3 ms·

If you read some older textbooks calculus is explained in a more intuitive way, to where you could see how someone like Newton might've arrived at it geometrica

If you read some older textbooks calculus is explained in a more intuitive way, to where you could see how someone like Newton might've arrived at it geometrically. Mathematics has gotten increasingly formalized/rigorized, which has trickled down into the teaching of calculus, at the expense of obscuring the origin of some of the ideas.

redis_mlc 5y ago
> Newton might've arrived at it geometrically That's funny, because calculus is derived from calculating the area under a curve. :)
rsj_hn 5y ago
It depends very much on how you define "calculus". If it's just a question of the fundamental theorem of calculus, then people like Isaac Barrow had already noticed that integration and differentiation were in some sense inverse operations, Fermat provided evidence of this by calculating integrals of polynomials, and I'm sure others did as well. But they did not supply a clear statement of a theorem with an accompanying proof, I believe mostly because of a lack of proper definition of the derivative, which was provided by Fermat, and almost immediately this was what Newton needed to prove exactly how differentiation and integration were related. I think it's no accident that once Fermat's definition (really, his method) of finding tangents was widely publicized, both Leibniz and Newton came up with the fundamental theorem. It was "hanging in the air", as all the pieces were on the table. Newton was not shy in crediting Fermat for giving him the definition he required. Math has many examples where the real breakthrough is identifying the precise definition of something that can be useful to form proofs. Once you have that, a flurry activity results in many proofs by other mathematicians. Cauchy's definition of limit triggered many results in analysis. Galois's discovery of a group or Enrico Betti's formulation of topology are other examples.

redis_mlc 5y ago

> Newton might've arrived at it geometrically That's funny, because calculus is derived from calculating the area under a curve. :)

rsj_hn 5y ago

It depends very much on how you define "calculus". If it's just a question of the fundamental theorem of calculus, then people like Isaac Barrow had already noticed that integration and differentiation were in some sense inverse operations, Fermat provided evidence of this by calculating integrals of polynomials, and I'm sure others did as well. But they did not supply a clear statement of a theorem with an accompanying proof, I believe mostly because of a lack of proper definition of the derivative, which was provided by Fermat, and almost immediately this was what Newton needed to prove exactly how differentiation and integration were related. I think it's no accident that once Fermat's definition (really, his method) of finding tangents was widely publicized, both Leibniz and Newton came up with the fundamental theorem. It was "hanging in the air", as all the pieces were on the table. Newton was not shy in crediting Fermat for giving him the definition he required. Math has many examples where the real breakthrough is identifying the precise definition of something that can be useful to form proofs. Once you have that, a flurry activity results in many proofs by other mathematicians. Cauchy's definition of limit triggered many results in analysis. Galois's discovery of a group or Enrico Betti's formulation of topology are other examples.