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The Paradox of the Proof

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Re: The Paradox of the Proof

#81

Post by ColinWright »

tzakrajs wrote: post #80

“The point is not to prove the theorem,” explains Ellenberg. “The point is to understand how the universe works and what the hell is going on.” School failed at conveying this to me in so many domains.

The point I often finish with when I give talks is this:

    People tell you that the point of being good at science
    and math is to help you understand the universe, and to
    understand what's going on.

    This is only the first step.

    The real reason for being good at math and science is to
    BEND THE WORLD TO YOUR WILL !!
The kids seem to like that ...

Re: The Paradox of the Proof

#82

Post by SatvikBeri »

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…

The key behind Math is to generalize from very specific rules like the concept of multiplication as n copies of addition to very general rules based on behavior. For example, in Ring Theory multiplication and addition are just functions that satisfy specific properties. By focusing on the class of all functions that satisfy these properties, we're able to prove very general theorems that have much wider applications than if we just focused on addition-based multiplication.

You see the same pattern in programming. Imagine if you had to have separate addition operators for doubles, floats, ints, and bigints. It's far more useful to have one function that only depends on the behavior of its input, rather than having its input be a very specific type of object.

Re: The Paradox of the Proof

#83

Post by pfedor »

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-pages long academic paper on top trees. Since it had been long established on the team that the guy was (a) a genius and (b) hundred percent reliable, it was generally assumed that the solution would work, however all code changes at Google need to be reviewed and nobody was able to review that CL. A number of people tried but they all dropped out, even though, unlike the ABC Conjecture professor, the author was absolutely willing to answer any questions you had.

Eventually the powers that be decided to trash the CL, not because there was any doubt regarding its correctness, but on the principle that you can't let anything that only one person can comprehend enter the codebase.

Re: The Paradox of the Proof

#84

Post by ics »

gwern wrote: post #60
msilenus wrote: post #49

Earlier quoted context omitted.

There is a micro-payment service called Flattr [1] which does something like this. It was founded by Peter Sunde, one of the pirate bay founders. It's actually quite successfull in the German web community especially for podcasters. There are German podcasters which earn about 1000 Euro per month via Flattr. [1] https://flattr.com

Someone actually mentioned Flattr to me today as a way to take donations for my own writings; I've always begged off because I've never wanted to take the time to set it up and clutter my site (more), but now here I see Flattr brought up again... Is there any way of even roughly estimating how much Flattr might bring in?

> Is there any way of even roughly estimating how much Flattr might bring in?

Of course: you try it, write up a massive future post about the results, and voila! But that probably wasn't the answer you wanted. I have no idea how many people are using it, but the idea sounds pretty great and if there's a good crowd to try it out on then HN readers wouldn't be a bad bet. I don't think anyone would mind another small button between PayPal and BTC.

Re: The Paradox of the Proof

#85

Post by mcguire »

brazzy wrote: post #21
crntaylor wrote: post #11

Earlier quoted context omitted.

I think there's a reasonable argument to be made that mathematical constructs are part of the universe. David Deutsch has a method for ascertaining whether something can can be said to exist or not - ask whether it "kicks back" when you interact with it, in the sense that simulating the response of the thing you're considering in a totally convincing way would involve an effort as large as building a new universe for…

> I think there's a reasonable argument to be made that mathematical constructs are part of the universe. Well, one could say that we "create" them using our minds, which certainly are, but that's deep into philosophical territory. > Mathematics "exists" because giving someone the genuine experience of doing mathematics when they really weren't would involve a simulation of almost unfathomable complexity. Sounds to m…

"By that standard, all fiction is 'a very real part of the universe'."

Strangely, I believe that statement supports crntaylor's argument, being the point of fiction as far as I can see.

Re: The Paradox of the Proof

#86

Post by magicalist »

voyou wrote: post #51
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…

It's cool that "Alex Knapp, a science writer at Forbes" has solved one of the major issues in the philosophy of mathematics by, erm, just asserting that Platonism is false. Of course, maths is a tool, but that doesn't mean that it is just a tool, and Knapp doesn't provide any arguments for that conclusion. I can use a rock as a tool, but that doesn't mean the rock has no existence independent of my mind. In fact, the…

Seriously. At least Leah Libresco had the intellectual clarity to declare herself a Platonist, as in, literally, a believer in or adherent to some part of Plato's philosophy.

Knapp latches onto a lack of empirical evidence for a Platonist viewpoint (though narrowly defines evidence as apparently needing to be extra-human to be valid, which isn't a well founded definition, but, then again, definitions are the fundamental problem here), and so he concludes that it must therefore be false, completely missing his own point.

Sorry. While there is a wealth of interesting discussion to be had on philosophy of mathematics, and even more discussion (and certainly more concrete discussion) to be had on the difference between how mathematics is actually created and how it is then defined in proofs and literature, this passage misses the boat.

Also, spitx: while fair use covers quoting from and discussing articles quite well, quoting the full article, even with discussion, could be problematic and certainly should be considered poor form. Take some quotes and then link to the rest.

Re: The Paradox of the Proof

#87

Post by tunesmith »

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…

Wow, that's horrible.

Re: The Paradox of the Proof

#89

Post by cscheid »

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…

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.

Re: The Paradox of the Proof

#90

Post by quanticle »

tunesmith wrote: post #87
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…

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 totally outweighed by the risks of the developer leaving.
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