There's more to mathematics than rigour and proofs (2007)
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There's more to mathematics than rigour and proofs (2007)
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Re: There's more to mathematics than rigour and proofs (2007)
#2Could modern AI help amateur mathematicians to build proofs?
Re: There's more to mathematics than rigour and proofs (2007)
#3The first time I really felt I understood math in depth was my uni linear algebra course. Distance and orthogonality were replaced with a more abstract but better inner product. It behaved like an IT interface: As long as some basic properties were fulfilled, aal of linear algebra came along. Half o the examples were the usual numeric vectors and matrices, the others were integrals, etc...
Re: There's more to mathematics than rigour and proofs (2007)
#4The kind of follows the standard midwit meme progression one sees in programming as well
1. Making stuff is fun and goofy and hacky 2. Coding is formal and IMPORTANT and SERIOUS 3. What cool products and tools can I make?
I feel like this pattern probably happens in many fields? Would be fun to kind of do a survey/outline of how this works across disciplines
Re: There's more to mathematics than rigour and proofs (2007)
#5> One can roughly divide mathematical education into three stages:
Similarly with programming.
1. Write programs that you think are cool
2. Learn about data structures and algorithms and complexity and software organization.
3. Write programs that you think are cool. But since you know more, you can write more cool programs.
If things are working as they should, the end stage of mathematics and programming should be fun, not tedious. The tedious stuff is just a step along the way for you to be able to do more fun stuff.
Re: There's more to mathematics than rigour and proofs (2007)
#6> One can roughly divide mathematical education into three stages: Similarly with programming. 1. Write programs that you think are cool 2. Learn about data structures and algorithms and complexity and software organization. 3. Write programs that you think are cool. But since you know more, you can write more cool programs. If things are working as they should, the end stage of mathematics and programming should be…
This is true but I think it's iterative, cyclic. It applies to any art and craft, really. You alternate between perceiving and projecting, receiving and creating.
Re: There's more to mathematics than rigour and proofs (2007)
#7Is there?
Re: There's more to mathematics than rigour and proofs (2007)
#8I want to be Terrance Tao when I grow up
Re: There's more to mathematics than rigour and proofs (2007)
#9Is there?
Absolutely.
We have machines that can crank out true theorems, rigorously proven, all day. It takes a mathematician to know what is worth working on. And that is fundamentally an intuitive decision. Computers don't care whether a proof is interesting or not.
Re: There's more to mathematics than rigour and proofs (2007)
#10I wish people had more exposure to building mathematical models of things. I am fairly convinced that the only real exposure I was given was to models that we knew worked. So much so, that we didn't even execute many.
Specifically, parabolic motion is something you can obviously do by throwing something. You can, similarly, plot over a time variable where things are observed. You can then see that we can write an equation, or model, for this. For most of us, we jump straight to the model with some discussion of how it translates. But nothing stops you from observing.
With modern programming environments, you can easily jump people into simulating movement very rapidly and let people try different models there. We had turtle geometry years ago, but for most of us that was more mental execution than it was mechanical. Which is probably a great end goal, but no reason you can't also start with the easy computer simulations.