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 incoherent. The reason we get the results we get is that they’re the only results that are mentally coherent. 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.
Math happens to have this amazing property of complexity in simplicity, which is why mathematical Platonists are duped into thinking that the “discoveries” of math are discovering actual things: it’s sometimes hard to impossible to predict what will be discovered due to its complexity. But what’s really being discovered is merely what thoughts are coherent. Another aspect of math that makes people think it has an existence of its own is the universality and functional immutability of mathematical concepts, such as numbers. I write about that later on.
One person who read this essay said that he sees things in the opposite way from what I said: it’s things that are discovered, i.e. that exist objectively, that can’t be any different, while things that are invented can be anything. To clear up such possible misunderstanding, when I say that something discovered could have been different, I mean from an epistemic point of view. You don’t know exactly what you’ll discover, and even if you could somehow predict the entire universe, what you discover would depend on where you are and where around you you look. And things invented can be anything in the general case, but in the case of math and logic, it can’t be any different because it’s all merely a reflection of coherent thinking, or something like algorithms applied to deriving or verifying “truths” from a set of assumed axioms.
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 existent thing. 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. Btw, I wrote more about the meaning of “exists” here.
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) “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, such as its Kolmogorov complexity; 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 reductive 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, or with other things? And how it interacts with the observer or other things is its function. And the physics of matter is its functionality.
The degree to which matter behaves according to mathematical principles is the degree to which matter behaves both consistently over time and space and coherently (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 coherently would be an incoherent imagining, and thinking incoherently is useless and irrelevant to reality, so of course matter behaves coherently. Also consider the possibility that every possible set of laws of physics exists in some universe “somewhere,” such as is predicted by string theory. And it’s true that the math is complex, but of course it’s complex because the behavior of matter is necessarily complex at our huge scale of reference (dozens of subatomic particles (including quarks) per atom, trillions of atoms per cell, trillions of cells in our bodies…).
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 coherent or self-consistent thinking, codified. (“As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality.” -Albert Einstein) See also my essay The Universe Is Neither Logical nor Illogical, considering that math ultimately reduces to logic, which I believe was shown by Gottlob Frege.
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 (e.g., anything that exists is clearly either a horse or not a horse). 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 that’s 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 it’s also to say nothing of the ubiquitous continuums between spaces of possible object forms where there are distinct islands of similarity and spaces 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, horses still in the womb ranging through all the phases of ontogeny, etc.
To say the same stuff as the previous few paragraphs a different way, here’s an excerpt my answer to a Quora question I recently answered at https://www.quora.com/Do-even-non-spatial-temporal-abstract-objects-evolve-according-to-evolutionists/answer/Richard-A-Nichols-III:
First, Platonism violates Occam’s razor, because it implies the existence not only of the object being categorized, but also of the universal category the object belongs to, or the “ideal form” or whatever, as something real, in addition to the given object, and there’s of course no way of actually empirically detecting this “ideal form” since it’s non-physical.
Second, if all horses and everything else belong to certain universal “ideal forms” or whatever, then why is there always the theoretical possibility (and often the actual existence) of objects “in between” a horse or whatever and some other given form, at any possibly place in the continuum between the two forms? For example, if “horse” is Platonic form and “zebra” is a Platonic form, why do we have zorses? And if you then suppose there must be a Platonic form “zorse,” then what happens if there is born something half-way between a zorse and zebra?
This applies even more to man-made objects. There are limited possibilities in the realm of cross-breeding, and there isn’t really anything between, say, “rocks” and “water,” but you could freely manufacture any item anywhere in the continuum between two objects in many cases—for example, between a “jar” and a “cup.” And also, even regarding flora and fauna, the existence of biological evolution means that there have existed species at every point in the continuum between any given species that exists today and the original primordial ooze.
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.
While writing the Quora answer mentioned above, it occurred to me that one could also argue for mathematical Platonism based on the apparent universality and immutability of mathematical ideas—in other words, that the core of the concept of, say, the number 2 is absolutely identical in every mind that holds it. I argued against that idea in the Quora answer as follows:
We may personally think of the number 2 differently over time; for example, it may evoke different emotions or associations, or a synesthete might or might not see it with a red aura, but it would probably be argued that the core meaning of 2 doesn’t include those extraneous attributes and is merely a matter of logic/definition. Personally, I would argue that there is no real part of a thought/idea that is unchangeable. The part of 2 that is thought to be static and universal is just its category, like the category “horse”; in other words, everything real about the concept of 2 is variable, and everything considered absolutely universal and immutable about it boils down to the fact that we put all these people’s concepts of 2 under the same category. And I’m a nominalist, not a Platonist, regarding horses and such, and I think the same ought to be applied to concepts. I guess I would say numbers and other mathematical concepts are “functionally” immutable and invariable, but not actually so.
This essay was loosely based on a much more awkward and obtuse essay I wrote 21 years ago that can be found here: http://inhahe.com/platomath.html
I wrote a little bit more about mathematical Platonism, particularly about why it’s not true that “pi is infinite,” here: https://exalumen.blog/2020/10/29/a-better-solution-to-zenos-paradox-of-motion/
