I'm currently working my way through Bertrand Russell's The History of Western Philosophy and I keep getting hung up on Russell's prose (or, should I say, his manner of thinking). Particularly troubling is his theory of the philosophy of mathematics, specifically that numbers are 'forms' rather than 'constituents'. I believe I read that for a time, Russell viewed numbers as a set of a sort of Platonic Ideas (i.e. universals), but that after discussion with Wittgenstein he abandoned this view.

Also interesting is the character or tone of his writing; by virtue of his erudition and disdain for thought incompatible with his own unique philosophy he comes across as a something of a parody of intellectualism. I cannot decide if his obfuscation of certain points is intentional, unintentional, or simply imagined on my part.

Despite these concerns (and others) I find him a fascinating read.

Interesting post!

Russell did believe at one point (around Principles) that arithmetic was synthetic a priori and known via direct access to some kind of Platonic Heaven. This surely conflicts with the idea the mathematics (as logic) is tautologous and analytic, which MIGHT be his view by the late teens/early twenties and does seem, in some sense, to be inherited from Wittgenstein.

At the very end of Introduction to Mathematical Philosophy, Russell introduces the idea of tautology as being somehow related to what logic will ultimately turn out to be. He writes that “…the characteristic of logical propositions we are in search of is the one which was felt, and intended to be defined, by those who said that it consisted in deducibility from the law of contradiction. This characteristic…for the moment, we may call tautology…”

Russell admits in the next several pages that he “does not yet know” how to define the term tautology as it is related to the propositions of logic . He claims that logical/mathematical statements, are, in the end, just expressions of tautologies, yet he himself makes it clear that he not entirely sure what it means to make this sort of claim, or, at least, that he does not yet have a grasp upon the proper way to put his understanding into word. While he asserts that mathematical claims are tautologies, he doesn’t really argue for this claim because he admits rather straightforwardly that he isn’t quite sure what the assertion boils down to.

Similarly, in the Lectures on the Philosophy of Logical Atomism, Russell also seems to assert (without any sort of fleshing out of the claim) that logical/mathematical propositions (boiled down to their underpinnings) are tautologies. He writes “Everything that is a proposition of logic has got to be in some sense or other like a tautology. It has got to be something that has some peculiar quality, which I do not know how to define, that belongs to logical propositions and not to others.”

The term tautology itself, as applied to logical propositions, was borrowed from Wittgenstein, and that Russell doesn’t appear to have fully grasped what it was that Wittgenstein meant by the term.

The question is whether or not Russell at this point fully dropped the synthetic a priori in relation to any of the logical underpinnings of arithmetic.

It's honestly hard to tell. Russell's Platonism, it seems, is born in large part out of his repudiation of Bradley style absolute idealism and I think he worries that if he gives up on Platonism across the board that idealism will be that much harder to refute. Even at the time of Introduction, propositional functions (which classes are reduced to) seem to keep on keepin' on up in Platonic Heaven.

He seems to want to deny the direct-access-Platonism of Principles, but on the other hand he doesn't know what to replace it with, so he seems to take on Wittgenstein's bullshit about tautologies as a promissory note. The idea being that, eventually, when the war is over and things are right with the world the great Wittgenstein will return from the trenches to save us all from the terror of a foundationless mathematics and the grip of muddleheaded thinking.

Unfortunately, like most great prophecies, the latter didn't quite come to fruition.

Thanks for the insightful response. I find Russell's dilemma regarding numbers perplexing; it seems clear to me that humans have a perceptual sense of "oneness" in that to focus one's attention on a subject is to "single it out", as it were, and of "twoness" as an extension of symmetry, etc.

As numbers grow larger (or as we move away from integers) the perceptual nature of our understanding gives way to abstraction, but in our system of arithmetic all larger or more complex numbers can be reduced to the perceivable integers and logical functions (i.e. addition, repeated addition (multiplication) etc.).

I get the sense that I'm missing something; I'm rather new at this and I haven't really done my homework on the topic. For the time being, I'll content myself that seeing the phrase "Wittgenstein's bullshit" appear in your post means that I haven't lost my mind completely yet.

You may be interested in Lakoff's Where Mathematics Comes From.

today there are things like category theory, which maybe he would have worked on, had it existed. in the early 1900s, there was some emphasis on Hilbert's program -- the idea that you could build mathematics from a small set of axioms. today that idea is largely rejected because of the famous works in the mid 20th century.

Could it be that there was no use of a computer to iterate equations leading to fractals? Or I am completely off?

Aerodynamics (3) re Wittgenstein (coprophiliac toro):
The Cambridge Companion to Wittgenstein
Wittgenstein's Vienna
Ludwig Wittgenstein: The Duty of Genius
Wittgenstein: A Life
Wittgenstein
Interpreting Wittgenstein: A Cloud of Philosophy, a Drop of Grammar
Tractatus Logico-Philosophicus by Ludwig Wittgenstein
Wittgenstein's Early Philosophy: Three Sides of the Mirror
Wittgenstein Reads Freud
Philosophical Investigations (3rd Edition) by Ludwig Wittgenstein
Culture and Value by Ludwig Wittgenstein
Pulling Up the Ladder the Metaphysical Roots of Wittgenstein’s
Wittgenstein: An Introduction (S U N Y Series in Logic and Language)
Zettel by Ludwig Wittgenstein
Wittgenstein and Political Philosophy
A Wittgenstein Dictionary (Blackwell Philosopher Dictionaries)
Wittgenstein on words as instruments: Lessons in philosophical psychology
Sorry about authors--copy and paste is such a hassle, however, Amazon might help or even Google