The Paradox of the Proof • Hacker News Board

Re: The Paradox of the Proof

#91

Post by andreasvc » Sat, May 11, 2013, 12:30 AM UTC

neogodless wrote: post #74

As a completely uneducated simpleton, it seems bizarre to me that addition and multiplication are considered "different" in the deeper explorations of math and number theory. It seems like multiplication is just an extension of addition. How many times do you want to add numbers together? The result is multiplication. Similarly, addition can be used to represent multiplication. You want to multiply, which can be repr…

> It seems like multiplication is just an extension of addition. [...]

> Similarly, addition can be used to represent multiplication.

Yet, addition cannot be represented as multiplication (AFAICT), so there you have a difference.

Re: The Paradox of the Proof

#92

Post by tikhonj » Sat, May 11, 2013, 12:30 AM UTC

cscheid wrote: post #89

tikhonj wrote: post #69

Earlier quoted context omitted.

To me, the most interesting observation about this came from computability theory, of all places. To some extent, in CS we study completely arbitrary constructs--Turing machines. Turing machines, indeed, are just tools: they're a simple model of a computing machine that makes sense just based on existing technology. Or perhaps we study the lambda calculus which, while less arbitrary than a Turing machine, is still ob…

This is a great post, but > they're a simple model of a computing machine that makes sense just based on existing technology Turing machines were invented before any working (almost-) universal computer existed.

Hah, yes. I was actually thinking of a literal tape made of paper, or something to that effect. As compared to the lambda calculus which is based entirely on logic rather than some allusion to the real world.

Re: The Paradox of the Proof

#93

Post by andreasvc » Sat, May 11, 2013, 12:36 AM UTC

raverbashing wrote: post #20

claudius wrote: post #13

Earlier quoted context omitted.

If you want general results, you need to abstract away from standard preschool algebra and build up models that then let you get said general results. That might appear ‘weird’, but I don’t see how else you could get even to calculus, not to mention, for example, the algebra driving a sensible description of quantum mechanics or differential geometry. You called it a rant, but could you maybe still make a suggestion…

" you need to abstract away from standard preschool algebra and build up models that then let you get said general results" Yes, of course. " but could you maybe still make a suggestion on how to better think about problems?" And that's what I meant. Thinking about problems in a different way (but still provable, and still working in a similar way) For example, for Peano arithmetic you have that equality is symmetric…

Your suggestion reminds me of typographical number theory (TNT), in Hofstadter's Goedel Escher Bach. TNT is introduced for didactic purposes, I don't see the point of using it for actual mathematics.

Re: The Paradox of the Proof

#94

Post by tunesmith » Sat, May 11, 2013, 12:54 AM UTC

quanticle wrote: post #90

tunesmith wrote: post #87

Earlier quoted context omitted.

Wow, that's horrible.

Why? Look at it from an organizational perspective. If the code can only be understood by one person, then that person leaves, you have a piece of code that cannot be maintained or modified without potentially months of effort. Moreover, because it's a particular algorithm, this effort cannot be shortened or distributed across multiple developers. I would argue that whatever efficiency gain Google would get is totall…

Just simply because the outcome was much less than the ideal outcome. Yes, accepting the reality that no one else could understand his output (assuming the output was correct), then that organizational outcome was the correct one. But it's a tragedy (again, assuming this guy was correct) that his knowledge couldn't be taken advantage of. I just think it would be really tough to feel the realization of being so powerful that you you are impotent.

Re: The Paradox of the Proof

#95

Post by csense » Sat, May 11, 2013, 1:14 AM UTC

lmm wrote: post #59

spitx wrote: post #26

"For centuries, mathematicians have strived towards a single goal: to understand how the universe works, and describe it." I cannot resist engaging that loaded premise. It is a thought that I've wanted - for some time - for someone to skilfully dissect and lay bare for easy comprehension. The closest I've come to seeing it, is the following - a concise and accessible yet well-rounded explanation of the relevance (or…

I feel like mathematics applies too closely to physics for it to be just a tossing of darts. With a bit of familiarity complex numbers are the "obviously best" numbers, the simplest (sort of) and most mathematically interesting; that they are so central to quantum mechanics is too much for coincidence. Then there's the fact that mathematical truths really are true in models to which their axioms apply (as far as we c…

> Hardy rejoiced in the idea that his work on primes would never be a weapon of war

What about RSA encryption, and the former US export controls thereon?

Also, what about more exotic mathematical objects, like finite fields for example, that have a lot of structural properties but don't seem to model much of anything in the real world, but are still useful theoretically?

Re: The Paradox of the Proof

#96

Post by _j5l3 » Sat, May 11, 2013, 1:20 AM UTC

pfedor wrote: post #83

This reminds me of something I witnessed when I worked at Google. There was this long-standing problem, I don't want to go into details but it had to do with websearch indexing and it had gone unsolved for years though it was regularly affecting search results in a negative way. Then one guy solved it. The changelist he prepared wasn't even that long but it was crazy complex. Helpful comments contained links to a 100…

As someone newly entering the workforce, this is a really depressing view into how bureaucracy can squash innovation.

Re: The Paradox of the Proof

#97

Post by tel » Sat, May 11, 2013, 2:13 AM UTC

tikhonj wrote: post #69

spitx wrote: post #26

"For centuries, mathematicians have strived towards a single goal: to understand how the universe works, and describe it." I cannot resist engaging that loaded premise. It is a thought that I've wanted - for some time - for someone to skilfully dissect and lay bare for easy comprehension. The closest I've come to seeing it, is the following - a concise and accessible yet well-rounded explanation of the relevance (or…

To me, the most interesting observation about this came from computability theory, of all places. To some extent, in CS we study completely arbitrary constructs--Turing machines. Turing machines, indeed, are just tools: they're a simple model of a computing machine that makes sense just based on existing technology. Or perhaps we study the lambda calculus which, while less arbitrary than a Turing machine, is still ob…

Category theory is beautiful because it's so unavoidable. We might critique Turing machines for being tied intimately to our particular mode of thought—both Turing tape and lambda calculus are a certain kind of specific, at least in comparison to category theory. (Not that I'm arguing that any computational models are somehow not touching on another kind of Platonic ideal)

Category theory arises from probably the simplest imaginable notions of "transformation". There's no way to not critique me for being anthropocentric, but it's hard to imagine an alternative theory where this doesn't exist.

Re: The Paradox of the Proof

#98

Post by jerf » Sat, May 11, 2013, 2:42 AM UTC

_j5l3 wrote: post #96

pfedor wrote: post #83

This reminds me of something I witnessed when I worked at Google. There was this long-standing problem, I don't want to go into details but it had to do with websearch indexing and it had gone unsolved for years though it was regularly affecting search results in a negative way. Then one guy solved it. The changelist he prepared wasn't even that long but it was crazy complex. Helpful comments contained links to a 100…

As someone newly entering the workforce, this is a really depressing view into how bureaucracy can squash innovation.

Do not be depressed, at least because of this. Few people will ever write anything that is correct, can not be meaningfully simplified, yet incomprehensible and essentially unreviewable. The only other thing that comes to mind are the top-grade encryption algorithms. (They are reviewed, but even after extensive, extensive review by the very smartest people in the field, there's still an irreducible part when selecting an algorithm where everyone still just has to sort of hope there isn't some fatal flaw in there somewhere. Often, years and years later, there is.)

Re: The Paradox of the Proof

#99

Post by dredmorbius » Sat, May 11, 2013, 2:42 AM UTC

calinet6 wrote: post #33

robinhouston wrote: post #5

This is a great article. The writer has obviously spent a lot of time speaking to Mochizuki’s colleagues, and has explained the whole strange situation in a way that is layman-friendly without being wrong. It’s interesting that it’s part of an experiment in donation-funded journalism: I was sufficiently impressed that I donated a few dollars, but I fear that is not likely to be a common enough response to be a viable…

I wonder what would happen if, say, I allocated $10 per month to compensating articles and journalists I enjoyed. And then it was distributed evenly by time I spent on the page, every single day. So if I spent 30 minutes reading this article, 10 minutes reading another, 2 minutes on 10 others, this author would get $0.16 ($0.33 per day, times (30 mins, out of 60 total mins of time = 0.5)). That's really not that much…

Readability tried something like this. Among the problems: authors / publishers didn't sign up, and money collected had to be distributed or reimbursed. Unclaimed funds eventually were awarded to a charity.

Though I believe Readability had good intent, they took a great deal of flack for this.

Money weirds things.

Re: The Paradox of the Proof

#100

Post by jerf » Sat, May 11, 2013, 2:57 AM UTC

raverbashing wrote: post #20

claudius wrote: post #13

Earlier quoted context omitted.

If you want general results, you need to abstract away from standard preschool algebra and build up models that then let you get said general results. That might appear ‘weird’, but I don’t see how else you could get even to calculus, not to mention, for example, the algebra driving a sensible description of quantum mechanics or differential geometry. You called it a rant, but could you maybe still make a suggestion…

" you need to abstract away from standard preschool algebra and build up models that then let you get said general results" Yes, of course. " but could you maybe still make a suggestion on how to better think about problems?" And that's what I meant. Thinking about problems in a different way (but still provable, and still working in a similar way) For example, for Peano arithmetic you have that equality is symmetric…

"For example, it may be possible to write Peano arithmetic as a grammar (so zero would be ' ', one would be I, two would be II, etc)"

I seem to recall doing this as a homework assignment in computer science, writing some unrestricted grammars that could be used to perform certain simple operations on some simply-encoded numbers.

But I wonder if you've gotten very far into mathematics. It's actually well known that there are many things equivalent to first-order grammar and it's completely common to choose different things based on the sort of thing you're doing, just as many things are equivalent to Turing Machines and one can freely choose the most convenient one for your local problem. Peano arithmetic is chosen merely for its convenience for the simplest of proofs, and the instant it becomes inconvenient a real mathematician abandons it for something more locally useful.