Basically, mathematical Platonists feel that, because there is so much complexity to math, it must be something “discovered,” as if from some Platonic mathematical realm. The problem with that is that the derivations in math cannot be any other way; they are what they are out of necessity (given math’s fundamental axioms, at least). That means that, even if there is some Platonic mathematical realm, the “discoveries” of math can’t come from it because that would imply that, were the realm somehow any different, the discoveries of math could be somehow different—that just working out the logic in our minds or by hand or on computer we’d come up with different results. But that would be incogent. The reason we get the results we get is that they’re the only results that are cogent. So, even if there were such thing as some Platonic realm containing all mathematical objects and relations, it would be completely superfluous because the discoveries of math can’t “come from” it, and therefore, given Occam’s razor, it makes little sense to assume its existence.
Another way to tackle the issue starts with an analysis of the meaning of the term “exists”. In order to coherently claim something exists, you must imply that it’s in some way, at least in principle, detectable or otherwise noticeable. If something is not noticeable under any potential circumstances, then what does it mean to say that it exists? To claim that something exists includes defining what the basic form is of the thing that exists; otherwise you’re not saying what it is that exists, and it might as well be the most contentless thing imaginable, with the limit being nothingness. And how can you imagine the form of something without imagining interacting with it in some way to see the form? (See my argument for “form is function” in my previous essay and here.) And if the thing you posit exists can’t be interacted with (or, more specifically, can’t affect you) even in principle then imagining this observation of it is self-contradictory when you take the whole context into account, i.e., the whole world, from you to the claimed extant. Not to mention that the idea that something that exists that doesn’t affect us is a) unfalsifiable, and b) in violation of Occam’s razor.
So, if to say that something exists is to imply that it can affect us, then it makes no sense to say that mathematical “objects” (or whatever they are) are “exist” in some Platonic mathematical realm, because if they actually affected us then it would be hypothetically possible for them to affect us in some other way, thus implying some other hypothetical nature in which they exist. Instead, mathematics is all tautology, as it all necessarily follows from its fundamental axioms. Interaction/affecting is a process of action in time, and the objects of mathematics are timeless and unchangeable, so they can’t affect us in order for us to observe them.
In his book The Emperor’s New Mind, Roger Penrose argues for mathematical Platonism on the grounds that a given point is or is not in the Mandelbrot set independently of what mathematician or computer is examining it. By “examining it”, of course, he means executing the algorithm that determines whether a point is in the Mandelbrot set. I would say that, since there’s no way for an independent truth of which points are in the Mandelbrot set to “make its way into” the results of a completely deterministic algorithm, that truth must be an aspect of, or an indirect reflection of, the algorithm itself (including the rules for multiplication of complex numbers). It is simply illustrated in a way by which it appears very complicated, while its abundant self-similarity across place and scale is one sign of its actual underlying simplicity. Basically, humans are not smart enough to see “through” the imagery to its underlying simplicity so our minds are tricked.
Let’s now tackle this problem from the opposite direction, starting with the fractal image and then deriving the algorithm. Let’s consider two reasonable suppositions: 1) The greatest measure of compressibility of a set of data is the smallest algorithm that can recreate that data, and 2) A set of data only actually contains as much information as its most compressed state; the rest is redundancy. If you made a program that could read a set of data and return the smallest algorithm that creates that data (though it might take a quintillion years to do that) and you fed it a Mandelbrot image, it would certainly (eventually) spit out the algorithm that created the image in the first place. Therefore, a Mandelbrot image actually, on a fundamental level, contains no more information than the algorithm that created it.
This thought experiment brings us to another interesting point: Penrose could have used for his argument any algorithm that produces an apparently complex set of data. For example, a pseudo-random number generator would generate an image with much more apparent complexity than a Mandelbrot image (in that it appears to be way less compressible, hence it appears to contain more information), yet Penrose doesn’t use a pseudo-RNG for his argument because it’s more obvious in that case that the only meaning in the data is in the algorithm that produces it. Yet the obvious structure of a Mandelbrot image is not any more evidence that the information exists in some Platonic realm than a pseudo-RNG-generated image is, because it’s no surprise that a simple algorithm could produce a structured image, since the image, being wholly a reflection of the originating algorithm, must therefore be a manifestation of complexity in simplicity. So, it’s apparent that Penrose was duped in this case by the mere interestingness (or whatever) of the patterns composing the Mandelbrot image.
Another argument for mathematical Platonism I’ve come across goes something like this: Math must exist prior to matter logically, if not chronologically, in order for matter to even exist because matter’s existence as such is wholly dependent upon mathematical laws. To this I have to say that mathematical laws aren’t something matter requires, as if they’re a separate thing from matter—the mathematical “laws” characterizing matter’s behavior are only ways to formally describe the behavior, and they’re merely abstractions. Reifying abstractions as something objectively existing is silly. In what form could they possibly exist?
A mathematical model of matter is basically a reduced simulation of matter. The math is merely a way of representing the matter’s behavior, and the matter is not separate from its behavior. Again, form is function. As I said in my previous essay, how can you know the form of something other than through how it interacts with the observer? And how it interacts with the observes is its function. And the functionality of matter and energy is the physics of it.
The degree to which matter behaves according to mathematical principles is the degree to which matter behaves both consistently and cogently (i.e. self-consistently). Of course matter behaves consistently, because it’s still the same stuff from one moment to the next, and the nature of its composition determines how it behaves. And to imagine that matter behaves in any way but cogently would be an incogent imagining, and thinking incogently is useless and irrelevant to reality, so of course matter behaves cogently.
Mathematical laws aren’t detectable even in principle except indirectly via the behavior of matter, so it’s unwarranted to assume that they have an existence independent of matter. And they’re not really even detectable via the behavior of matter because they could hardly have been anything different; they’re merely cogent or self-consistent thinking, codified.
Another argument (or perhaps merely a description) of mathematical Platonism I’ve seen briefly describes Platonism in general and then adds math to that realm in terms of some kind of basic or archetypal mathematical forms. The exact nature of these forms is irrelevant, because the premise of Platonism itself is silly.
Some Platonic forms, such as beauty, are merely abstractions derived from what many objects seem to have in common and then apparently reified as things-in-themselves by way of language. “Beauty” as being independent of anything beautiful exists only as a linguistic construct.
Other, more concrete Platonic forms, such as the ideal horse, are simply categories people hold in their minds as a result of seeing many similar objects which are given a common name, especially where there is not a smooth continuum of objects’ forms ranging from the ideal in question to completely different forms. There are many different reasons objects would take common forms in islands of similarity, and none of them is because there exists some Platonic form somehow supernaturally dictating their manifestations. For example, all horses are relatively similar to all other horses (and thus categorizable under one name) because of the evolutionary mechanics of speciation.
What’s more likely: That forms exist as templates in our minds used to categorize objects, created largely without our noticing over time through observation and teaching, especially in the early stages of learning; or that they exist in some unobservable, independent realm of abstractions without any conceivable sort of grounding, and that we psychically access a form in this realm every time we identify something? Especially considering how pragmatically useful it is to employ these categorizations, thus implying their likely arising from natural processes of cognition, and considering how naturalistically the islands of similarity in objects arise, thus making their definitions in an independent realm superfluous. And to say nothing of the areas of object differentiation where there are no islands of similarity, only continuums of object forms ranging between objects of completely different configurations, and also to say nothing of the ubiquitous continuums between areas of object forms where there are distinct islands of similarity and areas where there aren’t; for example, extruding from the island of horse forms are forms such as the zorse, a zebra/horse cross, a horse that just lost one of its limbs, horses with some sort of obvious genetic mutation, etc., horses still in the womb ranging through all the phases of ontogeny, etc.
Platonism is obviously a very naive and antiquated way of thinking characterized by a lack of self-reflection regarding language, abstraction and the process of identification, and mathematical Platonism is an even more problematic extension of that.
This essay was loosely based on a much more awkward and obtuse essay I wrote 21 years ago that can be found here: http://local.inhahe.com:8008/book/rough%20drafts%20%26%20notes/html/platomath.html.
I wrote a little bit more about mathematical Platonism, particularly about why it’s not true that “pi is infinite,” here: https://philosophy.inhahe.com/2020/10/29/a-better-solution-to-zenos-paradox-of-motion/