5 ms·

I know that the above was written in jest anyway :) but just for completeness (har), a contradiction would help you defeat many things (i.e., it's not specific

by wfn 10y ago

I know that the above was written in jest anyway :) but just for completeness (har), a contradiction would help you defeat many things (i.e., it's not specific to incompleteness theorems, so to speak), because given it, (at least in traditional logic) you can prove anything. This is trivial and known, but to spell it out, if we have a contradiction formalized by p and not p (given), we can prove that q: 1. p and not p (given). 2. p (from 1). 3. not p (from 1). 4. p or q (from 2 (disjunction introduction)). 5. q (from 3 and 4). So obviously this would upset quite a few things around :) that said, the interesting stuff is with Graham Priest's (et al.) paraconsistent logic systems wherein your system can tolerate a contradiction without exploding in whole. And (so the story goes) those systems may offer an actual insight into handling incompleteness (while still being usable). If anyone has looked into this more, would be interesting to hear about it!

jesuslop 10y ago
That behaviour was desribed by the old masters as 'ex falso quodlibet': from falsity whatever (when else am i having the chance to be this pedantic). In the uni I was taught that hippie-era AI dealt with that with things called non-monotonic reasoning, abduction, truth maintenance systems.
- dvt 10y ago
  Also known as the (much cooler-sounding) Principle of Explosion :)
brenderup 10y ago
Hofstadter's examples 'this sentence is false' blah blah aren't inconsistent at all: it just semantic nonsense likes Gödel's made up logic and Cantor's ordinals before that? Mathematics exist without us no?
- deepsun 10y ago
  Mathematics -- yes. Logic -- no. Logic is a physical framework mathematics is running on.
  - danbruc 10y ago
    Wait, you are saying mathematics exists independent of humans but logic does not. But mathematics is build on top of logic, how can it gain independence from humans if its foundation does not have such independence? Or am I just misreading your statement?
    - saghm 10y ago
      The way I understand it, they're saying that mathematics is a natural, universal thing, and logic is our attempt to explain/build a foundation for it.
      - danbruc 10y ago
        To me this seems backwards. Don't we define some basic objects with axioms and then use logic to find out the consequences of the axioms? We make up some axioms for natural numbers, then we define operations like addition and multiplication on them, and finally we investigate the consequence of those definitions, we discover for example that some numbers have very special properties and can be used to uniquely decompose all numbers, we call them prime numbers. I would therefore say that mathematics is a human invention and build on top of logic, it seems in no way universal. All the things we discover in mathematics are nothing but consequences of the axioms, the definitions, and the logic used to prove things. If there is something universal, at least so it seems to me, than it would have to be logic.
        jchrisa 10y ago
        Frege spent his career here. If there was one course from college I'd retake it was Frege.
    - naasking 10y ago
      > But mathematics is build on top of logic This assumption wasn't always accepted by mathematicians.
  - tacomonstrous 10y ago
leephillips 10y ago
“All difficult conjectures should be proved by reductio ad absurdum arguments. For if the proof is long and complicated enough you are bound to make a mistake somewhere and hence a contradiction will inevitably appear, and so the truth of the original conjecture is established QED.” — John Barrow
ratmice 10y ago
> If anyone has looked into this more, would be interesting to hear about it! I would have a look at e.g. Bob Harper's homotopy type theory lectures. In the 2nd lecture on Judgements, he goes through it at about 50 minutes in. http://www.cs.cmu.edu/~rwh/courses/hott/ http://www.cs.cmu.edu/~rwh/courses/hott/ And Andrej Bauer's paper/lecture on "5 stages of accepting constructive mathematics"
- wfn 10y ago
  Thanks, will look at both! (I've been interested in HoTT but never looked closer.) Paper looks good, too (for future reference / interested folks, full version available at http://www.ams.org/journals/bull/0000-000-00/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf http://www.ams.org/journals/bull/0000-000-00/S0273-0979-2016... )

by wfn 10y ago

jesuslop 10y ago

That behaviour was desribed by the old masters as 'ex falso quodlibet': from falsity whatever (when else am i having the chance to be this pedantic). In the uni I was taught that hippie-era AI dealt with that with things called non-monotonic reasoning, abduction, truth maintenance systems.

dvt 10y ago
Also known as the (much cooler-sounding) Principle of Explosion :)

brenderup 10y ago

Hofstadter's examples 'this sentence is false' blah blah aren't inconsistent at all: it just semantic nonsense likes Gödel's made up logic and Cantor's ordinals before that? Mathematics exist without us no?

deepsun 10y ago
Mathematics -- yes. Logic -- no. Logic is a physical framework mathematics is running on.
- danbruc 10y ago
  Wait, you are saying mathematics exists independent of humans but logic does not. But mathematics is build on top of logic, how can it gain independence from humans if its foundation does not have such independence? Or am I just misreading your statement?
  - saghm 10y ago
    The way I understand it, they're saying that mathematics is a natural, universal thing, and logic is our attempt to explain/build a foundation for it.
    - danbruc 10y ago
      To me this seems backwards. Don't we define some basic objects with axioms and then use logic to find out the consequences of the axioms? We make up some axioms for natural numbers, then we define operations like addition and multiplication on them, and finally we investigate the consequence of those definitions, we discover for example that some numbers have very special properties and can be used to uniquely decompose all numbers, we call them prime numbers. I would therefore say that mathematics is a human invention and build on top of logic, it seems in no way universal. All the things we discover in mathematics are nothing but consequences of the axioms, the definitions, and the logic used to prove things. If there is something universal, at least so it seems to me, than it would have to be logic.
      - jchrisa 10y ago
        Frege spent his career here. If there was one course from college I'd retake it was Frege.
  - naasking 10y ago
    > But mathematics is build on top of logic This assumption wasn't always accepted by mathematicians.
- tacomonstrous 10y ago

leephillips 10y ago

“All difficult conjectures should be proved by reductio ad absurdum arguments. For if the proof is long and complicated enough you are bound to make a mistake somewhere and hence a contradiction will inevitably appear, and so the truth of the original conjecture is established QED.” — John Barrow

ratmice 10y ago

> If anyone has looked into this more, would be interesting to hear about it! I would have a look at e.g. Bob Harper's homotopy type theory lectures. In the 2nd lecture on Judgements, he goes through it at about 50 minutes in. http://www.cs.cmu.edu/~rwh/courses/hott/ http://www.cs.cmu.edu/~rwh/courses/hott/ And Andrej Bauer's paper/lecture on "5 stages of accepting constructive mathematics"

wfn 10y ago
Thanks, will look at both! (I've been interested in HoTT but never looked closer.) Paper looks good, too (for future reference / interested folks, full version available at http://www.ams.org/journals/bull/0000-000-00/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf http://www.ams.org/journals/bull/0000-000-00/S0273-0979-2016... )