Stephan Wolfram: 100 Years Since Principia Mathematica

http://blog.stephenwolfram.com/2010/11/100-years-since-principia-mathematica/

57 37 16 years ago

blog.stephenwolfram.com

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.

slackenerny• 16 years ago

Amusing fact is it didn't take long at all, compared to some later approaches ;) http://mathoverflow.net/questions/14356/bourbakis-epsilon-ca...

And a twist you might find worthy a mention in your forthcoming essay, http://mathoverflow.net/questions/40920/what-if-current-foun...

mchouza• 16 years ago

An interesting project that is developing an updated version of this idea is Metamath: http://us.metamath.org/index.html

pbhjpbhj2• 16 years ago

That is interesting - I think the idea that looms over the whole essay is that we have computers that do logic and can fit those logical operations together, we should be able to give a computer some basic axioms and see what theorems it can construct. Et voila!, there it is. Mention of such a project would have fit the essay well.

zackattack4• 16 years ago

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!

xcthulhu3• 16 years ago

Check out the axioms of Peano Arithmetic: http://en.wikipedia.org/wiki/Peano_axioms#Arithmetic

If we use Dedekind's recursive definition of addition, and the definitions 1=S(0) and 2=S(S(0)), then:

S(0) + S(0) = S(S(0)+0) = S(S(0))

I know the computer proof assistant Isabelle/HOL, inspired by the Principia Mathematica, can prove simple theorems about arithmetic using a "rewrite" system similar to what I've done above.

jodrellblank• 16 years ago

Can you rephrase that question without using the word "why"?

I have no idea what "why does 1+1=2?" is trying to find out.

RiderOfGiraffes2• 16 years ago

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?

ionfish7• 16 years ago

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

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

pbhjpbhj2• 16 years ago

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

oiuyhgfthyujik• 16 years ago

So does -1 + 1 = 0 here?

Is there some book that proves this?

ez772• 16 years ago

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 [...]

joelburget4• 16 years ago

Volume 2 of Principia Mathematica, not Principles of Mathematics.

ez772• 16 years ago

Sharp reading... sorry!

joelburget4• 16 years ago

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.)"

ionfish7• 16 years ago

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

joelburget4• 16 years ago

Thanks for a better explanation than I could have written myself.

I also like this page as a more basic introduction to type systems: http://blogs.perl.org/users/ovid/2010/08/what-to-know-before...

xcthulhu3• 16 years ago

>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...

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.

weavejester2• 16 years ago

I'm impressed it took Wolfram until the third paragraph to mention A New Kind of Science.

ionfish7• 16 years ago

Whenever anyone mentions NKS I'm always reminded of Cosma Shalizi's brilliant and hilarious review, 'A Rare Blend of Monster Raving Egomania and Utter Batshit Insanity'.

http://www.cscs.umich.edu/~crshalizi/reviews/wolfram/

zackattack4• 16 years ago

From the review: "[Wolfram] has talent, and once had some promise; he has squandered them".

The creation of Mathematica counts as squandering?

zackattack4• 16 years ago

I also thought it was kind of a dick move to wait until the end to admit that he didn't understand the symbols in principia_1b1.jpg. This is actually a pet peeve of mine in mathematics writing in general, introducing ambiguity and not resolving it until several pages or paragraphs later. It's really alienating. =(

ionfish7• 16 years ago

Clearly he should have read the SEP entry on this very subject, 'The Notation in Principia Mathematica'.

http://plato.stanford.edu/entries/pm-notation/

ced2• 16 years ago

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.

wwortiz• 16 years ago

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.

Kototama• 16 years ago

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/

jackfoxy• 16 years ago

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.

Stephan Wolfram: 100 Years Since Principia Mathematica

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