Plato’s Realm Of Forms Tells Us Nothing About Mathematics

Summary

The article argues that Plato's Realm of Forms does not contribute to our understanding of mathematics, as it provides no concrete details about the nature of abstract concepts like numbers.

Abstract

The article "Plato’s Realm Of Forms Tells Us Nothing About Mathematics" critiques the philosophical concept of the Realm of Forms, introduced by Plato through the character of Socrates in his dialogues. The author contends that despite the popularity of the concept, it offers no substantive insights into the field of mathematics. The Realm of Forms suggests that abstract forms, such as numbers, exist in a perfect state, independent of the physical world, akin to the shadows seen by prisoners in Plato's Allegory of the Cave. However, the author points out that this metaphysical realm lacks any tangible description and presupposes an inherent understanding of the forms themselves. The article suggests that the Realm of Forms merely repackages our ambiguity about abstract concepts rather than elucidating their nature. It also notes that there is scholarly debate about whether Plato himself actually believed in the existence of this realm. The author concludes that the concept, as it stands, is unhelpful for understanding mathematics, as it does not provide any actionable knowledge or interaction with the physical world.

Opinions

The Realm of Forms is an interesting philosophical concept but lacks practical utility in the context of mathematics.
The existence of Plato's Realm of Forms is not only mysterious but also contested among scholars.
Abstract concepts like numbers are a mystery, and the Realm of Forms does not clarify their existence or nature.
The article suggests that understanding the Realm of Forms assumes prior knowledge of the forms, which makes it an ineffective tool for those without this innate understanding.
The author humorously notes that Plato's dialogues, particularly "Phaedrus" and "The Symposium," contain unexpectedly raunchy content, which posed editing challenges for the Victorians.
The article dismisses the Realm of Forms as unhelpful because it is an abstract concept without defined location, space, or time, and it lacks observable interactions with the physical world.

What is the Realm of Forms?

The realm of forms is something Socrates exposits in Plato’s writings. For those who don’t know, Plato didn’t directly write his thoughts, instead he wrote discussions in which other people debated and reasoned.

The forms, he argues, are like when a prisoner in a cave sees the reflection of an object from the outside. You learn something of it, but it never compares to seeing the object itself.

If you want to hear from the horse’s mouth, see this section from The Republic.

This sort of thing is a big deal, because it remains a mystery what abstract concepts like numbers are. If there were no humans, would numbers exist? In what sense do they exist? (The phrase ‘in what sense’ is quite illuminating, as we shall see that Plato merely packages up our existing ambiguity and tries to link it to our senses)

P.S. it is contested whether Plato himself believed the Realm of Forms existed.

P.P.S I think Plato’s dialogues are a very fun read. I think my favourites were Phaedrus and The Symposium, both of which are quite raunchy (surprisingly for a Philosophy book!). In fact, the Victorians had an absolute nightmare editing out all the homoerotic undertones and overtones.

Why it tells us nothing

Let’s think for a second about what it says.

There is some abstract realm, in which things like numbers exist.

Um, okay. But we have no detail on what this ‘place’ is like.

In fact, imagine you didn’t possess this innate knowledge of the form of a number. Then clearly this means nothing to you. If I said to a deaf person, there is a place where sound ‘exists’, they have learned nothing about sound. I have told them ‘hypothetical objects which you do not know exists in place I can tell you nothing about’.

So it basically assumes you know what these objects are.

Let’s run with him for a second. There’s some abstract idea of a number all other things are imperfect copies or statements of. Then what has he added? There’s a place for these numbers?

But this isn’t a place in any usual definition — it has no defined location, space, time, interaction with people. We don’t have any observable Platonic Particles drifting down with special mind dust.

So it boils down to: concept which you think exists, it exists, and let’s say it exists in a place because we say other things exist in a place.

Not. Very. Helpful.

The author is an Economics undergraduate student at Cambridge. His passions are Mathematics, Philosophy, music composition, and attempts at humor writing. He also enjoys writing flattering bios of himself in the third person.