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Come to think of it -- calculus wasn't really rigorous until a LONG time after Newton introduced it. Isn't Riemann sometimes credited as the guy who finally str

Come to think of it -- calculus wasn't really rigorous until a LONG time after Newton introduced it. Isn't Riemann sometimes credited as the guy who finally straightened it out?

pitaj 9y ago
Also, IIRC, many very important things like vector calculus, Fourier and Laplace transforms, and other differential equation stuff was added far later.
- qubex 9y ago
  Stuff is still being added nowadays: non-integer derivatives and integrals (”fractional calculus”), alternative formulations that rely not on the limit of the addition but on the limit of a multiplier (“L-calculus”), stochastic integration (Itō integrals), automatic differentiation (vital to estimating weights for neural networks in machine learning), derivatives of integers (???!!!!), and other things that might or might not be important going forward are all branches that have been developed since the 1960s.
  - somezero 9y ago
    What's "L-calculus"? And what does "alternative formulations that rely not on the limit of the addition but on the limit of a multiplier" mean? Any resources?
    - pitaj 9y ago
      An integral can be thought of as the infinite sum of infinitesimal areas below the graph (look up Riemann sums). I think what he is saying is that instead of summing these, you instead to the product. That is, multiply each infinitesimal quantity together.
    - qubex 9y ago
      A derivative is basically how a small addition to the input value changes the output value; this change in output value is also studied in terms of being something “additional” (though it could also be a subtraction). In L-calculus this approach is altered: how does multiplying the input value by a tiny value greater than one change the amount by which the output value gets multiplied? Here’s the relevant Wikipedia page to get you started down the rabbit-hole: https://en.wikipedia.org/wiki/Multiplicative_calculus?wprov=sfti1 https://en.wikipedia.org/wiki/Multiplicative_calculus?wprov=...
qubex 9y ago
Cauchy & Weiesraß reformulated ”infinitesimal calculus” rigorously in terms of limits and called it Real Funcional Analysis.

Come to think of it -- calculus wasn't really rigorous until a LONG time after Newton introduced it. Isn't Riemann sometimes credited as the guy who finally straightened it out?

pitaj 9y ago

Also, IIRC, many very important things like vector calculus, Fourier and Laplace transforms, and other differential equation stuff was added far later.

qubex 9y ago
Stuff is still being added nowadays: non-integer derivatives and integrals (”fractional calculus”), alternative formulations that rely not on the limit of the addition but on the limit of a multiplier (“L-calculus”), stochastic integration (Itō integrals), automatic differentiation (vital to estimating weights for neural networks in machine learning), derivatives of integers (???!!!!), and other things that might or might not be important going forward are all branches that have been developed since the 1960s.
- somezero 9y ago
  What's "L-calculus"? And what does "alternative formulations that rely not on the limit of the addition but on the limit of a multiplier" mean? Any resources?
  - pitaj 9y ago
    An integral can be thought of as the infinite sum of infinitesimal areas below the graph (look up Riemann sums). I think what he is saying is that instead of summing these, you instead to the product. That is, multiply each infinitesimal quantity together.
  - qubex 9y ago
    A derivative is basically how a small addition to the input value changes the output value; this change in output value is also studied in terms of being something “additional” (though it could also be a subtraction). In L-calculus this approach is altered: how does multiplying the input value by a tiny value greater than one change the amount by which the output value gets multiplied? Here’s the relevant Wikipedia page to get you started down the rabbit-hole: https://en.wikipedia.org/wiki/Multiplicative_calculus?wprov=sfti1 https://en.wikipedia.org/wiki/Multiplicative_calculus?wprov=...

qubex 9y ago

Cauchy & Weiesraß reformulated ”infinitesimal calculus” rigorously in terms of limits and called it Real Funcional Analysis.