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According to SEP:

Platonism in the Philosophy of Mathematics

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented.

The most important argument for the existence of abstract mathematical objects derives from Gottlob Frege and goes as follows (Frege 1953). The language of mathematics purports to refer to and quantify over abstract mathematical objects. And a great number of mathematical theorems are true. But a sentence cannot be true unless its sub-expressions succeed in doing what they purport to do. So there exist abstract mathematical objects that these expressions refer to and quantify over.

Frege’s argument notwithstanding, philosophers have developed a variety of objections to mathematical platonism. Thus, abstract mathematical objects are claimed to be epistemologically inaccessible and metaphysically problematic. Mathematical platonism has been among the most hotly debated topics in the philosophy of mathematics over the past few decades.

If mathematical platonism is true, how did biological brains, shaped by millions of years of evolution, develop the capacity for epistemic access to the Platonic realm of mathematical objects and their properties? Does this Platonic realm causally interact with space and time, and if so, how? How does the physical universe, governed by physical laws, interact with the Platonic realm, such that mathematicians can be said to "discover" rather than "invent" eternal mathematical truths that transcend space and time?

I'm asking this motivated by the answer I accepted in my previous question: Since mathematicians are physical beings, does this mean that mathematics ultimately reduces to physics?

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    This question is known as Benacerraf's dilemma. Most modern platonists, unlike Plato, do not envision a separate platonic realm and see mathematical (and other abstract) objects as perceptual and/or conceptual aspects of our ordinary "realm" accessed through various abstraction procedures, see IEP.
    – Conifold
    Commented Sep 29 at 6:01
  • @MauroALLEGRANZA Does this presuppose mind-body dualism?
    – user80226
    Commented Sep 29 at 9:29
  • If we assume Platonism about abstract non-physical objects, then...
    – Mauro ALLEGRANZA
    Commented Sep 29 at 9:59
  • @MauroALLEGRANZA philosophy.stackexchange.com/q/117585/80226
    – user80226
    Commented Sep 29 at 10:50
  • Does Frege's argument not just kick the can down the road? I suppose a non-platonist has a different notion of what the language of mathematics purports to refer to.
    – sdenham
    Commented Sep 29 at 15:03

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You are fallaciously assuming your own conclusion

You have, in your assumption here, and in the prior question, and in the answer you accepted, an unquestioned presumption that the material world is causally closed.

Make that assumption, and knowledge of mathematics, or any interaction with abstract objects, is of course inexplicable. You are just in conflict with your own circular presumption.

Matter is not causally closed

However, there is no justification behind this assumption. Matter is not fundamental per our modern physics. It can be created, and destroyed. What IS fundamental -- is not at all clear. We have math equations that seem to describe aspects of our universe that have no matter -- is the MATH fundamental? Some physicists think so. Most don't, but don't have any coherent alternative.

What we have with physics is not a fundamental ontology, we have a pragmatic practice. And that practice presumes several things:

  • Mathematics
  • Logic relations
  • Matter
  • Collections of matter into objects

Physics, therefore PRESUMES abstract objects: mathematics, logic relations, material objects (which are matter/abstraction fusions).

Summary

The interaction of abstractions with matter, is intrinsic to physics, and given physics, matter is not causally closed to abstractions.

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  • Physicists don't think "matter" is causally closed. Photons are not matter because they don't have mass. What is causally closed is physics - all the interactions between the fundamental particles according to the fundamental forces.
    – causative ♦
    Commented Sep 29 at 17:55
  • @Rushi -- if matter can be created and destroyed, it is not causally closed. The "not" is what physics says.
    – Dcleve
    Commented Sep 29 at 17:59
  • @causitive -- Photons are widely considered to be material in the understanding I have read among physicists. Probability fields, THOSE are not material. Relationships, THOSE are not material. But are essential parts of physics. This answers the abstraction matter-question -- physics assumes abstraction-matter interactions. Whether physics itself is closed or not, that is a different question, and I consider also to be a not only unjustified, but thoroughly falsified presumption.
    – Dcleve
    Commented Sep 29 at 18:04
  • @Rushi -- Einstein, then QM, showed the matter is not fundamental, nor causally closed. No, physics cannot consider mater to be causally closed. The physicalist claim that Physics is casually closed -- that is to deny mental causation, not abstract object causation. It is to deny agency to Popper and Frege's world 2.
    – Dcleve
    Commented Sep 29 at 18:09
  • @Dcleve Your thoughts here will be appreciated: link.
    – user80226
    Commented yesterday
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  1. We learn mathematics at school. Learning the concepts, the mathematical operations and some algorithms, e.g. how to multiply or how to divide in written form.

    The question, how we learn, is better addressed to experimental psychologists like Piaget, not to a philosopher and mystic like Plato.

  2. A main source where Plato presents his theory of forms is Socrates’ speech in the Symposion (Search for “tale of love”). Here Socrates describes the ascend by continued abstraction from physical beauty to the idea of beauty. The final step, recognizing the abstract idea, is an act of mystic intuition.

  3. Plato tells us a story and develops a metaphor. He does not explain the final mechanism. It is one of the characteristics of a metaphor to skip the explanation of the mechanism.

Hence I do not see that your title question finds an answer:

  • First, the existence of Platonic ideas is debatable. It is controversial how useful Plato's model of cognition is.
  • Secondly, Plato himself does not explain the mechanism underlying his theory. Probably he has no explanation.
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    +1 for final act of mystic intuition. As to "Plato's theory of cognition" I think Plato is generally more normative than descriptive even if he pretends to describe. ie the form-realm is not so much there as it's good to teach
    – Rushi
    Commented Sep 29 at 8:08
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    are you sure no mechanism is stated, rather than nothing contemporary
    – user71399
    Commented Sep 29 at 8:17
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    "We learn mathematics at school." But we learn from other people. Eventually this has to lead to people who discovered it independently.
    – Barmar
    Commented Sep 29 at 15:58
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This question, that question and your accepted answer partake of the same fallacy1 I elaborated viz. You assume the physical brain = the non physical mind. And then you straightforwardly arrive at the fact that only the physical realm is apprehendable by mind (= brain) and so real. IOW you assume physicalism find yourself in physicalist jail and then protest:

Why don't others enjoy life in my prison?

Well... as the saying goes it's different strokes for different folks.

So consider that for some of us, we feel claustrophobic especially in voluntarily chosen prisons and the boot is really on the other foot:

If you are a human being and have any concept whatsoever that is space time independent — math is just an intensively developed example — how can you not be a Platonist?3

So even the trivial belief in the color green as something other than trees, grass etc makes you a Platonist.

Here is an older question where by not allowing for the letter 'e' to have Platonic existence apart from:

  • what you see on your screen
  • ditto mine
  • the shape(s) of your handwriting
  • ditto mine
  • the ASCII (unicode2) number assigned
  • The bits transmitted around the World-Wide-Web
  • etc

one gets really wild and nonsensical conclusions.

1 There are many more rebuttals there, some better than mine.

2 Of course we all do have space time dependent thoughts also, in zillions: eg.

  • I'm hungry!
  • Where the &=@?* are my keys?
  • And so on

3 The early Unicode standard making the same point actually recommended to think of characters as Platonic entities apart from glyphs, codes etc. I'm however not able to find that statement now.

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    Did a +1 to neutralize the unwarranted -1 without explanation. Liked your "physicalist jail" analogy and the character set analogy. So all those 154,998 characters (so far) are Platonic entities, eh? Anyway, yes, the leap from sense-impression to concept STILL needs to be explained by neuroscientists, and I'm following the development of the discipline with fear and trembling because my soul may end being merely the phenomena of my brain despite some ingenuity of recent Thomists to relocate the "soul" as "configuration" in the mind of God though not as far as saying it's an emergent property.
    – GratefulDisciple
    Commented Sep 29 at 7:23
  • @GratefulDisciple tnx! Sure there is a correlation between the physical brain and the mind. Strong correlation. And many open problems there constituting an interesting and cutting edge field. But neither is correlation causation, still less is it identity. In fact CSists would call mind=brain a type-error. You can say char 'A' has ASCII value 65. You cant say 'A' IS 65. Likewise brain and mind. (Yeah in C 'A' == 65 but this causes all sorts of trouble)
    – Rushi
    Commented Sep 29 at 7:33
  • @GratefulDisciple Yeah unicode is going wild. When you add hundreds of emojis in each version which had no existence a couple years ago, Platonism becomes a bit hard to claim ☺️😀😇😜
    – Rushi
    Commented Sep 29 at 7:41
  • Indeed, the letter 'e' and in fact the whole alphabet are good examples of universals. We speak of the alphabet, not of Bob's or Jack's alphabets.
    – Olivier5
    Commented Sep 29 at 7:42
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    @Olivier5: are you referring to GOOG/Alphabet Inc.? 🙃 That's Larry's/Sergey's Alphabet.
    – tomasz
    Commented Sep 29 at 12:22
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The same way that fish "discovered" fluid dynamics and birds and insects discovered aerodynamics.

Evolution by natural selection tends to develop and enhance strategies that prove to be useful in the ecosystem of the organism. Abstract thought and mathematical abilities are simply two of those abilities that have become pronounced in humans. When you combine these abilities you get the ability to consider and discuss platonic ideals, among other things.

Is there really that much difference from our ability to discover physical laws, such as quantum mechanics and relativity? In both cases, we apply our abstraction and induction capabilities to imagine general processes and structures that explain what we've observed. When applying this to the physical world we get objects like atoms and subatomic particles; when applying it to math we get the platonic objects.

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  • There is a missing step in your argument, I think. Suppose that there is a platonic realm A and a physical realm B. A priori there is no reason for B to conform to A or instantiate eternal principles from A. So if evolution is a physical process that goes on inside B, then evolution will produce agents adapted to survival in B. In particular, brains will evolve to produce thoughts that are useful to survive in B.
    – user80226
    Commented Sep 29 at 17:19
  • But, again, a priori there is nothing enforcing that whatever happens in B conforms to the truths in A. A and B can be completely independent of and unrelated to each other. So it's not clear how a brain evolved in B will know anything about A, unless A and B were already entangled or linked beforehand somehow.
    – user80226
    Commented Sep 29 at 17:19
  • If you assume, for example, that B was designed according to mathematical principles from A, then it would make sense that evolution in B will benefit from learning about these instantiated truths from A in B.
    – user80226
    Commented Sep 29 at 17:22
  • I guess that step is the answer to the age-old question "Why is math so useful in explaining the universe?" Empirically, the universe does seem to work mathematically, so the ability to understand it proves useful to organisms and evolution has produced this.
    – Barmar
    Commented Sep 29 at 17:47
  • Quine, in "Two Dogmas", noted that the inference to abstract objects was identical in empirical and logical justification to the inference to material objects.
    – Dcleve
    Commented Sep 29 at 18:37
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If mathematical platonism is true, how do biological brains governed by physical laws access eternal platonic mathematical truths?

What they are really accessing is which thoughts can be formed and which thoughts cannot be formed. Thoughts are physical patterns of neuron activation. We cannot imagine a flat Euclidean triangle whose corners sum to more than 180 degrees; we cannot physically form such a thought.

Is it possible to build a bridge spanning this river, capable of supporting a 10 ton truck and made of steel weighing less than 100 tons in total? This is the same kind of question as whether we can imagine a Euclidean triangle whose corners sum to more than 180 degrees. They are both questions about the feasibility of a physical object.

The universe is filled with infinite patterns that arise naturally. Which way the river flows, the height of the ridges formed by water in the sand. Mathematics studies how some of these patterns can form - and how some cannot form - within the human brain.

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  • We cannot imagine a flat Euclidean triangle whose corners sum to more than 180 degrees. I think you've never had "weed". (Ive less knowledge about "graduate level stuff" like acid)
    – Rushi
    Commented Sep 29 at 18:22
  • @causitive -- the flatness would have to apply to the space the triangle is IN, as one can always define an absolute plane from the three points of the triangle. Given that space is not flat in our universe, ALL of our triangles actually sum to slightly different numbers than 180 degrees. Whether we can imagine that or not -- well I know the theory, and I am imagining it. Note this is a different question from whether thoughts are in world 2, or are "patterns of neuronal activation".
    – Dcleve
    Commented Sep 29 at 18:30
  • @Dcleve I said a flat, EUCLIDEAN triangle. That means a triangle in flat Euclidean space. Curved space - even the physically real curved space we live in - is not relevant to the question of whether a Euclidean triangle can be imagined with angles summing to more than 180 degrees. (It cannot be, because that violates Euclid's axioms.)
    – causative ♦
    Commented Sep 29 at 23:16

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