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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 migh

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 on how to better think about problems?

raverbashing 13y ago
" 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 (and transitive) Now, there are several ways to explain that, and it's usually explained more or less by "for all X and all Y, if X = Y then Y = X" Now, it would maybe be interesting to have a 'different explanation' that is as powerful as first order logic but works differently (and maybe easier to compute) 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)
- claudius 13y ago
  And after you spent ten years reformulating basic maths in your fancy new logic, people will look at your papers and won’t understand a word, which appears to be more or less what happened to our poor protagonist in the OP. Furthermore, I have to admit I don’t see the immediate advantage such a reconstruction would bring with it.
  - raverbashing 13y ago
    Well, the OPs reinvention looks like something more high level Well, there may not be immediate advantages, but in math you never know. There are several hard problems in one domain that are trivial in another domain, for example. An example from physics: http://en.wikipedia.org/wiki/Hamiltonian_mechanics http://en.wikipedia.org/wiki/Hamiltonian_mechanics
    - claudius 13y ago
      > There are several hard problems in one domain that are trivial in another domain, for example. Certainly, and this is pretty much what OP did, invent a new domain to solve a problem – Hamilton aka Lord Kelvin merely reformulated the problem slightly, and while I personally love Hamiltonian mechanics, I don’t think it is comparable to ‘inter-universal geometry’ or replacing first order logic with something else. So, yes, a different field may provide a different perspective and hence easier solution, but if you want to replace first order logic, you’re not looking at a different/new field in maths, you’re looking at rebuilding maths.
- andreasvc 13y ago
  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.
  - rickhanlonii 13y ago
    That's what I thought when I read his comment. Hofstadter emphasizes how unmanageable such a system becomes quite quickly. For example, here's what a proof that addition is commutative looks like in TNT: http://imgur.com/a/O63Ij http://imgur.com/a/O63Ij
    - raverbashing 13y ago
      Yes, I was thinking of it when I wrote the comment, there's also an arithmetic with P an M symbols - for plus and minus (but I can't find it on google, it's been a while, sorry) Thanks for sharing this snipped, one of the several fun things in GEB
      - rickhanlonii 13y ago
        No problem, the image at the beginning of chapter 10 is one of my favorite GEB gems: http://i.imgur.com/NahfzTp.jpg http://i.imgur.com/NahfzTp.jpg
- jerf 13y ago
  "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. See also http://en.wikipedia.org/wiki/Thue_%28programming_language%29 http://en.wikipedia.org/wiki/Thue_%28programming_language%29 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.

raverbashing 13y ago

" 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 (and transitive) Now, there are several ways to explain that, and it's usually explained more or less by "for all X and all Y, if X = Y then Y = X" Now, it would maybe be interesting to have a 'different explanation' that is as powerful as first order logic but works differently (and maybe easier to compute) 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)

claudius 13y ago
And after you spent ten years reformulating basic maths in your fancy new logic, people will look at your papers and won’t understand a word, which appears to be more or less what happened to our poor protagonist in the OP. Furthermore, I have to admit I don’t see the immediate advantage such a reconstruction would bring with it.
- raverbashing 13y ago
  Well, the OPs reinvention looks like something more high level Well, there may not be immediate advantages, but in math you never know. There are several hard problems in one domain that are trivial in another domain, for example. An example from physics: http://en.wikipedia.org/wiki/Hamiltonian_mechanics http://en.wikipedia.org/wiki/Hamiltonian_mechanics
  - claudius 13y ago
    > There are several hard problems in one domain that are trivial in another domain, for example. Certainly, and this is pretty much what OP did, invent a new domain to solve a problem – Hamilton aka Lord Kelvin merely reformulated the problem slightly, and while I personally love Hamiltonian mechanics, I don’t think it is comparable to ‘inter-universal geometry’ or replacing first order logic with something else. So, yes, a different field may provide a different perspective and hence easier solution, but if you want to replace first order logic, you’re not looking at a different/new field in maths, you’re looking at rebuilding maths.
andreasvc 13y ago
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.
- rickhanlonii 13y ago
  That's what I thought when I read his comment. Hofstadter emphasizes how unmanageable such a system becomes quite quickly. For example, here's what a proof that addition is commutative looks like in TNT: http://imgur.com/a/O63Ij http://imgur.com/a/O63Ij
  - raverbashing 13y ago
    Yes, I was thinking of it when I wrote the comment, there's also an arithmetic with P an M symbols - for plus and minus (but I can't find it on google, it's been a while, sorry) Thanks for sharing this snipped, one of the several fun things in GEB
    - rickhanlonii 13y ago
      No problem, the image at the beginning of chapter 10 is one of my favorite GEB gems: http://i.imgur.com/NahfzTp.jpg http://i.imgur.com/NahfzTp.jpg
jerf 13y ago
"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. See also http://en.wikipedia.org/wiki/Thue_%28programming_language%29 http://en.wikipedia.org/wiki/Thue_%28programming_language%29 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.