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"Wittgenstein, Mathematics, and Philosophy"
'A monk once asked Zhao-Zhao whether a dog had the Buddha-nature or not.
Zhao-Zhao replied, "Mu".'
Floyd's principal thesis is that Wittgenstein's philosophies are mathematical. Views that mathematics are not key to his work are mistaken, and views that he separated his philosophizing from his mathematical thinking are also wrong (1).
Floyd starts at the beginning. In 1914, she finds Wittgenstein rejecting as inadequate the best efforts of Russell and Frege and the achievements of the formalists. They are not wrong in their details, but they go too far with their claims. They simplify too much 'by presenting diverse mathematical techniques in the guise of just one sort of technique'(2). This one technique may be very general and powerful, but it is a hammer, and not everything needs to be pounded in.
Logic can be over-extended (3). It is a small, self-contained world, which excludes much. A formal derivation is not proof - 'proof' is some sort of psychological phenomenon. What makes a good proof is not following certain strict rules, but the result it achieves of convincing mathematicians, of providing 'forcible grounds' to believe something. The formalization of reasoning obscures the use of reasoning. Further, we cannot really search for something we already know (14).
For example, we can look at Indian logic, and we see that the earliest forms of logic, their syllogisms, make specific use of persuasion and appeal to open-mindedness: the earliest commentators cover the 10-part syllogism, of which 5 parts are issues like 'Do we actually have any doubt on this proposition?' 'Why do we seek to prove or disprove this proposition?' and so on, and only then does the syllogism lay out a major premise, a minor premise, and draw a conclusion. As time passed, the 10-part syllogism became 5 (the Indian philosophers and logicians assuming that the other 5 could be taken for granted - why would you discuss something you have no uncertainty on, or didn't believe could be answered one way or another), and critics like the Buddhist Diginaga would suggest that it be formalized and further reduced down to 2 premises (which seem to be almost identical to the Aristotelian syllogism) (4).
If Wittgenstein 'never equates "proof" with "formal derivation"', if 'no proof convinces solely in virtue of its capacity to be formalized in a logical structure' (5), then what is proof?
We have moved past the _Tractatus_ into the period of _Philosophical Investigations_. In the _Investigations_, we learn that languages 'are' inasmuch as they do, that their meaning is use or the 'form of life' they embody. Mathematics is another one of these languages (7). If 'A picture held us captive' (6) when it came to ordinary language, then it seems at least possible that we are held captive in other languages - such as mathematics.
Wittgenstein's approach to ordinary language did not involve the logical proofs (which were part of the problem), but sought to convince one that the picture-theory was inadequate, by having us consider knotty problems and things that did not fit. The problems do not disprove the theory, but we are nevertheless convinced (or at least disquieted). And that is the method and goal (8).
The genre of linguistic examples used in the _Investigations_ was characterized by being somehow on the edges or outside the language as presented; he shows us a language in which each name is an object to be presented, and asks us about what goes on when the object is not there to be presented. Or he brings up the many things we do in language (like saying, to an empty room and no-one, 'That was a nice nap!') which neither communicate any logical propositions nor clearly do anything.
The counterpart to these linguistic examples are various impossibility proofs. Clearly, these results are Mathematics with a capital 'm'. Clearly they are also solidly proven. What is not clear is how they can fit into any formalist straitjacket, and this is the issue Wittgenstein keeps hammering at. What system is a mathematician working in when he accepts one cannot double a cube with a rule and compass? Not Euclidean geometry, that is for sure. Nor the algebra which proves it. And his actions, his assent are not governed by some propositional logic, because the logic does not 'do' anything. It seems that he is motivated by aesthetics, by visceral likes and dislikes - by feelings (9).
So a mathematician is not a disinterested calculating machine. Aside from the reasons and motivations which are the real cause of him accepting or rejected a proof, he can change questions on the fly. How would one logically say that one is either looking for a proof/disproof of a conjecture, or a reason to not pursue the question? The mathematician might reply that he embedded the system (plane geometry, perhaps) into a different one (algebra), and he came up with a logical proof that the embedded system could not do something. But what should the original system say to that? 'I have nothing to do with that algebra over there, or that mess of symbols you say is me; give me a rational answer on my own terms.' But of course one can't.
There is no change in logic. The system doesn't evolve, because that would lack consistency. If we demand consistency of mathematics as it is actually practiced, then we are put in a funny position. Floyd covers at length the example of someone who 'wants' to find the trisection of the angle, and then finds out it is impossible and now no longer wants to find it. If we hold to a timeless view, then 'are we forced to say that for over 2,000 years trisectors engaged in an inquiry which made no sense, an inquiry which they didn't _really_ want to engage in?' (11). This is not a proof, of course, but is certainly casts doubt. The trisectors thought they wanted it, and we should be wary of any position which makes us say things like 'You didn't _really_ want to do that, you were just mistaken about yourself - I know better than you do.'
The trisectors raise another issue. They seem to be affirming some sort of contradiction or falsehood. It certainly was not because they didn't understand what they were doing - these were often extremely intelligent and learned mathematicians pursuing the trisection, and again, we should be wary of arrogance in saying 'they didn't really understand what they were doing and so accidentally affirmed what we know to've been a contradiction.' But of course, a contradiction or falsehood being held as belief is yet another difficultly for logical views. It's the rare and paraconsistent logic indeed that doesn't explode at a contradiction. It would seem to be, as Wittgenstein says, something 'unthinkable' (12). If we sincerely look for an object which turns to not exist, and even be outright contradictory, then how could it correspond to an existing object in our mind? ('One can ask, for example, how as it possible so much as to _look for_ the trisection of the angle?' (13))
If Euclidean algebra were to come up to you, and anthropomorphically demand a doubled cube, what could one do but shrug and be silent? The system can't change (it'd be a different system then) and its language just doesn't allow expressing concepts of impossibility and undesirability, doesn't allow for any changes 'in order to lead someone from one form of expression to another' (10).
If one thought in a language, and the language was 'crippled', and thinking really was as simple as formal logic, then how would one ever escape and conceive or accept any manner of meta-proofs. 'Can one think without speaking?', the _Investigations_ ask. Does one really have only one process, and not two which comment on and examine the other?
By this point, we think, not really. We no more accept mathematics as all logical than we did language after the _Investigations_. And the analogy went down deeper, to method and approach, and the flaws in the views were similar - they couldn't handle the actual practice, the self-referential moves. The problems came 'wearing fish net' (15), neither fish nor fowl; apt to be choked on and spat out as unacceptable. And Floyd finishes her case: impossibility proofs dispel illusions too, just like philosophical analysis can (16). The grand analogy is complete.
So then, what is my take on her analysis? Floyd suggests that we begin by revising our categorization of Wittgenstein. In particular, our class seems to have settled down into a binary system: we discuss 'early Wittgenstein', where language is the same thing as mathematics and the central text is the _Tractatus_; and we discuss 'late Wittgenstein', where he has recanted the idea that language is a unitary sort of thing with an isomorphism to some equally unitary logic, and the central text is the _Investigations_. It certainly seems fair to me to suggest that there is unity to his work; it's rare for one to abandon ideas and approaches entirely - rather than slowly modify them into something that looks different but in its core is still kin to the earlier.
We should revise the second clause: language is indeed a plurality of things that humans do, and so is mathematics. Neither is a unity, and neither is equivalent to any set or subset of the other except inasmuch as they are all things humans do (but nothing much is gained by that particular 'assimilation of expression'). Mathematics does not enforce itself. We may have rules for deriving new theorems, but there is no rule which makes us apply the previous rules (for we would need a rule to assure us that we are justified in following the rule-applying rule). We must not be Platonic and think that mathematics is something out there, something eternal. Rather we choose at every step along the
[Even so, it has been suggested that if there is extraterrestrial
intelligence, mathematics is our best bet for communicating with it;
certainly not English. Maybe even logic, since logic contains nothing
empirical, no facts.]
way to apply this rule or that, or to transform it into this equivalent expression or not. We choose also whether to stop and never take up the task again. There is nothing in the logic which speaks of how we are to use it or not.
Floyd elaborates at length on the Euclidean problems such as trisection of the angle or doubling of the volume of a cube, for the simple reason that they were so instructive for Wittgenstein. The interest of the problem is how they ultimately are rendered uninteresting because one can be convinced that they are impossible. There is nothing in the rules for playing ruler-and-compass which says to abandon a given task: it has no notions of possibility and impossibility. You simply are given a specification and are asked to provide the result. This is analogous to an _Investigations_ example of a simple language, where one asks for a hammer and gets it. Does a hammer not exist for you to get? The language doesn't cover it. What do you do? As it is, it is like Dr. Seuss's north-going Yak running into a south-going Yak. Neither one can give in. The rule-and-compass game can't change itself to be more 'powerful', because then it would not be itself - it would be some other game, and the original dilemma remains; the non-existent hammer can't will itself into being ('nothing from nothing'). What must give is the statement itself. One must answer 'mu', one must unask the question. The bottle is made up of a game and a fact: there's only one out for a fly, and that's going backwards.
One devises some more complicated mathematical game, and plays it, and discovers in it the old game. The old game is seen in the new light of the new system. It seems almost mysterious; in the course of playing with the new system, one ceases to do certain things with the old system. One may still construct shapes in rule-and-compass, but one simply no longer tries to construct a doubled cube. There is no logical necessity for this refusal, this self-discipline, but it happens anyway just like languages happen.
One unasks the question. This is a Zen concept, but Zen seeks therapy as well as Wittgenstein. Maybe using Zen is superficial, but somehow it seems to already
[No. Zen is great. Now you should move on to Chuang Tzu, much more
Wittgensteinian, I think.]
have a vocabulary suitable for talking about trying to get outside a system, for talking about how a system is brittle and unsuitable for living in.
'When one begins Zen, trees are trees, mountains are mountains; when one is advanced, trees are no longer trees, mountains no longer mountains; and when one is enlightened, one realizes: trees are just trees, and mountains just mountains.' When one goes into therapy, nothing changes. The world is the same as when you went in: no limbs have been removed, no dishes washed, the clouds scud across the sky just like they would have if you hadn't gone in; but the world is also very different.
[Splendid, Andy. You should submit the paper to one of the Undergraduate
Philosophy conferences that are held at different universities. Sometimes a
flyer about one is posted on the department bulletin boards, or you could
try googling.]
References
1. "But for over forty years the majority of Wittgenstein's readers, pro and con, have treated his discussion of mathematics as a sideline to his main philosophical work." pg 232
"As I see it, Dummett erred in claiming that for Wittgenstein, 'philosophy and mathematics have nothing to say to one another...'." pg 233
2. pg 234
3. "Many of Wittgenstein's writings on mathematics criticize what he took to be 'the disastrous invasion of mathematics by logic' - that is, what he took to be the claims of Frege, Russell, and others to have mathematically analyzed our notion of mathematics." pg 235
4. Richard King, _Indian Philosophy: An Introduction to Hindu and Buddhist Thought_; Arthur Berriedale Keith, _Indian Logic And Atomism_
5. pg 235
6. Wittgenstein quote on pg 232
7. "But what he calls throughout the _Investigations_ our 'ordinary' uses of language and forms of life include, and do not contrast with, mathematics." pg 235
8. '...his philosophical investigations do not, as do mathematical ones, issue into _proofs_. He has no wish to emulate [math] in philosophy...The sort of conviction and insight his philosophical investigations aim to produce is not to be generated mathematically or by means of what might be exhibited in a formally deductive argument." pg 236
'...Wittgenstein's investigations of (what to say) about mathematics, always partial, proceed by exploring and exploiting the power of such metaphors, and then using them to question traditional philosophical views [like formal proofs]' pg 237
'"The philosophical remarks in this book are, as it were, a number of sketches of landscapes which were made in the course of these long & involved journeyings. The same or almost the same points were always being approached afresh from different directions, & new sketches made.' Preface, _Philosophical Investigations_
9. 'We shall see that his treatments of such proofs inform (and are informed by) what he writes about philosophical insight into philosophical illusion: mathematical proofs of impossibility illustrate how it is that we may succeed or fail to extricate ourselves from a deeply felt but wrongly articulated sense of conviction and obviousness.' pg 238
10. Wittgenstein, quoted on pg 239
11. pg 241
12. Wittgenstein quote on pg 242
13. _Investigations_ #462, quoted on pg 243
14. 'That is: I cannot in any sense systematically search for, wonder about or hypothesize something which I am already aware that I possess. If I accept the proof as a proof, I cannot conjecture its outcome, for if I accept the proof I am convinced, and then I cannot at the same time doubt...' pg 249. I feel there is an interesting comparison possible with "Fitch's paradox of knowability" (see <http://plato.stanford.edu/entries/fitch-paradox/>).
15. Wittgenstein, first quote, pg 250, from _Lectures_
16. 'Nevertheless, the sort of perspicuousness...has a specific sort of application: it is used to stop someone who is under the illusion of making sense from speaking emptily, to wean one away from certain specific lines of thought, to part one from particular ways one might think that conviction can be expressed or produced. It is characteristic of the acceptance of a proof of impossibility that one will be able to use it against a specific illusion, to use it to stop others from trying to do that which they do not _really_ want to do.' pg 253-254
Bibliography
* Juliet Floyd, "Wittgenstein, mathematics and Philosophy" in _The New Wittgenstein_, ed. Crary and Read
* Ludwig Wittgenstein, "The Rejection of Logical Atomism", in Kenny, _The Wittgenstein Reader_.
* Wumen, "The Gateless Gate", in _Zen Flesh, Zen Bones_ ed. Paul Reps and Nyogen Senzaki