wtf

ps I might as well add this since it's on my mind and OP's not around at the moment. You don't need the axiom of choice to define the real numbers. That was an error in the paper although I don't think anything else depends on it.

I don't follow your point at all then. Are you claiming that SIA relates to anything that was happening in the 17th century? Lawvere's paper on synthetic differential geometry came out in 1998. And SIA is based on category theory. a subject that didn't come into existence till the 1950's.

Thank you for your detailed reply. Looks like I'll have to work through your paper and probably end up learning a few things. As I've mentioned I'm more familiar with Newton and not at all with Leibniz, so I evidently have some gaps in my knowledge. Your focus on nilsquare infinitesimals and denial of LEM reminds me of smooth infinitesimal analysis, is this related to your ideas? https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis Cool, I will check them out.The Wiki entry for Seki is very interesting. Thanks for the info.

> You have references to this? In addition to the link on the arithmetization of analysis that I gave above, for contemporaneous criticism of Newton's calculus, see the Berkeley's famous The Analyst, A DISCOURSE Addressed to an Infidel MATHEMATICIAN. WHEREIN It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith, https://en.wikipedia.org/wiki/The_Analyst. For a good history of the 19th century rigorization efforts see for example Judith Grabiner's The Origins of Cauchy's Rigorous Calculus, or Carl Boyer's The History of the Calculus and Its Conceptual Development (both of which I own), or any of the many other histories of the math of that era. https://www.amazon.com/History-Calculus-Conceptual-Development-Mathematics/dp/0486605094/ref=pd_lpo_sbs_14_t_0?_encoding=UTF8&psc=1&refRID=BN7J3SR7H6NX77YD4531, https://www.amazon.com/Origins-Cauchys-Rigorous-Calculus-Mathematics/dp/0486438155 > This bias towards one or the other european originator is common in articles. I happen to know a lot more about Newton than I do about Leibniz, but again, if there is a secret, suppressed, rigorous theory of infinitesimals, surely some kind soul would throw me a link, if only to show me the error of my ways, yes? > It is also common to entirely fail to mention Seki. A Google search did not turn up a relevant reference among the many disambiguations.

Is there an "original theory?" This would be new to me and of great interest. My understanding is that Newton could not logically explain the limit of the difference quotient, since if the numerator and denominator are nonzero, the ratio is not the derivative (what Newton called the fluxion). And if they're both zero, then the expression 0/0 is undefined. So Newton could explain the world with his theory, but he could not properly ground it in logic. He understood this himself and tried over the course of his career to provide a better explanation, without success. Fast forward 200 years and the usual suspects Weierstrass, Cauchy, et. al. finally rigorized analysis. The crowning piece was set theory; and in the first half of the 20th century the whole of math was reconceptualized in terms of set theory. This overarching intellectual project is known as the arithmetization of analysis. https://www.encyclopediaofmath.org/index.php/Arithmetization_of_analysis (The Wiki article is wretched, the one I linked is much better). Now OP suggests that there was actually a rigorous theory based on infinitesimals that got unfairly pushed aside by the limit concept. [Am I characterizing OP's position correctly?] I am asking, what is that theory? I've never heard of it and would be greatly interested to know if there's a suppressed history out there. I'll also add that in modern times we have nonstandard analysis, which does finally rigorize infinitesimals; and smooth infinitesimal analysis (SIA), which is an approach to differential geometry that uses infinitesimals. But neither of these theories are the "suppressed" theory, if such there be. Have I got the outline right? What is this suppressed theory? Who first wrote it down, who suppressed it, and why haven't I ever heard of it?

I stand by my remarks until corrected with actual facts.

> the claim that the theory which underpinned the subject for long after its creation was wrong I didn't click on the link but your error is that there was no theory underlying Newton's calculus till the late 19th century finally nailed it down. Newton himself perfectly well understood that he couldn't put his fluxions on a logically rigorous foundation and made several unsuccessful attempts. Took another 200 years to nail it down with the modern theory of limits. If you claim there was a rigorous theory of infinitesimals before nonstandard analysis, please reference it here. It would be news to me and I know a little about this subject. NSA is claimed by its proponents to offer some pedagogical advantages (which 40 years of practice since Keisler's book have failed to demonstrate) but nobody claims there are any theoretical benefits, since NSA is a model of the same first-order axioms as standard analysis. ps -- I gave your paper a quick skim. One thing that jumped out is that you say that "for some reason" infinitesimals went away and limits came into favor after 1900. Well duh, that's because nobody could make infinitesimals rigorous and the theory of limits DID make calculus and analysis rigorous. You might say that limits replaced infinitesimals for the same reason round wheels replaced square ones. They work better. And NSA is like Stan Wagon's square-wheeled bicycle. It proves that you can do it, but that doesn't mean we all should. https://www.math.hmc.edu/funfacts/ffiles/10001.2-3-8.shtml My take on this subject, clearly yours differs but I don't think you have made your case.

\[ y = \int f(x) dx \]

(Wiseguy kid): But isn't multiplication just repeated addition? How can you have apple-many oranges?

The discussion so far has been about the physics; but since this is the math section I'd like to say a word about the math. As a math-trained person when you tell me that 2 times 6 is 12, I believe that. I could drill it down to the Peano axioms. And it tracks a highly obvious and familiar fact of nature, namely that two rows of six are the same as six rows of two and there are twelve of them altogether. I can see the living proof of this in the world every time I buy a carton of eggs. However if you ask me what 2 feet times 6 pounds is, I know that's 12 foot-pounds and I can conceptualize it physically. But if I put on my formalist hat, I confess I have no idea what that means in math. I can't drill down foot-pounds to anything I know in set theory. I actually have no idea what it really is. As someone noted in this Stackexchange thread, we tell kids you can't add apples to oranges, and then we tell them to multiply feet times pounds. What kind of sense does that make? http://physics.stackexchange.com/questions/98241/what-justifies-dimensional-analysis No less a genius than professor Terrance Tao has blogged on exactly this subject. https://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis/ I don't have time to read this today otherwise I'd summarize as much as I understood. Hopefully I'll get to that later. Meanwhile I wanted to toss these links out there because this really is a good question. What is a foot-pound, really?