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added section on Russell’s relation to mysticism based on his essay Mysticism and Logic and the biography of Ray Monk.
Anyone want to take a crack at translating the Hegel quotes?
I fixed up the English in a few spots.
Thanks, Daniel. Nicely done!
I have added a few more hyperlinks to some keywords.
I’ll try to look into translating the Hegel quotes this evening. Or else somebody finds the paragraph in the translation of the “Encyclopedia” (my local copy died together with my old notebook, would have to search again.)
There’s an online translation but it’s not the easiest to navigate to the right place.
I’ve added the right section from this translation. I think I’ve ended at the right place. It continues:
As we have seen, however, the abstract thinking of the understanding is so far from being something firm and ultimate that it proves itself, on the contrary, to be a constant sublating of itself and an overturning into its opposite, whereas the rational as such is rational precisely because it contains both of the opposites as ideal moments within itself. Thus, everything rational can equally be called “mystical”; but this only amounts to saying that it transcends the understanding. It does not at all imply that what is so spoken of must be considered inaccessible to thinking and incomprehensible.
Thanks, David. Right, I should include the followup paragraph in German, too. I had truncated the quote with quite a cliffhanger.
Should we include a reference to Brouwer’s Life, Art, and Mysticism? I’m not sure it’s related, but I thought of it when I saw there was an nLab entry on mysticism.
Should we include a reference to Brouwer’s Life, Art, and Mysticism?
All good material concerning mysticism in relation with logic, mathematics or physics is clearly on-topic for a wiki on “Philosophy, Mathematics and Physics”, and deserves to be included.
I don’t know Brouwer’s thoughts on mysticism, but I’d be rather interested in learning about them.
good timing, I have just added a brief section on Brouwer and some relevant links. The van Atten & Trasseger paper covers Gödel too…
Re #6, I added in the extra paragraphs in German and English.
I put in a link from the Hegel section to [[Meister Eckhart}], and added this quotation to Russell:
“Continuity” had been, until he [Cantor] defined it, a vague word, convenient for philosophers like Hegel, who wished to introduce metaphysical muddles into mathematics. Cantor gave a precise significance to the word, and showed that continuity, as he defined it, was the concept needed by mathematicians and physicist. By this means a great deal of mysticism, such as that of Bergson, was rendered antiquated. (Chapter XXXI of “A History of Western Philosophy” (1945))
Was Cantor really the key figure in developing the sequential definition of continuity? As I recall this is attributed to Cauchy (though Cauchy messed up things sometimes, e.g. conflating continuity and absolute continuity). Wikipedia mentions Bolzano as the first to come up with the modern idea in 1815 and giving the modern formal definition in the 1830s thus preceding Cantor by several decades.
I thought that odd too, but even if he’s wrong, its a useful quote to characterise what Russell sees as problematic in Hegel and then Bergson. His belief in the importance of Cantor here goes right back to 1903 in The Principles of Mathematics.
David, thanks for the cross-link. I gave the subsection you linked to an explicit anchor name. That’s more robust against future changes.
From Chap 35, it seems he’s talking about continua rather than continuity of functions. Another chance for a dig at Hegel though:
The notion of continuity has been treated by philosophers, as a rule, as though it were incapable of analysis. They have said many things about it, including the Hegelian dictum that everything discrete is continuous and vice-versa. This remark, as being an exemplification of Hegel’s usual habit of combining opposites, has been tamely repeated by all his followers. But as to what they meant by continuity and discreteness, they have preserved a discreet and continuous silence; only one thing was evident, that whatever they did mean could not be relevant to mathematics, or to the philosophy of space and time
Another German mystic with mathematical connections was Nicholas de Cusa. I seem to recall reading of some of his thoughts on mathematics in a book on medieval mysticism. Does anyone know if this is relevant to Meister Eckhart? De Cusa was later than Eckhart. Here is a quote from the Stanford Encyclopedia:
Nicholas then proposes some geometrical “exercises” to provide his readers some object lessons designed to teach how we might reach for the unlimited even while we are aware that we cannot grasp what the infinite God may be. For instance, we are to imagine a circle and a straight line or tangent that meets the circle. From a certain perspective, as the diameter or circumference of the circle increases, its circumference approaches the straight line and appears less and less curved. If we then imagine and extrapolate the circumference to the infinite, we can almost “see” that both straight tangent and curved circumference should coincide—a kind of “coincidence of opposites” that is a figure of how we my think beyond limited things toward the transcendent One. All this is mathematically impossible, of course, but it demonstrates some metaphorical steps for moving beyond the finite toward the infinite that might be transferred from geometrical figures to created beings and their Creator.
There is more on his thought there.
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