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This question reminded me of a possibly stupid idea that I had a while back.

On page 2 of this paper, while discussing Euclid's axioms of plane geometry and spatial geometry, Manin makes an extremely interesting comment:

Euclid misses a great opportunity here: if he stated the principle

“The extremity of an extremity is empty”,

he could be considered as the discoverer of the

BASIC EQUATION OF HOMOLOGICAL ALGEBRA: d^2 = 0.

Ever since I read this, I've had a suspicion that the equation "d^2 = 0" of homological algebra is somehow related to the equation "epsilon^2 = 0" of (first-order) calculus (as in Newton)Newton)*, since the latter equation can be interpreted as saying "a very very small quantity is zero" which at least superficially seems similar to "the extremity of an extremity is empty". I once explained my suspicion to Dan Erman over beers, and he responded by asking another question: Can we do some sort of homological algebra using the equation d^n = 0 rather than d^2 = 0? Perhaps if d^2 = 0 can be related to first-order calculus, then d^3 = 0 can be related to second-order calculus, and so on...

I don't really have a specific question to ask -- I just thought I might put this idea out there. Maybe someone can tell me why this idea is stupid, or why it is not stupid.

*or the ring of dual numbers k[epsilon]/(epsilon^2) if you're an algebraist or an algebraic geometer.

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# Homological algebra and calculus (as in Newton)

This question reminded me of a possibly stupid idea that I had a while back.

On page 2 of this paper, while discussing Euclid's axioms of plane geometry and spatial geometry, Manin makes an extremely interesting comment:

Euclid misses a great opportunity here: if he stated the principle

“The extremity of an extremity is empty”,

he could be considered as the discoverer of the

BASIC EQUATION OF HOMOLOGICAL ALGEBRA: d^2 = 0.

Ever since I read this, I've had a suspicion that the equation "d^2 = 0" of homological algebra is somehow related to the equation "epsilon^2 = 0" of (first-order) calculus (as in Newton), since the latter equation can be interpreted as saying "a very very small quantity is zero" which at least superficially seems similar to "the extremity of an extremity is empty". I once explained my suspicion to Dan Erman over beers, and he responded by asking another question: Can we do some sort of homological algebra using the equation d^n = 0 rather than d^2 = 0? Perhaps if d^2 = 0 can be related to first-order calculus, then d^3 = 0 can be related to second-order calculus, and so on...

I don't really have a specific question to ask -- I just thought I might put this idea out there. Maybe someone can tell me why this idea is stupid, or why it is not stupid.