The exercises and quotations below are designed to help you gain a better appreciation of Plato's theory of ideas. Most importantly, if you answer the questions, and manage to see the difference between our intellective (thinking, understanding) and sensitive (perceiving, imagining, dreaming, remembering) cognitive acts, then you will have taken great steps toward grasping Plato's point.

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Exercise 1: if A and B are subsets of two non-intersecting sets, then how many members does the intersection of A and B have? Once you managed to find the right answer (which is really easy *if you understand the question*), try and see whether there could possibly be any other answer to the question. What is it that "makes you" give this answer? What do you have to be aware of in order to give this answer, and in what way are you aware of it (seeing, feeling, imagining, dreaming, remembering, thinking, understanding, etc.)? What are the things that you are thus aware of? Where are they? Can you change their properties at will (as you can do with objects of your imagination)?

Exercise 2: try to imagine a triangle, then a square, then a pentagon, then a hexagon, then a heptagon, then an octagon, etc. Where do you loose track? Then try to imagine a chiliagon (a polygon with 1000 angles). Can you imagine a polygon of 1001 angles, as distinct from a chiliagon? (Don't get frustrated if it doesn't work, since that's the point.) How many sides does a chiliagon have? How many sides does the polygon of 1001 angles have? How do you know?

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Modern Mathematicians on Platonism in Mathematics

Charles Hermite:
I believe that the numbers and functions of analysis are not the arbitrary product of our spirits: I believe that they exist outside of us with the same character of necessity as the objects of objective reality; and we find or discover them and study them as do the physicists, chemists and zoologists.

Rene Thom:
Everything considered, mathematicians should have the courage of their most profound convictions and thus affirm that mathematical forms indeed have an existence that is independent of the mind considering them... Yet, at any given moment, mathematicians have only an incomplete and fragmentary view of this world of ideas.

Godel:
despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e. in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perceptions will agree with them and, moreover to believe that a question not decidable now has meaning and may be decided in the future. The set-theoretical paradoxes are hardly any more troublesome for mathematics than deceptions of the senses are for physics.....They [intuitions] too, may represent an aspect of objective reality.

Roger Penrose in connection with Mandelbrot sets: though defined in an entirely abstract mathematical way, nevertheless it has a reality about it that seems to go beyond any particular mathematician's conceptions and beyond the technology of any particular computer...It seems clearly to be 'there' somewhere, quite independently of us or our machines.
Its existence is not material, in any ordinary sense, and it has no spatial or temporal location. It exists instead in Plato's world of mathematical entities... In general the case for Platonic existence is strongest for entities which give us a great deal more than we originally bargained for.
...Because of the facts that mathematical truths are necessary truths, no actual 'information' in the technical sense, passes to the discoverer.
All the information was there all the time. It was just a matter of putting things together and 'seeing' the answer! This is very much in accordance with Plato's own idea that discovering is just a form of remembering.

Georg Cantor:
The transfinite is capable of manifold formations, specifications, and individuations. In particular, there are transfinite cardinal numbers and transfinite ordinal types which, just as much as the finite numbers and forms possess a definite mathematical uniformity, discoverable by men. All these particular modes of the transfinite have existed from eternity as ideas in the Divine intellect.