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> It's all just flashes of algorithms, data structures, potential modifications, moving pieces, how they all affect each other and what happens to the entire en

> It's all just flashes of algorithms, data structures, potential modifications, moving pieces, how they all affect each other and what happens to the entire entangled web when you alter something. That's what doing math is like too - just substitute axioms, mathematical objects (whether numbers, sets, rings, or whatever is under discussion), potential lemmas and approaches, what bag of mathematical tools (theorems) you can use, and how much closer to a solution when you shift terms in your formulae around. Then you write it all down (if you haven't already), simplify it, and clean it up before showing it to others, just like you would code. Also, you can map programs to proofs and vice versa: https://en.wikipedia.org/wiki/Curry–Howard_correspondence https://en.wikipedia.org/wiki/Curry–Howard_correspondence

lioeters 7y ago
That sentence jumped out at me too. The intuitive thinking process - unverbalized or half-expressed leaps of logic making interconnections - I imagine mathematicians experience a similar rush of thoughts while solving or exploring. All code boils down to operations that can be described mathematically. Software is applied mathematics (with a sprinkle of art, perhaps). I think the reason why some people feel that programming is not closely related to mathematics, is that programmers are thinking and working on top of so many layers of abstraction, it's almost like working with the "stuff of the mind" itself, with models, processes, flows, transformations, events, composing behaviors. That said, I relate to what the grandparent commenter is saying. Software allows me to think with visible, malleable and "living" mathematics while building up a system, to ask questions and have a dialogue with it. >> there's more ways to think about problems and solutions than..applying formal mathematics I agree with this. Often a "looser" approach is needed to explore a problem space, and formal mathematics may not be the best medium for creative problem-solving. On the other hand, the qualities that are valued in software - types, functional programming, test-driven development, etc. - are all about proofs. Not necessarily mathematically rigorous, but the closer you get, the more reliable the logic.
- asark 7y ago
  > That said, I relate to what the grandparent commenter is saying. Software allows me to think with visible, malleable and "living" mathematics while building up a system, to ask questions and have a dialogue with it. Programming's friendlier to algorithmic thinking (versus equation/identity and proof). The former's really easy for me, and while on paper (aptitude test scores) one might think the latter would be too, it's very, very not. I've only relatively late in life realized I need to reframe any non-trivial math I encounter in terms of algorithms to have any hope of understanding it. It's probably why I bounce off—understand well enough, just strongly dislike—programming languages that try to make code more look more like a math paper (more focus on equality/identity and proof-like structures). And yeah algorithms are math, but lots of math's not really algorithms and when someone writes "think in math" that mostly means "think in proofs" to me. If they mean "think in algorithms" then that's close enough to programming—as I see it—already that it's a pretty fine distinction.
Sharlin 7y ago
Yeah. It is easy to people with no university-level math background to think math is a deterministic, conscious effort to execute what are basically algorithms, like long division with pen and paper. Whereas actually ”mathematical thinking”, like coming up with a proof, is an incredibly intuition-guided process, a parallel heuristic search in the solution space, a fundamentally creative endeavour. And as your intuition comes up with promising paths through the search space, you write them down, formalize them, probably discover some corner cases you have to handle, and either continue down that path or realize that it is a dead end and you have to backtrack. At least to me, this process is incredibly similar to programming effort. You come up with subsolutions, formalize them, fix issues revealed by the formalization, carry on with the next subsolution or realize that approach can’t work after all, and come up with something else.

lioeters 7y ago

That sentence jumped out at me too. The intuitive thinking process - unverbalized or half-expressed leaps of logic making interconnections - I imagine mathematicians experience a similar rush of thoughts while solving or exploring. All code boils down to operations that can be described mathematically. Software is applied mathematics (with a sprinkle of art, perhaps). I think the reason why some people feel that programming is not closely related to mathematics, is that programmers are thinking and working on top of so many layers of abstraction, it's almost like working with the "stuff of the mind" itself, with models, processes, flows, transformations, events, composing behaviors. That said, I relate to what the grandparent commenter is saying. Software allows me to think with visible, malleable and "living" mathematics while building up a system, to ask questions and have a dialogue with it. >> there's more ways to think about problems and solutions than..applying formal mathematics I agree with this. Often a "looser" approach is needed to explore a problem space, and formal mathematics may not be the best medium for creative problem-solving. On the other hand, the qualities that are valued in software - types, functional programming, test-driven development, etc. - are all about proofs. Not necessarily mathematically rigorous, but the closer you get, the more reliable the logic.

asark 7y ago
> That said, I relate to what the grandparent commenter is saying. Software allows me to think with visible, malleable and "living" mathematics while building up a system, to ask questions and have a dialogue with it. Programming's friendlier to algorithmic thinking (versus equation/identity and proof). The former's really easy for me, and while on paper (aptitude test scores) one might think the latter would be too, it's very, very not. I've only relatively late in life realized I need to reframe any non-trivial math I encounter in terms of algorithms to have any hope of understanding it. It's probably why I bounce off—understand well enough, just strongly dislike—programming languages that try to make code more look more like a math paper (more focus on equality/identity and proof-like structures). And yeah algorithms are math, but lots of math's not really algorithms and when someone writes "think in math" that mostly means "think in proofs" to me. If they mean "think in algorithms" then that's close enough to programming—as I see it—already that it's a pretty fine distinction.

Sharlin 7y ago

Yeah. It is easy to people with no university-level math background to think math is a deterministic, conscious effort to execute what are basically algorithms, like long division with pen and paper. Whereas actually ”mathematical thinking”, like coming up with a proof, is an incredibly intuition-guided process, a parallel heuristic search in the solution space, a fundamentally creative endeavour. And as your intuition comes up with promising paths through the search space, you write them down, formalize them, probably discover some corner cases you have to handle, and either continue down that path or realize that it is a dead end and you have to backtrack. At least to me, this process is incredibly similar to programming effort. You come up with subsolutions, formalize them, fix issues revealed by the formalization, carry on with the next subsolution or realize that approach can’t work after all, and come up with something else.