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A fairly standard approach to mathematical Platonism might start with the observation that any circle actually drawn is not a true circle; and one can imagine or have in mind a perfect circle - a universal circle, say; this is taken from Platos theory of Forms or Universals.

This is quite easy to actually imagine, but it does not remain the case, at least for me, when I consider numbers: here, there is two bottles, three stones, or five pebbles; yet I cannot imagine - I mean in the sense of visualise - the numbers two, three or five; though I can quite easily work with them.

Should one make a distinction then, between a geometric mathematical Platonism and an arithmetical version? I mean, is one actually made in mathematical platonism?

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I feel like this question is predicated on a misunderstanding of mathematical platonism. Mathematical Platonism is not, strictly speaking, part of Platonism. The name comes out of a kind of analogy to Platonism. Mathematical Platonism is the view that The language of mathematics refers to and quantifies over abstract mathematical objects, and that these objects are independent of any rational agent.

The mathematical ideal of twoness is not two pebbles or two bottles. It's (usually) the set {{}, {{}}}. You can use another definition, but this one is the most common construction of the cardinals. Likewise, the other natural numbers are defined (recursively) as follows:

0 = {}
1 = {{}}
2 = {{}, {{}}}
3 = {{}, {{}}, {{}, {{}}}}
n = {0, 1, ... , n - 1}

There's no sense in which a circle is more abstract than the number 2. They are all abstract object. Asking if they are objects of the same kind is complicated because in order to be meaningful, that must be done within a mathematical context, and mathematics is very careful about assigning structure to objects. For example, strictly speaking, the group of integers and the ring of integers are different objects. Some mathematical objects have arithmetic properties, some geometric, some both, and some neither.

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  • Upvote +1 for this clear reconstruction of numbers by set-theory. - Do you mean ordinals or cardinals? - Which mathematical objects have neither arithmetic nor geometric properties? Do you mean the topological structures, which are also considered fundamental by Bourbaki?
    – Jo Wehler
    Nov 14, 2015 at 8:07
  • For the finite case, ordinals and cardinals are the same thing. I chose to say ordinals because one usually defines cardinals out of the ordinals, though I suppose in the context of the "two" mentioned in the question one might consider cardinals more relevant. I'll edit to cardinals for consistency sake. There are all sorts of objects and structures which are neither geometrical nor arithmetical. The set of continuous functions from $\mathbb{R}\to\mathbb{R}$ when viewed as a group under composition do not have an arithmetic structure. There are a thousand examples on graphs and from CS too
    – Stella Biderman
    Nov 14, 2015 at 8:15
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Plato emphasizes that we have two ways to obtain knowledge, our senses and our thinking. To the domain of forms we have access only by thinking. Hence we grasp the idea of the circle as well as other mathematical ideas solely by our thinking.

In the symposion Plato describes the ascent to the domain of ideas as a kind of mystical intuition. But even when Plato employs words "to see the idea of beauty" he always means to intuit the idea by our mind.

In his late philosophy Plato starts some speculation about numbers. Possibly a lecture "On the Good" existed dealing also with numbers. Aristotle refers about Plato's conception of numbers in Metaphysics (Book 1, 987b15): Numbers are considered a third type of entities, between the sensuous entities and the forms. In addition, Aristotle refers: Numbers come into being by participating in the One (987b21f).

In any case, according to Plato we do not grasp numbers by our eyes but by our mind. Hence mathematical Platonism is not necessarily geometric.

In my opinion, all Platonic ideas and also numbers are formed by abstraction: From "two bottles, two persons, two trees" we abstract to the number Two.

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In addition to the above, I would note that there seem to be a number of meanings attached to the term "Mathematical Platonism," apart from what might be called Plato's mathematics and idealism.

Russell and Whitehead's Principia could be described as a Mathematical Platonism reduced to logical forms and including numbers. Godel wrote about his own mathematical Platonism, which was quite "Platonic." Penelope Maddy and a few other philosophers, on the other hand, have attempted to develop a "Mathematical Platonism," including numbers, based on set theory and a "naturalized epistemology" that correlates "sets" with "object identity" in child development and the creation of neural "cell assemblies." So the term is not even restricted to idealism.

Personally, I find it fascinating that geometry in Plato's day eschewed numbers and demonstrated proofs using only the compass and straight edge. This meant that the "Forms" were presumably "Performed" through hand motions, not unlike music, sign language, or carpentry. A very different, and perhaps superior, way of developing mathematical intuitions. A shame it is no longer taught this way.

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  • I can only agree - the charm of Euclidean geometry isn't just that it's axiomatic but also that it's geometric - it might be another good reason why Newton eschewed the use of his calculus for his book; interestingly, geometry has made something of a comeback during the last century - geometrising both number theory (schemes) and set theory (toposes); but not quite with ruler and compass, as such.
    – Mozibur Ullah
    Nov 14, 2015 at 17:14

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