Monday, July 28, 2014

What is Wrong With Wittgenstein: Or, What Is Wrong With Us?


Ludwig Wittgenstein was one of the 20th century's greatest philosophers, perhaps the greatest. his influence is so pervasive that those who consider themselves uninterested in philosophy frequently espouse them. The son of an immensely wealthy and troubled Austrian family, the Wittgensteins were raised by and expected their father to be geniuses, and several of them were. Wittgenstein originally trained to become an engineer, but started into philosophy out of an interest in Bertrand Russell (he was in England at the time). He made himself the student of Gottlieb Frege, the most innovative logician in the world at the time and a founder of a new, logic oriented philosophy. At heart, Witgenstein was a romantic pessimist, but he could not even begin to defend that world view against the precision of Frege's vision. As Wittgenstein matured philosophically, he would first develop an austere, but complete vision of Frege-Russellian logical atomism. He espoused this view in Tractus Logico-Philosophicus. This was easily the most innovative work in metaphysics for at least 100 years (50 forward and 50 back), making other apparently innovative works (such as David Lewis or Heidegger) seem like mere commentaries. Like the work of Hume, there is both positive (this is how large the world is) and negative (this is the edges of the world) readings. At the time the negative reading was poplar, but I think these days most people prefer the positive.

Wittgenstein later "abandoned" this philosophy, but that isn't to say that he stopped thinking that its propositions validated its conclusions. Instead, he began to feel as though he, Russell and Frege had avoided the big questions. They took a logical language as being metaphysically given. "The world is the totality of facts, not of things.", he says. But facts are defined as true propositions, but truth of a proposition is something given by a language. Based on his analysis of the ideal language, Wittgenstein comes to very severe limits on what can be expressed and known ("There is no possible way of making an inference form the existence of one situation to the existence of another, entirely different situation.", "What we cannot speak about we must pass over in silence.", etc.), without regard to non-ideal and imperfect situations. He left himself without any method of figuring out how this austere vision got into the dirtiness of the real world - at times it seems purposefully.

Probability was no help. He deliberately based probability on the ideal language already given: "The truth-functions can be ordered in series. That is the foundation of the theory of probability.".  In fact, Wittgenstein was always a subjectivist about truth and probability. "Agreement with the truth-possibilities can be expressed by coordinating with them in the schema the mark 'T' (true).", therefore there is no need for "such things as 'logical objects' or 'logical constants'...". Taking language as a given and taking priors as a given are subject to very similar criticism. It doesn't seem that Wittgenstein was ever worried about the evolution of beliefs of a speaker of an ideal language, but this idea of the foundation of probability would later be explored by Carnap and Ramsey and would help give birth to Bayesian philosophy.

"Darwin's theory has no more to do with philosophy than any other hypothesis in natural science.". This left him without resort to David Lewis/American Pragmatist arguments about the evolution of language. A Fregean ideal language would not evolve naturally - at least not ones close enough to perfect found metaphysics on, which is what Wittgenstein needed. He began another approach, one grounded in actual speech-acts and language games.

But before he did this, Wittgenstein began examining the foundations of mathematics.He had much to criticize in the standard methods, some of which precursed his later philosophy. For instance, he pointed out that despite Frege and Russell's claims that they were founding ordinary math, if their systems were found to contradict ordinary math, it is their systems that would be tossed out. Therefore, they can't really be said to be "foundations". But just as interesting was Wittgenstein's "finitism". Naive finitism is a very rare position in the philosophy of mathematics, for the same reason the denial of gravity is rare in physics and the denial of evolution is rare in biology. It denies far too much of what mathematics is about! How could Wittgenstein - an engineer, a logician and a mathematician - come to such a crazy position? The answer must be that Wittgenstein was a non-Naive Finitist. Let's figure out what that means.
G Cantor

The explicit study of infinite sets was introduced by Georg Cantor. Cantor was not working on an austere philosophical problem, but on Fourier Series - the mathematics of waves. The mathematician and physicist Fourier had made complex wavefronts amenable to simple mathematical analysis by considering them as being made up of a bunch of interacting simple waves. This theory is remarkably beautiful, it is perhaps the crown jewel of 19th century mathematics. The theory is extremely broad and wide, but also presents immediately measurable quantities to test (when applied to physics). But many statements about Fourier (and related) Series were quite difficult to answer with the primitive state of mathematics. One of the most basic questions "Is there one and only one Fourier Series for a given complex wavefront?" leads inevitably to the consideration of infinite sets. With modern mathematical machinery (Hilbert Spaces), the proof can be contained in a few lines. For Cantor, it required a staggering number of innovations.
K Weierstrass

To control the mathematics of infinite sets, mathematicians try to make them work as much like finite sets as possible without contradiction. There is a very clear history of this idea, which either began in Cauchy or was believed to begin in Cauchy by his contemporaries and simultaneously by Bolzano (explicitly). The methods then passed through Weierstrass into the wide mathematical world. This method was and is called "analysis". By the analysis (in the ordinary sense) of seemingly infinite statements, you convert them to limits of finite statements, then there is a simple, official way of turning limits into finite statements. The official way is by considering "Cauchy Sequences". Cantor was a student of Weierstrass. He thought - and probably correctly! - that his methods were elucidations of this technique. But there were difficulties, inclarities and even some contradictions in Cantor's methods, or at least in his explanations of them. Cantor fought off the contradictions with a technical trick, he said that while there may be infinite sequences of infinite sets infinitely larger than each infinite set before it, there were still somethings (now called "Proper Classes" in English) too big to be considered sets. But why? This was an arbitrary metaphysical statement made to correct an obvious contradiction, not something that was intuitively clear before the contradiction was realized. Frege arose to the challenge, he declared that he could and would find a logically clear system that would contain all the good and useful discoveries of Cantor.
G Frege

Incidentally, Frege's system contained a technical flaw that made it vulnerable to the above mentioned paradox. This flaw is so minor, it is somewhat surprising that he made it. If he did not, then mathematics could have been seriously damaged - formal systems could have been taken as a closed field instead of an open one!

Anyway, Frege's most powerful innovation was the notion of a Quantifier. Kant taught us that existence is not a predicate, but then what the hell is it? Frege figured that out. Doing so made almost all previous logics technically obsolete, and for this reason alone Frege would have and deserve a place among the philosophical pantheon. Wittgenstein realized that there was a difficulty in interpreting a quantifier, if we want to hold up the extremely fruitful analysis methods of Weierstrass and Bolzano. Wittgenstein was not a naive finitist. He could understand and believed in Cauchy Sequences. I'm going to teach you what Wittgenstein thought was wrong with Quantifiers. Let's suppose there is one quantifier, read "There exists". Then the sentence below says "There exists a p such that p has property D.":

$$\exists p. D(p)$$

This has obvious applications to mathematics and interacts beautifully with sets. For instance, we can say that "There exists p in a set S such that p has property D.":

$$\exists p \in S. D(p)$$

Now to get specific. Let's choose a set: \( S = N_n\) is the set of all natural numbers less than or equal to n (so that for instance \(N_3 = {1,2,3} \) ) and let \(D = D_n \) be the property of being the largest number in the domain (specified in the quantifier). Then for every n:

$$\exists p \in N_n. D_n(p)$$

Is read "There exists a largest number in every set of natural numbers less than or equal to n.". Since n has that property, this is always true. The method of analysis then suggests that we take a limit:

$$\lim_{n \to \infty}{\exists p \in N_n. D_n(p)}$$

And since this is true for every finite n, this statement is ... False! What madness is this? Wittgenstein was right to object. The problem is that quantifiers don't necessarily obey the rules of analysis that made the work of Cauchy, Bolzano and Weierstrass so powerful. There may be a largest number in every finite set, but not necessarily in an infinite set - indeed, there never is a largest number in an infinite set of integers! What rules were there? Many insisted on not giving rules, others (most importantly, Hilbert) used any possible rules they desired to make proofs!
A and K Goedel

Is there something wrong here? Not mathematically, as Goedel taught us in his completeness and compactness theorems that we can take these limits to mean what we choose and never come to contradiction - unless there was already one in the finite cases. Hilbert was right, if you could prove something like the (relative) consistency of one interpretation - one set of rules - you got the rest for free. This is now called "model theory". Wittgenstein objected to this, Frege did not offer us the foundations to an infinite number of differing models of arithmetic, he offered us the groundwork of ordinary arithmetic! Perhaps there is a deep philosophical problem here. If there is, we mathematicians, we Cantorians, we infinity-lovers do not and never will lose sleep over it. Wittgenstein was perhaps right to point out that standard arithmetic has this unintuitive property, and equally right to point out that quantifiers do not have the properties we might like - not if we also want standard models. But this is a philosophical (and perhaps pedagogical) problem. We have wandered our way out of the fly bottle. We don't ask to go back in for more.

1 comment:

  1. Hey Mr. T! Nice post! I must admit much of it is over my head. I very briefly did mathematics in university but as part of sciences and I dropped out pretty early. Really the whole thing started to lose me years earlier once imaginary numbers came into the picture. People described it as the language of the universe but I couldn't help wonder if it was all just an abstract construct that we made up to make sense of the universe and will bend it to our will where we see fit. Much like religion!

    And yet there is no doubting it has given us so much in the field of physics and other areas so it must indeed all work.