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Comment by comstock | Hacker News Reader

I love discrete math, it seems so much cleaner in general. I wish there were more reformulations of calculus, other numerical methods into discrete maths.

I think Knuth’s concrete mathematics might have been an attempt at this, but I’ve never found time to dig into it in depth. Perhaps I should try again...

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Discrete calculus is a thing [0], although it is not so much a reformutaltion of infinitesimal calculus as it is its own field that borrows heavily from calculus.

[0] https://en.wikipedia.org/wiki/Finite_difference

In some sense, don't the modern formulations of real analysis, etc. already start from as close to discrete maths as you can get (set theory)?

Sets -> Naturals -> Rationals -> Reals

I don't understand how you could reformulate study of continuous structures into discrete math in any sense other than the above.

Every mathematical object (ok, this is false but that's not the point here) can be constructed in ZFC (the standard axiomatic framework for set theory) so you can construct the real numbers in terms of sets (if you want more precise informations on this construction look up Dedekind cuts).

However this is irrelevant to, say, analysis, you could define the real numbers as the unique (up to isomorphism) complete, ordered, archimedean field and do analysis just as well, so I'd say that you are right in some sense and some formulation, but it's a bit of a stretch to consider analysis as starting from discrete maths.

I also don't see how set theory fits into discrete maths, apart from the basics it seems pretty far from the common structures studied in discrete maths.

Pick the Grothendieck-Tarski axiom instead, and use category theory to build ZFC via topos. This path is "big" enough to handle all the interesting sets; it can't deal with proper classes, but proper classes are kind of metaphysical anyway.

[0] https://en.wikipedia.org/wiki/Tarski–Grothendieck_set_theory

Sure, but ZFC by itself also deals with every interesting set, I was just being nitpicky of my own assertion.

I'm not familiar with TG, what's the relation between it and ZFC+some large cardinal axiom?

I understand that it's a stretch; thus the "in some sense" and "close".

See my last sentence. It's not clear to me how you could reformulate analysis, which in many ways is the study of the infinite, into discrete maths in any way other than the very loose sense of starting with ZFC.

I loved Knuth’s concrete mathematics too. But I don’t think it is an attempt at reformulating calculus into the discrete math framework. Instead in many places in concrete math, knowledge of calculus is assumed, especially in later chapters about generating functions etc.

He did elsewhere suggest teaching calculus by Big O notation: http://www.ams.org/notices/199806/commentary.pdf

I would be excited to see someone try that.

UPenn's Calculus I+II courses with Robert Ghrist uses this sort of notation right at the beginning (it takes the Talyor Polynomial as the natural starting point, rather than derivatives, with knocking off terms of the summation involves factoring them out into the O-notation block).

I found his original and not abbreviated version

https://www-cs-staff.stanford.edu/~knuth/calc

Now, how do I TeXify it on an iPad?

I've TeXed it a while ago; scroll down here: https://shreevatsa.wordpress.com/2014/03/13/big-o-notation-a...

What you're looking for probably either already exists, or doesn't make any sense, depending on how you clear up the ambiguities in what you've said.

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