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seen Dec 30 '14 at 15:37

Dec
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comment Which universities teach true infinitesimal calculus?
@Dima, here is what I found at wiki: "His foundational work on algebraic geometry is at a higher level of abstraction than all prior versions. He adapted the use of non-closed generic points, which led to the theory of schemes. He also pioneered the systematic use of nilpotents. As 'functions' these can take only the value 0, but they carry infinitesimal information, in purely algebraic settings."
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comment Which universities teach true infinitesimal calculus?
@Dima, it has been my impression that in the context of algebraic geometry one doesn't need hyperreal infinitesimals and Grothendieck-style nilpotent infinitesimals are sufficient. Is this correct in the context of real algebraic geometry?
Dec
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comment Which universities teach true infinitesimal calculus?
@DimaPasechnik, I am not familiar with the literature using a hyperreal framework to do real algebraic geometry. What are you referring to exactly?
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comment Which universities teach true infinitesimal calculus?
@Dima, that's precisely the point. When it comes to interpreting the procedures found in Euler, modern Weierstrassian frameworks are less suitable than modern infinitesimal frameworks, be it Robinson's, Lawvere's, or Bell's. Mostly scholars have been trying to understand Euler from the viewpoint of Weierstrassian frameworks, hence the routine claims of "non-rigorous" and even "dismal" as J. Gray put it. The source of the disdain for Euler lies in the inappropriateness of conceptual frameworks being applied to interpret his procedures.
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comment Which universities teach true infinitesimal calculus?
@Dima, Euler's computations are far more rigorous than they are reputed to be. See for example this text: ams.org/notices/201408/rnoti-p848.pdf
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Dec
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comment Which universities teach true infinitesimal calculus?
@DanFox, to respond to your comment about "non-standard analysis", I would like to share with you the reaction of philosopher Salanskis to the term "non-standard" from 1988: S'il a donc choisi d'emblée le nom d'analyse non standard pour le jeu qu'il définissait, c'est en référence à la notion de modèle non standard plutôt que pour suggérer le caractère divergent, subversif ou extravagant de ce jeu. De nombreux esprits, même les meilleurs, s'y sont pourtant trompés, de manière regrettable." What Salanskis is pointing out is that the term "nonstandard" is misleading.
Dec
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comment Which universities teach true infinitesimal calculus?
@Dima, the term "rigorous infinitesimal calculus" sounds precisely like epsilon-delta calculus, in other words the opposite of what is intended here. Similarly, "axiomatic infinitesimal calculus" sounds like you axiomatize the real numbers instead of building them, and then again do epsilon-delta.
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revised Which universities teach true infinitesimal calculus?
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comment Which universities teach true infinitesimal calculus?
Thanks, @BenCrowell. I found the following comment there: "I find it disheartening that the book was allowed to go out of print, and that there are now no (as far as I'm aware) current popular calculus texts using the infinitesimal-based approach. I, like the original poster, and like most students learning today, was always confused by what you could and couldn't do with dy and dx. How I wish I'd had this book 20 years ago." It is nice to see such enthusiasm about Keisler's book but one correction: the book is currently in print at Dover. Furthermore, there are today numerous TIC textbooks.
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awarded  Benefactor
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accepted Which universities teach true infinitesimal calculus?
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revised Which universities teach true infinitesimal calculus?
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comment Which universities teach true infinitesimal calculus?
@PietroMajer, since I see you are on the faculty at Pisa in Italy, could you comment on the recent interest in teaching calculus in highschool using infinitesimals? Are you familiar with Giorgio Goldoni's book?
Dec
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comment Which universities teach true infinitesimal calculus?
@FedericoPoloni, it is already.
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revised Which universities teach true infinitesimal calculus?
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comment Which universities teach true infinitesimal calculus?
@PietroMajer, nice to hear from you. Why don't you post this as an answer?
Dec
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comment Which universities teach true infinitesimal calculus?
@Dan, I added a note to the question to explain the use of the term. This does not mean that the approach limited purely to epsilon-delta and an Archimedean continuum is "false"; rather, it means that it does not use true infinitesimals.
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revised Which universities teach true infinitesimal calculus?
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answered Which universities teach true infinitesimal calculus?