[TheOtherHobbes]

I don't think it's a false dilemma, because the essence of mathematical platonism isn't really that forms exist, but that these forms are how the universe really works.

They're not an approximation, a metaphor, or a point of view. They're the real deal - the base mechanisms. And they can be discovered through the scientific process, with its combination of physical and speculative analysis.

Clearly this is nonsense. Math isn't truly self-consistent, physical research is limited to a range of lab-friendly experiments garnished with some astrophysical guesswork, and all of it gets filtered through consciousness, which we have no clue about.

What we have is a "looking for the key under the light" situation where can only explore the things we can see. We don't know what's in the dark, and it's actually very likely that our consciousness is extremely limited and unable to perceive essential detail.

But because (tautologically) we can't see it we just assume it's not there, and our tiny and contingent view is gloriously universal.

I find the cat metaphor very revealing. Cats share a space with us but they literally do not see the same objects we do. They perceive weight, texture, and dimensions, they're far more sensitive to smell, and they have some innate models for dynamics and mechanics.

But they have no concept of the meaning of a book, a laptop, a wifi card, a Netflix subscription, or a mathematical description of General Relativity. Unless we breed them specially for intelligence for a good few tens of millennia they never will, because cat consciousness is too small to contain those concepts.

It's ridiculously, almost comically naive to believe - purely on faith - that human consciousness isn't severely limited in some analogous ways.

We're quite good at the human equivalent of hunting for food - which includes manipulating physical materials and crude energy sources, with some meta-awareness of abstraction.

What are the odds that's all there is to understand about the universe?

[mannykannot]

> I don't think it's a false dilemma, because the essence of mathematical platonism isn't really that forms exist, but that these forms are how the universe really works.

Consider Euclidean geometry, which is fine, mathematically, but, it turns out, not how the universe works. If mathematical platonists were concerned about whether mathematics' forms are how the universe really works, surely they would insist on the verification of their axioms before proceeding? And then, would mathematical platonism not be just the uncontroversial parts of the physical sciences?

The author reprises the platonism / formalism issue in this article's final section, beginning with the paragraph "But there is still more to be said. Perhaps, after all some of those Big Picture questions do remain lurking in the mathematical background." He refers to Platonists as realists, but, I think, in the sense that the forms are real regardless of whether they are how the universe really works.

[techno_tsar]

I would consider the reverse (that mathematics is constructed) as just as nonsensical. One could argue that if mathematics were constructed, we are essentially taking on faith that mathematical properties in the physical universe just so happen (by coincidence) to correlate with the mathematical principles we have invented. But this seems backwards. The Pythagorean theorem makes more sense as something we have discovered, or the inner corners of a triangle add up to 180 degrees (half a circle); alien civilizations likely have arrived at the same conclusion. The simple answer is that math is simply a feature of the universe.

Even if mathematics isn’t truly self-consistent (it is not), that does not commit one to formalism or constructivism. The belief that abstract entities, if they are real, must be self-consistent, requires us to believe that self-consistency is a precondition for the realism of abstract entities to begin with. But there is no obvious reason to believe this.

As for the limits of our human consciousness: arguably there is a “floor” where we can have strong beliefs in the hypotheses we form about the universe (including those of mathematics). In fact, the essence of Platonism (and where it derives it’s name) is the very view that abstraction is realer than concrete or empirical particulars because it is more unchanging and absolute. It seems inconceivable to find a single world where 2+2 != 4, but we can conceive of worlds where say, Biden is not currently president, or where gravity had a different strength. In other words, the laws of logic (and perhaps many parts of mathematics) seem very fixed, but our other laws less so. Plato thought this told us something about the ultimate hierarchy of metaphysics; modern mathematical platonists like Godel think that we have a mathematical intuition that allows us to perceive mathematical objects; mathematicists like Max Tegmark thought that nothing other than mathematical objects exist at all.

It is this intuition towards the abstract as real, realer, or realist that motivates platonism. To committed platonists, the burden of proof is actually on the non-believers, partly as a preservation of logic and mathematics. If we dismiss that (a common logic), we might be incapable of having a real discussion in the first place. Whether or not that intuition is enough, or free of problems (it is not) is very debatable. However, platonism is not trivially or obviously false.

[defrost]

> The Pythagorean theorem makes more sense as something we have discovered, or the inner corners of a triangle add up to 180 degrees (half a circle);

These are literal examples of things proven to be true constructions as a result of Axioms (of a Euclidean Geometry).

> The simple answer is that math is simply a feature of the universe.

If (big if) in fact the physical universe is a Euclidean universe (and not, say, just a tiny bit hyperbolic [1])

[1] https://www.maa.org/book/export/html/115525

> If mathematics were constructed, we are essentially taking on faith that mathematical properties in the physical universe just so happen (by coincidence) to correlate with the mathematical principles we have invented.

Err, no.

The mathematics we as human have constructed is a superset of the physical universe - the mathematical world embraces much more than the mere confines of the physical world we can kick and observe across.

Indeed we have constructed various mathematic worlds that are at odds with each other - some my have application in this physical universe which then precludes others from also corresponding to the same.