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Mathematics as Ophelia

August 8, 2012

I will not go so far as to say that to construct a history of thought without profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him. That would be claiming too much. But it is certainly analogous to cutting out the part of Ophelia. This simile is singularly exact. For Ophelia is quite essential to the play, she is very charming … and a little mad.

–Alfred North Whitehead

I often have thoughts along these lines. Calculus with the early moderns, consolidation with the neoclassicals, non-Euclidean geometry with the Romantics, set theory and algebra with the onset of modernism… and all throughout the layering upon layering of new abstractions.

It would be interesting, I think, to think seriously about the relationship of structuralism and post- to abstract algebra. It is here that my thoughts always turn when I read Derrida.

I need to read Whitehead’s Science and the Modern World.

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9 Comments leave one →
  1. Thomas DePauw permalink
    August 9, 2012 12:35 am

    Alfred North Whitehead is a fascinating philosopher, to say the least. With regards to mathematical logic, I agree that the ‘progress’ achieved is ultimately just the expansion of the understanding of real or imaginatively derived relations.

    One thing I have noted about Derrida’s deconstruction project (and I will not claim that my observation is an entirely original one, though I happened to discover it before having read others point it out) is that his project is entirely dependent upon the assumption that his project is self-sufficient. This is a whale of an assumption because his project relies on the integrity of dialectic to itself be validated. Quite simply he tries to use thought to disprove thought, just as Hume tried (and failed) to call into question formal and objective causality by using them, as well as Descartes did in using error taken from experience with reality in order to doubt the reality upon which the error was based. In various ways they all attempted to call into question what is not questionable, but can only be observed with awe. Putnam’s Brain-in-a-Vat argument still seems to hold true, though the analogy he used is somewhat unjustified and only superficially applicable.

    I suppose the actual purpose of Derrida in deconstruction really consisted in trying to somehow to explain away the formal in reality (and thus its objective mental representations) by reducing it to prejudices of the human mind, carrying in a certain sense the entire critical project to its climax, which is also its own destruction. Quite justly the semiotic Thomist John Deely argues that Derrida’s philosophy (if it can be called that) is not truly postmodern, but rather the “last croak of modernity”. The problem of the critical method has always been its arrogance towards reality, its incessant frustration with reality for not revealing its inner nature, its impenetrable secrets.

    As a side note, I have found the Thomistic commentator John of St. Thomas to be an indispensable guide to rightly understanding logic, formal and otherwise (A portion of his Ars Logica has been translated into English by John Deely under the name Tractatus de Signis [Treatise on Signs]). John Deely is an excellent interpreter and developer of the philosophy of St. Thomas and John of St. Thomas, emphasizing the function of the sign as the key to epistemology and metaphysics. John Deely’s work, “Four Ages of Understanding”, is an excellent overview of the development of philosophy.

  2. August 9, 2012 5:43 pm

    I agree Derrida’s project deconstructs itself (which he freely admits, I think), but I don’t know if the proper response is to say that he “calls into question that which can only be observed with awe.” Not that that’s not true–it is–but I don’t see how it could satisfy anyone who doesn’t already believe it. (Most of what follows is cribbed from my (rather limited) understanding of Wittgenstein.)

    The deconstructive train of thought (which is just a version of the skeptical train of thought) runs like this: “The common sense understanding of language says X. But X implies not-X. Therefore language is nonsensical.” Of course, that doesn’t follow; only “the common sense understanding of language is nonsensical” follows. But to merely make this point is to say “well there’s this thing language, that isn’t nonsensical, but I can’t tell you what it is.” And the deconstructor responds, “I don’t see any such thing, and you haven’t told me what it is. I just see the common-sense description of it, and the common sense description is nonsensical. So I don’t believe you.” So you’re forced to describe language, to offer a replacement common sense understanding of it, and once you do, the deconstructor will try to find its inherent contradictions, and will succeed, unless you insist that your words are just meant to give you a general idea of what language is, not to define it precisely–but then the deconstructor says that he still doesn’t know what you’re talking about.

    The problem is that we can’t just dismiss the deconstructor because sometimes the skeptic is right–sometimes we think we see something, if only vaguely, but then when we get closer, it turns out to be a mirage. Take the sorites paradox; we think we know what it means to call something a heap, then it’s shown that we don’t. Is the solution to say “well we can’t say exactly what it means to call something a heap but we’re going to do it anyway don’t ask why it’s a mystery”? I don’t think so. (In this case the solution is perhaps to talk about Aristotelian causation–but that doesn’t mean there wasn’t a problem that the solution had to solve.)

    • Thomas DePauw permalink
      August 9, 2012 11:25 pm

      I agree that knowledge is something which must be self-reflecting in order for us to properly understand how we know, but I don’t think it is necessary (or profitable), as Descartes felt bound to do, to doubt all that we know in order to arrive at truth. Doubt cannot be a universal thing, because by its nature it is particular. All doubt presupposes a regularity to reality which has been (unusually) violated, for if a given system X is thought to be entirely erroneous then there is in fact no system X, since no regularity existed to begin with. We can only doubt relative to a given unified system, as the concept of error applies specifically to a breach in a system, not the non-existence of a system. Otherwise error as a concept would have no meaning. In this there is a profound connection between the Thomistic notion of evil and the notion of falsehood and error, as evil only ‘exists’ inasmuch as it subsists in a being. If no being existed, there would be no evil to begin with. Error, like evil, is a privation of a subject.

      Since all doubt presupposes some knowledge of the thing in question, to doubt and question the faculty of knowledge as such, in its entirety, is not only theoretically impossible but also destroys the reason for doubting. Doubt, too, is reliant upon the integrity of the intellective and sensitive faculties. Admittedly I am subscribing to the position laid out by Jacques Maritain in his book The Degrees of Knowledge. Inasmuch as I believe that absolute doubt is impossible and that any critique of knowledge must already assume the fundamental validity of the intellective and sensitive faculties, I am a dogmatic realist. I do believe that there is a meaningful exploration of the nature of knowledge, however.

      As for the problem of language, an inherent error in the position of many deconstructionists is that they consider logic to be, in a certain sense, identical to the form of the language in which it is expressed. Though language is supposed to be based upon rightly ordered logic, it should be clear that even when the structure of a language has some erroneous elements the logical truth can still be expressed, albeit with difficulty. There are ample cases of discussions in which language has become an extreme difficulty simply due to its limitations, but nonetheless overcome by secondary modes of clarification. One striking example of this is the discussion surrounding the Thomistic notion of God as Being, seemingly in conflict with the lingual (and even partially logical) principle of being as necessarily a predicate. A contradictory element in a given lingual system does not at all prove that logic is nonsensical, but only that the language is based on shaky foundations. And of course to doubt logic itself is to have one’s nail scratch itself, because it requires logic to doubt logic, just as it requires knowledge to doubt knowledge, perception to doubt perception, etc. I do think particular doubts can be very meaningful, but absolute doubt is a self-destructive plague that is self-evidently (at least when seen properly) wrong. I have to also admit that I am not entirely convinced by Derrida’s argument for the inherent contradictions of language using the concept of difference, but to argue that point would require more time than I care to take at the moment.

      When I said that the basic ontological and logical structures of objective and formal reality can only be viewed with awe, I precisely meant that in a certain sense they must be taken for granted in any investigation because they are self-evident, and as I have shown any attempt to doubt them always leads to metaphysical or logical self-destruction. This is true mathematically as well, as it necessary for at least one principle to be assumed in order to begin any investigation. Something is always presupposed, and always must be, because we are not the creators of reality but the receivers of it. One statement of Aquinas in particular seems to convey this point: “The essences of things are unknown to us.” This is not to say we have no knowledge, but that all of our knowledge is limited to glances at the secondary acts of beings, that we can never know the essence in itself but only through its acts. To attempt to exhaust reality, and then doubt its structure because we can’t fully comprehend it, is an act of arrogance towards the nature of things. The limit of inquiry is the proper beginning of faith and humility.

      Of course we must make sure that the things which we take for granted really are necessary, but once this has been established a reverent silence towards being must be observed.

  3. August 11, 2012 3:03 am

    I mostly agree with with you, but not entirely, and my disagreement can, I think, be summarized in my critique of this sentence of your response: “Something is always presupposed, and always must be, because we are not the creators of reality but the receivers of it.”

    My problem is with the words “something” and “presupposed.” What you’ve found yourself saying is that there is something, some specific thing we can isolate, which we are presupposing, that is, putting forward as true before beginning our investigation; as if we can point to what it is, and have a starting place without it from which to put it forward! I completely agree with the second half of your statement: we are not the creators of reality but the receivers of it, and we don’t know what reality is, and it doesn’t even make sense to talk about not accepting it.

    I suspect that you don’t immediately agree that there’s a contradiction between the first and second halves of your statement, so I’ll try stating it a different way. The first half makes it sound as if philosophy is like mathematics: we state our axioms, then derive our conclusions. This works perfectly for mathematics, because we can decide which axioms to begin with by seeing either which ones correspond best to physical reality, or which ones give the most interesting results. But we can’t do this for philosophy, because we can’t directly perceive the metaphysical in the same way as we can the physical, and we can’t just pick the axioms we like because we’re worried about truth here.

    Talking about premisses and conclusions in this way obscures the fact that most philosophical debate is about, if we want to keep using this terminology, determining which are the proper premisses. But that language is itself misleading, because philosophy isn’t about premisses or conclusions, it’s about trying to understand the regularity of reality when we can’t fully perceive it. Let’s “take for granted” perception, logic, language, and whatever else, if by “take for granted” you mean “assume that these words aren’t meaningless.” But let’s not assume we know what any of them are, and that we can just assume a reverential silence towards them.

    The difference between our positions comes down to, you want to say, “there is such a thing as a form,” and be done with it, because you want to be humble. Then, I suppose, you’ll go on to answer other questions, questions I probably wouldn’t consider strictly speaking philosophical. I want to say, “there is such a thing as a form. What does ‘form’ mean?” and then never stop talking, because that question can never be fully answered, and continual elaboration on the theme gives glory to God.

    But I don’t plan on being a philosopher, so I do stop talking, and go do other things, like read poetry. In doing so I “take for granted” certain concepts, e.g. that of formal reality, but in doing so I recognize that I’m making simplifications that could lead me to contradict myself, and whenever I detect such contradictions, I’m forced to do some ad-hoc philosophizing.

    The danger of all philosophy is that we’ll reduce reality to a simulacrum of itself, and then take our model of reality for the truth. This is where skepticism comes in–it brings out the contradictions that inevitably result. We can’t be silent; we can’t doubt that they exist, but we need to be willing to doubt that we know anything about them. At all. “I know one thing, that I know nothing.”

    (N.B.: This position is similar in many ways to, though ultimately different from, that Stanley Cavell outlines in The Claim of Reason.)

    • Thomas DePauw permalink
      August 11, 2012 3:53 pm

      I suppose some of my statements were not as clear as they should have been. I never meant to say that the method of philosophy is the same as that of mathematics; that would be to fall into a Cartesian error. What I did mean was that just as mathematics requires a given principle for any investigation to occur, so too is it necessary for us assume that the reality we encounter is intelligible and that our minds are equipped with the means necessary for encountering reality in order to be able to philosophize and truly inquire into reality. To assume this, however, is to necessarily assume immediately (even if it is not evident immediately, at least it will eventually be discovered that it was always so) that the four modes of Aristotelian causality exist. This has been borne up historically, as all attempts to disprove the necessity and fundamental grounding of reality in material (subjective idealism), formal (materialism), efficient (Hume’s billiards and his infamous “fork”), and final (atheism) causality have led to internal contradictions within the arguments themselves. The contradiction in Hume’s argument (which also show the necessity of potency and act) is probably the most obvious one, but the other modes of causality are equally necessary and by recourse to the actual, basic structure of reality can be shown to be so (in the case of defending material causality the problem of the juxtaposition [inter-situated-ness] of nonlocal being, in the case of formal causality the problem of matter as only viewable in relation to form, in the case of final causality the problem of finite natures without any origin or terminus, although there are many other contradictions in denying final causality). Since truth is by definition the adequation of mind to reality (Aquinas, De Veritate), there must be a reality that is intelligible for truth (and thus philosophy) to have any meaning, though it must be also pointed out that denying the existence of truth already presupposes the reality of truth due to some of the reasons I mentioned here and in earlier posts.

      When I said to “take for granted” intellection, perception and the existence of reality, I meant that we must assume that those things are real because they are the presupposition for a notion of the real. Subjectively speaking we may not realize that they are necessary at first, but objectively the only reason we can go on existing and intellectualizing is due to the fundamental reality of these things. Just because they are unknown does not mean they do not exist or are any the less necessary.

      As for assuming any knowledge of them, some knowledge must be present for us to even speak of the basic, objective presuppositions for truth. What I do agree with you on is this: we can never completely exhaust the reality of a thing with our concepts, but can only dimly perceive “through a veil”, as it were, the structure and majesty of reality as God’s creation. Josef Pieper rightly argues that though reality is only intelligible because God created it by giving it a participation in His Being, it is also unable to be completely penetrated by any finite mind because it has a mysterious share in the Being who is the never-ending fullness and plenitude of the act of existence.

      To reserve a respectful silence towards Being does not mean that one can never contemplate or further elucidate on the truth one has discovered to glorify God; on the contrary, truth requires us to continue discovering the various dimensions of reality for love of God and fulfillment of one’s “telos” and the common good. However, it does mean that we can never claim to possess a fullness of such knowledge; it also requires us to recognize the limits of our inquiry, such that we do not begin to exposit philosophically what can only be understood in the light of revelation.

      You said: “The danger of all philosophy is that we’ll reduce reality to a simulacrum of itself, and then take our model of reality for the truth.” This is precisely where humility requires us to recognize that we do not possess complete knowledge of anything, that we only have a limited understanding which must be constantly questioned so that we are sure that the little knowledge we have is true. But there is a difference between positive and negative doubt, just as there is a difference between being stabbed to death and being surgically healed. We should not and cannot doubt absolutely, namely we should not and cannot doubt that which makes our doubt and any truth possible. However, we must doubt that which is possibly erroneous, namely that which is based upon (but which nevertheless possibly undermines) reality and intellection.

      It is not possible for that which provides the possibility of doubt to be erroneous, for then the doubt would no longer be credible.

  4. August 11, 2012 5:18 pm

    I still think we mostly agree, but I also still think you’re underestimating the extent to which even definitions like “truth is the adequation of mind to reality” are just approximations, because they assume we know the meaning of a word, “reality,” whose meaning is exactly what we’re trying to understand by understanding “truth.” It’s always dangerous to slip into thinking that, for example, knowing that denying the importance of formal causes leads to Bad Things means we can neatly draw lines between the four modes of causality and think that in doing so we’re identifying a feature of reality itself. My problem with the scholastic approach to philosophy, then, isn’t with ideas like the four modes of causality, but with dogmatic statements like “To assume [that reality is intelligible], however, is to necessarily assume immediately … that the four modes of Aristotelian causality exist.”

    I would say, rather, that the four modes of causality have proven to be very good ways of speaking about things, and to have much explanatory power, and every attempted reduction in the number of causes has proven to be an oversimplification. But that doesn’t mean the causes “exist,” whatever that means, it means they’re a good way to understand the world, as far as they go–and I mean “understand” and “world” here in as unphilosophical a sense as possible. They’re good ways of answering the question “so what’s going on here anyway?” When we start asking if they “exist, if they’re “real,” what we’re doing isn’t answering some question about what the causes REALLY TRULY ACTUALLY ARE, what we’re doing is acquiring additional vocabulary that we find useful or interesting, and figuring out how this new vocabulary relates to the old. It is in this sense that philosophy is the search for truth: “truth is what philosophy searches for.”

    But there’s more than one way such vocabulary can work itself out (cf. Plato v. Aristotle on the question of the reality of forms), and while we can identify contradictions inherent in certain vocabularies, that doesn’t mean we can elevate our chosen vocabulary to the status of Truth. There are other plausible vocabularies out there. That we use ours, not someone else’s, is a historical contingency. I’m not proposing some sort of radical historicism here ( a la Rorty’s critique of final vocabularies); what I’m saying is, I think, something Aquinas would recognize, and something I think you’re not entirely against either (otherwise you wouldn’t talk as if it were meaningful that Aristotle’s causes have been historically validated). One consequence of this that I don’t think you accept, however, is that we need to stop conducting our metaphilosophical discussions–which are the only truly philosophical ones anyway–as if we already know what our vocabulary is (“truth,” “reality,” “intellect,” etc), and are just trying to get back to it. A bit of bafflement (as opposed to doubt) is necessary for good philosophizing.

    • Thomas DePauw permalink
      August 17, 2012 12:03 pm

      Here, however, is where I think I must differ with you. If philosophy is to mean anything but sophistry, we must already know what the pre-philosophical words we use refer to concretely, otherwise any dialectic will be completely fruitless and lead us through a variety of phantasms. Pre-philosophical words are worthless if they are not symbols which refer to actual realities encountered. Of course this does not mean a given pre-philosophical word corresponds to even a relatively complete understanding of a thing, as it is impossible for us to have such a conception until we have philosophized about it, and even then we do not have a full understanding of the thing. But such a word must at least refer to an image of a thing.

      Of course this argument can be taken to an extreme (as Wittgenstein did) in declaring that all abstractions from the actually known to something unknown are unwarranted and worthless. This is an incorrect judgement, however, due to the ability of the human mind to use the analogia entis, or analogy of being. If a dialectic leads us to necessarily posit something apart from what we concretely perceive and understand, we can understand this unknown thing in two ways:
      1. By the attributes it must have as the explanatory factor of some concrete real thing;
      2. By means of analogy, namely the predication of definitions that ordinarily signify a concrete reality but have some proportion with the unknown thing in question due to the aforementioned knowledge gained by means of method 1.
      These two methods are bound together in that both of them entail each other’s aid. These two are in fact quite justified in being considered together as the doctrine of the analogia entis. Therefore when I said that the four causes must exist, I meant that what our concepts of the four modes of causality actually signify must be real, even if we have a very imperfect knowledge of them, as it is evident that ontologically speaking they are not separate but unified in reality. We can only understand them, however, by separating the various necessary functions that the given nature performs.

      I must disagree that philosophical investigation merely gains us new vocabulary, though I agree that new vocabulary must be developed in order to designate new realities we have discovered, for we gain only indirect knowledge of the world through words, namely in what they signify. To say that we only gain new vocabulary is to look at philosophy from merely a lingual point of view, which is just as deprived of meaning as a merely material look at reality. Logic must be bound with language, just as corporeal forms are bound to matter. Not only do we gain vocabulary in a philosophical investigation, but we also gain some greater insight as to what reality our words represent.

      I quite agree that speaking of the truth or falsity of vocabularies is a meaningless discussion, except inasmuch as there are logical contradictions between concepts which are signified in the use of such vocabulary, and even here it has more to do with the use of a vocabulary than the vocabulary itself. However, coherent vocabularies are distinguished in dignity by the ambiguity or clarity which they possess. Vocabularies that more precisely and clearly signify the reality they are speaking of are better, even if another more ambiguous vocabulary also signifies such a reality. It is for this reason I generally choose to use the Aristotelian/Thomist vocabulary, as I find that it is one of the clearest and precise systems of signs there is. I will admit there are some weaknesses in it, especially in regards to the ambiguity of the term ‘accident’ when applied materially or formally, but overall it is a very clear and useful system. Having been a phenomenologist before being a Thomist, I can safely say that the lack of clear terms in phenomenology (and the ambiguous use of specific terms such as intentionality and categorial intuition) made it difficult to philosophize correctly regarding epistemology. I have, of course, taken some important concepts from phenomenology and used them in the context of Thomism because of their usefulness.

      Finally, as for the knowledge of what vocabulary signifies, I have already mentioned that it is necessary to have at least some concrete knowledge of what we are speaking about using such words before our investigation. Philosophy helps us not only to discover new realities, but to clarify already known ones. I believe I (mostly) answered your last point in the paragraphs above.

  5. August 11, 2012 5:35 pm

    Incidentally, if you’re wondering how this differs from Rorty’s critique of final vocabularies, recall that in his understanding an ironist must fulfill three conditions:

    1) She has radical and continuing doubts about the final vocabulary she currently uses, because she has been impressed by other vocabularies, vocabularies taken as final by people or books she has encountered;
    2) She realizes that argument phrased in her present vocabulary can neither underwrite nor dissolve these doubts;
    3) Insofar as she philosophizes about her situation, she does not think that her vocabulary is closer to reality than others, that it is in touch with a power not herself.

    Whereas I do not advocate accepting any of these without alteration as conditions for what a philosopher should be. My alterations are:

    1) There should not simply be radical and continuing doubts. There should be continuing doubt, but it should not be radical: it would be better to call it “humility.” There should be radical doubt sometimes, but not always: it should occur only (and always) as her first reaction to an impressive vocabulary she has just encountered for the first time, or to a realization of a contradiction in her present vocabulary.
    2) She should realize that doubt cannot be permanently dissolved, and accept it as inevitable, because even if the number of final vocabularies with no known inconsistencies is above one, the number of consistent final vocabularies is either one or zero, and the number of known consistent final vocabularies is exactly zero.
    3) She does think her vocabulary is in touch with “a power not herself,” but is willing to admit that the other vocabularies are also in touch with that power, that power being just “the world” or “reality” or whatever you want to call it. (The point being here that she does believe there to be something other than herself.)

    The difference between a philosopher and a believer is that the believer rejects even the revised (3). The believer does not reject (1) or (2).

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