I was already serendipitously planning to write about Wittgenstein in Infinite Jest and E&M when Tammy posted her questions for group discussion. The one I’m most interested in is Tammy’s third: In what way to abstractions ‘exist’?
It seems sensible to begin with Wittgenstein’s view of this question. He, like his teacher and friend Bertrand Russell (also an important mathematician/philosopher of mathematics, cited often in E&M), wants abstractions to truly exist. They exist as external abstract objects, considered as external and objective as things in physical reality. In Wittgenstein’s view, we can say that 3 exists in the same way as a rock or a house exists; it is external, outside any single individual. Despite 3′s abstract nature, it is objective in the sense that no opinion about it will change its characteristics. The complicating element here is that this abstract objectivity appears contradictory to the subjective-leaning Wittgenstinian notion of meaning as use. By this standard, wouldn’t different use of ’3′ change its meaning, and hence make it subjective? Strangely, though, it is precisely this principle that allows math to remain objective.
In paragraph 55 of Philosophical Investigations, Wittgenstein says the following: “What names in language signify must be indestructible; for it must be possible to describe a state of affairs in which everything destructible is destroyed. And this description will contain words; and what corresponds to these cannot then be destroyed, for otherwise the words would have no meaning.” (We are meant to understand that ’3′ is a name for 3.) What this means for physical objects is that even when the physical object itself is gone, the name still has meaning via that physical object’s concept, its abstraction. But since 3 is only an abstraction – it has no physical manifestation – it is entirely immune to destruction or alteration. It is unchangeable. When we prove new mathematical properties, we are simply discovering a property that already existed without our knowledge; we are not creating or adding anything to the abstraction itself, we add only to our personal understanding of that abstraction.
And so Wallace’s idea of abstraction (his abstraction of abstraction?) aligns with this Wittgenstinian/Russellian model. This view is necessary to consider math an objective enterprise. Without it, proofs are not proofs so much as just detailed strings of subjective reasoning. By this Wittgenstinian standard, post-Cantor, infinity exists in the way all numbers exist, as an abstract external object. If you think about it, it is just as hard to imagine 3 (the abstraction) as it is to imagine infinity.
On a completely different tac, as a side-note of sorts, I want to call attention to Bob Death’s joke on p. 445 of Infinite Jest. It is the same fish-joke that Wallace uses in his Kenyon Commencement speech. The joke is: “This wise old whiskery fish swims up to three young fish and goes, ‘Morning, boys, how’s the water?’ and swims away; and the three young fish watch him swim away and look at each other and go, ‘What the fuck is water?’ and swim away.” This joke is deeply Wittgenstinian in a number of ways. It is a play on the (more famously Orwellian) notion that to see what is in front of one’s nose needs a constant struggle. This struggle to see what is so obvious and basic as to never be noticed is the main struggle of Philosophical Investigations. Wittgenstein’s project is to break language down to its most basic elements and discover how it actually works, how it is actually used. This preoccupation with ordinary language is one of the things that makes Wittgenstein so revolutionary and original. He is trying to see water clearly, even while swimming in it. We already know from Lenore that we are all swimming in language.
That’s all for now. Thoughts?