Feser’s third proof is the Augustinian proof. Let’s see if this one does better than the other two. I suspect it shall not.
We are surrounded by particular, individual objects. You take a particular, individual pool cue, and with it knock a particular, individual billiard ball across a particular, individual pool table, then pick up a particular, individual rack to arrange the balls so that a new particular, individual game can begin. But each of these particular things is an instance of an abstract, general pattern.
Well, I think lots of things will share common properties, but I don’t think there’s a single general pattern. What makes something a ball is a complex cluster of traits, not an “abstract, general pattern.” Additionally, the distinctions we draw are arbitrary. We would not be wrong, for example, to reclassify soups to include cereal. It’s all a question of how something is defined.
They are also instances of even more abstract patterns, shared with even more kinds of things. Some of the billiard balls share the pattern redness in common with stop signs, fire engines, and strawberries; all of the billiard balls share the pattern roundness in common with basketballs, globes, and the moon; the billiard rack shares the pattern triangularity with pyramids, dinner bells, and dunce caps; and so on. Such patterns are called universals by philosophers, and they are “ abstract” in the sense that when we consider them, we abstract from or iguore the particular, individualizing features of the concrete objects that exhibit the patterns. For instance, when we consider triangularity as a general pattern, we abstract from or ignore the facts that this particular triangle is made of wood and that one of stone, that this one is green and that one orange, that this one is drawn on the page of a book and that one is metal, and focus instead on what is common to them all.
This is true.
Universals like triangularity, redness, and roundness exist at least as objects of thought. After all, we can meaningfully talk about them, and indeed we know certain things about them. We know, for example, that whatever is triangular will be three-sided, that (at least if it is Euclidean triangles we are talking about) its angles will always add up to the sum of two right angles, and so forth.
This is true by definition! We have series of things with the property of triangularity and then we can deduce certain common features that they share.
But unlike a wooden billiard ball rack or dinner bell, you can’t perceive triangularity through the five senses, can’t pick it up and put it on the table, or in any other way interact with it the way you would interact with a material object. If it is an object of some sort, then, it is what philosophers would call an abstract object.
Well, you can deduce patterns from thinking about triangles, much like you can figure out the best chess move, even though there’s no deeper universal platonic “best chess move,” that exists in the divine intellect.
Universals are not the only apparent examples of abstract objects. A second would be what philosophers call propositions— statements about the world, always either true or false, which are distinct from the different sentences we might use to express them. “John is a bachelor” and “John is an unmarried man” are different sentences, but they express the same proposition. “ Snow is white” and “ Schnee ist weiss” are also different sentences— indeed, one is a sentence of English, the other a sentence of German— but they too express the same proposition— namely, the proposition that snow is white.
Indeed, there are propositions, claims about what is the case. Different words can denote the same claim about the world.
Like universals, propositions exist at least as objects o f thought. But also like universals, they are not material objects.
True!
Then there are numbers and other mathematical entities. They too obviously exist at least as objects o f thought, as we know from our grasp o f mathematical truths and our carrying out o f calculations. But like universals and propositions, numbers are in no obvious way material things. The written numeral “ 2” isn’t the number 2 any more than the Roman numeral “ II” is, or any more than the name “Barack Obama” is the same thing as the man Barack Obama.
This is also true. Mathematical entities follow from certain presupposed axioms, some of which are necessarily true, meaning math accurately describes the world.
Finally, consider what philosophers call possible worlds. A possible world is a way that things could have been, at least in principle.
Possible worlds can either be fictional or abstract—I don’t have strong views about which one.
So, in some sense there are abstract objects such as universals, propositions, numbers and other mathematical objects, and possible worlds. But in what sense, exactly, do they exist? Are they merely objects o f human thought— purely conventional entities, sheer constructs o f our minds? Are they merely useful fictions? Or might they after all really be material things, but o f some more exotic kind than the ones w e’ve considered so far?
It depends. I think that mathematical facts likely just exist—no deeper account needed. Feser argues against nominalism next which is fine—I’m not a nominalist. Feser then argues against regular platonism—this isn’t my area of expertise, I’ll leave it to Schmid to defend against Feser’s bad objections.
Feser moves on to stage 2, writing
We have seen why, contra nominalism and conceptualism, some form o f realism vis-à-vis abstract objects like universals, propositions, numbers and other mathematical objects, and possible worlds must be true. Now, one implication o f the arguments was that, whatever mode o f existence these objects have, they do not (or at least a great many of them do not) depend on the material world. Material things are always particular.
I tend to agree with this.
So, the only sort of intellect on which these abstract objects could ultimately depend for their existence would be an intellect which exists in an absolutely necessary way, an intellect which could not possibly have not existed. Now, could there be more than one such ultimate intellect? Might we not suppose that such-and-such possible worlds, necessary truths, universals, and so forth exist in necessarily existing intellect A, and another group o f possible worlds, necessary truths, universals, and so forth exist in necessarily existing intellect B?
These can’t be features of an intellect for a bunch of reasons.
If they were features of an intellect then if the intellect were different, such concepts would have to be different. This is implausible, god’s brain being different couldn’t make 2+2=5.
An intellect can only grasp universals if they already exist. How the heck is god deciding upon the universals to think about? Why did he pick Fermat’s last theorem?
It seems like even absent god, 2+2 would still equal 4. Feser might object that this is metaphysically impossible, but he is making arguments about the impossibility of such concepts absent god, so if it is metaphysically impossible, it’s the type of metaphysical impossibility we can discuss, akin to asking about what one should do if utilitarianism were not the true morality. This question is coherent even though it’s metaphysically impossible for utilitarianism not to be the true morality.
First, an adequate theory must account for the fact that abstract objects exhibit objectivity insofar as they have reality independent of human minds. Second, it must account for the fact that they exist in a necessary rather than merely contingent way. Third, it must account for their intentionality, by which Welty means that abstract objects represent the world in something like the way thoughts do. For example, the universal triangularity represents triangles; the proposition that all men are mortal represents the state o f affairs of all men being mortal; possible worlds represent ways things might have been; and so forth.14 Fourth, an adequate theory of abstract objects must be relevant to explaining why there are the necessary and possible truths that there are. Fifth, it must meet what Welty calls a plenitude condition insofar as it must affirm the existence of a sufficient number of abstract objects to account for everything their existence is supposed to account for. Finally, it must at the same time respect a condition of simplicity by not positing more kinds of entity than is necessary. N ow, the objectivity condition can be met only by a realist theory o f abstract objects rather than a nominalist or conceptualist theory.15 The intentionality condition points in the direction o f an Aristotelian realist position, specifically, rather than a Platonic realist position, since it is easier to see how abstract objects could have representational content if they exist in an intellect than if they exist in a “ third realm” . The simplicity condition also points in the direction o f Aristotelian realism rather than Platonic realism, since the former view requires us to posit only two realms— the realm o f material objects and the realm o f intellects— whereas the latter requires a third. The necessity, plenitude, and relevance conditions, in turn, point in the direction o f Scholastic realism rather than a brand o f Aristotelian realism that does not appeal to the divine intellect. For human and other finite minds are contingent, and thus cannot account for the necessity o f abstract objects. And since there are universals, propositions, possible worlds, mathematical truths, and so forth, which have never been entertained by any human mind, Aristotelian realism, unless taken in a Scholastic direction, cannot meet the plenitude condition
Finally, the relevance condition points to Scholastic realism, specifically, in the following way. Again, there are abstract objects which cannot plausibly depend on human or other finite minds. O f course, Platonic realism can account for at least that much. But it is hard to see how possible worlds considered as the denizens of a Platonic “ third realm” would have any relevance to what might happen in the world. Consider (to borrow an example from Welty) a drawing o f Socrates pounding nails into wood. Suppose we allow that this at least represents the possibility o f Socrates being a carpenter. Still, Welty suggests, “ it makes little sense to think that a picture on a piece o f paper is a truthmaker for certain modal statements about Socrates, such that Socrates couldn’t have been a carpenter if that picture didn’t exist.” 16 But why, exactly, would a possible world in which Socrates is a carpenter, understood as an entity existing in a Platonic “ third realm” , be any more plausible a truthmaker than the picture? Even if (like the picture) the Platonic object would represent the possibility of Socrates’ being a carpenter, why would its existence (any more than that o f the picture) make it the case that Socrates could have been a carpenter?
Platonism accounts for 1 and 2. It also account for 3 because the universals supervene on all possible worlds. 4 is also met better by platonism—there are some self evident truths about mathematics, possibility, necessity, etc that can’t fail to be true. This is a better explanation—why the heck would god pick those axioms. It also accounts for 5 for the same reason. The platonic realm doesn’t lack simplicity—positing the reflexive axiom, symmetric axiom, transitive axiom, additive axiom and multiplicative axiom doesn’t lack simplicity. Those things can’t fail to be true. Asking whether adding the transitive axiom complicates our metaphysics just seems to be conceptually confused. Additionally, simplicity is only sometimes a virtue. If one theory posits A and requires not B, while another posits B and C, if A and C have equal plausibility, the first theory will only be plausible if the probability of B is less than 50%. If a theory posits that the transitive axiom depends on gods thoughts, rather than being a necessary feature of reality, that is a cost of the theory. Additionally, positing god has lots of random thoughts about logic, math, and redness is not simple—it’s a priori surprising that god would think about such things.
Additionally, we could posit truths without truth makers. This would posit that 1+1=2, but there’s no extra thing that makes it true. This would thus be a simpler theory.
Overall, Feser’s view rests on a questionable way of grounding universals and abstract objects, combined with sweeping arguments insufficient to establish that we should reject all other accounts. It’s thus like the moral argument—it fails to ground that which it seeks to ground and hastily dismisses all other accounts with simplistic objections.