It's quirky to think, for example, on the natural numbers that 'exists an operation + and a null element under that operation 0'
This is "very understandable" by humans, but very difficult to compute.
As such as this conjecture is stated in a way that looks qualitative (but has a good definition), still, usually it seems that proofs are even harder for theorems defined like that
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?
" 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)
"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.
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.
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.
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.
> 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.
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.
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
First-order Logic is a fascinatingly deep and interesting topic. The level of abstraction is a requirement - it makes it far easier to work with, prove, and understand.
I think mathematicians have a weird way of thinking about problems.
First-order logic for example: http://en.wikipedia.org/wiki/First-order_logic
It's quirky to think, for example, on the natural numbers that 'exists an operation + and a null element under that operation 0'
This is "very understandable" by humans, but very difficult to compute.
As such as this conjecture is stated in a way that looks qualitative (but has a good definition), still, usually it seems that proofs are even harder for theorems defined like that
http://en.wikipedia.org/wiki/Abc_conjecture