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May 16, 2006

Machinery

Samuel Lilley, on late 19th and early 20th century computing machinery:

The actual use of mathematics to solve problems about complicated machinery, or analogous problems in the electrical sphere, provided a new way of thinking for mathematicians. If a mathematical equation adequately represented the motion of the machine, then the motion of the machine equally represented the mathematical equation and could therefore be used as a means of solving the equation.
…

Fundamental to a very large class of mathematical problems is the differential equation. A large number of instruments called ‘integraphs’ were produced to solve special differential equations. But there are of little importance beside Kelvin's proposal for the mechanical integration of any ordinary differential equation. The principle of the thing is simple. The essential element, the integrator, is a continuous variable gear, which was already familiar in many integraphs and planimeters. The big step forward is to realize that by suitable mechanical connections between such integrators a machine can be produced to solve any differential equation. Kelvin pointed this out in regard to linear equations in 1876.

But he failed to overcome the difficulty that the friction drive from the integrator is not sufficiently powerful to drive the further mechanical connections which are required. And this difficulty was not overcome till 1931, when [Vannevar] Bush of the U.S.A. solved the problem by means of the ‘torque amplifier’. The principle of the torque amplifier is again extremely simple; it is essentially the same as the capstan of a ship. The capstan is kept rotating, a rope is wound around it, and it is found that a small pull on one end of the rope produces a much stronger pull on the other end. Kelvin, in fact, might have produced a complete differential analyser (as Bush's machine is called) in 1876, if he had happened to mention his difficulty to the deck-hand of a tramp steamer, or one of the many other workmen who use the capstan principle. In that way is scientific knowledge retarded in a society in which scientists and deck-hands do not usually move in the same circles.

While I had been dimly aware that electronic analog computers had once been relatively common, I hadn't considered that these had had mechanical predecessors. Consequently, the following nuggets were news to me:

[Come to think of it, I do recall learning a modelling notation in my undergraduate degree that produced diagrams that could have been mistaken for plumbing schematics.]

Posted by robin2 at 01:31 AM | Comments (0)

May 01, 2006

Notes on "Number & Numbers"

On reading Robin's translation of Badiou's ‘Number and Numbers’ I had a number of nagging thoughts, some of which I have tried to articulate below. Apologies in advance if the points raised have already been dealt with, contain technical errors, or are completely irrelevant.

The quotes here are confined to Chapter 1. This partly because I have read the beginning of the book more often, and so have had longer to think about it. It is also partly because this is where Badiou is setting out his stall, so to speak, and the biggest stumbling point for me is trying to get clear on what problem it is that he's trying to solve.

So anyway…

None of the modern thinkers of number […] have been able to offer a unified concept. […] We might expect a concept of number to subsume all these cases, at least the more “classical” among them, that is to say the whole natural numbers […] and the real numbers […] But nothing of the sort exists.

Why is this an issue?

Badiou points to the distinct characterization of natural numbers, integers, rationals, reals, complex numbers, quaternions, ordinals and cardinals. The concept of number is actually more frayed than he suggests: vectors, matrices and polynomials (for example) are all in certain respects number-like.

Furthermore, it isn't clear to me why we should expect an overarching concept of number, or why this expectation should be restricted in the way Badiou suggests. There doesn't seem any good reason why, say, complex numbers should be seen as poor cousins to the properly numerical. The only grounds for granting the natural numbers and the reals “classic” status is their intuitive relation with the activities of counting and of measuring respectively (as Badiou indicates): in which case a unified concept wouldn't be something we would expect at the outset.

[I]f the essence of “number” is only obtained through the specific nature of statements that constitutes its axiomatics, it is evident that the whole numbers and the reals have nothing in common with each other (as regards their concept) given that if one compares the axiomatic of the whole numbers with that of the reals, these statements are totally dissimilar.

This is over-stated. Informally we can say that the natural numbers are included in the integers, which are included in the rationals, and so on through the reals and the complex numbers. So an integer is a rational with a unit denominator, a real is a complex number with a zero imaginary part, etc. This “inclusion” cannot be understood as a straightforward subset relation if these number systems are understood as arising from a sequence of constructions (where, e.g., the integers are equivalence classes on pairs of natural numbers, and complex numbers are pairs of reals), but it is there nonetheless.

The whole thing could be turned around: accepting the integers, rationals, et al as given, but questioning why it is meaningful to say that these are all kinds of number. After all, if we can use the expression “the reals” in place of “the real numbers” then we might regard the term “number” as a mere relic. I think a good reason that we don't is due to the inclusion mentioned above. At each step in the sequence there is an injective mapping from the earlier system to the later which behaves in the way you would want. So for each integer there is a way to find a “corresponding” real, with zero mapped onto zero, unit onto unit, and arithmetic operations respected: if this were not the case then I think the reals wouldn't be accepted as numbers at all, and Dedekind would be seen as having constructed something else, whatever other axioms the reals might fit.

That said, these different kinds of number do have different properties, and so any description of them that did not account for this would be inadequate. For example: natural numbers can be factorized, whereas this concept doesn't really apply to complex numbers. It doesn't seem that strange that they should be different, as they have different jobs to do.

I mentioned earlier about the natural numbers and reals relating to particular tasks: counting discrete objects and measuring distances respectively. The cardinals are a different kind of number again relating to a different task: namely measuring the sizes of sets. Obviously if you didn't have any sets then you wouldn't need a measure of their sizes. And if you only accepted finite sets then the cardinals would be the same as the natural numbers (but then measuring sets and counting elements would be fundamentally the same). But if you accept finite and infinite sets—as Badiou does—then you can't very well dispense with cardinals.

In case it isn't clear why cardinals and ordinals are conceptually distinct, consider the ordinals ω and ω+1. Remembering that ω is the set {0,1,2,…} and ω+1 is set {0,1,2,…, ω}, we can define a function f from ω to ω+1 by

This is a one-to-one correspondence, and so demonstrates that, as sets, ω and ω+1 have the same size.

I don't like to harp on about the cardinals, but Badiou does not have them as subsumed by the surreal numbers, despite their importance for dealing with sets. Given the role Badiou gives to the surreals (Number as such) I conclude that for Badiou the cardinals do not fall under the concept of number. It is easy to suspect that the problem is being clipped in order to fit the solution at hand.

[I]s there a concept of number capable of subsuming, in a unique type of being, at least the whole natural numbers, the rational numbers, the real numbers and the ordinal numbers, finite or infinite? […] The answer is “yes”. It is here that we propose the marginal theory, which I wish to make philosophically central, of “surreal numbers”.

I'll point out here that the ordinals already subsume the natural numbers, and the reals subsume the rationals, and so the surreals subsume them all by virtue of subsuming the ordinals and the reals.

In devising the surreals, Conway's inventive step was to combine Von Neumann's construction of the ordinals with Dedekind's construction of the reals. (I think this is more apparent in the formulation of the surreals in terms of “left” and “right” sets than it is in Badiou's version.) The transfinite ordinals are in part given by an expansion of the concept of a limit, which comes from analysis. So, without meaning to take anything away from Conway, it makes sense that that such a unification should be possible.

Posted by robin2 at 07:59 PM | Comments (2)