Turing and Wittgenstein on Logic and Mathematics [pdf]
britishwittgensteinsociety.org
britishwittgensteinsociety.org
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!
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.
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.The Wikipedia article [0] does a better job of explaning it than I could.
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 Whitmanhttps://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).
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"
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...)
He was good at pointing out problems. The rest of his work consisted of hand-waving.
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
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... ;)
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.
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”.
Also, favourite quote:
"A philosopher who is not taking part in discussions is like a boxer who never goes into the ring."
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.
Am I seeing a biased sample or is LW out of fashion these days? If so, why?
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).
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.
2. Recursion is the foundation.
3. This is a true contradiction.
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.
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.
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."
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.
Academic fashions are weird though.
"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."
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 had read Godel, right?
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].
“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.”
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