Re: Modeling infinitesimals with 2x2 matrices
From: Steve Vickers <s.j.vickers@cs.bham.ac.uk>
I thought that was the whole reason for contemplating infinitesimals in the first place - 0/0 cannot be meaningful, but d/d is 1, ed/d is e etc.
Well, you're not alone in thinking that way. This was the basis for Robinson's invention of nonstandard analysis: the belief that a/b has to be defined for all nonzero b in order to make infinitesimals nonparadoxical. Instead of formulating division a/b as an operation, go back to its motivating formulation as a system in search of a solution, in this case the system consisting of the one linear equation a = bx in one unknown. Had the system been one of ordinary or partial differential equations, there would be no argument that the solution space could turn out quite oddly shaped. Now when a and b are reals the solution space is a rectangle: only the column indexed by b=0 is undefined. This remains true when a and b are extended to the complex numbers, or even to the quaternions. But if you extend the domain to the algebra R(2) of 2x2 real matrices, the columns indexed by singular matrices now lose some of their entries. But not all, and so the solution space ceases to be rectangular. Robinson believed that the way to make infinitesimals safe for analysis was to make the solution space for a = bx rectangular. Today's logicians are magicians with logic: if logic indicates the impossibility of a rectangular solution space, no need to abandon that goal, just bend logic until the solution space does become rectangular. The students will bend with you, at least those who've approached nonstandard analysis with the proper upbeat spirit about how much simpler analysis becomes when infinitesimals can be objectified. Power tools are wonderful. To answer your question (or comment), in the system of refined numbers I described, if b is infinitesimal and nonzero, a = bx is solvable if and only if a is infinitesimal. The division table is no longer rectangular. So what? One might grumble that a = bx can't have an infinitesimal part when a and b are both infinitesimals, but in the simple cases that's a plus. In more complicated cases, 2x2 matrices aren't enough, you need nxn matrices, with distance of nonzero entries from the diagonal measuring the degree of their infinitesimality (if that's a word). In this case d^n = 0 only for higher n's. After thinking along those lines for a bit more the other day, I decided that even though I liked this approach better than throwing ultrafilters at it, it still wasn't as good as doing analysis in Boole's finite difference calculus with h remaining unbound throughout, the approach I'd used since the early 1970's. That approach has the great advantage of being able to use the same analysis in classical and quantum physics by setting h=0 to interpret a result classically and setting it to Planck's constant to interpret the same result quantumly. As a case in point, the same integration formulas can deliver areas under smooth curves and discrete summations of e.g. n^3, the latter with h=1. (I already wrote a bit about that two or three messages ago.) The right power tools are even more wonderful. Vaughan Pratt
But if you extend the domain to the algebra R(2) of 2x2 real matrices, the columns indexed by singular matrices now lose some of their entries. But not all, and so the solution space ceases to be rectangular.
On reflection this is not so simple in the case when b in a/b is infinitesimal. First, noting that R(2) is noncommutative, the requirement should be phrased as two equations, a = bx and a = xb to prevent multiple solutions whose diagonal is not constant. But while this duplication then determines a unique real part (the diagonal), the two equations fail to pin down the infinitesimal part (the upper right entry). That's the sort of thing that happens with matrices of less than full rank. When there is no solution, certainly a/b should be considered undefined. But when there are multiple solutions, the question arises as to whether to punt completely (as with 0/0) or do something creative such as setting the undefined infinitesimal part to 0 (as with the ratio of two infinitesimals). One test is whether the "dominant" term is fixed, but this breaks down for 0/b. A better test would be to use the rank of b to decide how much of the quotient a/b to ignore---if b has rank 1 (a nonzero infinitesimal) then ignore the infinitesimal part of a/b. --------------- One virtue of Robinson's approach is its universality with respect to all first-order definable functions; this however is not sufficient to compensate for its more counterintuitive aspects. Now that I'm starting to see that the zero-divisor approach is less easily managed than I'd first thought, I'm not so sold on it either at this point (but maybe all its difficulties have been overcome somewhere...?). Meanwhile I remain convinced that Boole's finite difference approach to handling infinitesimals is superior (recall the trick here: setting h=0 instead of some positive quantity like 1 or Planck's constant introduces no artificial singularities with Boole's method). His 1860 *A Treatise on the Calculus of Finite Differences," substantially revised by J.F. Moulton for the 1872 edition after Boole's death, is 336 pages of inspired analysis. (You can get second hand copies for $10 from Amazon; my very second hand copy has "F.S. Curry, Trin. Coll., Feb. 1881" written on the inside cover.) The preface to the first edition starts out, "In the following exposition of the Calculus of Finite Differences, particular attention has been paid to the connexion of its methods with those of the Differential Calculus---a connexion which in some instances involves far more than a merely formal analogy. Indeed the work is in some measure designed as a sequel to my *Treatise on Differential Equations*. And it has been composed on the same plan." An updated version of this book incorporating the greatly matured perspective on linear algebra since then could be a worthwhile project for someone interested in improving on the existing explications of infinitesimals as real objects. While Boole's system beats the current crop hands down in principle (in my view anyway), in outlook it is showing its age. Category theory creatively applied might also help. I confess to having no idea how intuitionistic logic could be brought to bear effectively though. I can see that not cancelling certain double negations might preserve certain nuances that convey certain constructively motivated notions, but to my untrained eye these come across as nuances with a capital N when their contribution is assessed in the larger picture of alternative approaches to constructivizing infinitesimals. That makes me either a beer guzzler at a wine tasting or the owner of a screwdriver in a room full of hammer owners depending on one's outlook. :) YBMV (Your biases may vary.) Vaughan Pratt --------------------------
Vaughan Pratt writes:
Why not model d as the matrix
0 1 0 0 ?
This is a perfectly good quantity, adding and scaling just like any real, e.g.
2d = 0 2 0 0.
And obviously d^2 = 0.
Part of this idea is implicit in the usual algebraic geometry treatment of infinitesimals as nilpotents. In addition to the usual "point", such that complex functions on this space form the commutative ring C, algebraic geometers like to think about the "point with nth-order nilpotent fuzz", such that complex functions on this space form the commutative ring C[d]/<d^n = 0>. They visualize this as a space slightly bigger than a point: just big enough to tell the difference between the function 0 and the function whose first n-1 derivatives equal zero! To deal with this sort of "space" in a precise way, someone like Grothendieck invented the category of affine schemes, which is just the opposite of the category of commutative rings. But affine schemes are happier as part of a larger category of schemes... and thus topos theory was brought kicking and screaming into the world. To see how this led to a really nice treatment of infinitesimals, see: F. William Lawvere, Outline of synthetic differential geometry, available at http://www.acsu.buffalo.edu/~wlawvere/downloadlist.html or Anders Kock, Synthetic Differential Geometry, Cambridge U. Press, Cambridge, 1981. But, it's also tempting to embed the commutative ring C[d]/<d^n = 0> into the noncommutative ring of nxn complex matrices, by letting d be a slightly off-diagonal matrix, like this: 0 1 0 0 0 0 1 0 (in the case n = 4) 0 0 0 1 0 0 0 0 (Vaughan is considering the case n = 2.) And this is more like how Alain Connes thinks of infinitesimals: as part of the bigger world of noncommutative geometry! Best, jb
participants (2)
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John Baez -
Vaughan Pratt