Wittgenstein’s forgotten lesson

Prospect Magazine

Wittgenstein’s forgotten lesson

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Wittgenstein’s philosophy is at odds with the scientism which dominates our times. Ray Monk explains why his thought is still relevant.

Ludwig Wittgenstein is regarded by many, including myself, as the greatest philosopher of this century. His two great works, Tractatus Logico-Philosophicus (1921) and Philosophical Investigations (published posthumously in 1953) have done much to shape subsequent developments in philosophy, especially in the analytic tradition. His charismatic personality has fascinated artists, playwrights, poets, novelists, musicians and even movie-makers, so that his fame has spread far beyond the confines of academic life.

And yet in a sense Wittgenstein’s thought has made very little impression on the intellectual life of this century. As he himself realised, his style of thinking is at odds with the style that dominates our present era. His work is opposed, as he once put it, to “the spirit which informs the vast stream of European and American civilisation in which all of us stand.” Nearly 50 years after his death, we can see, more clearly than ever, that the feeling that he was swimming against the tide was justified. If we wanted a label to describe this tide, we might call it “scientism,” the view that every intelligible question has either a scientific solution or no solution at all. It is against this view that Wittgenstein set his face.

Scientism takes many forms. In the humanities, it takes the form of pretending that philosophy, literature, history, music and art can be studied as if they were sciences, with “researchers” compelled to spell out their “methodologies”—a pretence which has led to huge quantities of bad academic writing, characterised by bogus theorising, spurious specialisation and the development of pseudo-technical vocabularies. Wittgenstein would have looked upon these developments and wept.

There are many questions to which we do not have scientific answers, not because they are deep, impenetrable mysteries, but simply because they are not scientific questions. These include questions about love, art, history, culture, music-all questions, in fact, that relate to the attempt to understand ourselves better. There is a widespread feeling today that the great scandal of our times is that we lack a scientific theory of consciousness. And so there is a great interdisciplinary effort, involving physicists, computer scientists, cognitive psychologists and philosophers, to come up with tenable scientific answers to the questions: what is consciousness? What is the self? One of the leading competitors in this crowded field is the theory advanced by the mathematician Roger Penrose, that a stream of consciousness is an orchestrated sequence of quantum physical events taking place in the brain. Penrose’s theory is that a moment of consciousness is produced by a sub-protein in the brain called a tubulin. The theory is, on Penrose’s own admission, speculative, and it strikes many as being bizarrely implausible. But suppose we discovered that Penrose’s theory was correct, would we, as a result, understand ourselves any better? Is a scientific theory the only kind of understanding?

Well, you might ask, what other kind is there? Wittgenstein’s answer to that, I think, is his greatest, and most neglected, achievement. Although Wittgenstein’s thought underwent changes between his early and his later work, his opposition to scientism was constant. Philosophy, he writes, “is not a theory but an activity.” It strives, not after scientific truth, but after conceptual clarity. In the Tractatus, this clarity is achieved through a correct understanding of the logical form of language, which, once achieved, was destined to remain inexpressible, leading Wittgenstein to compare his own philosophical propositions with a ladder, which is thrown away once it has been used to climb up on.

In his later work, Wittgenstein abandoned the idea of logical form and with it the notion of ineffable truths. The difference between science and philosophy, he now believed, is between two distinct forms of understanding: the theoretical and the non-theoretical. Scientific understanding is given through the construction and testing of hypotheses and theories; philosophical understanding, on the other hand, is resolutely non-theoretical. What we are after in philosophy is “the understanding that consists in seeing connections.”

Non-theoretical understanding is the kind of understanding we have when we say that we understand a poem, a piece of music, a person or even a sentence. Take the case of a child learning her native language. When she begins to understand what is said to her, is it because she has formulated a theory? We can say that if we like—and many linguists and psychologists have said just that—but it is a misleading way of describing what is going on. The criterion we use for saying that a child understands what is said to her is that she behaves appropriately-she shows that she understands the phrase “put this piece of paper in the bin,” for example, by obeying the instruction.

Another example close to Wittgenstein’s heart is that of understanding music. How does one demonstrate an understanding of a piece of music? Well, perhaps by playing it expressively, or by using the right sort of metaphors to describe it. And how does one explain what “expressive playing” is? What is needed, Wittgenstein says, is “a culture”: “If someone is brought up in a particular culture-and then reacts to music in such-and-such a way, you can teach him the use of the phrase ‘expressive playing.’” What is required for this kind of understanding is a form of life, a set of communally shared practices, together with the ability to hear and see the connections made by the practitioners of this form of life.

What is true of music is also true of ordinary language. “Understanding a sentence,” Wittgenstein says in Philosophical Investigations, “is more akin to understanding a theme in music than one may think.” Understanding a sentence, too, requires participation in the form of life, the “language-game,” to which it belongs. The reason computers have no understanding of the sentences they process is not that they lack sufficient neuronal complexity, but that they are not, and cannot be, participants in the culture to which the sentences belong. A sentence does not acquire meaning through the correlation, one to one, of its words with objects in the world; it acquires meaning through the use that is made of it in the communal life of human beings.

All this may sound trivially true. Wittgenstein himself described his work as a “synopsis of trivialities.” But when we are thinking philosophically we are apt to forget these trivialities and thus end up in confusion, imagining, for example, that we will understand ourselves better if we study the quantum behaviour of the sub-atomic particles inside our brains, a belief analogous to the conviction that a study of acoustics will help us understand Beethoven’s music. Why do we need reminding of trivialities? Because we are bewitched into thinking that if we lack a scientific theory of something, we lack any understanding of it.

One of the crucial differences between the method of science and the non-theoretical understanding that is exemplified in music, art, philosophy and ordinary life, is that science aims at a level of generality which necessarily eludes these other forms of understanding. This is why the understanding of people can never be a science. To understand a person is to be able to tell, for example, whether he means what he says or not, whether his expressions of feeling are genuine or feigned. And how does one acquire this sort of understanding? Wittgenstein raises this question at the end of Philosophical Investigations. “Is there,” he asks, “such a thing as ‘expert judgment’ about the genuineness of expressions of feeling?” Yes, he answers, there is.

But the evidence upon which such expert judgments about people are based is “imponderable,” resistant to the general formulation characteristic of science. “Imponderable evidence,” Wittgenstein writes, “includes subtleties of glance, of gesture, of tone. I may recognise a genuine loving look, distinguish it from a pretended one… But I may be quite incapable of describing the difference… If I were a very talented painter I might conceivably represent the genuine and simulated glance in pictures.”

But the fact that we are dealing with imponderables should not mislead us into believing that all claims to understand people are spurious. When Wittgenstein was once discussing his favourite novel, The Brothers Karamazov, with Maurice Drury, Drury said that he found the character of Father Zossima impressive. Of Zossima, Dostoevsky writes: “It was said that… he had absorbed so many secrets, sorrows, and avowals into his soul that in the end he had acquired so fine a perception that he could tell at the first glance from the face of a stranger what he had come for, what he wanted and what kind of torment racked his conscience.” “Yes,” said Wittgenstein, “there really have been people like that, who could see directly into the souls of other people and advise them.”

“An inner process stands in need of outward criteria,” runs one of the most often quoted aphorisms of Philosophical Investigations. It is less often realised what emphasis Wittgenstein placed on the need for sensitive perception of those “outward criteria” in all their imponderability. And where does one find such acute sensitivity? Not, typically, in the works of psychologists, but in those of the great artists, musicians and novelists. “People nowadays,” Wittgenstein writes in Culture and Value, “think that scientists exist to instruct them, poets, musicians, etc. to give them pleasure. The idea that these have something to teach them-that does not occur to them.”

At a time like this, when the humanities are institutionally obliged to pretend to be sciences, we need more than ever the lessons about understanding that Wittgenstein—and the arts—have to teach us.

  1. October 27, 2012

    lamo

    What is being suggested here is that science and non-theoretical forms of understanding are alternatives each valid in its own domain and neither being complete. This cant be true. Many essays on this topic underestimate, how big a hold science claims on reality. In principle, all the non-theoretical behavior, in the scientific viewpoint, is reducible to simple mechanical events. Recognizing expressive playing would correspond to a loose pattern in the brains (not necessarily a symbolic representation).

    Sure, one can say that a pragmatic, non-theoretical understanding is the only practical understanding right now. Analyzing it in terms of brain events is too complex. Neuroscientists might not admit this complexity and say superficial things about music.

    However, this incompleteness in science is contingent on our present limitations, not a deep incompleteness. Potentially, a computer program might be able to play and also recognize expressive piano music as defined by any given culture.

    OTOH, if you wants to seriously question this radical scientific claim, then dont understate the challenge. Also, we need to work out how to reconcile this challenge with all the successful evidence that the reductionist model has in its favor.

  2. October 27, 2012

    lamo

    What is being suggested here is that science and non-theoretical forms of understanding are alternatives each valid in its own domain, neither being complete. This cant be true. Many essays on this topic underestimate, how big a hold science claims on reality. In principle, all the non-theoretical behavior, in the scientific viewpoint, is reducible to simple mechanical events. Recognizing expressive playing would correspond to a loose pattern in the brains (not necessarily a symbolic representation).

    Sure, one can say that a pragmatic, non-theoretical understanding is the only practical understanding right now. Analyzing it in terms of brain events is too complex. Neuroscientists might not admit this complexity and say superficial things about music.

    However, this incompleteness in science is contingent on our present limitations, not a deep incompleteness. Potentially, a computer program might be able to play and also recognize expressive piano music as defined by any given culture.

    OTOH, if you wants to seriously question this radical scientific claim, then dont understate the challenge. Also, we need to work out how to reconcile this challenge with all the successful evidence that the reductionist model has in its favor.

  3. January 25, 2013

    Al_de_Baran

    “Many essays on this topic underestimate, how big a hold science claims on reality.”

    No, I think that the commentators on the subject understand the power-play of Scientism perfectly well. They simply (and quite rightly) reject that overarching claim in terms that your response neither adequately addresses nor refutes.

    As for evidence-gathering, you need to do a little more of it yourself, especially with respect to the many powerful challenges to reductionism. I don’t want to spoil the fun of the search, however, so I’ll leave you to escape your echo-chamber on your own, assuming that the will exists to do so–which, frankly, I doubt.

  4. January 26, 2013

    Mike Stephenson

    We know very little about anything outside our experience; but using the discoveries of the past, the, exciting, never ending pursuit of truth goes on. The quest requires the combined strength of science and art. There is not a beginning in the past nor an end in the future; this is all we can know and understand.

    Mike

  5. January 26, 2013

    Martin Keaney

    There is a word, I suggest, to describe that aspect of non-scientific understanding that Mr Monk addresses – wisdom.

    As we can observe, many people, usually late in life and with the accumulated experiences of life’s joys, sorrows and surprises, have developed this quality. Father Zossima may be seen as an expression of this – a knowledge of the human condition and society.

    Science seeks to understand the natural world, wisdom offers us an understanding of the human condition. They are complementary, I think, not competing.

  6. January 28, 2013

    Tarun

    Can one have anything but a philosophical ‘understanding’ of science? Is a scientific question one that has a scientific answer?

    Symbolic representation in language aside, don’t scientism and philosophy both attempt to reduce synopses of trivial phenomenon to generalities and to symbolise this understanding?

  7. January 29, 2013

    Fox

    Would Wittgenstein have agreed to his thought being expressed in a simplistic ‘this understanding’ v. ‘that understanding’ binary opposition? No. ‘Science’ is just another language game. To complain Penrose cannot help us understand ourselves better, is to misunderstand the language game in which it’s being played. And obtaining your understanding from a self-help book is no more (or less) real – it’s just a different language game. Wittgenstein is important for the way he ‘cleared the ground’ (see Peter Winch) for social constructivist theories of meaning. Where he didn’t go far enough was in exploring the relationship between meaning and power; i.e. why is ‘scientism’ such an influential and widely played ‘language game’? I think that’s why Wittgenstein lacks celebrity – he doesn’t speak to the issues.

    • January 31, 2013

      Advocate

      Wittgenstein lacks celebrity? Of a sort possessed by social constructivist a? Dear Lord, what nonsense.

      • January 31, 2013

        Fox

        Sorry you don’t like my turn-turn-of-phrase Advocate. But it was only reflecting Monk’s line above: “In a sense Wittgenstein’s thought has made very little impression on the intellectual life of this century”.

        Derrida, Foucault, Bourdieu. These are the people we “celebrate”, and whose voices now cross our disciplines. Like it or no, Wittgenstein is B-list. Ray Monk laments that, but doesn’t help us understand how ‘Wittgenstein’s lesson was forgotten’.

        I’m suggesting it was because, to reuse Monk’s phrase, he didn’t “set his face” against what matters. The others did.

  8. January 30, 2013

    susan

    philosophy is trivial drivel.
    math is clarity.

    • January 31, 2013

      SDK

      Say people who give up on expressing anything interesting in non-mathematical language. Or did you try but were unable to do so? That’s not necessarily the fault of language or philosophy…

      • November 26, 2013

        peter

        “Math is an abstraction used only to compare things, a method for measurement and relationships.”

        seems to do a very good job of illustrating my statement much further down:

        “The problem is not …, but rather philosophers with almost no mathematical knowledge beyond one or two elementary undergraduate courses in ‘cookbook’ math or less, and yet who are quite full of opinions on what it really is that mathematicians are doing.”

        • November 27, 2013

          Blaggs

          Philosophers need only understand as much mathematics as is relevant to the scope and depth of their enquiry. Addressing the question “Is infinity something enormously big; or rather an unlimited technique?” can be illuminating even if one only knows how to count. It would make a very good philosophical question to engage young children with, e.g. when they’re playing “Bet I can think of a bigger number than you can”.
          If you take a Platonist view, that maths exists in its entirety awaiting discovery, then, by your account, when are we allowed to even start to make any philosophical enquiry? How much has to be discovered?
          Conversely from a Wittgensteinian perspective where we invent maths little by little, how much of the current state-of-invention do we need ingest before we are granted philosophical rights?
          Indeed, when can we even begin to consider whether we are Platonists or Wittgensteinians?
          In addressing transfinite set theory, it would have been pretty poor if Wittgenstein had no grasp of Cantor’s diagonals; but would we berate him for discussing FLT because he wasn’t up to speed on Wiles’ proof?

          Tegmark does pontificate on “Contemplative alien civilizations” btw! Seems we’re going to the wrong school!

           
    • November 25, 2013

      CB

      Math is an abstraction used only to compare things, a method for measurement and relationships.
      Philosophy seeks to look beyond the metrics into the essence of the things themselves.

  9. February 12, 2013

    SteveDGH

    “Whereof one cannot speak, thereof one must be silent.”
    Ludwig Wittgenstein. If only he he had taken his own advice more to heart.

  10. February 12, 2013

    Ben Cobley

    A lovely article. I do find it remarkable the similarities between Wittgenstein and Heidegger on the core points here, even though the two of them are meant to come from resolutely hostile opposing traditions. The conception of ‘understanding’ sketched out by Ray Monk here is almost a carbon copy of Heidegger’s use of the word, not as a theoretical understanding but as an ability to fit it to the world around us.

  11. February 25, 2013

    peter

    ‘In the Tractatus, this clarity is achieved through a correct understanding of the logical form of language….’

    Oh, really? But the present author has no time or inclination to explain this to us? About all I know about this Ludwig (nothing like the creator of the late quartets!) is that he completely misunderstood what Godel had done in his work on incompeteness, stated it utterly incorrectly, and just about everybody except third-rate philosophy Ph. D. students and maybe their supervisors have ignored him since about 1935 (especially Bertrand Russell, who had been taken in rather badly more than 20 years earlier).

    ‘In his later work, Wittgenstein abandoned the idea of logical form…’

    That was certainly a way of avoiding embarrassing questions about his ridiculous views concerning mathematics and science, such as the example above, very easily found in lots of places, such as the Stanford Encyclopedia of Philosophy, even articles written there by sympathizers (again, as described rather nastily by me above) Wittgenstein looks half-sensible compared to the postmodernists, but maybe that’s the only thing keeping him from being laughed off the stage, except by scientists and mathematicians mostly.

    ‘ “Yes,” said Wittgenstein, “there really have been people like that, who could see directly into the souls of other people and advise them.”…’

    Yes, say I, but can you give me even a single convincing example, not made up by a novelist?

    • September 16, 2013

      Annabella

      In August a very old lady looked into my eyes to and saw my soul. Her name was Ann Anderson. Is that the kind of eample you require

      • September 17, 2013

        peter

        Was that on the 30th anniversary of the death of this claimed-to-be Anastasia? Perhaps it is my asked-for example, but I thought Ludwig was referring to living persons, or at least living novelists who conjure up such a person.

        What did she see in your soul? Perhaps one of those missing socks, most of which supposedly make up the black hole at the centre of the Milky Way??

  12. March 17, 2013

    Ramesh Raghuvanshi

    Wittgenstein rightly wrote “Where does one find such a acute sensitive ?In one of great artists because they express their idea with deep deep unconscious mind.When we are in completely absorbed with any subject our unconscious mind upraise and we find out unique solution.Great artists expressed his idea through unconscious mind with acute sensitive way he find out truth.

  13. April 2, 2013

    joe neisser

    Well said. The new-found historical sensibility in the analytic tradition is a good thing. But giving Wittgenstein (or Heidegger for that matter) credit for the distinction between explanation and understanding is like giving George Washington credit for “Liberty equality fraternity” …

  14. April 2, 2013

    peter

    So none of the Wittgen-groupies here know of any such example as I asked for above?

    Instead I’ll ask for a single example of even one person who has made a real contribution to human knowledge in logic/math/science since Wittgenstein’s death and who would give any credit at all to Wittgenstein himself having made any useful contribution of any kind to human knowledge. Not commentators, bullshit or otherwise, but actually contributors is what I’d like to know an example of, who credit Wittgenstein with anything beyond entertainment of the intellectually slothful.

    • April 2, 2013

      Fox

      Richard Rorty.

      • June 1, 2013

        peter

        I assume Rorty is supposed to be the example of a person for whom I asked. But again not a single word of explanation from Fox of what that contribution to knowledge was, and why it depended on this Ludwig, who seems the only known personification as a practitioner of what mathematicians jokingly call ‘proof by intimidation’. I’m pretty certain Fox has no idea what he is talking about. Philosophers these days impress only each other unfortunately. Scientists and mathematicians are almost unanimous that Godel contributed much, and Wittgenstein virtually nothing except entertainment for the intellectually slothful.

        And Mr. Blaggs is simply confused. He seems to be dopey enough to think that some true proposition could be meaningful, but were it false, maybe it wouldn’t be meaningful. If he hadn’t confused himself into such a ridiculous sentiment, why would he bother to drag out Fermat’s Last Theorem to pretend to be illustrating that? I think that he is a perfect poster child for the Wittgen-groupies, exhibiting the intellectual laziness that the author of the article upon which we started to comment here has turned into a well-tuned fetish.

        • June 1, 2013

          Blaggs

          peter -”And Mr. Blaggs is simply confused. He seems to be dopey enough to think that some true proposition could be meaningful, but were it false, maybe it wouldn’t be meaningful.”

          Why not read what I actually said, and try to present an argument against that, instead of the complete opposite of what I said. And while you’re at it, you might like to point out where I invoked Fermat’s Last Theorem. I think you’ll find I didn’t, although I’m not holding my breath.

           
        • June 2, 2013

          peter

          Sorry, M. Blaggs, you’re correct on one thing. By taking x, y, and z all 0, and n to be 1, what you say is trivially true, and I mistook it for ‘Fermat …’ on a quick reading.

          On the other hand, one would hardly say, as you did:”… It also happens to be true, though it would still be meaningful even if it were false…”, unless one thought it was a serious possibility that a statement, which was meaningful, could somehow have failed to be meaningful, had it been false (but it happens to be true). Wittgenstain was surely trying to get at something with at least some depth there, not that silliness. But he still ended up as an abject failure when it came to much in the way of contribution to human knowledge. This is becoming clearer as time goes on. I suspect the main reason that weak-minded philosophers pay any attention to him now is some form of intellectual envy that their sentimental attachment to whatever non-(logical/mathematical/scientific) stuff they wish to be truth is being passed by and ignored. And they, perhaps subconsciously, realize what laughing stocks they are becoming. Wittgenstein is a kind of last refuge for trying to sow confusion, and avoid as much as possible that realization of others of that irrelevance of their ‘thought’.

          Mr. Monk wrote an entertaining biography of Wittgenstein, spoiled only by his pathetic beliefs in the ultimate value of what this bright but unfortunately unstable fellow had to say. And the article we comment on here is merely some more dreary book sales propaganda, repeating a bit of what he tried to say in the book.

           
        • June 2, 2013

          Blaggs

          Ahh, the intellectual sloth awakens! Good of you to actually read my post, and stir yourself up to n=1; think you have the vigour to think about n=2? And of course you won’t need reminding we’ve still not reached FLT :)
          And if you really fancy shaking out those synaptic cobwebs, v.a.v. truth, falsity and meaning – think you can see a distinction between “it’s raining outside, now”, “2+2=4”, “the speed of light is constant for all observers”, “the sequence 7777 occurs in the expansion of Pi”, “2+2=5”? Take your time over these, and I’d suggest reading Rodych, Matthieu Marion et al, you might also even see just how out of date your suspicions about today’s intellectual community are.

           
        • June 3, 2013

          peter

          Rather than regurgitating names of people who are likely soon to be forgotten, all you needed to do was give me one example. That would be of a statement where it is even the slightest bit conceivable that its meaningfulness actually depends on its truthvalue. Maybe it is one of the 4 or 5 you rhymed off. Maybe it is something else. But unless you can come up with one, plus a sentence or more which has even the slightest tendency to convince one of it having that property, I think you can be ignored.

          But I’ll respond briefly concerning those two people.

          Firstly, find me even one person who has actually contributed to mathematics, and who has something positive to say about Rodych’s musings on Wittgenstein’s almost laughable misreading of Godel or even his later and embarrassing sillinesses about set theory. I am assuming it is the Rodych in the phil dept. at the University of Lethbridge to whom you refer. If he is ‘listening’, maybe he could find such an ally to satisfy my curiosity. There are some intensely embarrassing entries in the Stanford Encyclopedia of Philosophy, at least to anyone who goes beyond self-congratulatory backslapping in the modern North American philosophy scene. That our friend from Lethbridge has done the bit there for Ludwig’s philosophy of mathematics is at most a neutral virtue, and less for me. And I think readers here will get an eyefull if they go to that article and scroll down to the section on Wittgenstein’s lamented writings on incompleteness.

          As to Monsieur Marion, let me just quote one sentence from the Dialogue review of his book on the subject as above. This quote is seriously supposed to be positive praise I believe. But the wording here should make just about anyone laugh who hasn’t wasted her or his time in the coffee lounges of Canadian phil departments (Marion is also from there, at least in this case a university with something of a math dept) :”…. offers a fresh and illuminating way of approaching these questions of how to read Wittgenstein’s remarks on mathematics…”

           
        • June 3, 2013

          peter

          Forgot to mention, but your more recent blathering about me moving on to ‘n=2′, etc. probably needs to have it pointed out (in this forum at least) that you (perhaps inadvertently) stated your silly little mathematical statement with the existential quantifier applied to ‘n’, and not only to ‘x’,'y’ and ‘z’. So my breathtakingly trivial example with ‘n=1′ is quite sufficient, despite you being unaware of that fact.

          Perhaps going back to study a few elementary courses in math would be helpful before blathering on about the philosophy of math. I rather suspect that could apply to Mr. monk as well, and perhaps even to the two Canadian scholars above.

           
        • June 3, 2013

          Blaggs

          You do seem to want a nice quick fix presented to you on a plate for these issues. It would save you the bother of actually reading what people have had to say about it, and maybe even engaging in some thinking of your own, – neither of which you seem to particularly care for, let alone providing reasoned argument to back up your delirious outpourings. I think enough’s been said on here for people to make their own judgement as to where any intellectual laziness might lie.
          For those who are inclined to make the effort, hopefully the pointers I’ve given may be of some little help, and there’s plenty of other stuff out there. Stanford seems to me a good enough place to start at – I’m sure there are many people with the intellectual vigour to judge it for themselves rather than relying on the ill-founded rants you have provided for our entertainment.
          The only thing I would say is that this stuff does require some effort, but you may be surprised to hear that some people actually enjoy that.

          As for “n=”; understandably after your first response I did wonder whether you were either being incredibly lazy, in spite of your groundless evaluation of others as being so, or if you needed help with FLT, or some of both. Your second post didn’t really alter this. You are quite right, n=1 is trivial, even if we don’t restrict the other three to 0 – in a way that n=2 isn’t (I’m hoping you can figure out why for yourself). As such it quite reasonably occurred to me that you maybe hadn’t grasped that n=2 would be a better example, and that it doesn’t involve FLT neither. That, or you were still being lazy. I’m still not sure which.

           
        • June 3, 2013

          peter

          Yes, good, leaving it at that is fine for me, modulo suggesting:

          (1) to non-English speakers here not to emulate the worse-than-awkward “…it doesn’t involve FLT neither..” (replace the last word by ‘also’), and

          (2) to amuse yourselves deciding whether that construction in (1) would or wouldn’t be considered a double-negative by a competent logician.

           
        • June 3, 2013

          Blaggs

          Two good points. I should have followed my urge to consult my “English Usage” library before posting.
          It’s good too that I quickly discover a high regard for your analysis of English language, even if we disagree in other disciplines :)

           
  15. May 31, 2013

    Blaggs

    It’s only been in the last two or so decades that it’s been realised just what a sound grasp of the foundation of maths Wittgenstein had. Prior to that, in respect of Godel, because Wittgenstein did make a mistake in his reading of GIT, most commentators dismissed him as being “utterly incorrect”. It has now been shown that this amounted to throwing the baby out with the bathwater. Wittgenstein’s grasp of the philosophical foundation of maths, including the relevance of Godel, is thorough and sound. In particular he realised that in maths as well as in everyday language, it is too, too easy to construct seemingly well-formed propositions that seem to say something, but actually say nothing; they are nonsense. We can take a perfectly coherent English sentence; “there are four numbers, let’s label them x, y, z, n, such that x to the power of n, plus y to the power of n equals z to the power of n”. This is a meaningful, perfectly correct English sentence. It also happens to be true, though it would still be meaningful even if it were false. It’s easy to “translate” this into a mathematical formalism, but if we claim that it thus becomes a mathematical proposition, worse that it is true in a mathematical sense, then we have been seduced by language. (cf Wittgenstein, Philosophical Remarks §150)
    This is one of the many problems with Godel’s IT that Wittgenstein recognised.
    Victor Rodych has written some fine essays on his approach to Godel and to maths in general. Recommended.

    Math indeed has clarity, but only providing one has a firm grasp of its philosophical foundations. Without this, one runs a serious risk of creating (seemingly mathematical) nonsense; Godel being a good case in point.

    • November 22, 2013

      Paul Johnston

      Interesting to hear that at least some people still think Wittgenstein’s views on mathematics are worth thinking about. I lack the mathematical expertise to have a real view of my own, but while I can imagine that he might have misunderstood some aspects of Godel, I would be very surprised if what Wittgenstein had to say did not at least give some very serious food for thought. I also fear that you can be a mathematical genus and have very elementary or confused thoughts on the philosophy of mathematics. I think that is just life – there is no correlation (or even possibly a slightly negative one) between being brilliant at maths and being brilliant about understanding the nature of the activity and what it involves.

      • November 22, 2013

        peter

        It is true that many professional mathematicians have little time for the philosophy of mathematics, and it is also true that anyone who spends little effort thinking about the latter is not likely to have an opinion on it that is worth much.

        On the other hand, it is ironic that someone who ” …lack(s) the mathematical expertise to have a real view of (his) own” should pontificate on “…be(ing) a mathematical genus and hav(ing) very elementary or confused thoughts on the philosophy of mathematics…”, especially with no example of such a person to provide. Maybe it was me, but Paul was too polite to say; and anyway, assuming the word “genus” was a minor misprint, it most certainly wouldn’t apply to me.

        I must say however that an older but still living mathematician who I certainly would regard as a genius seems to have a fundamental attitude concerning the philosophy of mathematics which is very far from the strong platonism which I would advocate. Referring to someone else, people here might enjoy the web site of the MIT physicist/cosmologist Tegmark, and try to come up with something beyond the possible future empirical falsifications he writes about of his suggestion that the physical world simply IS (not merely APPROXIMATED BY) a mathematical system. Belief in that suggestion would leave one with a choice between mathematical platonism and some ridiculous form of solipsism.

        The problem is not knowledgeable mathematicians who lack a coherent and defendable philosophy of mathematics, but rather philosophers with almost no mathematical knowledge beyond one or two elementary undergraduate courses in ‘cookbook’ math or less, and yet who are quite full of opinions on what it really is that mathematicians are doing. I have no knowledge at all about what intelligent life on a different planet are doing, and so I don’t pontificate about it.

        And what minimal response I got earlier by way of examples which might revise upwards my opinions re Monk and Wittgenstein on these matters was truly unimpressive to say the least. Such examples referring rather to Frege or Russell or Hilbert or Godel would be quite easy to come by. But no one has asked; maybe no one here would think to ask.

        • November 23, 2013

          peter

          Minor clarification:
          The solipsism would come only to someone believing that the reality of the external world implies Tegmark’s “level IV multiverse” is correct. Disliking the latter, and its essential implication of existence for mathematical systems, then forces the solipsist’s rejection of the reality of the external world.

           
  16. September 15, 2013

    Christopher

    I think Ian Dury captured the sentiments in this string well with his song ‘there ain’t ‘alf been some clever bastards’. Sorry, I’ll get back to my Mailonline lol

  17. September 16, 2013

    B

    I’d like to offer a triviality of my own. There is a lot of philosophy that is not ‘non-theoretical’, in Wittgenstein’s sense.

  18. November 23, 2013

    Ray Kohn

    As a practising musician I have always found Wittgenstein a more stimulating thinker than many of the late 20th century ‘celebrities’. Language as a “game” should invoke in us the question of the game’s function (and not assume, lazily, that it can all be explained under the term “communication”). Perhaps Wittgenstein did not explore this as critically as he might have.. but then the work he did provides many of the analytical tools for us to use. The ‘scientism’ described in this article then takes an appropriate place within a far wider panorama of understanding.

  19. November 25, 2013

    Mathieu Marion

    Dear “Peter”, I’ll skip your insult, given that my last three papers on Wittgenstein’s philosophy of mathematics have been co-authored with a mathematician. I’ll just provide here the one example as you requested: R. L. Goodstein, ‘Function Theory in an Axiom-Free Equation Calculus’, Proceedings of the London Mathematical Society, vol. 48, 1945, 401-434, in a footnote on p. 407, attributes to Wittgenstein the idea of replacing mathematical induction by a rule of uniqueness of a function defined by primitive recursion. Goodstein proved later on that such a rule implies mathematical induction for primitive recursive arithmetic (in Recursive Arithmetic, 1957, Theorem 2.8 & 3.7-3.81), thus that one can do without mathematical induction for PRA. I could elaborate more on the sources of Goodstein’s equation calculus in Wittgenstein. Granted, Goodstein did the work, and he is not exactly faithful to Wittgenstein’s ideas as we have them in his manuscripts, but if this is not a contribution to foundations, I don’t know what is. It is difficult to see this as a contribution if one imagines it to be limited to Fregean or set-theoretic foundations, but from the point of view of category logic, the point is a valuable one, albeit of historical value. That philosophers imagine for themselves what mathematicians think is one thing, what mathematicians actually do is another matter. Category theory and category logic is certainly something done more in mathematics and departments than, say, Neo-Fregean foundations with a second-order principle of mathematical induction. As one of my teachers, incidentally a mathematician, used to say: Frege was relevant to mathematics in 1879, not 1979. (If anything, mathematicians do not care about foundations, but this is no grist to the mill of philosophers focussed on foundations.) This being said, I grant to you that Wittgenstein does not add much to foundations, that he might have misunderstood some bits (that can be discussed), and that there are inflated claims made on his behalf, but none of this justifies the bad reputation he has. It was never his intention to contribute to foundations per se, he never tried (so it is wrong to blame him for having made no contribution), but reflected on philosophical issues raised in that field, his reflections deserve at the very least a fair trial.

    • November 27, 2013

      peter

      Firstly I assume the “..insult..”, referred to nonspecifically, was not to do with the quote in my recent reply above to SDK (should be CB), but rather to earlier remarks related to Canadian ‘defenders’ of Wittgenstein’s thought. Not intended as an insult, and my apology if it was taken that way. Pardon the juxtaposition of my ridiculous to his sublime, but Richard Dawkins seems to run into a lot of that kind of rejoinder, rather than opponents taking up his criticisms themselves. In any case, I’d still be interested to know whether the author does not take a summary of his book as ”…. offers a fresh and illuminating way of approaching these questions of how to read Wittgenstein’s remarks on mathematics…” not as real praise, but more to be damning with faint praise (but perhaps damning mainly a tendency in recent philosophy more generally).

      Now to Professor Goodstein, who did a great deal of excellent work in Britain for math education and even for making math logic more prominent (and whom I even knew slightly long ago despite our differences in era and interests). I got hold of his paper and a review, and will comment on that. The later book to which Marion refers should show up soon, but I’ll only come back on that if it revises upwards my opinion on the topic and Wittgenstein’s supposed contribution.

      In summary, the footnote concerns a mathematical fact of exceptional triviality, and seems only to say that Wittgenstein and Bernays had also noticed this obvious fact. Marion implies this is just one example of many, but surely he would at least choose something of substance. One might react as the tea lady to Feynmann : “Surely you’re joking. M. Marion!” However. the footnote’s wording might have been unfortunate, and really meant that those two had an inkling of how this was a fact of significance for foundations. However there seems no evidence at all that any version of Goodstein’s attempted mixture of formalism with finitism has had any effect whatsoever, even some that it was possibly partly of dubious correctness. In particular, it had no effect in motivating or advancing a modern movement which Marion calls “category logic”.

      In more detail, the footnote which M. Marion refers to is on p.407, and is this:

      “This connection of induction with recursion has been previously observed by both Wittgenstein and Bernays.”

      There is not a single other mention of Wittgenstein in the 34-page paper. The acknowledgement at the end of the introduction thanks Bernays for his help, but mentions no one else. I had initially suspected that this 1945 paper was at least part of his 1946 doctorate under Wittgenstein (who according the math genealogy had only two Ph.D.’s that he supervised, the other one’s thesis having the bizarre-sounding title “Some Philosophical Considerations about the Survival of Death”). But maybe not—it’s really a 1941 paper held up till 1945 for obvious reasons of the times and place. This footnote credit to W. and B. could almost be regarded as insulting, if they were as sensitive as some (see below).

      Let me quote the sentence to which it is a footnote:

      “Inductive proofs of the equivalence of two functions f(x), g(x) proceed by establishing, first the equation f(0)=g(0), and then the implication ‘(f(x)=g(x)—>f(x+1)=g(x+1))’; the basis of this implication is the expression of f(x+1) as a function of f(x) and g(x+1) as the SAME function of g(x), and so proof by induction consists in showing that two function signs satisfy the same recursive introductory equations.”

      So we are dealing with the triviality that recursion (in an especially simple form here) determines uniquely what it defines, or perhaps also the triviality that it is by induction that one sees this. Just to be clear about how trivial this really is, here is how one could define the function 2x+3 of x: f(0)=3 and f(x+1)=f(x)+2. The fact that one sees that (one and) only one function is determined by the latter two equalities is surely no more than what we call a ‘deepity’ in any math course beyond about 2nd year secondary school. And that it is inductively that we formally verify it, is the same, but maybe for 16 year olds rather than 14 year olds.

      So for this not to be insulting to W. and B., I imagine he must have meant that somehow these two saw that such an observation leads to something more substantial.

      But it didn’t, IMHO. Marion refers to “category logic”. There has been recently a serious area which perhaps is what he means. In the past year or so there has been a concentration at IAS in Princeton in what I take to be a large component of what he means by that phrase. It included such luminaries as Per Martin-Lof, and indeed Andre Joyal with whom he is likely familiar. The group, somewhat unexpectedly, produced a book, entitled “Homotopy Type Theory–Univalent Foundations of Mathematics” (2013). It is free and easy to download, all 460 pages. Neither the index nor the references contains a single reference to either Goodstein or to Wittgenstein, and likely the book does not either. Those are 18 pages and 8 pages respectively. There are historical references, including to Godel for example. In the index, Lawvere seems to beat Goodstein by 8 to 0, and Wittgenstein by the same margin. If I am, in my lack of expertise here, missing something, please explain. But I can only conclude that the two missing names have contributed rather little to the subject called “category logic”.

      As for the suspicion related to “dubious correctness” above, see the review of Goodstein’s paper: Journal of Symbolic Logic, vol 11, March 1946, pp. 24-26, especially the last page.

      But the book, when I get it, may revise my opinion on this topic upwards, for Goodstein, and maybe even for Wittgenstein. If so, I’ll say so here, in a mea culpa. But perhaps a reply to this will persuasive in making me see my errors.

      And yes, to reinforce what your mathematical friend said, Gauss was relevant to mathematics in 1820, and Newton even earlier. But maybe the assertion of irrelevance in more recent times would be easy to dispute.

      I find it peculiar that Marion writes “It was never his intention to contribute to foundations per se, he never tried”. That contrasts surely with Stanford Encyclopedia entry on the man, which includes “It is now widely agreed that the writings of the period from 1946 until his death (1951) constitute a distinctive phase of Wittgenstein’s thought. These writings include… parts of The Foundations of Mathematics.”

      • November 27, 2013

        Blaggs

        Just to comment only on peter’s final paragraph.
        I think the Stanford comment on “parts of The Foundations of Mathematics.” refers to the bibliography where the full title is given – “Remarks on the Foundations of Mathematics”. The Stanford article perhaps needs editing. This first point is that the title is not “Contributions to […]“. It is consistent with a reading of his work that Wittgenstein had no intention to contribute to ‘the foundations’ but merely to comment on others’ conceptions of what those foundations might be.
        Secondly, as far as I know, Wittgenstein never published a book with this title; the book consists of writings spanning decades, but published posthumously within a book of that title, edited and translated by others.

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