

As Math Grows More Complex, Will Computers Reign? - gtani
http://www.wired.com/wiredscience/2013/03/computers-and-math/

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breadbox
Interesting. I had assumed at this point in time that computers were
ubiquitous in math research. It's mind-boggling to me that there is still any
significant level of pushback against using them. (I suppose that getting a
CS-specific math degree gives one a skewed perspective on such details.)

Also, the point of not having standard programming classes leads to a lack of
standards in research coding is an important one.

~~~
lutze
To me, this is akin to asking "as building materials grow heavier, will cranes
reign?"

Computers have always been envisioned as levers for the mind. The fact that
mathematicians of all people aren't taking full advantage of that principle
utterly boggles my mind too.

~~~
fusiongyro
It probably doesn't help that we go around parroting the ludicrous notion that
you don't need math to program. It can hardly be true that you need to program
to do math when we've been doing math for millenia.

How long can you mock your older brother in front of your hip young friends
before he decides to ignore you and go play with his own friends somewhere
else?

~~~
enraged_camel
>>It probably doesn't help that we go around parroting the ludicrous notion
that you don't need math to program.

I thought the consensus was that it really depends on the type of problems one
is trying to solve, and that most programmers can in fact get away with
knowing little math.

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gems
How does a computer help when you have conjectures involving infinite
mathematical structures or extremely large numbers? You can't ask the computer
to enumerate the solutions for you. You have to reason in a roundabout way,
possibly building some abstraction.

Even if there are some branches of mathematics that will be affected by
computers, at the end of the day, there will still have to be someone who does
the conjecturing, someone with enough knowledge to pose the question to the
computer.

~~~
xyzzy123
> How does a computer help when you have conjectures involving infinite
> mathematical structures or extremely large numbers?

Basically, you hop up a level (go meta). Instead of working with the extremely
large numbers or sets, you symbolically manipulate statements about them.

~~~
gems
Give me an example of how this is done or can be done.

~~~
sharkbot
Check out Adam Chlipala's Coq tutorial, specifically the section on infinite
proofs: <http://adam.chlipala.net/cpdt/html/Coinductive.html>

I'll repeat what szany and xyzzy123 mentioned: you work at a level of
abstraction where infinite data structures are represented symbolically with
enough definitional scaffolding to allow proofs to go through.

In a sense, a properly typed program provides a proof of some theorem over
infinite data structures. For instance, an instance of a tree (in generic
Java) is usually a finite data structure, but the set of all trees
representable in Java is infinite. (Handwaving begins) The types prove that
certain operations can't happen, like a tree of Strings changing to a tree of
Arrays by a node search algorithm, which is a proof about an infinite set.

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walrus
If you find this sort of thing interesting, you may enjoy _A=B_ by Marko
Petkovsek, Herbert Wilf, and Doron Zeilberger. (Wilf was mentioned in the
article.)

It's available for download at <http://www.math.upenn.edu/~wilf/AeqB.html>

~~~
jonahkagan
Link seems to be down, but I'd be interested.

~~~
walrus
If you're still having trouble accessing it, here are direct links to the
mirrors:

<http://www.math.upenn.edu/~wilf/AeqB.pdf>

<http://www.math.rutgers.edu/~zeilberg/AeqB.pdf>

<http://www.fmf.uni-lj.si/aeqb/AeqB.pdf>

The license agreement is (copied and pasted from the download page):

    
    
      Copyright 1996 by A K Peters, Ltd.
    
      Reproduction of the downloaded version is permitted for any valid
      educational purpose of an institution of learning, in which case only
      the reasonable costs of reproduction may be charged. Reproduction for
      profit or for any commercial purposes is strictly prohibited.

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fractallyte
One of the reciprocal outcomes of this could be method(s) to mathematically
prove the 'correctness' of a computer program...

Although, perhaps this is verging on philosophy: is such a process even
possible? Do current (or future) computer programs involve leaps of intuition,
and abstractions that cannot be mathematically codified?

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d4vlx
No

<http://en.wikipedia.org/wiki/Betteridges_law_of_headlines>

