I’ve been writing a post explaining the practical difference between the synthetic and nonstandard notions of infinitesimals. It was getting a bit long, so I’m splitting it into two posts, of which this is the first.
Synthetic differential geometry (or smooth infinitesimal analysis) is a way to add infinitesimals to the reals; one that is an alternative to the nonstandard analysis approach. In SDC, infinitesimals can be nilpotent: their square or some higher power can be zero. This allows you to formalize arguments such as “this quantity is so small, we can treat its square as if it is zero”. You can also formalize these arguments in nonstandard analysis, but with more care (you can’t actually set the square to be zero, but you can treat it as an even smaller infinitesimal). Nonstandard analysis cannot have nilpotent infinitesimals directly, because it is required to preserve first-order theorems about the reals (which includes the theorem that the only nilpotent element of the reals is zero).
You can formalize infinitesimal arguments at the level of calculus equally well using either approach, so why would you ever want nilpotent infinitesimals? Here are some examples with a differential geometric flavor. Consider two points on the real line, and move them together so that they coalsce into one point. In ordinary differential geometry, that’s all they are — one point. In the synthetic approach, you can treat this as a double point, with defining equations x2 = 0. You can imitate this in nonstandard analysis; I’ll explain how in the next post.
Here’s an example that’s considerably harder to simulate in nonstandard analysis. Consider four lines in the plane, two horizontal, and two vertical. Collectively they intersect in four points. If we let the two horizontal lines move towards each other, they become a double line, which intersects each of the two vertical lines in a double point. If we now let the two vertical lines degenerate into a double line, we have two double lines intersecting in a quadruple point. But not just any quadruple point (there’s more than one kind), they intersect in the quadruple point with defining equations x2 = 0 and y2 = 0.
Thanks for posting this! I look forward to part 2 :D.
Are there any easily available references for the formal theory of nilpotent infinitesimals?
So are nilpotent infinitesimals is some sense the opposite of Ito increments?
In the sense that in the former case, squares of infinitesimals are smaller than we expect; in the latter case larger than we expect.
I wrote up some notes on it (the same notes that I mentioned in the comments on “Selling Infinitesimals”) and put them at:
http://arxiv.org/abs/0805.3307
There’s also John Bell’s “An Invitation to Smooth Infinitesimal Analysis” available at:
http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf
In addition to smooth infinitesimal analysis (which is essentially doing calculus with nilpotent infinitesimals) there is synthetic differential geometry (which is doing differential geometry with nilpotent infinitesimals. There’s an exposition available at:
http://www.math.uchicago.edu/~shulman/exposition/sdg/pizza-seminar.pdf
There’s also a book on synthetic differential geometry at:
http://home.imf.au.dk/kock/sdg99.pdf
There’s a general algebraic development of biplex numbers at http://www.rowan.edu/open/depts/math/osler/Biplex_Nos_Final_Corrected_Version_Aug_3.pdf including nilpotent infinitesimals in the form of dual numbers, along with the “perplex numbers,” which were apparently invented by a group of freshmen at St. Olaf College, with a norm ||h||=–1. These exotic beasties are useful for extending the usual formalism of special relativity to the case ||v||>c, as described in the American Journal of Physics — May 1986 — Volume 54, Issue 5, pp. 416-422.
You can almost define a useful nilpotent algebra in quantum mechanics, with exchange of coordinates as multiplication, and the product of superimposed fermions would vanish… but nature abhors a nilpotence, and the fermions remain stubbornly apart.
[...] This is a follow-up to this post. [...]