461 karma · joined July 15, 2009
I made a few contributions to Aleph as part of my PhD on doing transfer learning in ILP and really enjoyed working in Prolog.
Goldbach's conjecture is that "every even number bigger than 2 is the sum of exactly two prime numbers", so 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, etc.
For this statement to be *true* it just means that every even number there must exist two primes that add to that number. This is a statement about infinitely many integers.
A *proof* of Goldbach's conjecture consists of a finite number of formal reasoning steps that start with some axioms and end up at the statement of the result.
To this day, it seems as though Goldbach's conjecture is true. It holds for every number we've been able to test. However, no one has proved it is true or false yet (or proved that it is unprovable).
The proof of Gödel's result's involves very carefully formalizing what statements and proofs mean so that they can be encoded as statements about arithmetic. He then shows there is a statement with encoding G that says "The statement with encoding G cannot be proved" – if it is true, then it cannot be proved.
It's kind of confusing at first, but there is an easier way to get an intuition why there might be true statements that cannot be proved. Think of each statement about the natural numbers as a subset where each number in the subset makes the statement true. There are uncountably many subsets of the natural numbers (by Cantor's diagonalization argument). Proofs are finite chains of finite statements so there are only countably infinitely many of these. Therefore there must be subsets/statements that are true that do not have a matching proof.
The approach that Gödel's proof takes is not too different to the above argument – it is essentially a diagonalization argument – the complexity is in making the encoding of statements are numbers very precise.
It's a fascinating take on how those concepts relate to each other. You can find some of the papers that were synthesized into that book on his web page: https://sethares.engr.wisc.edu/pubs.html
Ok, I still think it’s an unusual example but you’ve convinced me.
I retract my earlier comment and replace it with a “today I learned” for me. :)
“Pour over” is the sense you describe would be a metaphor.
It’s a testament to DFW’s talents as a writer to get so much impact out of so few words but it’s an incredibly tough piece of writing to recommend. Especially if you have young children.
> Update 9.10.2015: exec.library base address is stored in > memory address $4. The library base address itself is not $4.
I learnt 68k assembler for the Amiga in high-school and made _exactly_ this mistake. After many "guru mediations" (the flashing red and black box) and much head-scratching I finally realized the problem.
It was a big breakthrough in my early days as a programmer and taught me a solid lesson about indirection and pointers.
She is an impressive and experimental writer (check out her "Visit from the Goon Squad") who does a great job with this form. The thought-sized chunks give the story a lot of space and cleverly invites the reader to imagine a lot of context.
[1]: http://en.wikipedia.org/wiki/Black_Box_(Jennifer_Egan_story)
I don't think anyone here is arguing the position that LSD will have a profound impact on anyone who takes it, just that it can give people a different way of looking at the world and sometimes that leads to novel and important insights.
Replace "LSD trip" with "university degree" in your example. Sure, there are many people who take a physics course and feel as though they have had profound insights but actually haven't. However, you can't use those examples to argue that everyone who claims their physics course has inspired their life and work is lying.
Secondly, no, John didn't say learning theory is "useful only in a crude way", he said, "learning theory is most useful in it’s crudest forms". Big difference. And besides, he says right at the beginning of the post that he believes "learning theory has much more substantial applications [than generating papers]".
To be convincing, theory needs to be precise – if you are not careful about what you are talking about it is easy to believe things that are not true. However, what I think John is saying is that value of theory comes from afterwards abstracting away the precision and understanding the main message (i.e., the "crude form") of a theoretical result. In general, I don't think it is possible to get to a convincing "main message" or "crude form" without someone having grappled with the details.
No matter how many experiments you run, you will only ever show that an algorithm works well or not in a typically small finite number of cases. What theory does is look those cases and ask something like, "It seems that every time X is true of my problem, an algorithm with property Y works really well. I wonder if that is always true?" This type of question gets carefully formalised and then (hopefully) answered. The process of formalisation (i.e., defining things carefully) can yield new ways of thinking about things (e.g., over fitting and bias-variance trade-offs), and having an answer to the general question means that you can be assured that future uses of your algorithm will behave as expected.
You seem to have an unshakeable opinion that mathematics/theory/"pencil-pushing" is inherently a waste of time. That's a real shame. Why do you believe the pursuit of precision, insight, and proof are somehow inferior to running experiments? I find both to be valuable and the interplay between the two extremely rewarding.
It's easy to dismiss what you don't understand but you should consider the possibility that it is significantly more difficult develop algorithms like those in VW and "bang out" implementations of them that are fast and correct.
Would you describe him as a "pencil pushing fraud" too?
Would we have GPS without relativity which in turn depends on non-Euclidean geometry, initially considered a curiosity? What about public key encryption, built upon number theory, proudly described as "useless" by the pure mathematician G.W. Hardy? There are so many examples like these.
Be careful of committing the fallacy in assuming that because a clear line of ideas can be traced back from the present day that finding that line was as easy as recounting it.
Out of interest, what is your position of the spending on sport, the arts, and other cultural endeavours?
I think you are underestimating just how intricate and challenging understanding a proof can be. Especially one of this length and by someone of this standing.
http://mark.reid.name/iem/warning-high-dimensions.html
Thanks to the comments, I learned this is a common warning, used in at least three textbooks.
I think you may be abusing Big-O notation a little here since, technically O(10) is equivalent to O(1000)—and both are O(1).