Hilary Putnam, who died a couple of months ago, had some interest in the Aristotelian-Thomistic tradition, even if in part it was a critical interest. One area where this interest manifested itself is the philosophy of mind; another is the philosophy of religion. I’ll address the former in this post and the latter in a later post. Let’s consider in particular an exchange between Putnam and the analytical Thomist philosopher John Haldane in the volume Hilary Putnam: Pragmatism and Realism, edited by James Conant and Urszula Zeglen.
Some background: In a series of books and articles, Putnam put forward penetrating criticisms of materialist attempts to explain the intentionality (or “directedness” or meaningfulness) of thought in causal terms. The basic idea of such theories -- greatly to oversimplify -- is that a brain process (say) will count as a thought with the content that the cat is on the mat if it is caused by the presence of the cat on the mat. As Putnam argued, a fatal problem with such theories is that they have no way of characterizing the causal relation in question without implicitly presupposing intentionality, when the whole point of such theories is to explain how intentionality enters the picture in the first place. More generally, the notion of causation which such materialist theories require is not one to which the theorists are entitled given their notion of what counts as “physical.”
I discussed Putnam’s critique in some detail in a post a few years ago, and also in my essay “Hayek, Popper, and the Causal Theory of the Mind” (reprinted in Neo-Scholastic Essays), wherein I noted that to some extent Putnam is recapitulating an argument presented in a more rudimentary way decades earlier by Karl Popper. (When I sent Putnam that essay he confirmed that he had “had no idea” that Popper had put forward such an argument. Great minds think alike.) As I also noted in these articles, Putnam argued that to salvage causal theories of the sort he criticizes would require returning to an essentially Aristotelian conception of nature of precisely the sort materialists suppose to have been supplanted by post-Newtonian science.
Now, while Putnam is sympathetic to the anti-reductionist bent of Aristotelianism, he too resists returning to an Aristotelian conception of nature. That brings us to his exchange with Haldane. Haldane’s essay has the memorable title “Realism with a metaphysical skull,” which is a play on the title of Putnam’s book Realism with a Human Face. Putnam wants to defend the reality of the everyday, commonsense world of human experience against reductionistic materialists who are willing to affirm the reality only of what can be described in the language of physical science. Haldane argues (correctly, in my view) that doing this successfully requires defending also some version of an Aristotelian conception of nature, which is what Putnam is unwilling to do. The “human face,” in Haldane’s view, requires an underlying “metaphysical skull” to hold it up.
One of the themes of Haldane’s article is that it is -- as Putnam himself emphasizes -- a mistake to try to explain the relationship between mind and world in terms of causal relations between inner mental representations (Lockean ideas, sentences in a “language of thought,” or what have you) and physical objects. One of the problems with this approach is that it opens up a gap between mind and world which can never be closed. Another is that it presupposes too narrow an understanding of causation. Modern representationalist-cum-causal theories of the mind confine themselves to what Aristotelians call efficient causation (and a desiccated notion of efficient causation at that). The right way to understand the relationship between mind and world, Haldane argues, is in terms of formal causation.
One application of this idea is that the right way to understand the relationship between a thought and its object is in terms of formal identity. When you judge that such-and-such an object is a triangle, what happens is that the mind takes on the very same form that the matter of which the triangle is composed has taken on. Precisely because there is, on the side of the thought, nothing material that has taken on this form, the thought is not itself a triangle (as any material thing that took on that form would be) but merely a thought about a triangle. But precisely because it has the very same form that the triangle has, the thought is a thought about a triangle rather than about something else. Again, thought and thing are formally identical, though not identical full stop. And because of this formal identity there is no gap between mind and world that needs to be bridged in the efficient-causal terms that causal theories of thought appeal to.
In his reply to this in the Conant and Zeglen volume, Putnam notes first that he is to some extent sympathetic with Haldane’s position. In particular, he notes that some analytic philosophers have tended to distinguish sharply between concepts (as constituents of thought) and properties (as entities entirely independent of thought). Putnam is sympathetic to the idea that this sharp distinction is mistaken and that talk of “concepts” and talk of “properties” are really just two ways of talking about the same things. To the extent that this is what the Aristotelian is saying, Putnam tells us, he is happy to go along with it.
But Putnam is not happy to go along with the whole metaphysical package associated with formal causation. As far as I can tell, he raises three objections in his reply to Haldane. The first is that Putnam thinks that talk about the mind taking on the form of a triangle when it thinks of a triangle (my example, not Putnam’s) “too much suggests” that the thinking is itself a triangle -- which is, of course, absurd, and not what the Aristotelian is saying -- but that Putnam doesn’t have “the foggiest notion of what [such talk] is supposed to mean” otherwise.
Now, it’s hard for me to know what to say about this objection other than that it seems to me that Putnam is just ignoring, without answering, the details of the Aristotelian hylemorphic analysis of physical objects -- including details Haldane summarizes in his paper. Because when those details are taken account of, it just isn’t mysterious what the Aristotelian is saying (whether or not one agrees with what he is saying). Hylemorphism, to a first approximation, distinguishes between the matter or stuff out of which a physical thing is made and the form or pattern that configures that matter or stuff into a physical thing of a certain specific type. It is only the two together that make up a physical object. Hence the form of a triangle -- that is to say, the form or pattern of being a closed plane figure with three straight sides -- is not itself a triangle or any other material object. It is only matter together with this form which is a triangle.
Now, in light of that, it is hardly fair to say that talk of the intellect taking on the form of a triangle absurdly “suggests” (“too much” or otherwise) that the thinking itself amounts to a triangle. It would suggest this only if it were suggested that the matter of a triangle as well as the form is taken on by the mind. But this is precisely what is being denied, since the claim is that the form of a triangle exists within the mind without the matter. Hence a thought about a triangle no more counts even prima facie as a triangle than the form of a triangle all by itself and without matter could count as a triangle. Hence for Putnam to claim that the Aristotelian position “too much suggests” that a thought about a triangle itself just is a triangle simply ignores the whole distinction between form and matter which is the heart of the theory.
Nor will it do for Putnam to say that it is mysterious what the Aristotelian is saying if the Aristotelian doesn’t mean that a thought about a triangle just is a triangle. For the Aristotelian is simply saying that the very same form or pattern -- that of being a closed plane figure with three straight sides -- is both what makes such-and-such a material object a triangle, specifically, and what gives such-and-such a thought its specific content. A certain material object has the form or pattern of being a closed plane figure with three straight sides; that’s what makes it a triangle specifically. You grasp the form or pattern of being a closed plane figure with three straight sides; that’s what makes your thought a thought about a triangle specifically. Because it is one and the same form in both cases, we have what Haldane calls formal identity. Because, in the case of the thought, it is the form of the triangle alone, and apart from the matter, that is present, the thought is not itself a triangle. Now, one can raise all sorts of questions and objections to this, but to suggest that we haven’t even the “foggiest notion” of what it means seems to me a real stretch.
Putnam’s second objection is that there is with at least some things no one pattern that could plausibly count as the “form” or essence of the thing. He gives the example of a dog, and he says that for a molecular biologist, the DNA of a dog would count as what is essential; for a population biologist, being part of a certain reproductive population would be what is essential; for a pet lover a wild dog might not count as a “dog,” while it might count as a dog from the point of view of a scientist; dingos might count as dogs from an Australian aboriginal point of view, but as members of a different species from an American point of view; and so forth.
The problem with this is that Putnam here runs together features and descriptions of very different types, which Aristotelian-Thomistic philosophers would carefully distinguish before trying to determine the essence of a thing. We need to distinguish, for example, between a thing’s essence and its “proper accidents,” which flow from its essence. (To take the stock example, the essence of a human being is to be a rational animal, and the capacity for amusement is one of the proper accidents that flows from this essence.) We also need to distinguish between these proper accidents (or “properties,” as that term is used in Aristotelian-Thomistic metaphysics) and merely contingent accidents. (For example, unlike the capacity for amusement, skin color is a merely contingent accident of a human being, not being something that flows from rational animality as such.) We need to distinguish what is “natural,” in the Aristotelian sense that it follows from an inherent or built-in tendency of a thing, from what is “artificial” in the sense that it is true of a thing only by virtue of some extrinsic or externally imposed pattern. (Spelling this out requires the distinctions between substantial form versus accidental form, and between immanent teleology versus extrinsic teleology.) And so on. The Aristotelian theory of formal cause can be properly understood only in light of all these distinctions.
Putnam essentially conflates all the different sorts of phenomena captured by these distinctions and lumps them together as equally good candidates for “the form” or “the essence” of a thing. And that simply gets the Aristotelian position badly wrong. An Aristotelian metaphysician looking for the essence of dogs would say, first of all, that cultural associations and linguistic practices of the sort that some of Putnam’s examples involve are simply not relevant to determining the essence of any natural substance like a dog. (Such assumptions and practices often mix together what are aspects of a thing deriving from its substantial form and those that involve merely accidental forms. Finding the essence is in part a matter of separating these out.) The Aristotelian would also note that reproductive capacities follow from deeper physiological facts about an animal, so that an animal’s status as a member of a reproductive population is less fundamental to what it is -- that is to say, to its essence -- than its DNA would be. But the Aristotelian would also caution against simply identifying the essence of a thing with some “hidden structure” like DNA. And so on. (All of these issues are dealt with in my book Scholastic Metaphysics, in David Oderberg’s Real Essentialism, and in writings by other Aristotelian-Thomistic philosophers.)
It is in any event no good for the critic of Aristotelianism simply to pull out of his hat some random example -- whether dogs or anything else -- note some complications or controversies in determining the essence of the thing, and declare that he’s identified some grave difficulty for Aristotelianism. This sort of objection is very common, but it is completely misguided. Not only does it typically ignore the sorts of distinctions just referred to, but it seems to assume that Aristotelians suppose that a thing’s essence can be determined fairly easily on the basis of a cursory examination.
Nothing could be further from the truth. Neither cursory inspection, nor even cursory inspection together with the complex theoretical apparatus I’ve alluded to (substantial versus accidental form, essence versus proper accidents, etc.), will typically suffice to tell us the essence of a thing. Detailed investigation of a physiological, chemical, or other scientific kind is often required in order to determine a thing’s essence, to determine what are really only proper accidents rather than part of the essence, and so forth. I know of no Aristotelian who denies this or supposes that these questions can be settled from the armchair. Nor does Aristotelian metaphysics as such ride on how such an investigation turns out -- as if the defensibility of the entire system of the four causes, the theory of actuality and potentiality, hylemorphism, etc. waits with bated breath on what the facts about dogs turn out to be! The most that would ride on the results of such an investigation is how we apply the system, not the soundness of the system itself. (For example, if it turned out even after detailed investigation that there simply was no plausible candidate for the essence of dogs, this might be interpreted as showing that “dog” is really a kind of accidental or artifactual category rather than reflective of a substantial form -- not that I think this is remotely likely.)
(Putnam also says that Haldane later suggested in response to the objection at hand that the essence of a dog might simply be the conjunction of all of the candidates for the form of a dog cited by Putnam. But Putnam does not tell us where Haldane said this, and I find it hard to believe Haldane did or would say that, since it is just a bizarre thing for an Aristotelian to say. I think Putnam must have simply misunderstood whatever remark of Haldane’s he has in mind here.)
Putnam’s third objection against Haldane is that forms do no real explanatory work. For to say that the form or pattern being a triangle is what makes some particular triangle a triangle “sounds either tautological or nonsensical.” But the answer to this objection should be clear from what was said above. The form being a triangle -- or, as I put it above, being a closed plane figure with three straight sides -- does not suffice to make something a triangle, because both triangles and thoughts about triangles have that form, and the latter are not triangles. Being a triangle requires both the form in question and matter. Hence the claim that “the form of being a triangle is what makes some particular triangle a triangle” is not tautological or trivially true. It could be tautological or trivially true only if the presence of the form of being a triangle sufficed all by itself for the presence of a triangle, and it does not. It might also sound tautological if the words “being a triangle” were thought to capture the entirety of what it is to be a triangle, but of course that is not the case either.
The claim in question is really shorthand for something like “The form being a closed plane figure with three straight sides, when combined with matter, is what results in a triangle (as opposed to a circle, a square, a dog, etc.).” And that is not a tautological claim. (Certainly not when we go on to explicate form and matter in terms of the theory of actuality and potentiality, etc. For the hylemorphic analysis so developed, far from being trivially true, is one that critics of Aristotelianism claim to be false, and a claim can’t be both false and trivially true.) Nor is the claim in question nonsensical. As I noted above, it is clear what it means, whether or not one agrees with the Aristotelian analysis. (The rote objection that explanations in terms of substantial forms and causal powers are tautologies is one I address at greater length in Scholastic Metaphysics, at pp. 43-46.)
So, Putnam’s objections fail. In fairness, though, it must be said that Putnam does try to engage with the Aristotelian-Thomistic position in a fair-minded way, and he sees its strengths -- and the ways in which neo-Aristotelian assumptions are implicit even in some of what contemporary naturalist philosophers unwittingly say -- far more clearly than these naturalists themselves do. And needless to say, Putnam was in any event one of the great figures of contemporary philosophy, whose work always repays careful study. Again, in a future post I’ll consider how he engaged with Aristotelian-Thomistic philosophy in the context of philosophy of religion.
When people object and say these Aristotelian notions are trivially true, do they basically mean that your claims are analytically true (like the way Kant defines analytic)?ReplyDelete
Ed, would you (or some other commenter) be kind enough to straighten me out here.ReplyDelete
One application of this idea is that the right way to understand the relationship between a thought and its object is in terms of formal identity. When you judge that such-and-such an object is a triangle, what happens is that the mind takes on the very same form that the matter of which the triangle is composed has taken on. Precisely because there is, on the side of the thought, nothing material that has taken on this form, the thought is not itself a triangle (as any material thing that took on that form would be) but merely a thought about a triangle. But precisely because it has the very same form that the triangle has, the thought is a thought about a triangle rather than about something else. Again, thought and thing are formally identical, though not identical full stop. And because of this formal identity there is no gap between mind and world that needs to be bridged in the efficient-causal terms that causal theories of thought appeal to. (emphasis added).
This is how I've always understood it, and it why I converted from Plato to Aristotle, many years ago. But I have been taken to task on this - in this combox - for understanding this literally, that is, that the forms of things, in the mind, and in objects, are indeed identical. Rather, I've been told that the correct AT position is that, e.g., the form of triangle in triangle A is different than the form of triangle B. (From this it would seem that the form in my mind, when thinking of A cannot be the same as that in my mind when I think of B). This leaves me utterly confused, and to me, it seems to defeat the point of hylomorphism. But then, I am no advanced student, so I don't want to be dogmatic. But I would like to be less confused. (Not confused at all is beyond my aspirations.)
@SK: I don't know if this helps, but IIRC, Kant's definition of "analytic" was taken from Aquinas's definition of "self evident", and acknowledged to have been.
Naturally, I screwed up the html tags. The whole 2nd paragraph was intended to be italics, except the highlighted bits. Oh, well.ReplyDelete
I have an objection. This is, of course, an internal dispute since I am likewise an Aristotelian. Take the following passage, where you say:
“The form being a triangle -- or, as I put it above, being a closed plane figure with three straight sides -- does not suffice to make something a triangle, because both triangles and thoughts about triangles have that form, and the latter are not triangles. Being a triangle requires both the form in question and matter.”
You make other statements similar to this one throughout your post. There problem is that the criterion you give for what makes something a triangle is inconsistent with the claim that material objects are formally indeterminate (a claim which you have often defended). If Ross’ argument is sound, it entails that geometrical forms (as well as, for example, arithmetical, logical and algebraic forms) cannot be determinately realized by any material object; thus, being intrinsically and genuinely of such a form cannot be a matter of having matter––since, again, there is no such thing as matter being a determinate instance of, i.e. intrinsically and determinately realizing, an abstract form. Matter can only approximate such forms, but it cannot be of such forms, which is why no material object can exclude its being a case of incompossible abstract forms (e.g. “triangle” or “quatriangle”?). So what it is to be a triangle cannot be a matter of conjoining (or realizing) the form and matter.
In other words, there is a categorial gap between the abstract form and the matter that is supposed to realize that abstract form. Ross’s argument, which you yourself have defended, entails that no kind or amount of matter can claim as its own a unique and definite abstract form. To sum up, what it is to be a triangle cannot be a matter of materially realizing the abstract form “triangle” since no such thing is possible––except, as Kripke says, in an observer-dependent (i.e. extrinsic) way. But if this is the case, then the criterion you give, on the basis of matter, for differentiating what it is to be a triangle from what it is to be a thought of a triangle collapses, since matter is not necessary for the former––indeed, it cannot be.
I have to say that I think there is something deeply misconceived about the question of why my thought of a triangle is not itself a triangle. I don't buy your answer, either.
First, let's consider your answer: my thought of a triangle is form without matter, whereas an actual triangle is form-plus-matter. Consider a seraph, who is a pure form. Since it belongs to the most exalted of all the choirs of angels, it is certainly capable of understanding what an ordinary angel is, since the higher is capable of grasping the lower. Thus its intellect contains the form of that angel. But since that angel is nothing but form, then why isn't the seraph's thought of an angel itself an angel?
You might respond to the above objection by invoking the essence-existence distinction: the seraph's concept of an angel is essence without existence, whereas an actual angel is essence-plus-existence. But if that's the line you're taking, then you no longer need the form-matter distinction to explain why my concept of an X is not itself an X. All you need say is that since my concept of an X lacks existence, it is not an actual X.
But there is a deeper problem, and this goes back to my objection to the way you framed the question in the first place. I would argue that you're guilty of reifying thoughts, or concepts. Thoughts, I would suggest, are not things, but actions: they are verbs (acts of thinking, or "thinkings," if you like) rather than nouns. There is no such thing as a thought.
When I entertain the concept of a triangle, my concept of a triangle is not in my head, or even in my intellect. Indeed, to talk of my concept of a triangle is absurd: there is only one concept of a triangle. Individuals may have different mental images of a triangle, but images are not concepts. What happens when I entertain the concept of a triangle is that my mind grasps the universal rule that defines something as a triangle: being a closed plane figure with three straight sides. The rule is not "in" my mind, even metaphorically; it only exists in real-world triangles themselves. The rule can be formulated in some ideal logical language which maps onto the physical features of a triangle. But whatever language it is formulated in, the rule itself is not a triangle, any more than a sentence about an X is itself an X.
A. R. Diaz: there is no such thing as matter being a determinate instance of, i.e. intrinsically and determinately realizing, an abstract form. Matter can only approximate such formsReplyDelete
But a material thing can easily be something determinate — what it cannot do is represent something determinately. So a physical brain-state has no problem being a particular brain-state with all the particular forms that entails; but it cannot represent, say, the Eiffel Tower or modus ponens, at least not determinately. And of course, your brain or any part of it is not actually the Eiffel Tower or modus ponens, hence your determinate thought thereof cannot be something [purely] material. But your brain is determinately a brain, and the Eiffel Tower is determinately an Eiffel Tower (at least, apart from any questions of part vs. whole, or artifact vs. substance, but that's a different matter (so to speak)).
@A. R. DiazReplyDelete
I think Feser would respond and say that an imperfect instantiation of something does not entail that the thing in question fails to instantiate it at all. For example human beings are rational animals and there are numerous instatiations of these in matter, but that does not mean each and every human being realizes the same potentials inherent in what a human being is. Having 2 arms is something human beings have, but not all human beings realize this potential. Yet a one-armed human being is still a human being, but just an imperfect one.
The same is true for triangles. The form can be present in the matter even if matter does not instantiate all the features of a triangle like being perfectly straight in its sides.
I think this all depends on one arguing from immanent universals too. So I think Feser's argument can work as long as you keep in mind about his arguments about universals that he makes in his writings.
Vincent Torley: But if that's the line you're taking, then you no longer need the form-matter distinction to explain why my concept of an X is not itself an X. All you need say is that since my concept of an X lacks existence, it is not an actual X.ReplyDelete
Yes, something like that would be the answer; but there's no problem there, merely something that Ed didn't happen to address (yet).
There is no such thing as a thought.
If that's the solution to a problem, then it strikes me we have a case where the cure is worse than the disease! But I don't see any problem to begin with. If my mind grasps a universal [rule], then yes, that universal, or form, is in my mind. That's exactly what we mean by saying that I have something in mind. To call it "my" concept just means that the concept is in my intellect; just as saying "my cat" refers to the form of felinity "in" my pet. Far from being a problem, that's just what the Aristotelian view is.
"But a material thing can easily be something determinate..."
Thanks for your response. Of course physical things are determinate, i.e. physically determinate. But that is not what is at issue here: what follows from Ross' argument (which Feser accepts and defends) is that physical things are not formally (i.e. mathematically and logically) determinate––in the specific sense in which Ross uses "form" and "formally" here mostly to refer to the abstract structure of logical or mathematical forms and functions and their properties, e.g. truth-preserving in virtue of form. A triangle, like any other geometrical object, is precisely an abstract form (unlike material forms).
Feser knows this argument well. Indeed, in one of his responses to objections made by Peter Dillard, he replies by saying that to suggest that mathematical and logical forms can be determinately realized by material things but that we don't know it is as silly as saying that geometrical forms are determinately realized in material things but that we just don't know it. So he knows (or, rather, is strongly committed to the claim) that material things cannot be of a definite abstract form, whether this be geometrical, logical, algebraic, etc. Perhaps they can be extrinsically determinate (say, by reference to an observer, as Kripke argued), but this just means that in themselves, i.e. intrinsically, they realize no such forms; only relative to us , to our interpretative conventions, our interests, or what have you, can they be said to realize any abstract function at all. But then this would be true of his criterion for what makes a thing a triangle (rather than a thought about a triangle). So I'm pretty sure he would not give your reply.
"I think Feser would respond and say that an imperfect instantiation of something does not entail that the thing in question fails to instantiate it at all. For example human beings are rational animals and there are numerous instatiations of these in matter, but that does not mean each and every human being realizes the same potentials inherent in what a human being is. Having 2 arms is something human beings have, but not all human beings realize this potential. Yet a one-armed human being is still a human being, but just an imperfect one.
The same is true for triangles. The form can be present in the matter even if matter does not instantiate all the features of a triangle like being perfectly straight in its sides."
Hello SK. I don't think Feser would give that response. It seems to me that there is an equivocation here with the adjective "imperfect". What I meant, in the context of Ross' argument at least, when saying that a material object imperfectly realizes a geometrical or logical form is to say that the material object does not realize such a form intrinsically, i.e. genuinely in itself, but only relative to something else (say, our interests). I did not mean that it realizes some parts but not others (what, at any rate, would could that possibly mean with respect geometrical figures, which are discrete not continuous units? Half a circle it not a circle). The Kripke-Ross-Feser argument entails that a material object can at most be said to approximates or simulate such forms, for nothing in its material properties determines a determinate function. It does not, for it cannot, realize them intrinsically and ontologically speaking. (See Feser's article on Kripke and Ross. You can also read Ross's on writings on the matter). It's not a matter of degrees (it's not that, say, a physical computing machine can realize the addition function only for some numbers but not others: it's that it cannot realize the addition function at all! Mutatis mutandis for anything geometrical). But in any case, since approximation is not realization (read: "intrinsic and determinate realization") it is likewise subject to considerations of indeterminacy (does this material object approximate the addition function or the quaddition function?).
At any rate. Feser's Rossean argument for the immateriality of the human intellect entails (and he is quite explicit about this) that mathematical and logical forms cannot be intrinsically realized by anything physical. I think this is correct. But then it conflicts with his requirement that for something to be of a definite abstract form (in this case, the geometrical figure of a triangle) it must be material or conjoined to matter. But that cannot be the case if Ross is right.
I'm trying to follow along with what A.R. Diaz started here. I have no specialized knowledge so bear with me and please correct me if I'm misunderstanding something.ReplyDelete
It sounds like he's saying that we cannot simultaneously hold that
1. Triangles exist
2. In order for a triangle to exist it must have the form of a triangle as well as matter
3. Matter can only approximate the form of a triangle
He then seems to say that all 3 have been defended by Dr. Feser. Is this correct?
I'm having a hard time seeing how anything can said to be an instance of a form that it only imperfectly represents. Maybe I'm muddling Essence and Form or something like that. My understanding is that a thing cannot lack any part of its essence or else it could not be considered that thing. For example, to say that you have a human who lacks a rational nature is to say that you don't have a human. I don't see how we could have something that is approximately a rational animal. I can see varying degrees of perfection to that rationality, but it seems like being rational or not is like being red or non-red. If the form that something instantiates is the same thing as its essence, then I don't see how the account of existence as matter attached to a form can work.
Can someone please help clarify this for me?
How can an imperfect instantiation of a triangle be said to be a triangle at all when triangularity is defined in terms of certain perfections?ReplyDelete
@A. R. DiazReplyDelete
"At any rate. Feser's Rossean argument for the immateriality of the human intellect entails (and he is quite explicit about this) that mathematical and logical forms cannot be intrinsically realized by anything physical. I think this is correct. But then it conflicts with his requirement that for something to be of a definite abstract form (in this case, the geometrical figure of a triangle) it must be material or conjoined to matter. But that cannot be the case if Ross is right."
I see your point. Although I don't think I know enough to make any final judgments. However could this arguments you are making about how forms are not realized determinately in matter apply only for math and logic? It seems that what your saying allows for biological, chemical, and basically not mathematical or logical forms to be realized in matter.
@A. R. DiazReplyDelete
Also if it is correct that mathematical and logical forms cannot be realized in matter, would this not mean that the triangles we see in the world are not really triangles at all? Since a triangle has 3 perfectly straight sides and no material triangle has that. Doesn't Ross's argument entail platonism of some kind for math and logic. Now of course the Scholastic can say the mathematical and logical entities are in the mind of God instead of being independent of God, so the Scholastic does not have to endorse full blown platonism.
To your first point, yes. The argument concerns logical and mathematical forms (as well as the form of intellectual acts, like judging, negating, asserting, etc.). Biological, chemical, and neurological forms, for example, are not abstract (pure) forms, so it does not apply to them (although it will apply to their theoretical and mathematical idealizations in scientific practice).
To your second point, yeah, it need not entail Platonism about logical and mathematical objects and properties since the Thomistic, as well as the Augustinian, response is to locate those forms primarily in the mind of God. This is not the only option and some would argue not even the most plausible one (Ross being one those to argue thus), but you're right in that it does not in and of itself entail Platonism. What it does entail is that metaphysical naturalism about logic and math as well as about human intellectual activity (if indeed human intellectual activity intrinsically realizes logical and mathematical forms) is false.
Hi Mr. Green,ReplyDelete
Thank you for your response. You write: "If my mind grasps a universal [rule], then yes, that universal, or form, is in my mind."
Are we all being misled by a spatial metaphor here? I know it's a common and natural-sounding way of talking, but I can't help wondering if it's a mistaken one - especially in view of the fact that the intellect is immaterial. Perhaps we should jettison the notion of the mind as a container of ideas.
I would also note that a universal is a kind of rule, and "receiving," "containing," "extracting" and "grasping" are not rule-following activities as such. They are spatial metaphors for intelligence, but I don't think they succeed in capturing its very essence.
I haven't resolved this question in my own mind, so I would welcome suggestions.
Are you a materialist regarding the mind?
I would also note that a universal is a kind of rule, and "receiving," "containing," "extracting" and "grasping" are not rule-following activities as such. They are spatial metaphors for intelligence, but I don't think they succeed in capturing its very essence.ReplyDelete
I don't think it makes much sense to argue that receiving, containing, extracting and grasping are normally spatial terms and thus should be jettisoned (because the mind is not spatial), but then to object that they aren't rule-following activities while universals are "a kind of rule". Why is linguistic looseness allowable with a thing that is not a rule (a universal) but not with a thing that is not spatial? Especially when we all know that the mind isn't literally a box with ideas spatially located inside of them. Call it a categorization if you prefer. Comprehension.
Hey, Dr. Feser! I'm stuck in the filter again. Thanks in advance. I'll go on a diet, I swear...ReplyDelete
Thoughts, I would suggest, are not things, but actions: they are verbs (acts of thinking, or "thinkings," if you like) rather than nouns. There is no such thing as a thought. and Are we all being misled by a spatial metaphor here? and They are spatial metaphors for intelligence, but I don't think they succeed in capturing its very essence.
I'm puzzled at this. Of course they use of "in my mind" is only analogous to "in my car". But don't we deal with that sort of thing all the time? We can be in our houses, in class, in the navy, in a quandry, in a flood of tears and a sedan chair. I don't see how that is a problem, per se, and I don't see where you've shown that it is.
Take the thought you are expressing here. Your thinking it, and expressing it here in the combox are activities, as are my reading and (I hope) understanding it. But note that over 10 hours passed between your second, and my activities. As I read and type, it is highly likely you aren't thinking about it at all, yet it is presumably the same thought we are speaking of. (Even if it's not the same - if I've gotten it wrong - it is still your argument I've misunderstood.)
Granted, I've never yet seen that Pegasizing gets us much of anywhere. But it really would need some sort of case made that it is so, rather than just a suggestion that it is. Surely (at a minimum), the presumption is in favor of Socrates's point that the universal is used in the rule, rather than being the rule itself.
In answer to your question: No, I'm not a materialist. Acts of understanding are non-bodily acts.
Thanks for the enlightening post.
I have a question about the intentionality of thoughts that I do not think I have seen an answer to in your other popular writings on the mind (I must confess I have not read your book Philosophy of Mind, just Aquinas and many of your blog posts on the subject). Your explanation of formal identity really made me understand how we can think about things in general. That makes sense to me. I'm curious, though, about how we go from thinking generally about triangles, say, to thinking about this specific triangle that is made in red ink on a piece of paper.
Does my intellect have to somehow interact with a specific thing's matter/existence/whatever individuates a substance to be able to think about that thing?
I am a philosophy newb so feel free to school me if I've gone off the rails here.
My guess would be that somehow perception plays a role in this story because perception lets one get in contact particulars. So while the intellect can think about universals, perception lets on get acquainted with particulars. So maybe the intellect plus perception could be how one gets to think of a particular triangle. This is jut a guess so take my answer with a grain of salt.
George L, I thought the standard A-T distinction is that the form "cat" in this cat and the form "cat" in that cat are identical in species and distinct in number. This works also - somewhat - for this cat and my thought of cat: they are the same in species. My thought of cat is not, precisely, different in number, for my thought of cat is not instantiated in a physical cat to be enumerated properly speaking. My thought of cat, and Ed's thought of cat, are the same in species and distinct as to the individual person's doing the thinking. My thought of cat today and my thought of cat yesterday are the same in species and, I would guess, quasi-distinct as to distinct operations of thought. There is no need to *enumerate* them as they are distinguishable even without material differences.ReplyDelete
A. R. Diaz: what follows from Ross' argument (which Feser accepts and defends) is that physical things are not formally (i.e. mathematically and logically) determinate[...] A triangle, like any other geometrical object, is precisely an abstract form (unlike material forms).ReplyDelete
Well, there are two different matters here: one is the determinateness of material things, and a given substance certainly is formally determinate according to whatever forms it happens to have — if it's green, then it is determinately green, not some unspecified generic colour; in fact, this is an example of a form that a material object must have determinately, as opposed to an intellectual abstraction which can be indeterminate (e.g. my understanding of a dog is not determined to some specific colour, even though any actual instance of a dog must be some colour or other). And this applies as much to mathematical forms as any other: a physical quadruped is quite determinate as to number of legs.
So it follows just from the fact that physical objects have forms, that they are determinate with respect to those very forms, including any mathematical forms that may apply. Or perhaps it would be better to say "qualitative forms", since "mathematical" implies abstraction as opposed to material instantiation. And that brings us to the other point, which is I think the one you're thinking of: when I talk about "a triangle", I don't mean a physical thing that is triangularly-shaped, I mean an abstract platonic triangle. And one of the reasons I mean this is because there's no such thing as a (perfect) physical triangle. Our lumpy, quantised matter just can't match the delicate mathematical fineness required for perfectly one-dimensional lines and zero-dimensional points, etc. But that's just our universe! There is no metaphysical reason why a different world might not allow for inking lines with no width at all, for example. (Dunno how you'd see a perfectly width-less line of ink, but hey, the laws of physics would have to be completely different in such a world, so who knows.) This isn't a problem for Ross's argument, because even a perfect triangle can represent something else only indeterminately (and of course abstractions like modus ponens couldn't be physically instantiated at all, no matter how queer the laws of physics were).
I actually thought of this when reading Prof. Feser's article, and would have preferred a different example just because there is no such thing as a "material triangle", but the objection is minor; partly because it's metaphysically possible for there to have been material triangles (if only God had made a somewhat different kind of universe), but mainly because it's a convenient simplification for the sake of example. If we want to be pedantic, we can simply substitute for "triangular" some rough, wiggly shape that a real physical object has — mathematically, or rather, formally speaking, a jagged irregular shape is just as much as shape as a simple triangle. It would make the text a bit more cumbersome, but it doesn't affect the point Ed is making either way.
Young and Rested: It sounds like he's saying that we cannot simultaneously hold thatReplyDelete
1. Triangles exist [...]
3. Matter can only approximate the form of a triangle
He then seems to say that all 3 have been defended by Dr. Feser. Is this correct?
Yes. …and no. Really, there is a pragmatic equivocation going on. When we say, "triangles exist", we really mean "approximate triangles exist" — but usually it's not necessary to elaborate on the approximation in ordinary context. For example, the correct answer to "What shape is the letter delta?" is "triangular", even though the most careful scribe could not draw a perfect triangle. A pedant might want to describe it as "roughly triangular", but that would make it sound like the Cyrillic letter de (Д), which has a deliberate flat top and tails. (Hm, sounds like a Fred Astaire song!) Of course, the letter de in fact evolved from the triangular delta, but a carefully-drawn delta would make a sloppily-drawn de, and vice versa. So triangular things do exist (traffic signs, pyramids, musical, er, triangles) in the everyday, practical, triangular-ish sense; and triangles do not exist (physically) in the strict mathematical platonic sense. So you're right that a thing cannot by definition lack its essence, as long as we're precise about exactly what that essence is.
Thanks for the reply, Tony, but it leaves me puzzled as ever. I don't see how forms themselves can be "identical in species and distinct in number." That is how we describe the substances themselves. And it is the identity of form which makes them identical in species, is it not? Equally, it is their matter which individuates, making them distinct in number, no?ReplyDelete
If that it correct, it doesn't seem to make sense to refer to the matterless forms as distinct in number. The only way forms can be said to be "distinct" is in being different forms entirely, as cat and dog, etc. Or so it seems to me. The alternative seems to me to be making of forms little particulars, secondary substances of a sort, one for each substance and the form it exemplifies. (And of course, it looks as if these forms, as secondary substances, would have their own forms - tertiary substances of a sort - which makes them resemble the similar but not really identical forms in other members of the species. Harry Lyme returns.)
It also seems odd to call them identical in species, as being identical in species just means "having the same form", or so I thought.
When I gave up Plato for Aristotle, the single biggest reason was the point that Plato had made his forms out as, not really universals, but another kind of particular. I still remember in class when that came up. I didn't like it, but could find no answer. And I still cannot.
A question on the indeterminate nature of matter and material things: When Ed is speaking of this, is he referring to material things as he regards them (under hylomorphism), or as the standard modern view does, mechanically? Much of the time, it seems to be the latter; that Ed is using the argument to show the inadequacy of the modern view, rather than putting forward his own. (OK, strictly, as prelude to putting forward his own.)ReplyDelete
In any case, doesn't this question entail two different answers to Diaz's question. Or at least, the possibility of two different answers?
I am absolutely sure that Feser would not give your response (or even find it remotely plausible).
Let me clarify something. My objection to Feser’s characterization of what it is for something to be a triangle (and mutatis mutandis any other abstract, pure, form) presupposes a proper understanding of Ross’ argument. Since Feser knows this argument as well as anybody, there was no need for me to go into the details of how it works, how it establishes what it establishes, and what exactly is it that it establishes. For example, there should be no confusion as to what I mean by “formal indeterminacy”, since I am using it in exactly tin he same sense in which Ross used it (that is, “form” is restricted to the pure abstract structure of mathematical and logical objects and relations. The distinction logicians make between the “form” of an argument and the “content or semantics” of argument is but one instance of this use of “form”. Ross’ argument shows that no physical thing can be a genuine case of a determinate logical form, say, AAA-1 Barbara or modus ponens. Mutatis mutandis for mathematical forms). All of this is well known to Feser. I do not plan to rehearse it.
I will not re-hearse Ross’ argument. I will only, on the basis of it, point out the main problems with your response:
1.. “So it follows just from the fact that physical objects have forms, that they are determinate with respect to those very forms…” Ross’ argument does not gainsay this. It is obvious that he is not talking about material forms. As I said before, this is not even the issue at hand.
2.. “…including any mathematical forms that may apply.” This is where you beg the question, since the whole question is whether any mathematical forms apply DETERMINATELY to material objects (approximately, of course; no one denies that. Ross’ point is that they only apply approximately, which means non-intrinsically and non-observer-independent. Your talk about it being “metaphysically possible that other material worlds” so different from ours that they might intrinsically realize geometrical forms seems to me (a) to make no sense (what content would “material” then have?) and (b) to be a conceivability thesis; but, of course, A-T philosophers reject that conceivability is a guide to real, metaphysical, possibility, so this is not a response Feser could make). For the problem of formal indeterminacy of the physical is that nothing physical can exclude its being a case of an indefinite number of incompossible pure forms (plus or quus? triangular or qutriangular?), and thus since realizing intrinsically in a given case a determinate form means that in that case a unique form is realized to the exclusion of any other incompossible one, physical things cannot consistently be said to determinately realize any such forms. (Again, I would refer you to both Ross’ writings on the matter and Feser’s ACPQ article).
3. The fundamental problem here is that according to Feser, at least in this post, in order for something to be a triangle it must be materialized, that is, the pure geometrical form or figure “triangle” must be realized in matter in order for something to be a triangle. But this is impossible, as shown by Ross’s argument, for no material object realizes intrinsically a determinate and unique pure (i.e. mathematical and logical) function or form. (It is also proven false by the fact that geometrical forms are complete without matter, which is why we can carry out such things as the axiomatization of geometry. But this is a separate objection. My objection here was purely dialectical: that Feser’s comments here are inconsistent with his Rossean argument for the formal indeterminacy of the physical.). If this is the case, then being a triangle cannot be a matter of being materially realized; otherwise, there would be no such a thing as a triangle since no material thing can determinately realize it.
4. Your example of a quadruped person is likewise subject to the consideration of formal indeterminacy. Ross’ argument can be run with numbers as well, not only with functions, since these are likewise pure asbtract forms (is it a case of four or of quafour? Cf. Kripke’s ‘plus’ and ‘quus’ considerations).ReplyDelete
Now, let us use Feser’s criterion with your case. It entails that what it is for something to be (a genuine instance of) the number 4 is for it to materially realize the pure abstract form or object referred to by the numeral ‘4’. To this I would simply respond by saying that that makes no sense (and if it did, it is plainly and demonstrably false.) The number four, not even a genuine instance of the number four, does require matter for it to be what it is. Your legs are like numerals, or dots on a sheet of paper, with respect to formal entities and realities: mere approximations, even if very good ones. Body parts qua physical are both semantically and formally indeterminate. A genuine instance of a pure form contains that form (to the exclusion of any other incompossible one), together with all the properties that come with having that form. But is a quadruped the square root of 16? Can it be given a prime factorization? Is a quadruped the four-hundredth part of four-hundred? Is it a member of the set of all the positive integers? Of course not. But regardless of this, we know from Ross’ argument that a material realization of the number four (or any other pure form) cannot be anything more than an approximation (an probably a very good one) of that formal reality: that’s the most matter can do. If material objects realized determinately mathematical and logical forms, then both mathematic and logic could be studied and assessed by means of the study of the material processes that purportedly realize them. Again, we know this is false.
At any rate, from Ross’ argument we know that what it is for something to be of a determinate pure form is not a matter of having matter, for matter cannot possibly partake in the formal determinacy of any formal reality. That is the objection I raised.
Lastly, you say:
3. “So triangular things do exist (traffic signs, pyramids, musical, er, triangles) in the everyday, practical, triangular-ish sense; and triangles do not exist (physically) in the strict mathematical platonic sense.”
Then this is precisely the point. Feser’s criterion for what it is for something to be a triangle was not in terms of the “everyday, practical, triangulari-ish sense” (which is nothing but a placeholder for “in an approximate and observer-dependent way”) but rather in terms of what makes an object, intrinsically and really, a triangle (as opposed to a thought of a triangle). Otherwise, if he were using it in the “everyday, practical, triangulari-ish sense”, then he would not be answering Putnam’s objection, which was that A-T metaphysics does not have the resources to distinguish between something’s being, intrinsically and really, a triangle from something’s being a thought about a triangle. Feser answered that it did have the resources because what it is for something, intrinsically and objectively (i.e. in an observer-independent way) to be a triangle is for the geometrical form to be realized in matter. But this is inconsistent with the results of Ross’ argument. And this is far for being a “harmless” problem.
Sorry for misspelling your name, and for the long response. I shall probably leave it at that.
I am sure Feser knows what is at stake here, since he knows Ross' argument better than anybody else probably. Hopefully, he will address the problem (either in this blog or in print), since it seems to me to be of paramount important.
Putnam doesn’t have “the foggiest notion of what [such talk] is supposed to mean” otherwise.ReplyDelete
A vector can be interpreted as a directed line segment from point A to point B. You are point A, the mind is the vector, and the thought you are thinking about is point B. Just like how the vector doesn't "become" the point even though the point determines its orientation, likewise the mind doesn't "become" the thought even though the thought determines its orientation.
I'm wandering into this discussion to ask a few questions, since I am doing work on quantifying incongruity between objects as part of my research in humor. Be kind.ReplyDelete
1. Vincent Torley writes:
"What happens when I entertain the concept of a triangle is that my mind grasps the universal rule that defines something as a triangle: being a closed plane figure with three straight sides. The rule is not "in" my mind, even metaphorically; it only exists in real-world triangles themselves."
Well, it can be encoded in your brain. We know this because computers can decode the spike trains that always fire whenever the concept of a triangle is brought to thought. The encoding is not a triangle, of course, but it is a proxy for the concept.
2. What about topological deformations? An inner tube can be smoothly deformed into a coffee cup. At when point does the one form end and the other one begin? Is this a continuous change or does it occur catastrophically (in the mathematical sense) at some threshold?
3. What about the case where three piles of sand form an enclosure in the shape of a triangle? Technically, it is not a closed figure, because sand has holes in it and the straight lines are an illusion. Is this a real triangle or not?
4. What about the case where the triangle sits on top of a mound with very gradually sloping sides that asymptotically approach infinity, so that the plane is only enclosed in the limit of infinity and, in fact, there is no outside of the triangle?
What I am getting at is can one actually make a simple, pure definition of a triangle. If so, what are the essential attributes?
Well, it can be encoded in your brain. We know this because computers can decode the spike trains that always fire whenever the concept of a triangle is brought to thought. The encoding is not a triangle, of course, but is it a proxy for the concept?
Has anyone actually interpreted on a computer the neuronal firings associated with conceptualizing a geometric shape? If so, could you provide a link. That has a distinctly piscene aroma to me.Delete
I have a problem with reducing the mental to intentionality, as it seems to me that intentionality is not the essence of the mental. Non-mental things can participate in intentionality, e.g. a sign. It may be the "final cause" of the mind, but the formal cause of the mind seems to be something else - the best definition I have heard is that the mind is a form that can hold other forms without losing its own form. This seems to get right to the essence of the mind. No other form is like this, it exhibits the peculiar form of the mind. My mind is not a mind because it is "about" or "directed" to something, but because it contains forms that are not itself and, indeed, potentially contains all forms other than itself; my mind can hold the form of a fish without becoming a fish, the form of a number without becoming a number, the form of an idea without becoming an idea. So the mind in effect reproduces the world by holding it within itself while retaining its own form.
"IT IS POPULAR AMONG AQUINAS SCHOLARS to present esse intentionale as the mode of being that distinguishes cognizant from noncognizant beings. St. Thomas says something is cognizant just in case it is able to possess, in addition to its own form, the form of some other thing."
Robbie Moser, Thomas Aquinas, Esse Intentionale, and the Cognitive as Such, https://www.questia.com/read/1G1-261632110/thomas-aquinas-esse-intentionale-and-the-cognitive
And from St. Thomas:
"To prove this, we must note that intelligent beings are distinguished from non-intelligent beings in that the latter possess only their own form; whereas the intelligent being is naturally adapted to have also the form of some other thing; for the idea of the thing known is in the knower."
St. Thomas, Summa Theologica, Part I, Question 14, Article 1, http://www.newadvent.org/summa/1014.htm#article2
This agrees with what I said about the mind being principally a form that contains other forms.
"To cut a long story short, for Aquinas, intentionality or aboutness is the property of any form of information carried by anything. If we look at his remarks about esse intentionale in this way, all will make good sense. After all, it is not only my perceptions and my thoughts that carry information about my environment, but also the medium carrying this information to my senses. Furthermore, even if I never receive any of this information, the information is there, and qua information it certainly is about the thing that produces it, when the information is encoded by a natural effect of the thing. This is how, for example, the tracks, the scent, or the sounds of an animal, or the light reflected from its body carry information about the animal whether these are actually perceived by another, say, its predator, or not. Or, to use Aristotle‟s famous example, this is how the impression of a signet ring in a piece of wax encodes information about the shape of the ring itself."
Gyula Klima, Three Myths of Intentionality vs. Some Medieval Philosophers, http://faculty.fordham.edu/klima/FILES/3M.pdf
This agrees with what I said about things that are not mental (like signs) having intentionality.
George LeSauvage: Much of the time, it seems to be the latter; that Ed is using the argument to show the inadequacy of the modern view, rather than putting forward his own.ReplyDelete
Yes, I think which one will differ in different contexts. In this case, I think the context indicates that he's using "triangle" in the everyday sense (as applies to traffic-signs, etc.) rather than the mathematically perfect sense.
Zach: I'm curious, though, about how we go from thinking generally about triangles, say, to thinking about this specific triangle that is made in red ink on a piece of paper. Does my intellect have to somehow interact with a specific thing's matter/existence/whatever individuates a substance to be able to think about that thing?
It is by the imaginative part of your mind that you can be aware of the individual particular; but of course you can also narrow down abstractions: from an abstract platonic triangle or abstract redness, you can form the concept of an abstract red triangle, or a red inky triangle, or a red inky paper triangle, and so on. You can add more (abstract) detail to your concept of something but it still remains — insofar as you are contemplating or reasoning about it conceptually — a form in your intellect.
The Masked Chicken: At when point does the one form end and the other one begin? Is this a continuous change
Since the forms we're referring to here are the (accidental) forms of shape, then whenever the shape or structure changes, even by the slightest amount, then a different form applies. Which is really just to say that when the shape changes, then the shape changes.
3. What about the case where three piles of sand form an enclosure in the shape of a triangle?
It's a real triangle in the sense that sand is real, and in the "everyday" sense of more or less approximating a mathematically perfect triangle. (And it's unreal in the sense that piles of sand are not a thing at all, that is, not a single substance. But that's probably getting into a different area from the one you are interested in here.)
4. What about the case where the triangle sits on top of a mound with very gradually sloping sides that asymptotically approach infinity
I'm afraid I didn't follow this.
What I am getting at is can one actually make a simple, pure definition of a triangle.
In the pure mathematical sense, certainly. But if you mean in the "everyday", practical, material sense, we run into issues of vagueness: when does an approximate triangle (like a traffic sign) get too approximate and not triangular enough? Well, a practical answer is good enough: when Quality Control at the traffic-sign factory says so!
A. R. Diaz: Sorry for misspelling your name, and for the long response. I shall probably leave it at that. [...] I am using it in exactly tin he same sense in which Ross used it (that is, “form” is restricted to the pure abstract structure of mathematical and logical objects and relations.ReplyDelete
No worries, on either account. I'll make one point since I think it gets at the crux of the matter: Is Ross in fact arguing about mathematical objects? Or just mathematical reasoning? He talks explicitly about judgements:
"But now let us look at the argument: Some thinking (judgment) is determinate in a way no physical process can be."
This is surely why he keeps referring to functions — the problem for materialism is not mathematical objects per se but mathematical operations. Even if there were laws of physics that allowed for perfect triangles, there would be no way to instantiate an operation or an argument, which is all Ross needs (and, as far as I can see, all he explicitly argues for).
I'm in a combox as a complete layperson here, and am trying to learn as much as I am trying to challenge some of the thinking here (probably unsuccessfully)...ReplyDelete
Can you reach out, and with your hands, touch the thought of a triangle before it's expressed in some form (e.g. drawing it fully on paper)? No.
Then I posit that until that expression takes place fully (as in, it's finished being drawn here), the triangle itself does not exist yet. It is still only a thought that is in the process of manifesting itself into a triangle.
Without expression, that thought may not completely manifest into a triangle EVEN IF we are holding the thought of its complete manifestation in our head.
From my perspective, our use of language describes our interpretation of reality through expression (drawing, words, images etc...), and to me, we can dissect this semantically all day long but never get anywhere. In the end though, what we know (and have historically described) as a triangle does not exist for us to interpret, until it is expressed.
Another way of thinking of this ... what if the triangle could never be (and never had been) expressed and only held as a thought in our head? Would we know it is a triangle? Would we ever have had the thought in the first place?
After all of that, could someone then please tell me how an incomplete expression could be equal to the full object intending to be expressed in our thought (e.g. a triangle in our thought not being expressed as our language describes a triangle ... isn't a triangle yet)?
“I'll make one point since I think it gets at the crux of the matter: Is Ross in fact arguing about mathematical objects? Or just mathematical reasoning? He talks explicitly about judgements:
"But now let us look at the argument: Some thinking (judgment) is determinate in a way no physical process can be."
This is surely why he keeps referring to functions — the problem for materialism is not mathematical objects per se but mathematical operations. Even if there were laws of physics that allowed for perfect triangles, there would be no way to instantiate an operation or an argument, which is all Ross needs (and, as far as I can see, all he explicitly argues for).”
You make a fine point.
Let me begin by stating the dialectical point I made earlier: Feser accepts Ross’s argument for the formal indeterminacy of the physical. In defense of Ross’s argument from an objection made by Peter Dillard, Feser responds by suggesting that what applies to geometrical objects or figures applies to mathematical and logical forms (say, functions) in general, and what applies to geometrical figures is that (1) they cannot be determinately realized in matter and (2) we know this. This is why Feser retorts by saying that to claim that mathematical functions might be realized determinately in matter, but that we just don’t know it, is as silly as suggesting that geometrical objects or figures, say, a triangle, is determinately realized in matter, but that we just don’t know it. So, on Feser’s own account, yes, the argument concerns both mathematical functions and objects. (Perhaps Feser would say that it does not concern all mathematical objects, but only a subset of them, namely, geometrical ones. But that is sufficient for my objection to Feser. In addition, he would also have to give an account as to why it is applicable to only these but not other mathematical objects.)
Now, in my opinion, I think Feser is right in that the formal indeterminacy of the physical concerns all mathematical and logical phenomena, and that Ross’s argument can be shown to be extendable to all of them––even to music scores. (And mathematical and logical objects are per se indeed a problem for materialism, since one can argue that if we were entirely material, we would not even be able to think and grasp them, let alone carry out determinately functions over them. Also note that by "abstract" I do not mean "Platonic". I am leaving this open, although I am inclined to think that both the Platonic and Augustinian ways of understanding the metaphysical status of mathematical "objects" are mistaken.) There's also the further question of what exactly is the relationship between mathematical objects and mathematical functions, but that's a huge and difficult issue. So yeah, Ross is primarily arguing about mathematical reasoning but his argument applies mutatis mutandis to all mathematical structures (actually, Ross thinks that mathematical structures are in se constituted by human intellectual activity or at least dependent on it, in part because of his argument for the formal indeterminacy of the physical. This is already crossing over to the field of philosophy of mathematics, but the point is that Ross' argument if sound has implications far greater than what has been hitherto acknowledged.)