I've read some of Aristotle but I'm no expert. I am wondering what an Aristotelian response might be to abstract objects such as the principles and axioms of mathematics? Are those pure abstractions to an Aristotelian? — DS1517
The best way to conduct an investigation in every case is to take that which does not exist in separation and consider it separately; which is just what the arithmetician or the geometrician does. — Aristot. Met. 13.1078a
Thank you for sharing that. I found the article very helpful. Some Platonists accuse Aristotle of laying the groundwork for nominalism. I don't think it is fair to accuse Aristotle of nominalism. Are there other good Aristotelian responses to nominalism? — DS1517
I think that is correct. Aristotle believed that mathematical properties are immanent within concrete objects. I'm wondering how he would account for the laws of logic and the principles of mathematics that make geometry and math possible in the first place? I'm guessing they are just abstractions of concrete objects or fundamental principles of being? — DS1517
Another thought ... Perhaps a Platonic objection but I was wondering what you thought. From an Aristotelian perspective, if I could destroy all the circular objects in the world, would I have successfully destroyed the essence of circularity? What might an Aristotelian response be? (I've read Aristotle but I can't remember if he addresses this question.) — DS1517
As you may know, Aristotle was an immanent realist, not a Platonic realist. He regarded mathematical objects as an aspect of the world that could be investigated (albeit in a more abstract sense), not as existing apart from it (in the sense of Plato's Forms which he rejected).
The best way to conduct an investigation in every case is to take that which does not exist in separation and consider it separately; which is just what the arithmetician or the geometrician does. — Aristot. Met. 13.1078a — Andrew M
As I understand it, the essence or universal of circularity is in the circular object, because for Aristotle, concrete objects demonstrate mathematical properties (weight, volume, extension, etc.) The essence of circularity is not floating around in a Platonic heaven somewhere. — DS1517
The above quote from Aristotle's Metaphysics seems to indicate that he didn't think mathematics exists in the same way other things exist (which I think is intuitively correct). However, does that make Aristotle a conceptualist or nominalist? — DS1517
Not object, but concept. Such a concept doesn’t exist anywhere but in a mind — Wayfarer
But I think in Platonic philosophy generally, and in this context this includes Aristotelianism, there is an at least implicit conception of what is called the ‘intelligible object’, whose existence is purely intellectual, but which is real in own right — Wayfarer
Not according to Aristotle. From the point of view of Aristotle, that's nominalism; closer to Lycophron's thesis than to his own. — Two
As Aristotelians and Thomists use the term, intellect is that faculty by which we grasp abstract concepts (like the concepts man and mortal), put them together into judgments (like the judgment that all men are mortal), and reason logically from one judgment to another (as when we reason from all men are mortal and Socrates is a man to the conclusion that Socrates is mortal). It is to be distinguished from imagination, the faculty by which we form mental images (such as a visual mental image of what your mother looks like, an auditory mental image of what your favorite song sounds like, a gustatory mental image of what pizza tastes like, and so forth); and from sensation, the faculty by which we perceive the goings on in the external material world and the internal world of the body (such as a visual experience of the computer in front of you, the auditory experience of the cars passing by on the street outside your window, the awareness you have of the position of your legs, etc.).
That intellectual activity -- thought in the strictest sense of the term -- is irreducible to sensation and imagination is a thesis that unites Platonists, Aristotelians, and rationalists of either the ancient Parmenidean sort or the modern Cartesian sort. — Feser
First, the concepts that are the constituents of intellectual activity are universal while mental images and sensations are always essentially particular (hence the remark by Gerson above, paraphrasing Aristotle, that 'reason is a universalising activity'). Any mental image I can form of a man is always going to be of a man of a particular sort -- tall, short, fat, thin, blonde, redheaded, bald, or what have you. It will fit at most many men, but not all. But my concept man applies to every single man without exception. Or to use my stock example, any mental image I can form of a triangle will be an image of an isosceles , scalene, or equilateral triangle, of a black, blue, or green triangle, etc. But the abstract concept triangularity applies to all triangles without exception. And so forth.
Second, mental images are always to some extent vague or indeterminate, while concepts are at least often precise and determinate. To use Descartes’ famous example, a mental image of a chiliagon (a 1,000-sided figure) cannot be clearly distinguished from a mental image of a 1,002-sided figure, or even from a mental image of a circle. But the concept of a chiliagon is clearly distinct from the concept of a 1,002-sided figure or the concept of a circle. I cannot clearly differentiate a mental image of a crowd of one million people from a mental image of a crowd of 900,000 people. But the intellect easily understands the difference between the concept of a crowd of one million people and the concept of a crowd of 900,000 people. And so on. — Feser
What's important is what Aristotle says about the way universals exist and nowhere does he say that universals exist "purely intellectually", as far as I know. — Two
Everything in the cosmic universe is composed of matter and form. Everything is concrete and individual. Hence the forms of cosmic entities must also be concrete and individual. Now, the process of knowledge is immediately concerned with the separation of form from matter, since a thing is known precisely because its form is received in the knower. But, whatever is received is in the recipient according to the mode of being that the recipient possesses. If, then, the senses are material powers, they receive the forms of objects in a material manner; and if the intellect is an immaterial power, it receives the forms of objects in an immaterial manner. This means that in the case of sense knowledge, the form is still encompassed with the concrete characters which make it particular; and that, in the case of intellectual knowledge, the form is disengaged from all such characters. To understand is to free form completely from matter.
Moreover, if the proper knowledge of the senses is of accidents, through forms that are individualized, the proper knowledge of intellect is of essences, through forms that are universalized.
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