The Search for Mathematical Roots, 1870-1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel: Logics, ... from Cantor Through Russell to Godel [Paperback]

I hoped that reading this book would give me a better understanding, in an historical context, of the issues involved in the controversies about the foundations of mathematics a century ago. I found this book fairly interesting, and it was a quick read, but it seems to be written for those who already have an essentially complete understanding of those issues, since the ideas themselves were addressed only tangentially. The focus of the book is much more on: who published what paper when, to what journal did he send it, who was the editor of the journal, who refereed the paper, to whom were offprints sent, in what archives can the manuscript be found, who read whose paper when, who met whom at what conference, who used what notation in writing which paper. This is very much a documentary history, and historians of mathematics will probably love it, but I am probably not the only mathematician who will not find this book completely satisfying.

I have owned a copy for over a year, and not a week goes by in which I do not consult it. The 50pp bibliography alone is worth the price.

Modern foundational mathematics emerged around 1840, with the work of Boole, De Morgan, and Bolzano. In the 1870s, Cantor, Peirce, Frege made their appearance. In the 1890s, Peano, Hilbert, Russell and Whitehead came on line. The author is an authority on Cantor, Peano, the rise of set theory, on Russell, and Principia Mathematica, and these are covered in great detail. The era closes in the 1930s, with the negative metatheorems of Goedel and Church, and the rise of Quine. All this makes for an exciting human adventure, and this book is the best narrative we have of that adventure.

The book is a gold mine of details little known to most philosophers and to nearly all mathematicians. Here I learned that Husserl was trained as a mathematician, and that much of foundational mathematics can be seen as a meditation on bits of Kant. I should grant that IGG is not fair to everyone: Skolem, for instance, is slighted. Also, this book is far from definitive about Polish logic, which deserves a book of its own.

You'll notice that although GG still lists Cantor's "paradox" in his index, in the text he doesn't quite bring himself to say that there is such a thing. Why not? Because he has read Garciadiego's BERTRAND RUSSELL AND THE ORIGINS OF THE SET-THEORETIC 'PARADOXES,' which shows quite clearly that there is no such thing as Cantor's paradox, or the Burali-Forti paradox, or Russell's for that matter. The so-called "set-theoretic paradoxes" were for the most part inventions of Russell, and not a single one from the period, comes out as anything but a meaningless formulation.

The problem this creates for GG is that so-called "set theory" is nonsense, and not much worth wasting time on. Apart from Cantor's own pathetic inability ever to define what a set is, the history is a farce of the blind leading the blind--trying to "avoid" formulations which are not paradoxes or anything else. This is worth writing about? Worth listing 1900 items in a bibliography, about? It's sad, but a good study in how wastes of time and resources occur.

So GG goes ahead and talks about these "paradoxes" as if they really were such, and about people's "responses" to them as if there was anything to respond to. GG still hasn't quite weaned himself from the "paradoxes," although he cites Garciadiego and should have known better. The gist of the book is that the "paradoxes" which led to Godel's argument (and those of the Intuitionists, the Logicists and Formalists as well as their successors), are not paradoxes at all--they are meaningless formulations. This undermines most, if not all, of twentieth-century mathematics, and in particular destroys Godel's very sloppy argument.

Garciadiego cites Richard's own formulation of this "contradiction" (Richard's term) in a letter to Poincare. He also cites Richard reducing the argument to meaninglessness. What does this have to do with Godel? It's simple. For Godel, Richard's "paradox" means that truth in number theory cannot be defined in number theory. On this basis, he distinguishes truth from provability. He combines his idea of Richard's "paradox" with the idea that provability in number theory can be defined in number theory. He arrives at the conclusion that if all the provable formulae are true, there must be some true but unprovable formulae. However, since Richard's "paradox" is without meaning, since it has no logical content whatsoever and is simply letters pulled out of a bag, there is no basis in Godel's argument for distinguishing truth from provability. It turns out that there is no logical content in the idea that if all the provable formulae are true, there must be some true but unprovable formulae.

People are having a hard time getting over the notion that Godel didn't do his homework, and has nothing to say, but really you have to grow up. Get over it. The problem is that Godel was a terrible scholar, and did not apply himself sufficiently to the details of the development of set theory.

Garciadiego's book has implications for all twentieth-century mathematics. Here are just a few examples of horrendous errors which explain a lot about why mathematics today is regarded as the province of clowns. For example, Brouwer based the idea of an infinite ordinal number on the idea that Cantor had proved well-ordering of the ordinal numbers. But not only did Cantor never prove this, but also, he never said he had done so, and never used the term infinite ordinal number. Turing never examines the "paradoxes" in order to determine whether they are simply meaningless formulations. Thus, in an attempt to "prove that there is no general method for determining about a formula whether it is an ordinal formula, we use an argument akin to

that leading the Burali-Forti paradox, but the emphasis and the conclusion are different." As Garciadiego reveals, there is no Burali-Forti paradox. In the context of an attempt to prove the Trichotomy Law, Burali-Forti tried "to prove by reductio ad absurdum that the hypothesis [involved in his own argument] was false and this method required supposing the hypothesis true and arriving at a contradiction. The employment of the hypothesis, as an initial premise, generated the inconsistency. But once the hypothesis is seen to imply a contradiction it is thereby proved to be false." Turing purported to distinguish completeness from decidability, not realizing that the absence of a contradiction made the distinction insupportable. Turing claimed justification for his definition of a computable number in a "direct appeal to intuition." This is not a cavalier reference to intuitionism. In fact, it provides the basis for Turing's use of binary numbers. This base2 system is a metaphor which traces itself back through Turing's own bifurcation of the mathematical process to Brouwer's own bifurcation of the operation of the human mind ("the connected and the separate, the continuous and the discrete")-all in an attempt to "avoid" the "paradoxes." Brouwer's complaint is that the "paradoxes" deprive us of distinctions. Turing's entire apparatus of calculability is designed to "restore" "distinctions." The binary number, and Turing's later restoration of a modified form of completeness in the form of decidability, are assertions by way of distinction.

However, there is no problem against which to assert it. Operating on these numbers with "finite means" (Turing's definition of a computable number) merely takes us back to Richard's response to his own contradiction and no recourse to intuition can rescue us from the consequences of that response: the computable number only has meaning if finite means are defined in totality, and this can only be done with infinitely

many words.

It turns out that Richard's discussion of his own "contradiction" serves as a useful template for evaluating, and then discarding, putative "paradoxes." There is much work to be done in that field. But as for twentieth-century mathematics, to the extent it is based on already-discredited "paradoxes," it loses any logical content. This, unfortunately, is certainly true of Godel's argument. It is even more glaringly true in the case of now-trivial figures such as Carnap and Tarski. After Garciadiego, these names go from the headlines to the footnotes. In general, Garciadiego's book is an indictment of twentieth-century math academics.

The real story is the insidious advance of intutionist-style mathematics through the other disciplines during the twentieth-century. This idea that mathematics is a "natural" part of human experience--an idea nowhere tested or even rigourously used as a hypothesis--provides a crutch for a lot of investigators who were unfamiliar with contemporary mathematics but needed a mathematical expression for their ideas. Thus Sraffa, the economist, whose work came to be expressed in intuitionist math, and Kimura, the biologist, who got his intuitionist math from Malecot, a protege of Boel. Einstein also fell victim to it. Note this passage for his book RELATIVITY:

Up to now our considerations have been referred to a particular body of reference, which we have styled a 'railway embankment.' We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated....People travelling in this train will with advantage use the train as a rigid reference-body (co-ordinate system); they regard all events in reference to the train. Then every event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises: Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative. When we say that the lightning strokes A and B are simultaneous with respect to the embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length A -> B of the embankment. But the events A and B also correspond to positions A and B on the train. Let M' be the mid-point of the distance A -> B on the travelling train. Just when the flashes (as judged from the embankment) of lightning occur, this point M' naturally coincides with the point M, but it moves towards the right in the diagram with the velocity v of the train.

This translation is accurate (the French and Italian are not). Einstein really does say "fallt zwar...zusammen." That is, he says that one point "naturally" coincides with another. The "naturally" reveals the intuitionist expression of the concept, for it reflects the belief that the formulations of geometry do not express facts.

Obviously, the logical problem with it is that, regardless of what Einstein may "feel" about mathematical expressions, nowhere in Einstein's writings--either in the 1905 papers or after--is any meaning assigned to 'naturally.' The failure to do so, destroys the idea, and it is easy to see why. If we retain the concept without meaning there is no logical basis on which to proceed beyond it. If we eliminate it, we wind up with a contradiction: the two assumed coordinate systems collapse into one. What is more, when we place this train experiment next to the various other thought experiments, we see that they are simply translations of the same problem into other terms, just as the false 'paradoxes' turn out to be subject to the same problem Richard indicated (reference to an infinite domain which destroys the meaning). In special relativity, natural coincidence can only be defined by infinitely many words. So the distinction collapses.

Thus, GG's account is merely the beginning of an attempt to get out from under bogus "paradoxes" and the ridiculous intuitionist-style mathematics designed to "avoid" them.