

Stephan Wolfram: 100 Years Since Principia Mathematica - RiderOfGiraffes
http://blog.stephenwolfram.com/2010/11/100-years-since-principia-mathematica/

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weavejester
I'm impressed it took Wolfram until the third paragraph to mention _A New Kind
of Science_.

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ced
The post by Wolfram is a great discussion of logic and the philosophy of
mathematics throughout the 20th century. Yet the top comment on HN is a snark,
tribal bashing of the author.

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wwortiz
These snarks appear because of Wolfram's writing style, he comes off as a kind
of Steve Jobs who he himself is the product he is selling. Though at least
this one is lighter than usual.

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ced
_Though at least this one is lighter than usual._

That's the whole point. If you read it without thinking "Wolfram wrote this
_therefore_ it will be self-centered", then it sounds _perfectly fine_. The
GP's snark was merely appealing to people's confirmation bias. It's a high-
level version of that high-school classic: "I'll pick on that kid to show I'm
part of the group".

Besides, it's depressingly ironic that a community producing a deluge of "How
my startup achieved X in Y months" complains about this post being _self-
centered_.

Read it! It's a good essay!

~~~
weavejester
I don't particularly object to Wolfram being self-centered. Nor is there
anything wrong with advertising one's own achievements.

But Wolfram seems incapable of talking about _any_ subject without shoehorning
in some reference to his own works. Even when expected, it's a jarring,
unnecessary interruption; a wild swing from a reasonably interesting article
on mathematical history to unabashed self-promotion.

The snark wasn't intended as an offering to Hacker New group-think, but rather
borne out of a genuine frustration at Wolfram's style of writing. I find it
very difficult to read anything he says, because I'm continually awaiting that
moment when he will pause, smile cheesily at the proverbial camera, and
explain how amazing his products are.

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joelburget
It's interesting that Wolfram makes a quick foray into computer science to
bash type systems. (I think he couldn't be more wrong)

"To resolve this, Russell introduced what is often viewed as his most original
contribution to mathematical logic: his theory of types—which in essence tries
to distinguish between sets, sets of sets, etc. by considering them to be of
different “types”, and then restricts how they can be combined. I must say
that I consider types to be something of a hack. And indeed I have always felt
that the related idea of “data types” has very much served to hold up the
long-term development of programming languages. (Mathematica, for example,
gets great flexibility precisely from avoiding the use of types.)"

~~~
ionfish
It is a strange comment. The type theories used in programming languages,
certainly in those based on the λ-cube, descend from simple type theory (via
the simply-typed λ-calculus), which Russell discarded in favour of his
ramified theory of types. I also wonder why Wolfram, originally a
mathematician, doesn't even mention the Curry-Howard correspondence, which
seems to me a fairly important result linking mathematics and computation.

No one uses ramified type theory these days, at least not that I am aware,
although Russell's predicativism lives on in e.g. Feferman's programme [1]
(there are some interesting results in reverse mathematics relating to this;
see Simpson's book [2]).

Anyone missing the background I and the parent comment allude to might want to
take a look at Thierry Coquand's article on type theory in the SEP [3].
Coquand is the originator of many ideas in this area, including the calculus
of constructions [4] and the proof assistant Coq based on it.

[1] <http://math.stanford.edu/~feferman/papers/predicativity.pdf>

[2] <http://www.math.psu.edu/simpson/sosoa/>

[3] <http://plato.stanford.edu/entries/type-theory/>

[4] <http://en.wikipedia.org/wiki/Calculus_of_constructions>

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xcthulhu
>No one uses ramified type theory these days, at least not that I am aware

In the 1920s Frank Ramsey proved that the theory of ramified types + "The
Axiom of Reducibility" is equivalent to the theory of simple types:
[http://en.wikipedia.org/wiki/Type_theory#Simple_theory_of_ty...](http://en.wikipedia.org/wiki/Type_theory#Simple_theory_of_types)

The history I've heard is that ramified types were abandoned after this, since
simple type theory is easier and has the same expressive power.

~~~
ionfish
That's certainly always the way I've heard it, although I did come across an
interesting lacuna when I was reading the the SEP article earlier, which is
that the effect of the axiom of reducibility was first noticed by Polish
logician Leon Chwistek [1]. His article 'The Theory of Constructive Types' was
published in 1924, while Ramsey's paper dates from 1926.

José Ferreirós in _The Princeton Companion to Mathematics_ mentions his name
as well, albeit without much detail. Chwistek seems a fascinating character:
like his contemporary Witkacy, he was an artist as well as a logician, and
according to the biography of Alfred Tarski by the Fefermans [2], was
appointed to a professorship at Lvov in 1930 which Tarski was also in the
running for; apparently a letter of recommendation from Russell was the
decisive factor (see p. 67 of the aforementioned).

Bernard Linsky seems to have written a chapter on Chwistek and type theory in
_The Golden Age of Polish Philosophy_. You can read the first page [3] but I
haven't been able to find the entire thing online.

[1] <http://en.wikipedia.org/wiki/Leon_Chwistek>

[2]
[http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=052...](http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521802407)

[3] <http://www.springerlink.com/content/v078l3n714589036/>

~~~
xcthulhu
> Bernard Linsky seems to have written a chapter on Chwistek and type theory
> in The Golden Age of Polish Philosophy. You can read the first page [3] but
> I haven't been able to find the entire thing online.

Try gigapedia.com ;-D

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RiderOfGiraffes
In case you're interested, about halfway down is the usually quoted result
(*110.643) that 1+1=2. I'm often asked why it took so long (it's over 80 pages
into volume 2) to prove something so trivially, and obviously true, and
recently I've come up with an example that demonstrates the idea.

I'll try to write about it later when I get a bit more time.

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zackattack
Please do...

In my freshman year of college, I asked my real analysis professor why 1+1=2
and he failed to provide an edifying explanation. He did, however, commend me
for asking -- I think it earned me some brownie points, which I redeemed by
asking for extra clarification on more course-related topics later in the
quarter.

Anyway, it's always bothered me so if I can learn something about it then I
would love to!

~~~
RiderOfGiraffes
As a taster ...

What do you mean by "2"?

What do you mean by "1"?

What do you mean by "+"?

What do you mean by "="?

There's more than one way to get to the number 7. You can start at 0 and count
upwards, or you can "add" the numbers "3" and "4". Why should it be that you
end up in the same place?

Slightly more complex/general ...

Consider the number line, and divide the stretch between 0 and 1 into 9 equal
pieces. Start from 0 and move along two of these pieces. Call the place you
get to "T".

Now consider the stretch from 0 to 2, and divide that into 9 equal sized
pieces. Take just the first one, and call where that gets to "S".

Why are they the same point?

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zackattack
>Now consider the stretch from 0 to 2, and divide that into 9 equal sized
pieces. Take just the first one, and call where that gets to "S". Why are they
the same point?

This seems to me like it's simply rephrasing the question why does 1+1=2. Or,
at least, I can't answer it without invoking the field axioms, or perhaps only
the ring axioms. Please forgive me if my terminology is awkward.

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Kototama
There is a nice comic book about the life of Bertrand Russell: Logicomix: An
Epic Search for Truth

The story is well-written, it's fascinating. Here the link:
<http://www.logicomix.com/>

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ionfish
There's a Principia Mathematica anniversary symposium at Trinity College,
Cambridge this weekend.

<http://www.srcf.ucam.org/principia/>

~~~
pbhjpbhj
Sorry I miss clicked and downvoted you instead of upvoting; the interface
prevents me from rectifying this error.

~~~
oiuyhgfthyujik
So does -1 + 1 = 0 here?

Is there some book that proves this?

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jackfoxy
Wolfram ends the essay on a tantalizing note _...another level of
automation.... [I]nventing and developing [systems using other axiomatic
systems] to respond to some particular purpose._

I see this _almost_ happening daily on HN with the huge interest in new
languages especially functional languages. By _almost_ I mean these are mostly
descendants of either Lisp or ML and they are based on the same logic. I'm not
sure if you would gain anything by hard-wiring a language into one of the
other multitude of axiomatic systems. It would be an intellectual challenge to
first select an appropriate system, and then construct a useful language. And
it would be hard to recruit people to use it enough to provide useful
feedback.

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ez77
First:

    
    
      In 1903, he published The Principles of Mathematics: Volume 1 (no volume 2 was ever published) [...]
    

Later:

    
    
      [...] it did not hurt the whole impression that it took until more than 80 pages into volume 2 [...]

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joelburget
Volume 2 of Principia Mathematica, not Principles of Mathematics.

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ez77
Sharp reading... sorry!

