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Realism vs. Anti-Realism III: The anti-realist response — Guest Post by G. Rodrigues

St. Thomas Aquinas from ‘The Demidoff Altarpiece’ by Carlo Crivelli

We have no space to follow St. Thomas through all these negative heresies; but a word must be said about Nominalism or the doubt founded on the things that differ. Everyone knows that the Nominalist declared that things differ too much to be really classified; so that they are only labelled. Aquinas was a firm but moderate Realist, and therefore held that there really are general qualities; as that human beings are human, amid other paradoxes. To be an extreme Realist would have taken him too near to being a Platonist. He recognized that individuality is real, but said that it coexists with a common character making some generalisation possible; in fact, as in most things, he said exactly what all common sense would say, if no intelligent heretics had ever disturbed it.

— G. K. Chesterton, Saint Thomas Aquinas, The Dumb Ox

In this post I will consider the anti-realist (Chesterton’s “intelligent heretics”) response to the realist challenge. Anti-realism comes in several varieties as one would expect: we have nominalism (universals do not exist) and conceptualism (universals are mere beings of reason), and within these major genera there are then several species[1]. For what concerns me here, the disagreement between conceptualists and nominalists is much less important than their common denial of the extra-mental reality of universals, so I will purposely lump the two together.

After a short introduction, I will divide anti-realist strategies in two major trends: linguistic and class-based ones. Ideally, these two sections should be read side by side to get a feel for the parallel arguments made against the two strategies, as the same problems tend to recur in all nominalist accounts. I will not probe this phenomenon in any depth, although I will drop a hint here and there. A third section will be devoted to the form of nominalism that is in better shape, trope nominalism. I have looked extensively for ways to present it and its criticism, in my judgment conclusive, but found it impossible to do it fairly within the constraints of a blog post, so the comments will be brief. For more information, look up J. P. Moreland’s Universals, chapters 2 and 3, the book that will, once again, be my main source. For a bird’s eye view of nominalism, see Nominalism in Metaphysics.

Before proceeding two important caveats. First, this is the longest post in the series. There is a frightening amount of ground to cover and I will be moving at a fairly brisk pace. Second, I will be arguing against (some) anti-realist positions. In many cases, the objections can be responded to and the dialectic continued. Since the post must be finite and of a reasonably short length, I have to cut off the dialogue at some point. It is a judgment call, biased in favor of the realist. So if you want the whole story, you know what to do: hit the books.

The bottom metaphysical question is: what makes P-things P? Realists account for this by an appeal to universals. Anti-realists deny the extra-mental reality of universals and related abstract objects such as propositions and relations, and only accept the existence of particulars. They want to connect language and thought directly with reality without the mediating link of universals. The reasons why they want to do it can be stated in the form of objections against realism and will be considered in the next post. Here, I will just gauge their response to the realist challenge.

Recall four basic examples of sentences from the previous post:

  1. Fido is green.
  2. Rover is green.
  3. Socrates is green.
  4. Greenness is a color.

To be successful, the anti-realist has to offer a paraphrase of sentences like these that either entirely eliminates or reduces universals to the objects allowed by his ontology. Two major strategies are linguistic nominalism, in which universals are eliminated in favor of words or concepts with general application, or class nominalism, where a universal is reduced to its extensional content or class of instances[2]. Trope nominalism is a beefed up version of class nominalism.

A. Linguistic nominalism

Linguistic strategies seek to eliminate universals by employing translation devices, some of them fairly sophisticated such as those developed by W. Sellars, to show that the realist’s claim that universals are needed is illusory. Starting with predication, the nominalist can paraphrase (1) as:

  • 1a. The term “green” correctly applies to Fido.

As for similarity, the nominalist can say that Fido and Rover resemble in the aspect of greenness because the term “green” correctly applies to both.

There are several problems with the linguistic strategy. First, linguistic predicates are neither sufficient nor necessary to specify a universal. They are not sufficient for there are contrived predicates that correspond to no universal whatsoever (exercise to the reader[3]). They are clearly not necessary, for “Fido is green” would still be true even if no human being thought it or uttered the corresponding sentence, or if humans never existed in the first place. The simple fact is that universals are far more numerous than linguistic predicates; the former are infinite in number but the linguistic predicates thought or thought-able, uttered or utter-able in principle by the whole humanity, past, present and future, is finite.

Rover?
Second, the linguistic nominalist owes us an account of what it means to say that a term applies correctly to explain predication and similarity. The term “green” correctly applies to Fido and it does not apply to Socrates, but what accounts for the difference? A first answer is to simply say that it is a brute, primitive fact. “Green” correctly applies to Fido because Fido is green and that is it. If the realist complains that this is trivial and uninformative, the nominalist will concede it is and then reply that the realist explanation is only superficially more informative, with the disadvantage of dragging in a boatload of extra objects. But how cogent is this?

For starters, it commits one to a staggering amount of brute facts, so much so, that this has been called “Ostrich nominalism”. If you refuse to play the explanatory game, you can hardly say that you have won it. Also, contrary to what the nominalist suggests, correct application of a term does not seem to be a primitive fact but analyzable, for
we say that “green” applies correctly to Fido because Fido is green, not the other way around. But there are other problems.

Whether or not the predicate nominalist gives a non-circular account of correct term application, he is in a very awkward position. For green lizards existed long before human language came unto the world, so it seems he is committed to say that language creates properties and that Fido was not green until the first human thought it or uttered it, a barely comprehensible suggestion. As a bonus, this makes clear that an account of the modal status of necessary truths like (4) above is beset by severe problems.

Finally, words are universals too. For I utter this particular word “green”. Since my first language is Portuguese, I can also utter the particular word “verde“. And Socrates also uttered a “green” word, presumably in Greek. And now I will again utter “green”, a different utterance from my first one. But all these utterances (or thoughts, if you go the conceptualist route) all express the same thing, or in the philosophical jargon, are different tokens of the same word type, so we again have a problem of unity within plurality.

In fact, how is even communication possible in the first place if all these utterances do not express the same thing? But if they do, as they surely must, what can it be other than the corresponding universal word type? So how can the nominalist disentangle this self-imposed knot? He cannot offer a reductive analysis of word types in terms of words, for that would be circular. Maybe he can say that it is because words resemble each other. But in what sense? If he says that it is because these words correctly apply to the same objects, he is just going in circles. And even if some sense can be attached to it, a vicious regress looms large in the horizon because the nominalist is appealing to the typed relation of resemblance, and thus to a universal (see below for more details). It seems then that reference to universals has not been eliminated.

B. Class nominalism.

Class[4] nominalism takes its cue from set theory and replaces universals by their extensional content so that (1) gets paraphrased as “Fido is a member of the class of green things”. Class nominalism suffers from similar ills that plague linguistic nominalism. As in linguistic nominalism, classes are neither sufficient nor necessary to specify universals. In the literature, this goes by the name of the companionship and imperfect community problems.

An easy example of the first is given by two obviously different universals with empty extension, e.g. unicorn-ness and griffin-ness. If the nominalist objects that these name nothing at all, and even if we buy this retort, the problem still persists because we can find examples of different, instantiated universals with the same extensional content. Consider a possible world in which there is only one object, a green ball. Then greenness and roundness have exactly the same extensional content. Conversely, a distinct class of particulars is not a sufficient condition for there being a distinct universal. This is left to the reader as an exercise[5].

How does the class nominalist account for predication and resemblance? We can ask in virtue of what is Fido in the class G of green things. If the nominalist says because Fido is green, then this is uninformative and circular. If he answers because it has the property green, he concedes the point to the realist. The nominalist can always assert that class membership is a primitive, unanalyzable relation, just as the realist asserts that the instantiation relation is primitive. But in order to assert that Fido is in G, we must be given the class G and scan it in order to see that Fido is in it; but this is false to the facts for in order to assert that Fido is green we only have to check Fido and not anything external to it.

If the nominalist wants to construe membership in G as membership in {x: G(x)} where G(x) is the predicate “x is Green” then of course, it suffices to assert G(x) but this concedes the point to the realist that class membership is not primitive. A related problem is that the identity conditions for the class G change in time as green things come and go out of existence. The universal itself does not change, independently of how many times it is exemplified, but the class of green things does change because its identity conditions change.

Maybe one way out is to partition the concrete particulars in resemblance classes and then the question becomes how to non-circularly pick them. This form of nominalism is called resemblance nominalism. One possible way is to select a paradigmatic exemplar. The class of green things will then be the class of things that resemble the green exemplar. The problem with this is what is the criterion to select the paradigmatic exemplar if not itself an exemplar of circular reasoning? And if there are different paradigmatic exemplars, how do we know that we obtain the same class independently of the exemplar picked? Maybe the nominalist can pick a maximal class such that any two particulars in it resemble each other (and if you are a mathematician and want to sound cute, you add, apply Zorn’s lemma). But this does not work[5]. What the class nominalist would like to say is that Fido green-resembles Rover and not green-resembles Socrates. The problem with this answer however, is that what the nominalist is trying to account for is greenness, so to use it to separate the “bad” resemblance cases from the “good” ones is circular.

Taking the cue that resemblance is resemblance in some respect, we can construct a vicious regress in the following way. Unless the resemblance relation is specified the class is not specified either, in other words, the resemblance relation is a universal and has a type. Therefore, resemblance in respect to something—color —is once again appealing to a universal. Maybe, we can eliminate this universal by appealing to its resemblance class; let us throw a bone to the nominalist and allow him to proceed unencumbered without having to explain in what this higher-order resemblance relation amounts to. But then this itself will make appeal to a higher order relation of resemblance, and the nominalist either has not eliminated universals or he has a vicious regress in his hands.

Things do not fare much better on accounting for abstract reference or the necessary modal status of universals and the truths about them. In the first place, taking classes as the referents of abstract singular terms fails because as already observed, classes are neither sufficient nor necessary to specify universals. Maybe the nominalist can construe the class in terms of scattered objects. Applying this to (4) we get:

  • 4a. Necessarily, the scattered object of all green things is a colored thing.

This is indeed a true statement, but is it is a faithful translation of (4)? In other words, are the truth conditions preserved? Colors and colored things are not the same kind of entities and different predicates apply to them, so by the indiscernability of identicals, (4b) is not saying the same thing as (4). But even if we accepted this translation, this strategy does not work for other universals: for an example, just replace greenness for humanity. Two further problems with translation strategies in general are first, that they tend to be ad hoc, with no clear pattern emerging on how to uniformly account for all cases and second, for those that pursue the eliminative route, it commits them to the implausible position that things like triangularity or humanity do not exist and talk about them is ultimately talk about words.

C. Trope nominalism.

In the previous section we have seen that the class nominalist would like to say that Fido green-resembles Rover and not green-resembles Socrates. This is precisely what a trope nominalist will say. The trope nominalist replaces the realist’s universal greenness by a multiplicity of green tropes, one for each green instance. Fido has a particular green trope and Rover its own distinct particular green trope and so on. On this view, tropes[6] are simple qualities and an object is a collection or bundle of tropes.

Before tackling predication and resemblance, it is best to see what the trope nominalist makes of abstract reference. As we have seen, a realist takes abstract singular terms to be the proper names of existing things, universals. Eliminative nominalists have to scrounge all sorts of convoluted strategies to deal with abstract reference. The trope nominalist could be a denialist, but instead most trope nominalists, upon recognizing the difficulties facing such strategies, will say that abstract singular terms do name something, just not universals but sets of resembling tropes.

This image is green (and black)
From this, the account of predication follows. If the universal greenness is replaced by the set of green tropes, to say that Fido is green is just to say that Fido has a trope that belongs to the set of green tropes. As for resemblance, the trope nominalist will say that Fido and Rover resemble each other in the aspect of green because they each possess a green trope that qualitatively resemble each other.

As a first remark, note that since tropes are simple qualities, resemblance classes are correctly formed and their formation is immune to the companionship and imperfect community problems. On the other hand, the price to pay is not only the introduction of a new category of entities, tropes, but also a commitment to the paraphernalia of set theory. And the trope nominalist still owes us an account of what a trope is if he is to evade the charge of avoiding realism by positing an extravagant ontology, what is the exact nature of the relationship of a trope with the object that possesses it, the nature of the qualitative resemblance relation between tropes, etc. As an example, consider the latter.

The trope nominalist could follow his strategy to its natural conclusion and posit resemblance tropes, that is, to treat relations the same way as other universals. This immediately raises the prospect of a vicious regress. One strategy is to back down and say that it is just a brute fact that green tropes resemble each other more than blue tropes. But then, for all the bells and whistles, tropes have not done the explanatory work they are supposed to do. So at best this is unsatisfactory, at worse, it generates a vicious regress parallel to those of linguistic and class nominalism.

Much more could be said, in both response and counter-response. My suggestion is for the reader to consult the references. I cannot however, resist the temptation to lodge one objection first raised by Wolterstorff. How innocent is opening the door to sets in one’s ontology? Well, sets have their members necessarily. Accordingly, given any set, it is impossible that it have members other than those it in fact has. Since greenness is the resemblance set of green tropes, it follows that this set necessarily has the members that it has. In particular, it follows that the green trope possessed by Fido necessarily is a member of this set.

There are two options now: if the green trope possessed by Fido individuates it, that is, only Fido could have possessed such a green trope, then it seems we are committed to assert that Fido existed necessarily. If the green trope possessed by Fido is not necessarily possessed by Fido and Fido alone, the tie between tropes and their possessors is loose and accidental. But this suggestion seems incorrect as it makes the identity conditions of tropes, and a fortiori the identity conditions of sets of tropes, unintelligible. But even so, the necessity problem still remains, for it still is the case that the set of green tropes could not have failed to be what it is. In particular, the number of green things (and human things and…) could not have failed to be what it is. But this surely is false.

Scanning the argument, the nominalist could dispute that sets have their members necessarily. The problem with trying to deny this is that, contrary to the class nominalist, the trope nominalist does not have the resources to cash out set talk in terms of concrete particulars, since tropes and sets were introduced precisely to overcome the difficulties with class nominalism. So he is stuck with sets of tropes, and sets, being abstract objects constructed out of its members and with identity conditions fixed by said members, have their members necessarily.

D. Conclusion.

If I could summarize this (long) post in one sentence, it would be: universals cannot be eliminated. I do not expect the unsympathetic reader to fully agree with me, but it should at least be clear that evading them is notoriously very hard to do.

———————————————————————————–

Endnotes:

[1] D. M. Armstrong has, with the veritable patience of an entomologist, surveyed various varieties of nominalism in Universals and Scientific Realism, vol. 1: Nominalism and Realism.

[2] One could object that the anti-realist is making an appeal to abstract objects, namely sets. The objection can be circumvented, since he can hold that a class of concrete particulars is itself a concrete particular in one of several ways: by identifying the class with the scattered concrete object composed of all the concrete particulars in the class, by translating set talk in terms of their members, etc. Quine and most trope nominalists bite the bullet and do admit sets into their ontology. Although this is a concession, the concession is to the existence of abstract objects not universals, and particularly well behaved ones, so the realist cannot proclaim victory. Not immediately, anyway.

[3] Virtually every introductory course or book on metaphysics contains a discussion of the problem of universals. One reference is E. Lowe, A Survey of Metaphysics, chapters 19, 20. M. Loux, Metaphysics, a Contemporary Introduction dedicates chapters 1 and 2 to the problem. The first chapter gives one example of a self-referential predicate that does not correspond to any universal. A less exotic example can be found in J. P. Moreland, Universals, pg. 29.

[4] For technical reasons (e.g. Russel’s paradox) it is important to maintain a distinction between sets and classes. Here, I will not bother with such technicalities and will use the two words interchangeably.

[5] An example of the imperfect community problem is in E. Lowe, A Survey of Metaphysics, pg. 357 ff. Lowe is working with the more complicated maximal definition of resemblance class, but the problem arises just the same with simpler definitions.

[6] Also called abstract particulars. This can be confusing, depending on how exactly the trope nominalist conceives a trope, but abstract usually is taken in the epistemological sense.

26 thoughts on “Realism vs. Anti-Realism III: The anti-realist response — Guest Post by G. Rodrigues Leave a comment

  1. I have not finished reading this post, but I do appreciate what I have read so far. Thank you for your efforts.

    A physical (causal) process gives me the experience of “seeing something green”. So, how does the supposed existence of abstract universals or abstract mathematical objects help various realists explain why we believe in them or how we come to know or see mathematical axioms such as two points may determine more than one line in non-euclidean geometry. Mathematical proofs is the essential part of mathematics. How is the existence of abstract objects affecting our thinking or related to proofs?

    I hope my questions make sense.

  2. @JH:

    Mathematical proofs is the essential part of mathematics. How is the existence of abstract objects affecting our thinking or related to proofs?

    Mathematical proof is *one* crucial component of *doing* mathematics.

    As far as your question, if I am understanding it right, there is no direct relation. The particular question of realism vs. anti-realism, at least at the level of generality I have construed it, has little to no bearing on the actual practice of mathematics. Not all philosophical questions on the foundations of a given discipline have a direct influence on the practice of said discipline — that is why there can be deep philosophical disagreement (just think of the widely divergent philosophical positions within the mathematical community), and yet business goes on as usual.

    As far as your question related to non-Euclidean geometry, the question is misguided, since it rests on a conflation of two different meanings of the word axiom.

  3. @G. Rodrigues

    I kind of put this on the last thread, but I am going to restate it here.

    I am not sure if this falls under the conceptuallist approach or not.

    The whole realist argument rests on the premise that the universal truths must exist as extra-mental entities because human intelect is contingent.

    Assuming as argued in previous posts that the human intelect is not stricly housed in the physical body, what makes it contingent?

    Additionally, given that argument, the existence of even a single non-contingent intelect (God) would cause the realist argument to fail as this non-contingent intelect could house universal mental constructs.

  4. @MattS:

    For the sake of simplicity I will merge your two posts in the two different threads and respond everything here:

    For your 2 + 2 = 4 argument this is actually contingent. It is contingent on the numeric base being > 4. In a base four system 2 + 2 = 10, in base 3, 2 + 2 = 11, in base 2 it is meaningless but 10 + 10 = 100.

    You are confusing numbers with the particular strings of characters denoting them in a specific representation.

    There is one more premise unstated that needs proving. That human intelect is contingent. There is strong metaphisical evidence that the human mind is not strictly a product of the physical brain so what makes it contingent?

    Whether or not the human intellect has an immaterial component (which in fact I agree it has) is irrelevant for whether it is contingent or not. A contingent being just means that it is metaphysically possible for it not to come into existence, and that is certainly true of every human being.

    Even if human intelect is contingent these truths should not need individual exta-mental entities to exist, all they need is a single non-contingent intelect (God) for all these truths to be universal.

    This depends on the specific branch of realism you espouse, but a realist of Thomistic persuasion will tell you that, yes, universals *do exist* in the intellects. God has an intellect, but only in an analogical sense. The universals exist in Him *ante rem*, literally *before the thing*, and they are the archetypes of creation pre-existing from all eternity in His mind. But this fact does not imply that they do not exist *in re*, literally *in the thing*. In fact they better exist that way, or otherwise true human knowledge would be impossible without some sort of divine illumination a la St. Augustine. Maybe you will want to wait until the next post, as I will adumbrate on this issue, and it might make things clearer.

    The whole realist argument rests on the premise that the universal truths must exist as extra-mental entities because human intelect is contingent.

    First, it is not a premise, it is a conclusion. Second, it is far from being the “whole argument”, it is in fact only one argument, although in my judgment a particularly strong one. Third, let me ask you, do you understand why mathematical truths are necessary? And you understand why their being necessary implies that they cannot be reduced to a thought in the human mind?

    all they need is a single non-contingent intelect (God) for all these truths to be universal.

    What do you mean by “universal truths”?

  5. @G. Rodrigues

    “But this fact does not imply that they do not exist *in re*, literally *in the thing*. In fact they better exist that way, or otherwise true human knowledge would be impossible without some sort of divine illumination a la St.”

    I disagree. Any truths that exist only in Gods mind are meaningless to us. Any truths that matter (to us) must be embodied in what he has created and can there fore be discovered by studying that creation.

    “Third, let me ask you, do you understand why mathematical truths are necessary? And you understand why their being necessary implies that they cannot be reduced to a thought in the human mind?”

    I am not sure I understand what YOU mean by necessary. To my thinking, if they don’t apply universally to physical instances of what they describe they aren’t necessary. The realist position seems to argue that these universals exist, but not in either the physical realm or the mental realm. If these universals are fully embodied in their physical instances I do not understand why the entities you describe are necessary.

  6. G. Rodrigues,

    I am not sure what you meant by “the question is misguided.’ My question is meant to find out whether any realists have any explanations as to why we have mathematical intuition such as every natural number has a successor. (How do we know it?) Naming it ‘a priori” or saying ‘we just know’ is not satisfactory to me.

    Yes, business goes on as usual.

  7. @MattS:

    I disagree. Any truths that exist only in Gods mind are meaningless to us. Any truths that matter (to us) must be embodied in what he has created and can there fore be discovered by studying that creation.

    What are you disagreeing with, if I said something close to this? I have a few quibbles with what you say, though. Besides some potentially equivocating or misleading expressions, the last claim of yours is false as there are plenty of truths we can arrive at that are not “embodied” in creation in any reasonable sense of the word, mathematics once again providing the most obvious example. A mathematician does not study “creation”, he studies the mathematical realm (no implied adherence to Platonism).

    I am not sure I understand what YOU mean by necessary. To my thinking, if they don’t apply universally to physical instances of what they describe they aren’t necessary.

    The nature of necessity is a complicated subject (Plantinga has a book with this title). The way I am using the word is, a necessary truth P is a truth that obtains in all possible worlds, that is, there is no possible state of affairs into which the whole reality could have unfolded in which P was false.

    The realist position seems to argue that these universals exist, but not in either the physical realm or the mental realm. If these universals are fully embodied in their physical instances I do not understand why the entities you describe are necessary.

    No, that is *NOT* what the realist is saying.

    Either I am failing to see what you are getting at or you are confused the whole problem (or a combination of both), anyway, there is some misunderstanding going on here.

  8. @JH:

    My question is meant to find out whether any realists have any explanations as to why we have mathematical intuition such as every natural number has a successor.

    It was misguided because axiom has a descriptive meaning in mathematics, that is, sets of axioms or more precisely, formal theories (e.g. non-Euclidean geometries), do not fix a single object of study but rather *define* a family of objects which for all sorts of reasons have been found useful. It can happen that the theory is categorical, but that is the exception rather than the norm.

    As for your second question, at the level I have construed realism, no. The mathematical intuitions about what natural numbers are can be justified by all sorts of considerations, but never proved in the normal sense of the word.

  9. @G. Rodrigues

    I am probably the one who is confused but I find the topic interesting so I will keep pluging away in the hope that either the light comes on for me or you figure out where I am confused.

    “The way I am using the word is, a necessary truth P is a truth that obtains in all possible worlds, that is, there is no possible state of affairs into which the whole reality could have unfolded in which P was false.”

    I don’t see how this could possibly be said of any truth. The simple fact that we lack the capacity to imagine a state of affairs where P is false does not constitute sufficient evidence that such a thing would be impossible.

  10. Sory for the multiple posts but the version of IE on the computer I am doing this from has troubles handling longer posts.

    Going back to the universals themselves, You describe a universal triangularity and say that this must exist as an entity in it’s own right. Where exactly does this entity exist?

    Here is how I am perciving the issue of the universal triagularity. Every possible physical or mental instance of triangularity no matter how flawed must fully embody every aspect of the universal triangularity. If a specific instance fails to do so it is either NOT an instance of the universal triangularity or the universal triangularity is not in fact universal.

    If every possible instance of the universal fully embodies all aspects of the universal, I fail to comprehend why it is necessary for the universal to exist as an entity in its own right.

  11. Previously I had stated that 2+2=4 is actually contingent on what numeric base you are using (I will admit that this was more than a bit tounge in cheek) You replied that I was confusing the actual numbers with the symbols used to depict them.

    However, now you have at least implied that what mathematitcians do is wholly separate from physical reality. Here I will object on a serious not that when the work of mathematitians is wholly separate from physical reality then all they are doing is manipulating symbols and 2 and 4 have NO more meaning than X or Y.

  12. @MattS:

    I don’t see how this could possibly be said of any truth. The simple fact that we lack the capacity to imagine a state of affairs where P is false does not constitute sufficient evidence that such a thing would be impossible.

    The “do not see” or “cannot imagine” is irrelevant; can we come up with *arguments* that it is so? Yes.

    First, by pushing axioms of a formal theory into the premises of an if-then conditional, we get rid of every non-logical axiom and every mathematical truth is true as a pure matter of logic. So the question is reduced to: do the laws of logic hold in all possible worlds?

    1. Hypothesis: there is a possible world in which the laws of logic do not hold.

    2. From 1, it is possible that the laws of logic do not hold in the actual world.

    3. To know that the logical laws do hold in the actual argument an argument is needed.

    4. But every argument relies on the laws of logic for its validity. But this would make the argument fallaciously circular and thus invalid.

    5. It follows that we cannot rationally justify our belief in the fact that the laws of logic hold in the actual world.

    6. It follows that we cannot rationally justify *any* argument whatsoever in the actual world.

    This is absurd, so we must reject 1.

    Going back to the universals themselves, You describe a universal triangularity and say that this must exist as an entity in it’s own right. Where exactly does this entity exist?

    I did not say that.

    Every possible physical or mental instance of triangularity no matter how flawed must fully embody every aspect of the universal triangularity. If a specific instance fails to do so it is either NOT an instance of the universal triangularity or the universal triangularity is not in fact universal.

    You have a flawed conception of universal instantiation; no realist is going to claim that, precisely because there can be *no* perfect instantiations of triangles in the material world. It is a different story with the intellect, but triangularity “is” in the intellect in a different mode of being than it is “in” the things that instantiate them and in yet another different mode of being in the Platonic realm — or in God’s mind if you make the neo-Platonic move of identifying the two. As I said, I will say something more about this in my next post.

    However, now you have at least implied that what mathematitcians do is wholly separate from physical reality. Here I will object on a serious not that when the work of mathematitians is wholly separate from physical reality then all they are doing is manipulating symbols and 2 and 4 have NO more meaning than X or Y.

    I am not going to address the connection between mathematical activity and physical reality, because that is a thorny issue in which realists are not in agreement. What I will say however, is two things. First, “wholly separate” is false; insofar as mathematicians study a specific sort of universals and these universals are instantiated in physical reality (however imperfectly), they are studying physical reality.

    Of course, there are large swaths of mathematics whose connection to physical is from tenuous at best to positively non-existent (this also depends on what means exactly by “connection”). Here, your second claim simply betrays an ignorance of mathematics. Pick a supremely abstract part of mathematics: say Topos theory. Now take a significant result result from it, Diaconescu theorem: every Topos satisfying the internal version of the axiom of choice is Boolean. What all this means is besides the point, the point is that the mathematician is *NOT* doing mere symbol pushing. He is rather producing understanding; understanding of a corner of reality which we can call the mathematical realm. In what exactly this realm consists of, is a point of contention even within realists as I already said. But saying that because the objects a mathematician deals with have no instantiation in physical reality then they are devoid of any reality is a non-sequitur and it makes the rather presumptuous assumption that all reality is reducible to physical reality. But why should we think so? In fact, that is a claim I violently disagree with and if the traditional realist account is right, we have an argument for it (there are others).

  13. @G. Rodrigues

    When I said:

    “Every possible physical or mental instance of triangularity no matter how flawed must fully embody every aspect of the universal triangularity. If a specific instance fails to do so it is either NOT an instance of the universal triangularity or the universal triangularity is not in fact universal.”

    I was not saying that a realist would claim this. I am claiming this. It is not possible for an instance of a universal to not fully embody the universal. To me it makes no logical sense for such a condition to be possible. If you have an physical object that does not fully embody the triangularity universal it either isn’t an instance of the triangularity universal or triangularity isn’t a universal.

  14. @G. Rodrigues,

    Mathematics at its most fundamental level is a language of symbol manipulation. To the extent that a mathematician produces under standing, the mathematician is either applying mathematics outside of the real of mathematics or is producing understanding ABOUT symbol manipulation.

  15. @Matt
    It is precisely the other way around: if an instance of triangularity fully embodies every aspect of triangularity the universal triangularity is not in fact universal. One of the aspects of triangularity is to allow for any shape that is still a triangle. If you would want to embody triangularity you need a moving triangle that is continuously changing. I am not saying this will embody *every aspect*, merely directing you attention to something you seem to overlook.

  16. @rembie,

    I disagree. Trangularity is a shape. Motion is irrelivant to shape. If it is changeing but the aspects that are changing are irrelevant to shape the change is irellevant. If the shape itself is changing then it isnt an instance of triangularity.

    If what you say is trully what realists believe then I call the whole thing nonsense, becuase if that is the case you only trully need one single universal which encompasas and instatiates all things.

  17. @Matt
    Then please draw triangularity for me.
    Triangularity is not a shape,it is a principle of construction that is common to a range of shapes called triangles.

  18. @Matt:

    It is not possible for an instance of a universal to not fully embody the universal. To me it makes no logical sense for such a condition to be possible. If you have an physical object that does not fully embody the triangularity universal it either isn’t an instance of the triangularity universal or triangularity isn’t a universal.

    That is because you are using expressions like “fully embody”, possibly with wholly incorrect connotations and imagery attached to it. A realist speaks about *instantiation* or *exemplification*. Since there is no argument in here (the “to me” is a dead give a way) all this says is that you do not really understand what the realist is claiming.

    To the extent that a mathematician produces under standing, the mathematician is either applying mathematics outside of the real of mathematics or is producing understanding ABOUT symbol manipulation.

    So Diaconescu’s theorem is a result about symbols? The understanding produced by it is about squiggly marks drawn on a page? Right. And how a result about markers on a page gets subsequently incorporated in physical theories and becomes an application to the “real world”? What you are saying does not even make sense.

  19. Matt,

    I like your implied idea that it’s the various applications of mathematics, such as the numerical quantifier, that (possibly) account for why we believe in them.

  20. @G. Rodrigues

    So explain what the realist means by instatiation and exemplification.

    What exactly is the relationship between the universal and its instances?

    Does or can a universal have properties not fully realized in all instances? How would such a thing be possible? If not what is it about the universal that makes it neccesary for the univeral to exist as a thing in its own right?

  21. “So Diaconescu’s theorem is a result about symbols? The understanding produced by it is about squiggly marks drawn on a page? Right. And how a result about markers on a page gets subsequently incorporated in physical theories and becomes an application to the “real world”? What you are saying does not even make sense.”

    I come from a computer programing background. While a squiggly mark drawn on a page would be a symbol but not all symbols are marks on a pice of paper.

    I looked up Diaconescu’s theorem. It is a theorem of mathematical logic which is also known as symbolic logic. So yes, this theorem is entirely about how to manipulate symbols.

  22. @MattS:

    While a squiggly mark drawn on a page would be a symbol but not all symbols are marks on a pice of paper.

    You are making the same mistake you made when reducing numbers to a specific representation.

    I looked up Diaconescu’s theorem. It is a theorem of mathematical logic which is also known as symbolic logic. So yes, this theorem is entirely about how to manipulate symbols.

    Sorry, but you simply do not know what you are talking about.

    But if this particular example is tripping you up, consider the latest greatest triumph in mathematics: the proof of Pincare’s conjecture by Perelman that if a 3-manifold is homotopy equivalent to the 3-sphere, then it is homeomorphic to it. Or if you dislike geometry, let us go to analysis and consider say, the closed graph theorem: a linear operator is bounded iff its graph is closed. Or maybe algebra and we can quote say, Hilbert’s basis theorem that every ideal in the ring of (multivariate) polynomials over a Noetherian ring is finitely generated. Or…

    These are all theorems about symbols, whether you view them as squiggly marks on the page, or as sequences of characters in the memory of a computer, or whatever. Right.

    I repeat myself, you simply do not know what you are talking about.

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