What is Reddit's opinion of A Primer of Infinitesimal Analysis?

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Nilsquare infinitesimals are the main idea in smooth infinitesimal analysis which is different from nonstandard analysis, see the guidebook. Neglecting higher powers of the increment is not 'hand waving' because f(x + E) = f(E) + Ef'(x) always allows us to cancel first order increments, so their size can be decreased without affecting their ratio. But if their size can be decreased indefinitely then the higher powers which have remained can be rendered indiscernible (by whatever external method is being used). Also note, if you give the increments a value which is discernible and leave them in the equations you get a form of finite element analysis.

If you go deep enough smooth infinitesimal analysis is based on category theory according to the main text on the subject.

Traditional calculus, based on set theory, is plagued by antinomies stemming from the inherently punctiform nature of set theory. It forces you to pick an encoding of a "continuum" as a "set of points." Naturally, this is both tortuous and torturous!

You might find An Invitation to Smooth Infinitesimal Analysis more straightforward, and the full book actually refreshing.

There are frameworks within which you can have analysis developed on top of infinitesimals. You might want to look for smooth infinitesimal analysis and nonstandard analysis. Both systems are well within 20th century mathematics, so they can't quite be compared with Newton's or Leibniz's ideas, though.

Here you have a calculus book with hyperreal (nonstandard analysis) infinitesimals, and this book works with smooth infinitesimals.

>I'm (perhaps unduly) suspicious of constructive mathematics, so it might be nice to see a counter point.

Why though?

The smooth analysis book I know best (that is to say, at all) is Bell's <em>A Primer of Infinitesimal Analysis</em>. It looks like the second edition has an expanded introduction which will touch on philosophical issues, but the book is mostly a textbook.

Following up on the Hellman/Shapiro suggestion, some papers on the subject from them:

• &quot;Towards a Point-free Account of the Continuous&quot;
• &quot;The Classical Continuum Without Points&quot;
• &quot;Frege Meets Aristotle: Points as Abstracts&quot;
• &quot;Aristotelian Continua&quot;
• &quot;Regions-Based Two Dimensional Continua: The Euclidean Case&quot;

There's specific PhilPapers sections on various areas of philosophy. Most of these aren't curated, so may not be super accurate.

Lastly: do you check Philosophia Mathematica? That's the best journal in philosophy of math and will definitely publish stuff on specific areas of math.

thanks for the reply. i thought i might give some references here in case you or someone else is interested in smooth infinitesimal analysis (SIA) and synthetic differential geometry (SDG).

• an invitation to smooth infinitesimal analysis by john l. bell

• an introduction to smooth infinitesimal analysis by michael o'connor

in the introduction by o'connor, he explains what i was getting at in my question (which was originally typed on mobiel without easy access to references). bell does the same in the below referenced primer. and that is, smooth infinitesimal analysis (and thus synthetic differential geometry) is a system which is a collection of some axioms, the adoption of intuitionistic logic, and then a model which says that the prior two things are consistent. with the logic and axioms, one can explore a lot of SIA and SDG without getting into the details of the model theory. but it's always been a curiosity of mine what that model theory is. i guess i could stop being lazy and learn it, but i'm almost not for sure i need to. i'm happy with someone saying that things are consistent and one get can on with doing SDG without having to worry about it. but there is some curiosity of what's actually going on, which i don't really understand.

further references that aren't immediately available online:

• a primer of infinitesimal analysis by john l. bell

• toposes and local set theories: an introduction by john l. bell

• models for smooth infinitesimal analysis by moerdijk and reyes

For the physics perspective you need something like A Primer of Infinitesimal Analysis.