
Turing and Wittgenstein on Logic and Mathematics [pdf] - tosh
https://www.britishwittgensteinsociety.org/wp-content/uploads/documents/lectures/Turing-and-Wittgenstein-on-Logic-and-Mathematics.pdf
======
voidhorse
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!

------
frereubu
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...](https://www.goodreads.com/book/show/12079.Ludwig_Wittgenstein)

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.

~~~
Emma_Goldman
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.

~~~
frereubu
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!)

------
odomojuli
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.

~~~
mettamage
I don't get it.

~~~
seanhunter
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.

------
ukj
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](https://doi.org/10.1017/S096012950100336X)

2\.
[https://en.wikipedia.org/wiki/Programming_language_theory](https://en.wikipedia.org/wiki/Programming_language_theory)

~~~
30f0fn
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.

~~~
ukj
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”.

------
random3
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](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...](https://www.goodreads.com/book/show/11797471-the-idea-factory)

------
codeulike
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."

~~~
kingkawn
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.

~~~
idoubtit
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.

~~~
kingkawn
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.

------
braindongle
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?

~~~
canjobear
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.

~~~
ukj
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](http://nuprl.org/Intro/intro.html)

2\.
[https://en.wikipedia.org/wiki/Systems_theory](https://en.wikipedia.org/wiki/Systems_theory)

~~~
QuesnayJr
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...](https://plato.stanford.edu/entries/type-theory-
intuitionistic/)

------
ofrzeta
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."

------
Kednicma
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.

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

------
chunsj
Too heavy for me, just curious.

Wittgenstein had read Godel, right?

~~~
canjobear
He rejected Gödel's arguments.

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

~~~
ProfHewitt
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](https://papers.ssrn.com/abstract=3603021)

------
ProfHewitt
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](https://papers.ssrn.com/abstract=3603021)

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

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

~~~
ProfHewitt
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](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](https://papers.ssrn.com/abstract=3603021)

