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Of course we can't see a perfect circle through our eyes, but can I imagine a perfect circle in my mind? But Plato said that the ideal forms can't be seen. I don't know whether he meant that even imagining its shape is a sort of seeing. So should I only consider it as an abstract definition, such as πr2?

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    What color is the abstract mathematical object you're imagining? Do abstract mathematical objects have colors? What does that tell you about whether you're actually imagining an abstract mathematical object, or a physical object which approximates an instantiation of the abstract mathematical object?
    – g s
    Feb 9 at 2:56
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    Plato's perfect circle is a perfectly circular form, an immaterial "intelligible" thing outside of space and time. Whatever you can imagine in your mind, if it is any sort of image, is probably vaguer and more deficient than what can even be drawn on paper with a good instrument. It is only through your intellect that you can know this or some other form, not see it through senses or imagination. And even the intellect, normally constrained by sensible bonds, requires quite a bit of meticulous sharpening and purification, before it can truly know forms.
    – Conifold
    Feb 9 at 4:41
  • Perhaps Platon would rather consider it as an abstract definition than imagining its shape as a sort of seeing though such intuitive empirical seeing is usually useful anyway since the perfect circle dependently arises from the said visual sensibility of those perceptible circle like images. As an abstract yet worldly form of idea, circle is always perfect implicitly by its very definition and construction in contrast to the varieties of imperfect circle like images... Feb 9 at 6:25
  • A perfect circle is just an abstract ideal, like an integer number. It is not the symbol. It is not a size, weight, a group, a count, a representation, etc. It is a number.
    – RodolfoAP
    Feb 9 at 6:54
  • Do you know Plato's Theory of Forms? A Form is aspatial (transcendent to space) and atemporal (transcendent to time). For example, the Form of beauty or the Form of a triangle. The idea of a perfect circle can have us defining, speaking, writing, and drawing about particular circles that are always steps away from the actual being. The perfect circle, partly represented by a curved line, and a precise definition, cannot be drawn. Feb 9 at 7:01

3 Answers 3

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A perfect circle is an idea, hence an abstract entity which cannot be seen by our eyes but can be captured with our mind.

Plato exemplifies the ascend from the world of concrete visible objects to the domain of ideal entities in his work “Symposion”. His example is “the form of the beautiful”, but one can carry over his words to geometric forms like “the form of the circle”.

Socrates recalls a speech, the teaching of Diotima, about this subject. One can already read-off from the words chosen by Plato, that in this passage Plato leaves the domain of clear and well-defined concepts and enters the world of mysticism with metaphorical concepts.

Concerning the act of vision Plato says: ”he sees the beautiful through that which makes it visible” (Symposion 212a), see Plato symposion.

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You cannot see a perfect circle. Having in mind the idea of a perfect circle is not seeing it. Even if you can conjure up an image in your mind's eye, it won't be a perfect circle, in the sense that every point on the circumference of the imagined shape will not be exactly the same distance from its centre, since your mind does not have the resolving power to discriminate tiny differences.

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    Ha! You underestimate the power of my mind's eye. My visualised perfect circle comes with a visualised certificate that every point on the circle is exactly the same distance from its centre.
    – Stef
    Feb 9 at 18:02
  • @Stef You must have a laaaarge uncountable store of certificates 😉. More seriously, I was thinking of answering starting with an old comment of yours on what exactly π is.
    – Rushi
    Feb 9 at 19:12
  • @Rushi Hahahahahah I can't remember what I was thinking when I said that the definition of pi was circular :D
    – Stef
    Feb 9 at 19:30
  • @Stef Yeah That 'circular' is both circular and meta circular. I've quoted it on occasion as a classic. [See my comments also following]
    – Rushi
    Feb 9 at 19:32
  • @Stef when I said 'you', I meant you the person who posted the question, not you Stef. Obviously you have the exceptional ability to imagine pi exactly, as befits a person who managed to read an 'Awful Truth' book to its very end!! Feb 9 at 21:34
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Preliminaries

  • In order to keep this conceptual and impersonal lets try to keep Plato — the person — out of it to the extent possible, beyond the key Platonic idea that applies here — the theory of forms. We immediately have an issue here since Plato didn't speak English very well and I (and I presume you) dont speak Greek very well.
  • The words that get translated to form are eidos (εἶδος), idea (ἰδέα) which are used near interchangeably in the Greek original.
  • Plato was not really a mathematician even if his academy had on its gateway: Let no one enter here who does not know geometry. This was more a reminder that in order to philosophize effectively, one needs to have a clear head and — Plato believed — math to be a suitable preparation for this. Now the εἶδος that Plato was primarily interested in were Justice and Truth and Beauty and... ultimately the Good, not points, lines, numbers etc. For the former, applying the English word form literally, clearly seems absurd. But it turns out to be at least as problematic for form-al math entities like circle as well.

So lets take a different point on the εἶδος translation spectrum
form → idea → ideal → essence → soul
I'll use essence.

And then the question can be rephrased:

Whats the essence of circularity or circle-ness?

You then ask whether it could be πr2?

That immediately pushes the question into:

What is the essence of π?

From an old comment thread on this SE somewhat expanded

Comment 1: Now I show you a circle and ask: What is the ratio of the perimeter and the diameter? The answer is Pi. Now I ask: What is Pi? The answer is: It is the ratio of the perimeter and diameter of any circle. That's circular. You could also say that Pi is approximately 3.14, but it's not exactly 3.14, so there really is no better answer than the circular Pi is the ratio of the perimeter and the diameter, and the ratio of the perimeter and the diameter is Pi.

Comment 2: Ah, but there are many other formulae for calculating Pi, and you might answer in terms of those.
eg
contfrac1
Or
contfrac2

Comment 3: Yes, sometimes one writes π as 3.14, or 22/7 or 3.1415926535 sometimes as the more correct and sophisticated formulae in Comment 2 above. And sometimes we leave π as π! (Not to mention sometimes making Pi into π as I've done or π into Pi as you've done 😉).

We could continue this conversation:

Mathematician: Those continued fractions look so much more beautiful than the messy 3.1415926535! It occurs to me that the messiness is really a result of being stuck on the decimal form perhaps?

Engineer: You kidding?! I need π regularly in my daily work. 3.142 does just fine almost always, the ten digit form almost never. And as for your continued fractions — you must be joking if you think we engineers will give them a second look!

Computer Scientist: BTW your circular is both circular and meta-circular

So then the thought that we know exactly what π is may not be quite as unproblematic as we may think?

π is fraught

Lets see some different angles to why π may be fraught

τ more fundamental than π?

There is a contention among a few mathematicians that 2π is a more natural constant than π. Setting τ = 2π we get that if for a circle, d is diameter, r is radius and c is circumferance we get:
c = πd
c = τr

Alternatively one could say τ is a full circle whereas π is only a half circle.

This may seem like hair splitting but the people who claim that τ is better than π for doing math give many examples:

Formula original π modified τ
Normal dist pinormal taunormal
Planck constant piplanck tauplanck
Sterling's approx pisterling tausterling
Euler pieuler taueuler

And many more

So is the essence of circle-ness captured better by π or by τ?

This question is more philosophical than may be apparent. Every circle is similar/congruent to every other... geometrically.

Likewise one can use any multiple of π in formulae. The tau-ists choose 2π. But why not √2π? Even in the small selection of formulae above we have two occurrences.

In short there is an arbitrariness in algebra which becomes canonicalized in geometry.

Wigner/Gauss

In the Unreasonable Effectiveness of Math, Wigner starts with an amusing story:

There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."

So to recollect. The standard normal distribution looks like this geometrically

stdnormal

Or like this algebraically

pistdnormal

Your question itself is: Is a circle this?

circle

Or these?

circleformulae

So your question multiplies!

Geonetric Algebraic
circle circleformulae
Standard Normal pistdnormal

If you go across the columns it becomes Is the essence of circleness algebraic or geometric?

If you go down the rows it becomes Is π mathematical or statistical?

Given the ubiquity of the normal distribution perhaps the second view should be given more credence?

Euler

Euler's formula comes from the very prosaic fact that a complex number has a cartesian (rectangular) form: x + iy and a polar (circular) form: (r, θ)

z = x + iy = r (cos θ + i sin θ)

From where.. voila!

pieuler

Often cited as the most beautiful equation in math — unifying algebraic and geometric, small basic integers and the two most will known transcendental numbers.

Maybe the Euler identity should be the π definition rather than a derived result?

Geometry v Algebra? Math v Stats? Queen v Servant?

Starting from the (seeming) Platonic question:

What is a perfect circle for Plato?

Via the question:

What is the essence of π?

We've seen the question morph into:

Is geometry or algebra the heart of math?

Lest we are too easy with the algebraic manipulations it may be worthwhile to remember this comment found on math-SE

Geometry excites insight; whereas algebra elides it

A more deeply philosophical version of the question becomes

Does math come first or statistics?

After all we dont find perfect circles in nature but we find the Gaussian/normal distribution all over wherever there are large numbers!

An even deeper philosophical form of the question then is

Is Mathematics/Statistics the queen of science or the servant of STEM?

Gestalt Thinking

In the early 20th century the German psychologists Wertheimer, Koffka and Köhler seeking to push back against the dominant reductionist school of psychology based on English empiricism created a new school of psychology — which sought a whole-person, whole-context view of psychology not tied down by by reductionist presumptions. Thus was born Gestalt Psychology.

A Gestalt is an integrated, coherent structure or form, a whole that is different from the sum of the parts. Gestalts emerge spontaneously from self-organizational processes in the brain.

Philosophy Empiricism Rationalism
Outlook reductionism holism
Psychology Structuralist Psychology
based on sensation-ism,
atomism, associationism
Gestalt Psychology
based on reification,
multistability, invariance

Back to Platonism

Perhaps we need a term within the εἶδος translation spectrum:
form → idea → ideal → essence → ????? → soul
for representing that we not only have the essence of a question but have fully grasped and digested it, and yet not go all the way to the spiritual realm entailed by soul.

Perhaps the real issue with Plato who is arguably the most important philosopher in the western canon yet much disapproved/disliked is that Plato did not know German.

Maybe if he had he would have used gestalt for εἶδος, the misunderstandings around the English form and suspicions around Platonic heaven would not have occurred??

Yeah... I am being quasi ironic out here but only quasi: Platonism is too central to all philosophy to have us derailed by glitches in Greek translation!

We started with your question:

What is a perfect circle for Plato?

Which we may now restate in a »new« translation:

What is the gestalt for circle-ness?

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