Stephan Wolfram: 100 Years Since Principia Mathematica(blog.stephenwolfram.com)
blog.stephenwolfram.com
Stephan Wolfram: 100 Years Since Principia Mathematica
http://blog.stephenwolfram.com/2010/11/100-years-since-principia-mathematica/
7 comments
I'm impressed it took Wolfram until the third paragraph to mention A New Kind of Science.
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.
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.
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!
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!
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.
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.
I, for one, feel its news when a raving egomaniac and lawsuit happy control freak like Wolfram doesn't sound like one. Indeed, I probably would not have read the article if someone hadn't pointed out he wasn't being an egomaniac.
Your analogy works, except its Wolfram who does the bullying, not the other way around.
Your analogy works, except its Wolfram who does the bullying, not the other way around.
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/
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?
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.)
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.)
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. =(
Clearly he should have read the SEP entry on this very subject, 'The Notation in Principia Mathematica'.
http://plato.stanford.edu/entries/pm-notation/
http://plato.stanford.edu/entries/pm-notation/
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.)"
"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.)"
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
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
>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.
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.
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...
[3] http://www.springerlink.com/content/v078l3n714589036/
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...
[3] http://www.springerlink.com/content/v078l3n714589036/
> 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
Try gigapedia.com ;-D
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...
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/
http://www.cis.upenn.edu/~bcpierce/tapl/
Agreed.
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.
I'll try to write about it later when I get a bit more time.
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...
And a twist you might find worthy a mention in your forthcoming essay, http://mathoverflow.net/questions/40920/what-if-current-foun...
An interesting project that is developing an updated version of this idea is Metamath: http://us.metamath.org/index.html
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.
Well, the problem is that first-order theories have an infinite number of theorems, so while one can prove as many theorems using a computer as one likes, there is still nothing to say whether they are of any interest or not. Most results are trivial, and in order to find the interesting ones we would have to go digging through a heap mainly composed of uninteresting ones.
This presumes, of course, that one can tell interesting from uninteresting by examining result statements. I am not certain that this is in fact the case. Our appreciation of the importance of a result stems from our ability to connect it to others; for example, König's lemma allows us to prove completeness and compactness (amongst many other things). If we simply saw it stated in symbolic form, amongst millions of others, would we recognise its importance?
This presumes, of course, that one can tell interesting from uninteresting by examining result statements. I am not certain that this is in fact the case. Our appreciation of the importance of a result stems from our ability to connect it to others; for example, König's lemma allows us to prove completeness and compactness (amongst many other things). If we simply saw it stated in symbolic form, amongst millions of others, would we recognise its importance?
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!
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!
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.
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.
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?
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?
>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.
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.
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.
I have no idea what "why does 1+1=2?" is trying to find out.
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/
The story is well-written, it's fascinating. Here the link: http://www.logicomix.com/
There's a Principia Mathematica anniversary symposium at Trinity College, Cambridge this weekend.
http://www.srcf.ucam.org/principia/
http://www.srcf.ucam.org/principia/
Sorry I miss clicked and downvoted you instead of upvoting; the interface prevents me from rectifying this error.
So does -1 + 1 = 0 here?
Is there some book that proves this?
Is there some book that proves this?
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.
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.
First:
In 1903, he published The Principles of Mathematics: Volume 1 (no volume 2 was ever published) [...]
[...] it did not hurt the whole impression that it took until more than 80 pages into volume 2 [...]Volume 2 of Principia Mathematica, not Principles of Mathematics.
Sharp reading... sorry!