Mathematics, Ideas and the Physical Real by Albert Lautman & Simon Duffy

Author:Albert Lautman & Simon Duffy
Language: eng
Format: azw3
ISBN: 9781441123442
Published: 2011-08-03T23:00:00+00:00

SECTION 2

The Schemas of Genesis

CHAPTER 4

Essence and Existence

1. THE PROBLEMS OF MATHEMATICAL LOGIC1

The problems of the passage from essence to existence, which will occupy us henceforth until the end of this essay, belong to questions that have been raised for a long time by the development of mathematical logic. It doesn’t seem to us however that logic has the benefit in this regard of a special privilege. It is in effect only one mathematical discipline among others, and the geneses that are manifested there are comparable to those observed elsewhere. The presentation that we will make of what could be called the metaphysics of logic therefore has above all the value of an introduction to a general theory of connections that unite the structural considerations to assertions of existence.

Two periods can be distinguished in mathematical logic: one, the naive period, ranging from the early work of Russell until 1929, the date of the metamathematical work of Herbrand and Gödel which marks the beginning of what could be called the critical period. The first period is that where formalism and intuitionism are opposed in discussions that extend those raised by Cantor’s theory of sets. Proponents of the actual infinite claim the right to identify, for a same mathematical entity, the essence of this entity, as a result of its implicit definition by a system of non-contradictory axioms, with the existence of this entity. We know, on the other hand, the attitude adopted towards an entity whose construction would require an infinite number of steps, or a theorem that is impossible to verify, by those that Poincaré called the pragmatists: in his Last Essays, the famous mathematician says that ‘they see in it only unintelligible verbiage’ (Poincaré 1913, 66). Poincaré relies mostly on the assertion of existence contained in the Zermelo theorem: ‘there is a way of well ordering the continuous,’ to show how this statement could be meaningful only if the manner in which it is necessary to proceed for well ordering the continuous was really known. Asserting the possibility of an unrealizable operation is to assert something which is either meaningless, or false, or at least unproven.

In his essay on the Infinite, Hilbert (1926) justifies the introduction of transfinite elements in mathematics in putting them side by side with the ideal elements introduced by Kummer in algebra, which were mentioned in Chapter 3. Ideal numbers are no more parts of the numbers of a field than transfinite elements of axiomatized mathematics are determinable in a finite number of steps. But, just as the consideration of ideal numbers is essential to generalize the theorem of decomposition into prime factors, the consideration of transfinite elements is necessary to generalize the application of the excluded middle. Requiring their elimination would imply abandoning the rule of contradiction in logic. Here’s how, in an essay from 1923, the connection is presented between the transfinite axiom of choice and the excluded middle for infinite sets. The axiom is introduced as follows: