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The article is delightful, but regarding the title "Will Computers Redefine the Roots of Math?" No. Math is about human understanding, elegance and beauty. C

The article is delightful, but regarding the title "Will Computers Redefine the Roots of Math?" No. Math is about human understanding, elegance and beauty. Computers may be useful tools, and perhaps they will be of even greater assistance in mathematics moving forward, but Alan Turing and others have shown us they can't do it all. Math is a fundamentally human endeavor. We will have to transcend modern computers with something like strong AI before that can change.

wamatt 11y ago
This line of reasoning feels similar to that which says AGI will never be possible. It's also a bit unclear to me, how one can justifiably hold this position with such confidence?
- ColinDabritz 11y ago
  Sorry, I didn't mean to imply that strong AI isn't possible, I believe it is very possible. I was reacting strongly to the headline specifically, as it was invoking a "magic computers" image that I dislike. I only meant that the computers of today are tools we use to explore math, but that the work today is very human. The "computer" that transcends that will look very different. What Alan Turing showed us was that you cannot turn all of "mathematics" into an algorithm. The beauty and elegance and human experience of the exploration of mathematics are important. Some day we will probably be able to teach a machine those traits. The title reads more like someone wrote an app that solved math, which is silly. The article itself is fortunately much better.
  - kefka 11y ago
    I think an interesting question here is: How do we humans identify and solve the halting problem?
    - ColinDabritz 11y ago
      In the general case, we can't of course, as Turing proved. How we untangle special cases and categories of algorithms around this is part of the wonder of mathematics for me.
      - eli_gottlieb 11y ago
        >In the general case, we can't of course, as Turing proved. Not quite. Turing proved that no single algorithm cannot solve the halting problem for all machines; it has always been trivial to show that we can answer the halting question for many machines we care about, often because they are specifically designed to always halt. Chaitin elaborated this into a detailed theory of quantified incomputability, showing (roughly) that any N-bit axiomatic system could decide the halting problem only for machines composed of strictly less than N bits of information (Kolmogorov complexity). So the question is either, depending on your philosophy of mathematics, how we locate ontologically special cases where halting proofs are possible, or how we obtain the information we put into our axioms that allows us to write halting proofs for increasingly complex programs.
        kefka 11y ago
        I'm sorry that my understanding of set and computation theory is lacking. Please excuse my ignorance while I study up on the topic further. But, I do believe the original commenter misunderstood my question. Why is it that we humans, when armed with how a computer works, can identify halting problems? Why do we not get stuck in a loop when studying a potential halting problem? What allows us to make the intuitive leap past that? And to follow up, what are the implications about "brains are 10^10 200Hz computers in parallel" hypothesis?
        eli_gottlieb 11y ago
        >But, I do believe the original commenter misunderstood my question. Why is it that we humans, when armed with how a computer works, can identify halting problems? Why do we not get stuck in a loop when studying a potential halting problem? Well, given my current understanding, I'd say it's because human beings reason inductively. We pick up new axioms by observing our environment, which includes things like equations and programs. Since we reason inductively, we're only bound by incompleteness theorems for phenomena too random/complex for our current understanding. We "catch up" eventually to be able to solve computability problems, at least to some degree.
agumonkey 11y ago
Maybe they can stir you into direction that wouldn't be explored without...
- cbd1984 11y ago
  They've already done this, in fact. Computer-assisted proofs are well-known, such as the proof of the four-color theorem.
  - agumonkey 11y ago
    It's not completely new then.
gertef 11y ago
> Alan Turing and others have shown us they can't do it all Please explain where Alan Turn shown that computers cannot do they something that human can.
- ColinDabritz 11y ago
  Apologies, I didn't mean to make that implication. I was referring to the history of the original computability paper. There was an era in math where we believed it may be possible to determine the rest of mathematics via algorithm, that the game was up. Turing showed that this was not possible, and in doing so invented the general purpose computer. Computers may be able to keep up, but the title was invoking "magic algorithm" suggestions to me. At the moment it may be unclear, but you are correct that it appears that computation can mimic the brain. For computers to be used as more than just tools in humans exploration of mathematics, to "redefine the rules" I think it would have to look quite different than the computers we have today.
j2kun 11y ago
I don't know why this comment is so downvoted. At its roots (which is what the headline is referring to), mathematical inquiry is an activity invented by humans for their own amusement. We mathematicians do what we do because it brings us aesthetic pleasure to understand the nature of some mathematical thing. If the use of computers removes that pleasure then mathematicians will stop doing mathematics.

wamatt 11y ago

This line of reasoning feels similar to that which says AGI will never be possible. It's also a bit unclear to me, how one can justifiably hold this position with such confidence?

ColinDabritz 11y ago
Sorry, I didn't mean to imply that strong AI isn't possible, I believe it is very possible. I was reacting strongly to the headline specifically, as it was invoking a "magic computers" image that I dislike. I only meant that the computers of today are tools we use to explore math, but that the work today is very human. The "computer" that transcends that will look very different. What Alan Turing showed us was that you cannot turn all of "mathematics" into an algorithm. The beauty and elegance and human experience of the exploration of mathematics are important. Some day we will probably be able to teach a machine those traits. The title reads more like someone wrote an app that solved math, which is silly. The article itself is fortunately much better.
- kefka 11y ago
  I think an interesting question here is: How do we humans identify and solve the halting problem?
  - ColinDabritz 11y ago
    In the general case, we can't of course, as Turing proved. How we untangle special cases and categories of algorithms around this is part of the wonder of mathematics for me.
    - eli_gottlieb 11y ago
      >In the general case, we can't of course, as Turing proved. Not quite. Turing proved that no single algorithm cannot solve the halting problem for all machines; it has always been trivial to show that we can answer the halting question for many machines we care about, often because they are specifically designed to always halt. Chaitin elaborated this into a detailed theory of quantified incomputability, showing (roughly) that any N-bit axiomatic system could decide the halting problem only for machines composed of strictly less than N bits of information (Kolmogorov complexity). So the question is either, depending on your philosophy of mathematics, how we locate ontologically special cases where halting proofs are possible, or how we obtain the information we put into our axioms that allows us to write halting proofs for increasingly complex programs.
      - kefka 11y ago
        I'm sorry that my understanding of set and computation theory is lacking. Please excuse my ignorance while I study up on the topic further. But, I do believe the original commenter misunderstood my question. Why is it that we humans, when armed with how a computer works, can identify halting problems? Why do we not get stuck in a loop when studying a potential halting problem? What allows us to make the intuitive leap past that? And to follow up, what are the implications about "brains are 10^10 200Hz computers in parallel" hypothesis?
        eli_gottlieb 11y ago
        >But, I do believe the original commenter misunderstood my question. Why is it that we humans, when armed with how a computer works, can identify halting problems? Why do we not get stuck in a loop when studying a potential halting problem? Well, given my current understanding, I'd say it's because human beings reason inductively. We pick up new axioms by observing our environment, which includes things like equations and programs. Since we reason inductively, we're only bound by incompleteness theorems for phenomena too random/complex for our current understanding. We "catch up" eventually to be able to solve computability problems, at least to some degree.

agumonkey 11y ago

Maybe they can stir you into direction that wouldn't be explored without...

cbd1984 11y ago
They've already done this, in fact. Computer-assisted proofs are well-known, such as the proof of the four-color theorem.
- agumonkey 11y ago
  It's not completely new then.

gertef 11y ago

> Alan Turing and others have shown us they can't do it all Please explain where Alan Turn shown that computers cannot do they something that human can.

ColinDabritz 11y ago
Apologies, I didn't mean to make that implication. I was referring to the history of the original computability paper. There was an era in math where we believed it may be possible to determine the rest of mathematics via algorithm, that the game was up. Turing showed that this was not possible, and in doing so invented the general purpose computer. Computers may be able to keep up, but the title was invoking "magic algorithm" suggestions to me. At the moment it may be unclear, but you are correct that it appears that computation can mimic the brain. For computers to be used as more than just tools in humans exploration of mathematics, to "redefine the rules" I think it would have to look quite different than the computers we have today.

j2kun 11y ago

I don't know why this comment is so downvoted. At its roots (which is what the headline is referring to), mathematical inquiry is an activity invented by humans for their own amusement. We mathematicians do what we do because it brings us aesthetic pleasure to understand the nature of some mathematical thing. If the use of computers removes that pleasure then mathematicians will stop doing mathematics.