.

The crisis in Logic was equally profound, initiated by the work of Frege, who developed the modern system of formal logic. (Frege's logic, now referred to as the predicate logic was a great advance of the traditional Aristotelian syllogism. Indeed it could be shown that the syllogism was not universally valid, despite the fact that it had been accepted virtually without question, for over two thousand years [#!Gil92a!#]. Frege appeared to be unaware of this consequence of his work, or afraid to make it explicit.) Frege started a project, the so-called Logicist project which attempted to eliminate all the unjustifiable assumptions in mathematics and to reduce it to pure logic. Here any theorem in mathematics was to be proved using the simple rules of logic from axioms that were self evident truths. The study of set theory was particularly important again, because numbers were defined as sets³ and the theorems of arithmetic were to be derived from some axioms of set theory. Frege wrote

The firmest method of proof is obviously the purely logical one, which, disregarding the particular characteristics of things, is based solely upon the laws on which all knowledge rests. (Preface to [#!Fre72!#] page 103)

He then continues, in the same article, to attempt to demonstrate that arithmetic and probably geometry, differential and integral calculus can be handled by this very rigorous method of deduction. To quote Frege again, `arithmetic is a branch of logic and need not borrow any ground of proof whatever from experience or intuition.' This project was taken up by the English philosophers Russell and Whitehead, who produced a massive tome Principia Mathematica which tried to prove all the important theorems of mathematics in formal logic. This was an attempt to secure the foundations of Science by showing that mathematics dealt only with an unchanging and absolute truth.

This project failed, but it is interesting to see that it was not because of a philosophical attack, but a contradiction within the subject, in the language of logic and pure mathematics. The failure came in the form of the Incompleteness theorem of Kurt Gödel which can be stated roughly as follows.

THEOREM  

In any consistent, recursively enumerable, formal logic sufficient for arithmetic there will be true statements for which there exists no proof.

`Sufficient for arithmetic' means that the language should have symbols for addition and multiplication together with appropriate axioms for these. The theorem demonstrates the inadequacy of formal logic because no formal system is capable of proving all the true formulas.

Work related to this also showed that basic notions, like `set' or `number' cannot be identified by a formal definition. This is because for any axiomatisation of set theory (for example the Zermelo-Fraenkel axioms) apart from the intuitive model of set theory there will also be non-standard models satisfying the axioms but either including as sets objects which you had not intended to count as sets, or missing out some sets which you wanted included. The philosophical implications of this turn out to be far-reaching [#!Put83!#]. (Incidentally, this philosophical problem about non-standard interpretations of the axioms has been put to good use in the subject known as Non-Standard Analysis which uses models containing, as well as all the ordinary real numbers, infinitely large and infinitesimally small numbers. So, for example in calculus the differential is usually defined to be limit of a sequence of fractions as tends to zero. The concept of a limit can be handled rigorously, but it is not entirely straight-forward. In non-standard analysis dx is taken to be an infinitesimal, i.e. a non-standard real number, smaller than all the `standard' positive reals but greater than zero. is then just a non-standard fraction and the `standard part' of is defined to be the differential. In this way we obtain a mathematical respectability to Leibnitz's notion of the infinitesimal,or fluxion, in calculus.)

A vast collection of new problems arose: the undecidability of the halting problem in algorithms, the independence of the axioms of set theory, even the undecidability of the consistency of arithmetic.

Suffice it to say that the foundations were no longer considered to be quite so solid. It is perhaps worth adding that as far as formal logic is concerned it is possible to reconstruct a Platonist framework for mathematics while avoiding the contradictions [#!Bre92!#]. It is rather like the struggle between the Ptolomeic and the Galilean view of the solar system. Given enough cycles and epi-cycles the Ptolomeic system could explain everything that the Galilean system could. But the latter system benefited from its relative simplicity and turned out to be a better framework for later developments like Kepler's elliptical motion and eventually Newton's theory of gravitation. Similarly, in the philosophy of mathematics the Platonist model lost its authority partly because it became increasingly complicated to defend but, equally important, because a revolutionary momentum built up in which attempts to defend the old system were largely ignored. With the foundations of mathematics and logic cracked open, Logicism was defeated and the Platonist philosophy was also weakened. But what was to be put in its place?