths in a mathematics are completely set by the axioms and the operant logic upon those axioms, but they're mostly unknown. It's like an entire abstract space, which contains ideas (not Platonic Forms), and mathematicians genuinely engage in a process of discovery therein.

So, the process of doing mathematics is mostly a process of discovery, though it also includes invention: inventing new abstractions that can be understood in the terms of the mathematics in play. This isn't the same thing as suggesting it's a "social construction," though.

Why isn't mathematics a social construction? Because it doesn't really depend on what social group constructs it. It somehow transcends that, and the reason most of us accept for that transcendence is because of the underlying realism that defined the axioms and logic in play.

The world, thus its basic axioms of enumeracy and logic, is utterly impersonal; it is even a category error to say that it doesn't care about us or our vanities because it can't even care or not care. It's something outside of our subjective perspectives of the world.

One could, of course, invent other "mathematics" that have nothing to do with the basic axioms and logic of the universe we find ourselves in, and maybe that's an interesting intellectual exercise, but I don't think it's what most people mean by "mathematics."

So, if you have a bunch of different social contexts, they're going to find roughly the same axioms of enumeracy and measurement and develop roughly identical mathematics. We all used to know, accept, and marvel at this and see it as a great universalizing truth, even off Earth.

If we met intelligent extraterrestrial life, we'd pretty reliably be able to figure out how to communicate with them first through mathematics because the likelihood that theirs matches ours with different symbols and "discovered" theorems is overwhelmingly high. Probably 1, a.s.

So, you could say that because mathematics is something people express in language and symbols that are local to the speakers and agreed upon by the people using them to mean what they mean and arose through social processes that it's a "social construction," but this is crap.

Calling mathematics "socially constructed" is another trick of strategic equivocation. The people saying it move the ball by letting you believe the banal thing above that all theories of realism aren't bogged down by, especially modern ones. They mean something else, though.

What the critical constructivists fruitfully, thus really, mean by saying mathematics (or science) is "socially constructed" is that it is *politically constructed* and politically constructed in a certain way that requires Critical Theories to remake. https://newdiscourses.com/2020/09/no-science-isnt-social-construct/

Critical Theories say things are "socially constructed" to mean that they were constructed by social groups who held power and ordered ideas in such a way that their power would be maintained by advancing favorable ideas to their power and excluding and marginalizing others.

When a Critical Theory steps in and declares that something is "socially constructed," they mean that it is fatally biased with cynical self-interested power that doesn't even have the capacity to admit to itself that it is cynically self-interested power or even biased.

Critical Theory then posits that only someone outside of those biases, particularly someone using a Critical Theory, particularly THEM, can understand the cynical self-interested power that's baked into the thing it's calling "socially constructed." This is a power grab.

Critical Theorists contend that their method is the only way to understand the powerful self-interested biases. Thus, the Critical Theory is asserting itself into a domain in which it doesn't belong (i.e., intellectual colonialism) while pretending that it isn't doing that.

The trick they use to pull this off is to say that the thing is already made out of power, so they're offering a critique of power instead of an application of power, but they're really just making this up (for the most part), picking at seams to create demand for themselves.

In reality, the Critical Theory is the only one really interested in applying political power through claims of knowledge, especially in fields like math and science, which take great pains to minimize that. It's really disgusting. Stop falling for it.

The long and short here is that the critical constructivists are using a strategic equivocation on the meaning of "socially constructed" to sever the tie between science/math and the real world. They're attacking realism in favor of their anti-realism, where they have the power.