There Is No ‘Proper English’: Ignore grammar scolds. If people say it, it’s the right way to speak

Wall Street Journal ^ | 03/17/2015 | OLIVER KAMM

[Tolerance Sucks Rocks]

So it is OK to speak in Ebonics, then?

[dr_lew]

Of course it’s OK to speak in Ebonics!

Don’t you just wish you could?

[FredZarguna]

It was well articulated by the Greeks in terms of plane geometry. That's about all, except for Archimedes, whose grasp of calculus was probably [in some ways] better than Newton's. All the more remarkable because analytical mathematics was a wasteland in ancient times, and thanks to the Greeks remained so for two millennia.

And it largely was in Newton's time as well. Your claim that Newton's understanding was simply tweaked by later mathematicians is preposterous. George Berkeley's critique of the conceptual underpinnings of calculus was devastating. It took more than a hundred and fifty years to satisfactorily answer his charges in The Analyst.

[dr_lew]

But Newton explains the whole thing. “Passing to the limit” is my characterization of his explanation. Anyway, the proof of his method is in the Newtonian pudding. Are you saying Newtonian physics is null and void since it’s not founded on Weierstrass? To me, your arguments are all rhetorical. I don’t see any substance to them.

[FredZarguna]

Like Feynman, who considered the Path Integral Formulation conceptually fine but mathematically suspect, Newton was not entirely comfortable with the rigor of his arguments, so you're placing more faith in Newton's grasp of what he was doing than Newton himself probably did. Newton did not claim that he "explained the whole thing." Far from it.

Like all physicists, he was mostly concerned with answers to practical questions, and believed in "justification by works" [of experimenters] rather than "justification by faith" [in his own arguments.]

[dr_lew]

That's about all, except for Archimedes, whose grasp of calculus was probably [in some ways] better than Newton's.

Absolutely not. Based on my own studies, I can see that Archimedes did not arrive at a general method, such as calculus is. In particular, his formula for the surface of a sphere is not an integration.

He made two constructions: Of a figure inscribed in the unit sphere consisting of conic figures whose surface ( given exactly ) is less than 4 pi squared, and of a circumscribed figure whose surface ( also given exactly ) is greater than 4 pi squared. Hence, by the obvious logic, the surface of the sphere is 4 pi squared. This is not calculus and it is not infinitesimal reasoning. It IS brilliant.

[dr_lew]

Oops. For “4 pi squared” read “4 pi”. I was trying to get rid of “r”, you see.

[FredZarguna]

Archimedes method of exhaustion between limiting figures was a general method. It was not the Calculus, but he applied it to the solution of several different problems, not just one.

It's not infinitesimal "reasoning" because infinitesimal "reasoning" is not reasoning. And it's EXACTLY why I say Archimedes understood calculus better [in a sense] than Newton did. He is not waving his arms frantically in the air about "reaching a point where the small quantity o is zero." He is actually showing you how to do integration, and in the course of doing the construction, he is asserting that

The intersection of an infinite number of nested non-empty closed subsets of a compact space is not empty.

That is the fundamental basis of Calculus, and does not involve the "reasoning" of "infinitesimals."

[dr_lew]

That is the fundamental basis of Calculus, and does not involve the "reasoning" of "infinitesimals."

Archimedes constructions depend on finding exact expressions for the limiting values. This makes them idiosyncratic, because, supposing you can't?

The calculus derivation for the surface of a unit sphere is most definitely based on infinitesimal reasoning. The notation, d theta, in the integral is an expression of this. We have Int (d theta 2 pi sin theta) from 0 to pi, where d theta is the infinitesimal width of each strip of the surface in the integration, and 2 pi sin theta is the length of each circular strip. Of course this evaluates to 4 pi.

You have to use infinitesimal reasoning just to write this down!