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I recently started reading Amir Alexander's 2014 book "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". It's a really great read, th

I recently started reading Amir Alexander's 2014 book "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". It's a really great read, this is pre-Newton, pre-Leibniz. People were doing calculus more than two hundred years before those guys! They struggled with paradoxes to do with infinitely many zero length points making up a line, and the Jesuits just outright banned that stuff. Partly because it was protestant mathematics, but mostly because they wanted a version of mathematics that was somehow eternally unchanging and crystal clear (like how God is supposed to be.) Boy did that not work out. Anyway, these early mathematicians used imagery like the threads in a cloth to describe how infinitely many lines form a plane surface, or the pages of a book to help understand how infinitely many planes form a solid. So cool! In the beginning it was Italians working this stuff out (including Galileo). But then the Jesuits totally banned it thereby casting a pall over Italian mathematics for hundreds of years! For a modern treatment of infinitesimals, check out "smooth infinitesimal analysis" [1]. The way to avoid all the contradictions is to weaken the underlying logic: in this case they remove the law of excluded middle, so it's not the case that something is either true or it is false. I think the Jesuits would definitely have trouble with that! [1] https://xorshammer.com/2008/08/11/smooth-infinitesimal-analysis/ https://xorshammer.com/2008/08/11/smooth-infinitesimal-analy...

maaku 11y ago
Removing the law of the excluded middle strengthens the underlying theory. It makes it constructivist...
erichocean 11y ago
> The way to avoid all the contradictions is to weaken the underlying logic Pet peeve: I would argue that constructive logic is stronger, in the sense that it allows more reliable (read: usable in the real world) results than classic logic allows. But that's a value judgement, and there's no definitive right answer here. Totally agree Re: Infinitesimals. Dual numbers[0] are crazy useful in computer science (c.f. Google's Ceres Solver[1]). [0] https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Dual_number [1] http://ceres-solver.org/ http://ceres-solver.org/
- mathgenius 11y ago
  I agree with you, there's going to be more shades of grey using intuitionistic logic. I guess I was taking the point of view of the Jesuits! So actually I understand about Dual numbers, but from what I see this is not the same thing as smooth infinitesimal analysis (there's no playing around with the logic.) I feel like if we scratch this a bit more we are going to understand what Grothendieck was on about with topos theory [1]. [1] http://www.oliviacaramello.com/Unification/Unification.htm http://www.oliviacaramello.com/Unification/Unification.htm

maaku 11y ago

Removing the law of the excluded middle strengthens the underlying theory. It makes it constructivist...

erichocean 11y ago

> The way to avoid all the contradictions is to weaken the underlying logic Pet peeve: I would argue that constructive logic is stronger, in the sense that it allows more reliable (read: usable in the real world) results than classic logic allows. But that's a value judgement, and there's no definitive right answer here. Totally agree Re: Infinitesimals. Dual numbers[0] are crazy useful in computer science (c.f. Google's Ceres Solver[1]). [0] https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Dual_number [1] http://ceres-solver.org/ http://ceres-solver.org/

mathgenius 11y ago
I agree with you, there's going to be more shades of grey using intuitionistic logic. I guess I was taking the point of view of the Jesuits! So actually I understand about Dual numbers, but from what I see this is not the same thing as smooth infinitesimal analysis (there's no playing around with the logic.) I feel like if we scratch this a bit more we are going to understand what Grothendieck was on about with topos theory [1]. [1] http://www.oliviacaramello.com/Unification/Unification.htm http://www.oliviacaramello.com/Unification/Unification.htm