Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Stephan Wolfram: 100 Years Since Principia Mathematica

blog.stephenwolfram.com

Post reply on HN
Print view View original

31–38 of 38 posts

Re: Stephan Wolfram: 100 Years Since Principia Mathematica

#31

Post by jodrellblank »

zackattack wrote: post #14
RiderOfGiraffes wrote: post #2

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.

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!

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.

Re: Stephan Wolfram: 100 Years Since Principia Mathematica

#32

Post by ionfish »

joelburget wrote: post #24
ionfish wrote: post #12

Earlier quoted context omitted.

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…

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...

From a PL perspective, you can't beat Benjamin Pierce's Types and Programming Languages.

http://www.cis.upenn.edu/~bcpierce/tapl/

Re: Stephan Wolfram: 100 Years Since Principia Mathematica

#33

Post by xcthulhu »

ionfish wrote: post #29
xcthulhu wrote: post #27

Earlier quoted context omitted.

>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 pow…

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 m…

> 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

Re: Stephan Wolfram: 100 Years Since Principia Mathematica

#34

Post by RiderOfGiraffes »

zackattack wrote: post #14
RiderOfGiraffes wrote: post #2

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.

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!

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?

Re: Stephan Wolfram: 100 Years Since Principia Mathematica

#35

Post by joelburget »

ionfish wrote: post #32
joelburget wrote: post #24

Earlier quoted context omitted.

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...

From a PL perspective, you can't beat Benjamin Pierce's Types and Programming Languages . http://www.cis.upenn.edu/~bcpierce/tapl/

Agreed.

Re: Stephan Wolfram: 100 Years Since Principia Mathematica

#36

Post by zackattack »

RiderOfGiraffes wrote: post #34
zackattack wrote: post #14

Earlier quoted context omitted.

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!

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. St…

>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.

Re: Stephan Wolfram: 100 Years Since Principia Mathematica

#37

Post by zackattack »

ionfish wrote: post #15
weavejester wrote: post #11

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

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/

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

The creation of Mathematica counts as squandering?

Re: Stephan Wolfram: 100 Years Since Principia Mathematica

#38

Post by hga »

zackattack wrote: post #37
ionfish wrote: post #15

Earlier quoted context omitted.

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/

From the review: "[Wolfram] has talent, and once had some promise; he has squandered them". The creation of Mathematica counts as squandering?

For "someone who was once a respectable physicist", yes.

Especially to the extent he's putting resources into this instead of Mathematica.

(Note: the above is from the author's (and others) point of view; I've not read the book and have no first hand opinion on it or Wolfram, besides the law-suit happy thing and such, e.g. I don't use Wolfram Alpha because they own whatever is the output of it, I don't and am restricted in how I can use it.)

Post reply on HN