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Turing and Wittgenstein on Logic and Mathematics [pdf] (britishwittgensteinsociety.org)
95 points by tosh 22 days ago | hide | past | favorite | 53 comments



Like all great philosophers Wittgenstein had a knack for pointing out the nuances in the things we take for granted--I think this is what he's driving at with his emphasis on analysis over proof, and indeed, he was brave enough to claim there are certain things in life we can't say--things we can't master, things that may never be captured by our epistemological games--unknowables.

If nothing else, reading Wittgenstein always induces thinking, which is something I think it's easy to take for granted. Reading other philosophers or intellectuals, I think it's easy to agree or disagree with their propositions, Wittgenstein, at least for me, mastered the peculiar art of pushing his reader beyond agreement or disagreement into the realm of active thought--whenever I read his aphorisms, I don't find myself going "yes!" or "no!" as I do with some other writers, but rather being gently pushed into a reflective mode, into an active consideration.

Thank you for sharing!


This lecture was by Ray Monk, who wrote a fantastic and very accessible biography of Wittgenstein: https://www.goodreads.com/book/show/12079.Ludwig_Wittgenstei...

Wittgenstein's work can be pretty difficult to wrap your head around, but Monk does a great job of getting its main thrust while covering his life too.


Monk's biography is fantastic. But I have to disagree, it gives an intimate picture his life, relationships and general outlook, but only glancing snippets of his philosophy. If you are interested in his thought, you would do far better just reading Wittgenstein himself, or reading one of the many fantastic overviews of his work - I would recommend Hans Sluga's book, the Routledge introduction to the Philosophical Investigations, and the Cambridge Companion to Wittgenstein.


I think we're probably talking about different audiences. If you're an undergrad studying a philosophy minor then sure, go to the source. But if you've read a couple of Virginia Woolf novels and are interested in the Bloomsbury set, it gives you enough of a flavour. I studied both early and late Wittengstein at university and I certainly wouldn't recommend the Tractatus to anyone except as a novelty, and although the Philsophical Investigations are more approachable, they quickly become tiring for someone not motivated to invest some time in reading them slowly and thoroughly. (I know this from the time when I was reading the Investigations and badgered other people to read them because I thought they were so fantastic!)


+1 on the Monk biography. I also really enjoyed Andrew Hodges' biography of Turing (Enigma). Even if you didn't like the movie that was based on it (I didn't), the biography itself is excellent :-)


thanks for the recommendation!


From the Cambridge seminars:

    WITTGENSTEIN: I won’t say anything which anyone can dispute. Or if anyone does dispute it, I will let that point drop and pass on to say something else.

    TURING: I understand but I don’t agree that it is simply a question of giving new meanings to words.

    WITTGENSTEIN: Turing doesn’t object to anything I say. He agrees with every word.

    TURING: I see your point.

    WITTGENSTEIN: I don’t have a point.


These lines have become a bit of a meme in philosophy circles, but, for the record, while all of these lines are attested to in Cora Diamond's edition of "Wittgenstein's Lectures on the Foundations of Mathematics," Wittgenstein and Turing did not have a back-and-forth conversation like this.

https://www.google.com/books/edition/_/d4YUZVq1JSEC?hl=en&gb...

The first line, "I won't say anything which anyone can dispute" is from Lecture 2.

The next two lines ("I understand but I don't agree" / "Turing doesn't object") are from Lecture 6.

The "I see your point" / "I have no point" was a remark amid a deeper conversation during Lecture 10.

Juxtaposing the lines like this makes Wittgenstein seem comically insane (which is the joke, and I do get it).


The joke is that Wittgenstein's statements are the axioms/rules for the language game he intends to play.

And then he proceeds to play the language game he said he is going to play given his interpretation of the rules he set forth.

It's the rule-following paradox in practice.

  "This was our paradox: no course of action could be determined by a rule, because any course of action can be made out to accord with the rule"
He's taking the piss of axiomatic/formal reasoning which is the epitome of pointless rule-following.


also noted in https://existentialcomics.com/comic/321

I find the unreasonable effectiveness of formal systems to be this: just as the shaman crosses into the "spirit world" and uses their experience there to predict happenings in our world, we can turn statements about bridges and dynamic loads in our world into formal statements, arrangements of symbols, and manipulate them mindlessly according to a formal system, yet the resulting safety margins do indeed predict happenings in our world.

(A city once gave an engineering school the contract to demolish an old bridge. The date was agreed upon, but that afternoon the city had to sent representatives out at lunchtime to insist that even though it hadn't been formally specified in the contract, their intent —and the neighbour's expectations— had always been that the bridge would be blown up all at once, not that little bits be blown off all morning to see how much structure could be removed before collapse...)


This reminds me of the Socratic method that Socrates would use to refute someone's claims. He would first make them agree with a series of statements ("truths") one after the other and then run into a contradictory result thereby showing them how their original thesis was wrong.

The Wikipedia article [0] does a better job of explaning it than I could.

[0] https://en.wikipedia.org/wiki/Socratic_method


This is what I do. Is that normal?


You've noticed that other people's narratives/beliefs are internally inconsistent.

Perhaps you haven't noticed that this technique can be applied to your own narratives/beliefs? Because your beliefs are inconsistent too.

Everybody's beliefs are inconsistent - it's a systemic issue. We know about it. The root cause is the fact that language is recursive and it succumbs to Russel's paradox/liar's paradox.

Philosophers have been using inconsistency as a crutch for guilting people into changing their minds for... ever.

  Do I contradict myself? Very well, then I contradict myself, I am large, I contain multitudes. --Walt Whitman


Who knows? But, it is Socratic.


All I'm getting is pops and buzzes.


I don't get it.


It didn't really happen so there's nothing to get. Those are seperate quotes taken from different lectures and strung together as though they are a dialogue with a sort of Zen character. See one of the sibling comments by dfaubulich for the actual origin of each of those lines.


Don't bother, it's just the usual nonsense by Wittgenstein, this time related to the rule following discussion. Wittgenstein has not published anything of substance and is the most persistently overrated philosopher of 20th Century. Thanks to his obscurity and an ability to invent interesting puzzles, which he scribbled on little notes, he has gathered a large number of fanatic followers. Ironically, if Wittgenstein was still alive, he'd make relentless fun of most of these followers.

He was good at pointing out problems. The rest of his work consisted of hand-waving.


like the zen teacher W is


The crux of Wittgenstein can be summed up thus (and I am totally stealing this from Jean-Yves Girard).

It's not about the rules of logic, it's about the logic of rules [1]

The axioms and deductive rules of ALL formal languages are invented/designed by humans, not discovered, so it begs the question: Who invents the rules and why?

Logic (when seen as a formal language itself) is a subset of Programming Language Theory [2]

1. https://doi.org/10.1017/S096012950100336X

2. https://en.wikipedia.org/wiki/Programming_language_theory


The remark from Girard may not be so far from some things Wittgenstein said at this or that point. But it's not clear that Girard's remark means anything like that "logic is invented/designed by humans". For example, it might also mean something like what Wittgenstein wrote in the Tractatus:

We have said that some things are arbitrary in the symbols that we use and that some things are not. In logic it is only the latter that express: but that means that logic is not a field in which we express what we wish with the help of signs, but rather one in which the nature of the absolutely necessary signs speaks for itself.


If you trawl Girard’s papers you will find him saying at least these two things:

Logic is subjective.

Logic is implicit - will never be explicit.

“The symbol speaks for itself” is the notion of denotational semantics mathematicians use. I am in the camp of “symbols mean whatever you interpret them to mean”.


> Who invents the rules

Anywho is:

they could be someone with states of mind, bending over endless amount of paper, paging, writing and deleting symbols on them with their pencil for some time (or forever)

Otherwise how could anyone invent rules?

It would be so great to get out some general&universal results starting from this setup... ;)


"Computer" used to be a job description before it acquired its modern meaning...

https://en.wikipedia.org/wiki/Computer_(job_description)


To whom is interested in history - "A Mind at Play: How Claude Shannon Invented the Information Age" - https://www.goodreads.com/book/show/32919530-a-mind-at-play is an excellent book. A good companion to this is The Idea Factory: Bell Labs and the Great Age of American Innovation https://www.goodreads.com/book/show/11797471-the-idea-factor...


Something that sticks in my head about Wittgenstein (someone correct me if I'm getting this wrong) was that during World War 2, Wittgenstein felt that being a professor of philosophy at Cambridge was 'intolerable' and went to work as a hospital porter instead, incognito.

Also, favourite quote:

"A philosopher who is not taking part in discussions is like a boxer who never goes into the ring."


He also volunteered for military service in world war 1, and when he reached the front line volunteered for the extremely dangerous assignment of crawling into no-man’s land alone to make observations of the enemy. He said it seemed absurd to be a volunteer among draftees and not take on the most dangerous missions.


His World War I journal portrays him has an very egoistical man, with a violent contempt to everyone, including his comrades. He wrote: "It is almost impossible to find a trace of humanity in them [the men of his unit]" and, about marines, "I often cannot discern the human being in a man". As an extremely wealthy and high-class officer, he despised the low ranked soldiers ("stupid", "malicious", "pigs"...), but also the other officers ("pigs", "utterly limited"). He volunteered for the army of Austro-Hungarian empire and expected others to follow; he failed to understand that the e.g. Serbian soldiers whose people was oppressed by the empire did not share his views.

I don't care about his personal life, but since this thread was turning into an hagiography, I just wanted to draw a more complex portray.


He was a self-critical upper class twit who continued to evolve and transform his view of the world and himself over the course of his entire life.


Maybe someone here can explain: I've come across more than a few philosophers, of the PhD/academic flavor, who are dismissive of Wittgenstein's work. I have a gist-level understanding of his work, and a hobbyist's knowledge of the history of Western thought up through, say, Foucault.

Am I seeing a biased sample or is LW out of fashion these days? If so, why?


Here's my take on it. Academic philosophy in the US is highly focused on making completely clear claims with a rigor approaching that of mathematical logic. It is more or less pursuing the program Wittgenstein sketched in his first book, the Tractatus, creating a collection of concepts and network of relationships among them in which apparent philosophical paradoxes vanish. This is the analytical tradition. It is nice because it is indeed rigorous, but it can be limiting because it severely constrains the topics you can talk about. Something as big and multifarious as, say, Heidegger's notion of Dasein does not fit into this mold. And certainly nothing written as poetically as Nietzsche would pass muster.

In his second book, Philosophical Investigations, Wittgenstein completely rejected this approach, and his earlier work. He claimed that....well, no one's really sure what he claimed, or that he really claimed anything, and that's exactly the problem academic philosophers have with him. To a first approximation, he claimed that the whole idea of language as a formal system was either wrong or a waste of time, and that language is better thought of as some kind of game.

The thinking, then, is that later Wittgenstein was not making a clear point, was not interested in making a clear point, and possibly was not even serious at all. Philosophical Investigations is an enigma, and modern academic philosophy doesn't deal in such things.


I've never felt enticed to read a book by any philosopher, but that kind of made curious!

Especially given the comment about language could be seen as some kind of game, about which I'd say he's at least then partially been shown to be right? Being understood is very much a game, as you without anticipating your counterparts expectations and knowledge are often hopefully lost. Though I don't know if that's even close to what he was referring to, so yes, a but curious.


The idea of a "language game" was merely an analogy. It wasn't meant to trivialize the subject. The point is that language is an activity that operates according to conventional rules. (This also relates to his argument against "private language", though I'm personally not as convinced by that.) If you were to give a short summary of Wittgenstein's philosophy, I think it's better to say that he claimed meaning is use. "For a large class of cases—though not for all—in which we employ the word "meaning" it can be defined thus: the meaning of a word is its use in the language." This is in opposition to traditional analytic philosophy which holds that the meaning is specified by external references and truth conditions.


Somebody should tell Philosophers about the Curry-Howard-Lambek isomorphism.

Because that's all there is to the Mathematical notion of "rigorous proof".

And the 'next step' in scaling up this process is the mission undertaken by the NuPRL project [1] well on our road towards internalising systems theory as the mode of scientific discourse [2]:

  Starting with the slogan "proofs-as-programs," we now talk about "theories-as-systems."
1. http://nuprl.org/Intro/intro.html

2. https://en.wikipedia.org/wiki/Systems_theory


What makes you think that philosophers haven't heard of it? Here's the Stanford Encyclopedia of Philosophy entry on it: https://plato.stanford.edu/entries/type-theory-intuitionisti...


It's not surprising, because Wittgenstein himself was in a sense dismissive of academic philosophy. He turned against his former self, of the Tractatus, which was highly regarded among so-called "analytic" philosophers. The controversy began almost immediately: Bertrand Russell, who had previously been Wittgenstein's biggest supporter, had nothing but scorn for the Philosophical Investigations.

Wittgenstein was a kind of anti-philosopher, as was Richard Rorty. Their goal, as I see it, was not to "solve" traditional philosophical problems as such, but rather to dissolve them. They believed that many philosophical problems were misunderstandings, projections of our human languages onto reality, false anthropomorphism. Rorty also started in academic philosophy and left the field later in his career, while generating similar controversy. If they're right, then it's unclear whether philosophy as it had traditionally been practiced has a proper place in society. Academic philosophers deem their own projects to be "foundational", but Wittgenstein was a threat to that way of thinking.


1. There are no foundations - it's turtles all the way down.

2. Recursion is the foundation.

3. This is a true contradiction.


You can't be a philosopher and posit "most philosophy is non-sense" at the same time. If language is the product of ephemeral transactions between subjective people on a need to communicate basis, then does it possess the objective rigor needed to accurately investigate the true essence of the world? If philosophy can only be expressed with language, where does that leave you? Setting aside Wittgenstein while you work on your philosophy may do the trick.

I've been listening to a lot of George Carlin recently. I feel much of his late best work is linguistic absurdism. If we are to take Carlin's assertion that most of us are dumb and society is glued together by bullshit, then so would be our language. If that is the case, then Carlin's stand up is the perfect example of taking language for what it truly is and applying objective rigor and logic to it (which he explicitly claims to be doing in many of his routines), only to reveal the true essence of the world. If anything, Carlin proved that reality is just as absurd as the language we use to describe it.


I'm not up on current PhD philosopher's tastes, and I'm biased since I'm a major fan of Wittgenstein, but, assuming you don't have a biased sample, I'd say this phenomenon is explainable through some of the Frankfurt School's theorizing on specialization, the professionalism of philosophy, and the general bureaucratic turn of society.

Personally, I like to view philosophy and academic philosophy as two distinct things. The former is represented best by the famous philosophers, many of whom would never make it past a peer review, e.g. Wittgenstein himself, Kierkegaard, Schopenhauer, Heidegger, Nietzsche, Deleuze, Guattari, Bataille, Adorno, even Foucault to an extent, etc.--I think the reason for this is that all of these famous, epoch shaping philosophers have a certain mysticism or poeticism about them--their work doesn't really hold up to the standards of academies, since these are institutions with very specific mechanisms and rules. All the great philosophers have a certain creativity and ingenuity to them that defies the confines of convention and reason.

Wittgenstein is in many respects a key player in shaping our modern though on logic, yet he was also an undeniable mystic who went so far as to say certain things escape representation and codification as formalized knowledge altogether, which is not amenable to the motives of academies--producing formalized knowledge that they can sell. I say this, again, as someone who entirely lacks context but who can imagine certain structural tendencies that would disfavor a philosopher like Wittgenstein.

Academic philosophy is, in my opinion, quite a different beast that's focused on solving very specialized and particular abstract problems deemed to be foundational, meta, or novel enough to escape analysis in the fields of application they'd otherwise belong to (the foundations of mathematics, of great concern to Wittgenstein and Turning as this post illustrates, is a good example of such a topic--it's too meta to be the concern of mathematics proper, too narrow to be the concern of a philosopher in the classical or "true" sense, who is supposed to concern herself with the broad problems of existence (like Wittgenstein points out, the living (assuming to philosophize still means to theorize on what it means to live a good life don't really need to concern themselves with such issues), so it falls to specialized academics).


It sounds to me that what you are labelling as 'Academic philosophy' (solving abstract problems) is very much the general process of meta-linguistic abstraction.

https://en.wikipedia.org/wiki/Metalinguistic_abstraction


Obviously just one anecdote, but my brother has a PhD in philosophy from Oxford and now teaches philosophy (albeit at a small college, not one of the major ones). Wittgenstein is one of his three big philosophy heroes (the others being Thomas Aquinas and a medieval Islamic scholar that I can't remember the name of).

Academic fashions are weird though.


Seeing that facsimile of "Mind" reminds me of Wittgenstein's quotes:

"If I read your mags I often wonder how anyone can read 'Mind' with all its impotence & bankruptcy when they could read Street & Smyth mags. Well everyone to his taste."

"How people can read Mind if they could read Street & Smith beats me. If philosophy has anything to do with wisdom there's certainly not a grain of that in Mind & quite often a grain in the detective stories."


I wonder whether Wittgenstein would be satisfied with how things have turned out. We now know that what he considered "useless language-games" are in fact the building blocks of the universe, or as close as we're going to get to building blocks while still being human. We also know that every logic is either formal or trivially useless. Formalism ended up being the only path forward.

It's fun to see part of the origins of what we'd now call constructive logic in the question of whether a bridge might fall down if built from double-negation and classical logic. The idea that the bridge proves its sturdiness by holding itself up under its own weight is exactly the same sort of constructive-deconstructive idea.

I can't help but compare Wittgenstein to Confucius in their insistence on the meanings of words and their usage. Confucius wanted people to play fewer language-games and use simpler language because it was part of his legalist philosophy; by being direct and plain, a government could be more transparent and its people could find more harmony in their social interactions. But Wittgenstein was concerned with human ability to perceive truth, logic, and abstracta, and his desire for simpler language was so that people could see the world as it really is, with a simpler map for a more intuitive territory.


Wittgenstein tried to describe an (emotional + logical) universe with logical rules. That stuff don't work!


Too heavy for me, just curious.

Wittgenstein had read Godel, right?



He rejected Gödel's arguments.


I would say he rejected the Gödel’s interpretation of his theorem but not the theorem itself.


Wittgenstein wrote about the incorrectness of Godel's famous 1931 argument for incompleteness of Russell's Principia Mathematica because Wittgenstein showed that the [Godel 1931] proposition I'mUnprovable leads to inconsistency in mathematics.

For more information, please see the following: https://papers.ssrn.com/abstract=3603021


I would say each of them had a different implementations of the eval() function.


Wittgenstein was one of the first to realize incorrectness of Gödel's famous 1931 article on the inferential undecidability (incompleteness) of Russell's system Principia Mathematica. For more information please see the following: https://papers.ssrn.com/abstract=3603021

Professionals (including Gödel) ridiculed Wittgenstein's work on logic and ignored his correct important argument against [Gödel 1931].


Can you elaborate a little more on how Wittgenstein showed the incorrectness of incompleteness theorem?


Wittgenstein was one of the first to argue that adding the [Gödel 1931] proposition I’mUnprovable results in contradiction by providing the following correct but somewhat convoluted argument [Wittgenstein 1937-1944] which has been explained by adding explanations of steps in brackets:

    “Let us suppose [Gödel 1931 was correct and therefore] I prove the unprovability (in Russell’s system) of [Gödel’s proposition] P [that is, ⊢⊬P where P⇔⊬P]; then by this proof I have proved P [⊢P because P⇔⊬P]. Now if this proof were one in Russell’s system [⊢⊢P] — I should in this case have proved at once that it belonged [⊢P] and did not belong [⊢¬P because ¬P⇔⊢P] to Russell’s system. But there is a contradiction here! [⊢P as well as ⊢¬P] 
  …[This] is what comes of making up such propositions.”
However, Wittgenstein did not point out that the [Gödel 1931] proposition I’mUnprovable is not allowed by the rules of Russell’s system because of restrictions on orders of propositions.

Nothing of practical importance depends on the existence of Gödel’s proposition I’mUnprovable. As discussed in https://papers.ssrn.com/abstract=3603021 , the important property of inferential undecidability (incompleteness) of Russell’s system can be proved in a different way without using I’mUnprovable. Furthermore, having Gödel's monster proposition I’mUnprovable comes at the heavy cost of introducing another monster, namely, “A powerful theory cannot prove its own consistency” as discussed here: https://papers.ssrn.com/abstract=3603021




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