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According to the summary of Platonism (ie the Forms) by Aristotles Metaphysics:

Besides sensible things, and the Forms, there are mathematical objects; of the first (the sensible) they share in multiplicity; of the second (Forms) eternality & immutability.

First, why would there be 'multiplicity' of mathematical objects - surely there is only one perfect circle?

Secondly, why this domain of 'mathematical objects'; why not render them simply as Forms - this is guise, for example we have Mathematical Platonism; which is this philosophy, where the only Forms are Mathematical Objects.

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  • The Plaonism of Plato is different from the Platonism of mathematical platonists ... For Plato "the Good" is a Form: it is a "single entity". If we assume that numbers are "entities" of some sort, they share with Forms eternality and immutability, but not "singularity" : we have "the number 2", "the number 3", and so on. The same, I think, for geometrical shapes : why only one circle (of course the "ideal") ? we have the circle with radius 1, the circle with radius 2, ... – Mauro ALLEGRANZA Jan 27 '15 at 16:13
  • @MauroALLEGRANZA: Ok, I see where you're driving at; I was thinking of the multiplicity of the number '1' ie lots of 1's; is there a specific reference in Plato for this - Aristotle doesn't elucidate; he wants to push and place the Forms into sensibility. – Mozibur Ullah Jan 27 '15 at 21:01
  • Related question: Is mathematical platonism compatible with Platonism? – Artem Kaznatcheev Jan 29 '15 at 0:59
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The multiplicity of mathematical objects is meant to account for scenarios such as the following:

  1. Geometry: Let triangle a intersect triangle b ...

  2. Arithmetic: 2 + 2 = ... (count your fingers)

In both cases we tend to use concrete objects and drawings. But these are only practical aids, not the real mathematical objects. The multiple imperfect triangles on the blackboard stand for multiple perfect, exact, pure triangles.

The source is Plato's Analogy of the Divided Line in the Republic through which he presented four different kinds of objects, corresponding to four different kinds of knowledge. Here is an excerpt where Plato relates to the mathematical objects:

You are aware that students of geometry, arithmetic, and the kindred sciences assume the odd and the even and the figures and three kinds of angles and the like in their several branches of science; these are their hypotheses . . .
Yes, he said, I know.
And do you not know also that although they make use of the visible forms and reason about them, they are thinking not of these, but of the ideals which they resemble; not of the figures which they draw, but of the absolute square and the absolute diameter, and so on— the forms which they draw or make, and which have shadows and reflections in water of their own, are converted by them into images, but they are really seeking to behold the things themselves, which can only be seen with the eye of the mind? (Republic Book VI)

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I don't have ancient references, but my best understanding is an interpretation from Schoepenhauer:

Mathematical necessity, according to the principle of sufficient reason of being, in virtue of which, every relation which is stated in a true geometrical theorem, is as that theorem affirms it to be, and every correct calculation remains irrefutable.

So I think you are right; 'circular' is formal. Take the example of a triangle contains 180 degrees. It is necessary due to the all the constraints in the representation; it cannot be otherwise, therefore what is 'responsible' for this is a formal idea (eidos, causa formalis).

However, I am guessing Aristotle is wanting to put a fundamentally empirical or pragmatic outlook to work on this. There is still debate over what exactly a generalized reference is (how it is or isn't related to the collection of representations it references), so it has not been settled.

Firstly, a geometric understanding of 'circular' contains a set of relations that must be observed and can take multiple appearances and representations related to the circle's being. What is it that you think about when you think of a circle? It might be its roundness (can you conceive it perfectly round?), or the proportion of its radius to circumference, a curved line joining itself, a plot of sinusoids, a collection of innumerable points, a highly symmetrical object. If you think of its symmetries, which of them? What size is the circle in your mind? In reference to what? Its position? The circle we observe or imagine is relatively not far from a form, but it is still a relation of finite constraints, and it cannot be represented perfectly. The circle will therefore have a 'logical necessity' as well as a formal one.

Any representation, drawn or imagined, will make certain concessions and could be considered simplified in some sense or wrong in some senses, so it is not the ideal form. It is represented by definite, possibly finite materials that at any rate describe it and caused it to be. Therefore, it has a 'material necessity'.

We have certain understandings of why a circle exists, in what context, where it comes from, what it is meant to achieve, what it reveals or is of interest, therefore it has a 'moral necessity'.

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  • There's an interesting analogue here in formal mathematics, which isn't widely understood because of the 'scariness' of Category Theory; the idea there is similar to the jumping between numbers individually understood as 1,2,3..; and when they're understood as being defined by an axiomatic system. – Mozibur Ullah Jan 27 '15 at 21:03
  • So, in CT, one defines the property of the number one, and then one doesn't care about how they're constructed, because they're all isomorphic; I was thinking about this idea of 'multiplicity' when I wrote the above question; similarly for the circle, in the usual differential geometry one specifies a differential structure and 'completes' to all possible compatible ones, to get uniqueness; but in CT, you can drop that consideration and just have lots of circles with different but diffeomorphic differential structures. – Mozibur Ullah Jan 27 '15 at 21:08
  • The latter part of what I wrote is how formally in CT, we can concieve of 'multiple appearances and representations' to the 'circles being'. – Mozibur Ullah Jan 27 '15 at 21:10
  • This question was very interesting. I wonder how well this CT concept of 'the circle' can be understood, experienced, or connected to other concepts? (loose terms, but I think you get the gist) If the other CT forms were also connected 'mathematical' forms in the ideal sense, then all would have to originate simultaneously, it would seem, a sort of causal necessity. (how can a permanent ideal 'arise' anyhow?) If they would not be 'logical', they would have to have an apparent unity of understanding as well. If they would not be 'moral', they would need to to be free from intention or purpose. – dwn Jan 27 '15 at 23:24
  • The ontological philosophy that is associated with CT is the standard one in mathematics - mathematical Platonism; causality is a good question, and isn't normally touched upon; it was this that pushed Aristotle to modifying Platos theory; I think though the 'causality' for what its worth is through the participation of the ideal mathematical objects in the Forms; and through the Forms in the One; this is the Pythagorean element in Plato and leads to the Neoplatonism of Plotinus; I can see why, now Pythagoras was considered as a religious leader rather than as a mathematician. – Mozibur Ullah Jan 28 '15 at 16:04

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