Reviews for Logical number theory

The average rating for Logical number theory based on 2 reviews is 3 stars.

Review # 1 was written on 2014-07-24 00:00:00 Marjean Hellman French Arithmetic There is always something of the tongue-in-cheek about French philosophy. An ironical tone as if communicating to a colleague over your shoulder about whom the reader should have known but doesn’t. As if the writer is making history, perhaps, and giving the reader a privileged seat before the prizes are handed out. What seems like hot air may only be heavy breathing. Badiou considers that numbers dominate our lives. Not just in economics where numbers are obviously important, but in everything from medicine to culture. They constitute our practical existence. What is counted counts. And yet most of us feel alienated from numbers. Mathematics is arcane wisdom and aside from making change for the purchase of a newspaper or a restaurant tip, we have no interest in these numbers. But Badiou thinks we should: “if we don’t know what a number is,... we don’t know what we are.” Fair enough. So, Badiou is on a quest - to overcome “the despotism of number,” to correct the condition that “we have at our disposal no recent, active idea of what number is,” to create “a second modernity,” the route to which “... constrains thought to return to zero, the infinite, and the One. A total dissipation of the One, an ontological decision as to the being of the void and that which marks it, proliferation without measure of infinities: these are the parameters of such a passage. The amputation of the One delivers us to the unicity of the void and to the dissemination of the infinite.” Who knew? What motivates Badiou is that the leading lights of 19th century mathematics - Dedekind, Cantor, Frege and Peano - couldn’t come up with a theory that included all the various kinds of numbers - natural, real, integers, rational, ordinal, etc. Each used a distinct but incomplete method - axiomatics, set theory, logic, etc. But the result has been disappointing: “The thinkers of number have only in fact been able to demonstrate how the intellectual procedure that conducts us to each species of ‘number’ leaves number per se languishing in the shadow of its name.” I can’t help but feel deep sympathy for such numerical languish. One is reminded daily of the numbers who suffer but never of the suffering numbers! Some of the ideas of these mathematicians are arresting. For example, Frege’s definition of Zero as that which “is not identical with itself.” Since by definition something not identical to itself does not exist, voila Zero appears. Even more remarkable, Frege goes on to define "One as the number that belongs to the concept ‘identical to Zero’". Which makes perfect sense since Zero is not identical to itself. Confused yet? Nevertheless there are also some enlightening observations. For example, Badiou recognises that the Russell Paradox* which blew Frege’s logical boat out of the water has a profound implication: “It is impossible, says the ‘paradox,’ to accord to language and to the concept the right to legislate without limit over existence.” In other words, the map of language is not the territory of reality, even when the language is extremely precise. Dedekind takes a very different approach. For him the issue is not building up a number system but selecting from an already established infinite set so that the unit of analysis, as it were, is “not ‘a’ number, but N, the simply infinite ‘system’ of numbers.” **Using this tack, Dedekind is able to show that when it comes to infinite numbers, the Euclidean maxim that the whole is always greater than the sum of its parts is in fact bunk. Unfortunately for Dedekind, he uses a variant of the Cartesian ‘cogito’ or ego which is outside the infinite system it hypothesises and so ends up in a paradox similar to that of Frege. It is in discussing Peano that Badiou gets up a head of intellectual steam and plies into his real foe - language-based philosophy: “We see here, as if in the pangs of its birth, the real origin of that which Lyotard calls the ‘linguistic turn’ in western philosophy, and which I call the reign of the great modern sophistry: if it is true that mathematics, the highest expression of pure thought, in the final analysis consists of nothing but syntactical apparatuses, grammars of signs, then a fortiori all thought is under the constitutive rule of language.” Yup. And if it’s the case with mathematics, what chance does even the French language have for claiming precision in representing reality? I feel Badiou’s pain. And I understand why he quickly exits that field of battle in order to engage his remaining Mathematicians, who are at least gentlemen in their ignorance of linguistics. Cantor was also concerned with the infinite, but with the infinite in a grain of sand rather than the whole beach. His insight was that any subset of real numbers, no matter how limited, was also infinite. Put another way, no matter how precisely two numbers might be expressed, there is always a number between them. This leads to the counter-intuitive conclusion that there are more parts to any set than there are elements. I can’t follow Badiou’s digressive criticism of Cantor but it appears to be summarised in the phrase “We do not want to count; we want to think counting,” and somehow this has to do has to do with cats. Eventually Badiou comes to rest on a concept he believes addresses all the various issues of what numbers really are. They are, he believes ‘surreal.’ Not being a mathematician, Surreal Numbers are new to me despite the fact they have been around for some time. From what I understand in Badiou’s presentation (not that much), I am sympathetic to the general idea. The concept is presented, appropriately enough, not in a professional paper but in a novel from 1974, Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness, which begins “In the beginning everything was void, and J.H.W.H. Conway began to create numbers. Conway said, ‘Let there be two rules which bring forth all numbers large and small...’” And on it goes. The two rules, the syntax of/for surreal numbers, are enough to generate almost all the sorts of numbers Badiou is worried about. There is of course one small problem: these two rules are the apotheosis of Lyotard’s ‘linguistic turn;’ they are the fundamental grammar of what Badiou considers a coherent mathematics. And indeed they do mean that all thought is ‘under the constitutive rule of language.’ It seems we had already arrived at the destination before we departed on the journey. As I said: French philosophical irony. See here for a more or less parallel argument: *Regarding the membership of the set of all sets with itself. ** This he calls “secularisation of the infinite,” a lovely Badiouan flourish.

Review # 2 was written on 2014-10-30 00:00:00 Sheal Recto I don't have the background to entirely understand this thing, or develop any arguments for or against Badiou's work here, but for the most part I found it pretty fascinating and mind-bending. As usual for a philosopher of his ilk, he insists on making blanket, ill-considered political statements even in a book that is dealing primarily with set theory and the like. I don't think we need to believe that numbers are necessarily implicated in a system of political hegemony to want to answer the question of what a number is.