Russell, in "Introduction to Mathematical Philosophy," says:

"It has been thought ever since the time of Leibniz that the differential and integral calculus required infinitesimal quantities. Mathematicians (especially Weierstrass) proved that this is an error; but errors incorporated, e.g. in what Hegel has to say about mathematics, die hard, and philosophers have tended to ignore the work of such men as Weierstrass."

The philosophy of math class I'm taking now will probably show me what the work of a mathematician like Weierstrass has to offer philosophers. (I don't really know.) However, this quote made me wonder what errors Russell had in mind. What errors were incorporated into what Hegel said about mathematics? Did Hegel talk about infinitesimals? Russell doesn't specify what the errors are, which is too bad. I hope he doesn't mean the alleged quip about the necessity of the number of planets being 9. Whether or not Hegel actually said that, it is a stretch to call that an error in mathematics.

## Friday, September 05, 2008

### Russell on Hegel and math

Posted by Shawn at 9:23 AM

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## 3 comments:

Hi Shawn,

Hegel has written lot on math as part of the Science of Logic, including infinitesimals.

However I'm not sure that Russell is right in what he says about Hegel. The way I read that part in Hegel, I'm inclined to think that Hegel might be closer to Russell than Russell thinks. But then, with the whole dialectical exposition in Science of Logic, where a category is presented as a solution, only to be dismissed as contradictory in the next step, I think Hegel is particularly easy to misunderstand if things are taken out of context.

Interesting quote. I bet Russell had something like the following in mind (if I may take a stab):

1. Leibniz's calculus took the notion of an infinitesimal differential

duas primitive. So, for example, when you integrate the area under a curve f(x), you are adding up a collection of rectangles with height f(x) and "infinitesimal width" dx. This is actually a powerful formalism for an old idea: calculations involving infinitesimal quantities were studied even during the time of Galileo. I suppose it really an error, just a poorly understood mathematical tool.2. But: in the 19th century, this idea was given a more rigorous basis, which doesn't deal with infinitesimals. Instead, differentials were defined in terms of limits of linear maps (which in turn hinge on quantifier-logic). Weierstrass in particular contributed to the famous delta-epsilon definitions of these limits.

Maybe there are some interesting lessons to learn from this history, but I'd say Russell was bit too quick to write off infinitesimals. The non-standard analysis developed by Abraham Robinson (and others) in the 60's brought back the notion of an infinitesimal -- this time rigorously defined as a special element of an ordered field. But of course, Russell couldn't have known about that.

Hope that helps!

I wouldn't know about Hegel, but we can look back at the idealist Russell to make sense of these sorts of remarks. As explained in Griffin's book, Russell's Tiergarten program involved linking the sciences in a series, beginning with the more abstract and ending with the social sciences. What drives the series are the internal contradictions revealed in each science. This is a broadly Hegelian idea, filtered through Bradley.

The first step came in Russell's fellowship essay, where spatial points are deemed incoherent. See, e.g., here. But thankfully Russell soon moved away from this perspective to the extreme realism of Principles.

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