Here is my difficulty: The abstract object which is a triangle for example is not inherent in any concrete reality. In concrete reality, there are only forms which approximate a triangle. However the abstract object which is the triangle is perfect. So how is it that we pull out or drag away from concrete reality something which is not actually in concrete reality? 

Hello Jimbo, about Ari and triangles... He goes into the nature of mathematical objects most fully in Metaphysics M (book 13).2-3. M.2 opposes Plato's separation of mathematical objects from perceptible objects. Basically he attacks Plato for reduplicating objects. His attack is on the tendency to treat the ideal solid or plane as though it were a solid or a plane. In chapter 3, his thrust is to say that math is not based on what is distinctive of its subject matter but on math's way of treating a subject matter, like a physical object, which math may share with other disciplines. The cook might say that one piece of toast is hotter than another; the geometer might say that it is more triangular than the other. The geometer studies physical objects "as" shapes. The shape is not something distinct, over and separate from the piece of toast. The geometer also ignores properties of the object that are incidental to his discipline. So perhaps irregularities in the outlines of the piece of toast are to be ignored. So he can say, "So if one posits objects separated from what is incidental to them, and studies them as such, one will not for this reason assert a falsehood, any more than if one draws a foot on the ground and calls it a foot long when it is not a foot long; the falsehood is not part of the premises." 1078a17-20. 

Here I think it's important to recall that Aristotle usually predicates some F of a substance under a certain description. Under the description, "plane surface," he'll allow himself to attribute geometrical properties of a piece of toast. Under the description, "product of the cook's art," he might say that the sides are not cut evenly. [I'm making up this latter example.]

Julia Annas, whose translation I used above, goes into the many difficulties in Aristotle's account of the relation between the geometrical property and the material substance. If you're really into the problem, I suggest working through her commentary on Metaphysics Books M and N. 

Some people think that Aristotle's synthesis, brilliant as it was, fails to split the difference between Platonism and what are later forms of nominalism. I'm inclining to think this, but it's not easy to come to conclusions about many things in Aristotle because of the nature of the texts that we have. You'll find things in one work that seem to contradict things in other works, or aporiae that are not really resolved satisfactorily. A glance at the history of commentary on Aristotle reveals the efforts his followers made to harmonize all the stuff found in his surviving writings. An industry that keeps on giving us puzzles!

ETA: in the Physics (ΙΙΙ.1), Aristotle famously says that in a genus, one thing is only in act (entelecheia), and other, partly in potentiality and partly in act. In one piece of toast, heat can be more fully actual than in the other. Ari might say that triangularity is more fully actual in one object than in another.

It would be interesting to see Aquinas' take on the math difficulties, but Aquinas' commentary on the Metaphysics ends with book Lambda. Perhaps someone else on here knows what Aquinas said about mathematical objects.

Last edited by ficino (11/03/2017 7:48 pm)

Jimbo28 wrote:

My understanding of Scholastic realism is that abstract objects do not exist in any kind of mind-independent "third realm". Rather, they only exist in contingent minds and in the eternal intellect of God. This view is generally construed as a realist ontology. But isn't it in a very obvious sense an antirealist ontology? That is, according to this view, abstract objects don't exist independent of observation. 

What is the philosophical motivation for construing this view of abstract objects as a realist one? 

Mind happens to be such that it can observe itself. So things in mind are not really independent of observation.

Jimbo28 wrote:

Here is my difficulty: The abstract object which is a triangle for example is not inherent in any concrete reality. In concrete reality, there are only forms which approximate a triangle. However the abstract object which is the triangle is perfect. So how is it that we pull out or drag away from concrete reality something which is not actually in concrete reality? 

From concrete reality we get a reflected image into the mind, but this is only part of the story. The mind itself is something, and it already has some prior things in it, such as laws of thought, which will determine how the abstraction goes. There can be images that are just distortions of concrete objects, the mind can extract irrelevant or accidental aspects of objects, but it can also extract essential or fundamental aspects. All this is testable by oneself as thoughtful observation and experience of empirical reality continues. Perfect or at least relevant abstractions surely withstand the experiential test better than others.

Then there's also Creator's mind. Creation can be viewed as a crystallization of the perfect forms in Creator's mind, i.e. the process is reversed compared to the human being. Humans abstract from the concrete reality to perfect ideas that are applicable to reality, but the perfect ideas in Creator's mind, analogous to the abstractions that human being can come up with, provide the formal[*] foundation for concrete reality. Bodily we live in the concrete reality, but mentally we are closer to Creator, when we think of concrete reality in that particular way.

[*] As in Aristotelian or Platonic forms, probably what Aquinas called exemplar forms.

This is a problem I'm very interested in. I've been having a long debate on The Philosophy Forum about the reality or otherwise of abstract objects. I am advocating a (classical) realist view, i.e. the abstract objects are real but not material.

I think the stumbling block in accepting that idea is that numbers and geometrical forms have a different mode or kind of existence to phenomenal objects. I would put it like this: that natural numbers are real but they're not, strictly speaking, existing things. 

You might say, hang on, there's the number 7, carved in granite - that's real, isn't it? But that is a symbol, not a number. The same number can be represented by any number of symbols. What is being represented is the actual number. And that is a 'mental object', although the term 'object' is misleading, because it's not really 'an object' in the sense that a billiard ball or a pen is an object. 

Now I think in Platonic epistemology, this is reflected by the view that knowledge of number and geometric form, dianoia, is of a higher order than knowledge of sensory objects, pistis or doxa. It's not the highest form of knowledge, but it's higher than merely sensory knowledge. But the problem is, the sense in which it's 'higher' has now been lost in contemporary discourse. And that's because there is no 'vertical dimension'. So everything that exists, does so in the same way: chairs, tables, the number 7, and triangles, exist; unicorns, the square root of 2, and square circles don't.

Whereas in the ontology of classical philosophy, there is an assumed hierarchy, with 'nous' being higher than matter. //edit// This means there are 'degrees of reality'.// The corporeal eye receives sensations, but the immaterial intellect perceives the Forms, and thereby understands what the thing it is percieving is, and also what it is for. However, that has been lost in the transition to modernity.

So, I think the problem with the expression of 'the third realm' (and I have read about Frege's conception of the third realm) is that it creates an irresistible tendency to ask: where might that be? And then you've objectified the whole question, i.e, treated intelligible objects as amongst phenomena. I say that us moderns do that instinctively and unconsciously, due to the inherent naturalist bias that is embedded in our universe of discourse. There's no 'up there' for the forms to reside in - there's only the objects of the astronomical sciences 'up there' or 'out there'. Cosmos is all that exists, we say.  Hence the problem!

Last edited by quotidian (11/12/2017 10:05 pm)