"The whole is more than the sum of its parts"

Aristotle thought empty vacuous matter united with natures, combining to be objects in the world. So things are part material part spiritual. But the spiritual natures were tailored made for matter, to be instantiated. Thus there is the finite world. Then there is his Prime Mover(s). Besides the finite world and the divine, Aristotle thought nothing existed. So maybe math for him was true only in our minds. Unless he would say the Prime Mover(s) guaranteed the truth of them

Also important is that Aristotle thought God or Gods were ahead of instead of behind Aristotle's eternal universe. His God acted as a final cause instead of an efficient cause. This may have some relation to the distinction between this Greek philosophy and Christianity, and to Truth as seen by Aristotle

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

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

There is no afterlife for Aristotle. Just an eternal succession of temporal states. All things must pass away he thought, except God or Gods who were in the never reached future. So it's an interesting question if Aristotle thought mathematics was true because of matter or true in itself

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?

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

James Franklin is your man.

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?

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

You're welcome. (Actually decades ago I was manager of a University computer store, and Jim Franklin was one of my customers!)

Have a listen to a couple of minutes of this lecture starting from where I've bookmarked it. 'Thinking as a universalising activity....literally, you could not think if materialism was true' :clap: . When we recognise kinds, types, species, they're all essentially manifestations of form. I think this is a reference to the famous passage in De Anima about the 'active intellect', the faculty which grasps 'the forms' and is able to reason on that basis.



(If you don't know Lloyd Gerson, by the way, he's probably one of the leading academics in Platonist studies. That lecture he's reading is Platonism vs Naturalism.)

Nominalism proper only took root with William of Ockham, Francis Bacon, and others, in medieval times. I believe that the debate between scholastic realists (who accepted the reality of forms) and the nominalists was a watershed in Western thinking. The nominalists - precursors of later scientific empiricism - won the day, and, as is said, 'history is written by the victors'. So much so that in this particular matter, that it is very hard for us moderns to even understand what the argument was about. But the upshot was, in my view, that with the victory of nominalism, the possibility of a real metaphysic was lost, as this depends on there being degrees of reality, which neither nominalism nor later empiricism can accomodate.

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

To take the law of non-contradiction as an example, Aristotle regards it as a fundamental principle of being ("It is not possible for the same thing at the same time both to belong and not belong to the same thing in the same respect" - Met. 1005b19-20). That distinguishes his view (immanent realism) from both Platonism (that the LNC exists in separation from being) and Nominalism (that the LNC is just a law of thought).

Excellent. Thank you so much for all the helpful comments. I really appreciate it.

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.)

Thanks again for the help!

Thank you! That is very interesting and enlightening.

Yes, that makes sense. Thanks!

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

I think demonstrating the potential for circular objects is sufficient to ground mathematical circles. And since mathematical circles can be considered in separation from circular objects anyway (see the earlier Aristotle quote), the contingency of circular objects has no effect on mathematical practice.

Absent a demonstrable grounding, circles might be regarded as mysterious or dubious, just as negative numbers and complex numbers have been in the past before constructive visualizations were found for them.

I think that it should be kept in mind that mathematical principles are not the same kind of thing that mathematical objects (i.e. triangles or numbers) are. Same with logic. Whatever principles (or laws or axioms or posits) are, which is not at all clear as is usually the case with Aristotle, they do not seem to be treated or function similarly to "proper" abstract objects. For example, it is usually said that such axioms, laws or principles are intuitively or naturally grasped and they have to be grasped in order to be able to know anything. They are also said to be unprovable or indemonstrable, yet indubitable. Instead of "objects" or "things", these principles seem to be more like relations between such "objects" or "things". The debate around principles such as PNC seems to be whether Aristotle takes them to be metaphysical or just logical/mathematical.

When it comes to the way Aristotle thinks of universals I find it useful to think about it from the POV of "priority". It's usually accepted that Aristotle recognises different senses of priority. For example, priority in definition/account, priority in time, priority in nature and substance. A common view is that, according to Aristotle, universals have definitional priority compared to substances, but concrete objects are prior in nature and substance compared to universals. There's much literature around this and it gets really complicated really fast, but I think that the following quote from Stephen Menn's The Aim and the Argument of Aristotle's Metaphysics illustrates neatly the nuances between the platonic, the aristotelean and the nominalist approaches. The issue of priority is also discussed in terms of ways of existing, but it's also related to the issue of archai. Contra Plato and the the Pythagoreans, Aristotle argues that universals cannot be archai, that is to say, the cause of all being.

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.

I think it is correct to say that Aristotle believed we could understand mathematics in a more abstract sense, as mathematics and logic are derived from being and particular objects. He also mentions in the Posterior Analytics that the mind is so constituted that we can apprehend and understand these more abstract principles. 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? (I know conceptualism and nominalism are later philosophical phenomena. However, I had a professor tell me that Aristotle laid the intellectual foundation for nominalism and I'm trying to figure out for myself if that is really true.)

Thanks again for all your insight and help. I know I have a lot more to learn about this!

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

Not object, but concept. Such a concept doesn’t exist anywhere but in a mind, but it is nevertheless the same for all who think. I think the big underlying difficulty in our thinking about this, is that we are ‘born and bred naturalist’, in that we can only conceive of reality in terms of what exists in space and time. Whereas concepts such as geometrical forms, and even natural numbers, are not locatable in terms of space and time. That is the sense in which they are ‘transcendent’ to space and time.

I am not a fan of Edward Feser in all respects, but some of his Blog posts are very good. Have a read of Think, McFly, Think

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

Important point, and a contested point, in that almost nobody else on this forum will agree with it: that numbers, universals, and so on, don’t exist in the same way that the objects of experience do.

Nearly everyone will object that things either exist or not, and that it is unintelligible to say that some categories of things exist in a different sense to some others. 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; even, in some sense, of a higher degree or domain of reality than the objects of common sense. The reason that is controversial, is that modern culture tends overwhelmingly towards some form of philosophical materialism, which holds that ultimately only material objects or forms of matter and energy exist. It holds that intelligible objects such as concepts are the activities of brains, and that brains are physical, thereby grounding such things as abstractions in ostensibly physical (i.e. neurophysiological) processes generally shaped by evolutionary development.

So, it’s not really just a coincidence that the typical advocates of Platonism in modern culture turn out to be Catholic, on the whole, because Aristotelian philosophy offers a model of dualism, called hylomorphism (matter-form dualism) which accommodates a broadly Platonist philosophy (see, for instance, Peter Kreeft's Youtube lecture series on Platonism.) Whereas the majority view is that only one half of the Cartesian dualism of mind and body, namely, body, has any ultimate reality. What is mental or intellectual is in some sense either subjective or socially constructed but possessed of no inherent reality.

As I say, mine is a minority view on this forum, as I’ve never been able to accept the premisses of scientific materialism, but that is the way I see it.

Would Aristotle say geometry is true because of matter or because of the Prime Mover?

Not object, but concept. Such a concept doesn’t exist anywhere but in a mind — Wayfarer

Not according to Aristotle. From the point of view of Aristotle, that's nominalism; closer to Lycophron's thesis than to his own.

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

Again, that's not what Aristotle says. Universals' existence is certainly not "purely intellectual" according to him. It might be true of "Aristotelianism" though, who knows.

Not according to Aristotle. From the point of view of Aristotle, that's nominalism; closer to Lycophron's thesis than to his own. — Two

How does nominalism account for the nature of concepts, then? Isn't it the case that, according to nominalism, concepts (and the like) are simply names given to like things? Did you perchance glance at the Feser blog post I linked to about it?

For the sake of clarity, then, an edited passage from Ed Feser, with my comments on it:

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

I would say, that what Feser designates 'thought', I would designate 'reason' or 'judgement', and that what he designates 'intellect' is clearly descended from the Greek term nous.

He gives some examples:

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

Depends on the specific branch of nominalism. The whole talk of nominalism is already steps away from Aristotle, it's parasitic to his work. I referred to it because I found it paradoxical that you want to argue against the idea that Aristotle was a "nominalist", yet what you ascribe to him (by way of "Aristotelianism" or "Platonism"), is closer to ancient "nominalists" like Lycophron. 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. That's one of the theses he argues against. I guess you're taking as a given that the matter in which we come to know, grasp or understand something, inevitably leads to how it's supposed to exist, i.e. if we know universals through the intellect, then universals exist in a purely intellectual manner. That's not what Aristotle says though. Neither the logic nor the conclusion makes justice to him. Even talk about knowing universals through the intellect can't be taken seriously as an interpretation of Aristotle, since it discounts numerous distinctions both regarding "universals" and the "intellect".

I skimmed through Feser's article. It's not about Aristotle's discussion of various ways of existing. It's entirely free of it. It's also free of any substantial discussion of various ways of existing simpliciter. For example, the part that you quoted only refers to various human faculties. There's no reference (let alone argument), to the various ways things exist. I'm somewhat familiar with Feser's work: I've read multiple blog articles in the past. Nothing scholarly in them, it's mostly cultural commentary (aka polemics). I would never recommend them to someone who wants to understand Aristotle. I've also read his introduction to scholastic metaphysics and his book on Aquinas. Better than his blog, still in no way a sound recommendation for someone who wants to understand Aristotle (or Aquinas for that matter).

In my first post I linked a few quotes from Stephen Menn's draft of "The Aim and the Argument of Aristotle's Metaphysics" - that's what I consider Aristotelian scholarship. Some related books and articles that I consider worth reading on this and related subjects (irrespective of the interpretation they provide):

Aristotle on the Many Senses of Being and The Question of Seperation -- Stephen Menn

Priority in Aristotle's Metaphysics -- Michail Peramatzis

The Priority in Being of Energeia -- Jonathan Beere

Ontological Priority and Grounding in Aristotle's Categories -- Riin Sirkel

thanks, good references, and I shall read them. //edit//although as often with these papers, knowledge of ancient greek is assumed, which poses a rather high bar to entry. Several are behind academic paywalls, and the last one returns an error. But thanks all the same.//

I'm very encouraged by the first sentence in the first reference, to whit, 'Aristotle thinks that serious philosophical errors have been made, from Parmenides down to his own day, as a result of failing to draw distinctions between different senses of "being". '

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

I am the very first to admit the scanty nature of my knowledge of Aristotle and indeed the classics generally. But there's a very interesting concept in Aristotle that, as you appear to be so well versed in the matter, perhaps you like to comment on. This is that in Aristotle's hylomorphic dualism knowledge comprises a union of both sensory and intellectual elements, whereby the sense detect the material substance, but the intellect detect the form. The senses receive the material form, but the intellect perceives the intelligible form:

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.

From Thomistic Psychology: A Philosophical Analysis of the Nature of Man, by Robert E. Brennan, O.P.; Macmillan Co., 1941.

So, this lends to support to the 'intelligible' nature of universals - they're the 'intelligible forms' that the mind discerns in order to understand what a thing is.

they're the 'intelligible forms' that the mind discerns in order to understand what a thing is. — Wayfarer

How would you say that the "mind" does this (as far as Aristotle is concerned)?

Have a look at the Gerson video above - it’s been bookmarked to the passage about this point.

It doesn't seem to be bookmarked, it starts from the beginning.

Have you considered Aristotle's metaphysics viz time?

"It can be said that the world of mathematics exists in an eternal present, a state in which neither the past nor the future have any meaning; there is no significance to the questions of what came before, or of what will happen next... Within the sphere of mathematics, the moment of time is always 0. In other words, time has neither meaning nor significance within mathematical operations."

https://www.oxfordscholarship.com/view/10.1093/0199247900.001.0001/acprof-9780199247905

http://www.torahscience.org/mathematics/time1.html

And of course the so-called infamous Aristotelian paradoxes of same:

In his first published work Hegel, after the the preface and introduction, immediately starts expounding on the "paradox" of time. I personally don't see the problem. The present merely moves forward constantly within metaphysical nothing. Time exists but past doesnt. The future doesn't exist except as a present. So I don't see a true paradox. Whether the present is in our heads or outside i don't find to be a fruitful topic of discussion.

I don't see a problem with nominalism either. Are two men more similar or dissimilar? That's the only aspect in which the question has meaning. Feser is an idiot. He insists he can prove God exists separate from us and the wheat hylomorphism thing. When asked to prove it he makes up a bunch of empty categories. He is also arrogant