wtf

As an alternative, may I suggest running a webserver, running your Python program as a CGI script, and using simple HTML forms for data input. Different approach to give a Python program a GUI.

I don't know what it is but it's pretty cool!

I think the OP is not around but I read through the paper a couple of times and have some thoughts. There are two things going on in the paper. One, the OP is making the point that there are striking similarities between infinitesimals as they were used in 17th century math; and the nilsquare infinitesimals of Smooth infinitesimal analysis (SIA). This point of view says that, say, if we went back to the 17th century but knew all about category theory and differential geometry and SIA, we could easily show them how to logically found their subject. They were close in spirit. Ok. That might well be, and I don't agree or disagree, not really knowing enough about SIA and knowing nothing about Leibniz (being more a Newton fan). So for sake of discussion I'll grant the OP that point. But the other thing that's going on is that the OP seems to feel that the history itself supports the idea that they somehow understood this, or that they had a rigorous theory of infinitesimals that was shoved aside by the theory of limits in an act more political than mathematical. That's the second thesis of the paper as I understand it. But the OP presents no historical evidence, none at all, that there was any kind of rigorous theory of infinitesimals floating around at the time. On the contrary, the history is that Newton himself well understood the problem of rigorously defining the limit of the difference quotient. As the 18th century got going, people noticed that the lack of foundational rigor was causing them trouble. They tried to fix the problem. In the first half of the 19th century they got calculus right, and in the second half of the 19th and the first quarter of the 20th, they drilled it all the way down to the empty set and the axioms of ZFC. That is the history as it is written, and there isn't any alternate history that I'm aware of. If there were, I would be most interested to learn about it. The OP makes a historical claim, but doesn't provide any historical evidence. That bothers me. So to sum up: * From our modern category-theoretic and non-LEM and SIA perspective, all of which is math developed only in recent decades, we can reframe 17th century infinitesimals in modern rigorous terms. I accept that point for sake of discussion, though I have some doubts and questions. * But on the historical point, you are just wrong till you show some evidence. The historical record is that the old guys KNEW their theory wasn't rigorous, and that as time went by this caused more and more PROBLEMS, which they eventually SOLVED. They never had a rigorous theory and they never thought they had a rigorous theory. But if they did I'd love the references.

@dasnulium, Can you please explain this passage? "Mathematicians could however always claim that they were not assuming that the so-called law of excluded middle (LEM) applies to the continuum, and that nilpotency is a corollary of this. But as the supporters of LEM gained influence in the late nineteenth century this position became less tenable; ..." * What does it mean that LEM does or doesn't "apply to the continuum?" That makes no sense to me. LEM applies or doesn't apply to propositions. * How is LEM or its denial a corollary of nilpotency? * The supporter of LEM gained influence in the 19th century? Are you making the claim that 17th and 18th century mathematics was a hotbed of LEM denial? That flies in the face of the written history, doesn't it? My understanding is that denial of LEM came into math via Brouwer in the early 20th century, and not before then; and that it's making a contemporary resurgence due to the computational viewpoint. But to say that the supporters of LEM gained influence in the 19th century doesn't seem right. My understanding is that LEM had universal acceptance in math until Brouwer. Would appreciate clarity on these points, thanks.

I'm taking another run at your paper. I just read the intro. Some of this is sinking in. I agree with your point that infinitesimals in the hyperreals are not nilpotent hence aren't quite the right model for the powers of epsilon that go away. Am I getting that? I think you are clarifying the distinction between an approach like SIA and the nonstandard analysis model. I think you have a good point. Now what I am not too sure about is what you are saying about the status of the infinitesimal approach. I always thought it was a search for rigor; but I think you're saying they already had rigor and got unfairly demoted. Am I understanding this right? My point earlier was that SIA is very recent and quite modern in the sense of being based on category theory. They did not have that point of view in the 18th century. As far as I know. Is that the case you're trying to make? *

"The third possibility is that the crisis was a side-effect of the introduction of Georg Cantor’s theory of transfinite numbers. The theory depends on the Axiom of Choice, which implies LEM for the continuum ..." Sorry you didn't claim the reals require choice, you claimed Cantor's theory of transfinite numbers does. Equally wrong. And what does implying "LEM for the continuum" mean?

That's as true today as when I wrote it a few weeks ago. But you surely don't need choice to define the reals. See any modern textbook on real analysis for a construction of the reals using only the axioms of ZF. I'm not sure why you took my correction of a minor and inconsequential error in your paper, and doubled down to a demonstrably false claim. It seems like digging a hole deeper where a simple "Thanks for the clarification" would be appropriate.

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.

What kind of patterns are you looking for? It would help if you were specific. Remember that there are many domain registrars out there, and they each have their own policies for how they handle expired domains. Rather than look at the data 50 at a time, if I had this problem I would write a scraper to hit this site and collect a database of thousands or (if project has a serious budget, for example if I were doing this for a company) millions of records, along with the registrar, the date/time, the phase of the moon, whatever else you think might be of interest. Then I'd datamine the heck out of it, looking for ... whatever it is I'm looking for. What exactly are you looking for?

>I apologize, I do not understand English well, I speak Russian. From Google Translate: Значит ли это, что ваши идеи зависят от того, как люди изменяют свое поведение?

> it is only important that the country behaves as a native, and not as a stranger, then the value will not flow abroad. So, does that mean your idea depends on human beings altering their behavior? Secondly, although you are presenting the technical implementation details of your idea; isn't the idea itself essentially political? That is, normative. "What should be," rather than based in what actually is. Thirdly, by contrasting native behavior to stranger behavior, are you arguing for some form of nationalism as opposed to globalism? As you know, this is the core issue of the day. It's what the Yellow Vest riots in France are about.

"I only have a vague idea of what the syntax should look like right now ..." That would be the place to start. First you figure out WHAT you want to build; then you figure out HOW to build it. Projects that reverse the order are doomed, since you have no basis for design decisions. You don't know what you want so you end up with something else, as they say. So first spec out your language. Doesn't have to be complete in every last detail, but get the broad outlines and main syntactic features nailed down on paper. Then the implementation will be far easier, since you are harnessing tools and techniques to a particular objective.

You're entitled to that opinion. You'll have to convince the entire worldwide community of mathematicians, along with all of the physical scientists (biologists, physicists, chemists, ...) whose work is grounded in infinitary math. Biology? Yes. How are you going to replace the importance of differential equations in the life sciences? Are you really going to recast everything in the world that depends on diffEq with the theory of finite differences? A huge intellectual project with ZERO practical payoff. Akin to recasting astronomy by taking the earth to be the center of the universe. It COULD be done with great difficulty, at the cost of making everything incredibly convoluted. But why? I don't disagree that neo-intuitionism is making a comeback via automated proof checking software, but it's a long way from that to overthrowing LEM in mainstream math and logic. But as I noted, you are raising interesting points that would be sensible in their own thread. Denial of LEM, advocacy of constructive math, denial of noncomputable reals (what do you make of Chaitin's constant then? Even in computability theory they prove the existence of noncomputable problems), etc., are all interesting. But you are hijacking this thread to grind your constructivist axe. Nothing you've said bears on the thread topic. Start a new thread titled, "What do you think about constructive math?" or "Down with ZF and the evil Cantorians," or "The hell with Mrs. Zermelo and her pro-Choice views." If you did that we could have an interesting discussion. But in this thread? Just a thread jack by someone pushing an agenda.

You're wrong about this. Cantor's theorem is a valid theorem of first-order ZF. It's true in any model of ZF, even a countable model. In set theory we are not required to "construct" anything, as I'm sure you know. https://en.wikipedia.org/wiki/Cantor's_theorem Point being (I do hope you understand this subtle point) that even in a countable universe, there is no bijection between a set and its powerset. And the simple and beautiful proof can be appreciated by a high school student. But if you are using Skolem's theorem to resolve pengkuan's issues, this is surely far off the mark. Perhaps a separate thread on Skolem's paradox would be interesting, but in the context of the present thread I can't see how it sheds any light.

Completeness is a second order property. You're reading too many Wiki pages and too little actual math. As I'm sure you would understand IF you understood, even in a countable model of the set theory, the reals are uncountable. If you don't understand why that is, you don't understand Skolem's result. Even in a countable model, there is still no bijection between the naturals and the reals. You haven't troubled yourself to reply to my observation that the computable real line is full of holes and fails to satisfy the Intermediate value theorem, making it a poor representation of anyone's idea of a continuum.

The constructible real line doesn't satisfy the Intermediate value theorem. Hell of a poor model of the continuum, don't you agree? Contrary to your claim that there are no holes, the constructible real line is full of holes, one hole where each noncomputable real used to be. There are many Cauchy sequences that do not converge. Worst model of the continuum ever.

ps -- The larger point is that OP seems to believe that there are natural numbers that are infinite; and can't distinguish between the fact that there are infinitely many natural numbers, but each one is finite. On that basis, I don't think the ordinals are going to reduce the confusion in this thread. If as @studiot says I sounded "vitriolic" my apologies once again. I am staying out of this thread from now on.

Oh my. You don't believe in \(\omega + 1\)? Perhaps you meant the smallest non-finite ordinal. And by order, perhaps you meant well-order. I'll leave it here as to not appear to be piling on. ps -- Ok I'll pile on just a little bit more. > So it is fairly easy to see that, for any subset of N (it's ordered recall) if there exists a largest ordinal n then this corresponds to the cardinality of our subset Really? That's fairly easy to see? It's not even true as you expressed it, and I'm not even sure what you're trying to say. The smallest ordinal larger than any of the elements of {2, 4, 6} is 7, but the cardinality of that set is 3. The largest ordinal in the set is 6.I couldn't understand what you're getting at. "If there exists a largest ordinal? There is no largest ordinal. Can you clarify your thoughts please?

https://math.stackexchange.com/questions/588/what-functions-can-be-represented-as-power-series