From our school days we are taught to look upon mathematics, with its self-evident truths “axioms” and its rigorous logical deductions as the last word in scientific exactitude. In 1900, all this seemed certain, although in the International Congress of mathematicians held that year, German mathematician David Hilbert set forth a list of the 23 most significant unsolved mathematical problems. From that point things have got steadily more complicated, to the point where it is possible to talk of a real crisis in theoretical mathematics. In his widely read book, Mathematics: The Loss of Certainty, published in 1980, Morris Kline describes the situation thus:

Creations of the early 19th century, strange geometries and strange algebras, forced mathematicians, reluctantly and grudgingly, to realise that mathematics proper and the mathematical laws of science were not truths. They found, for example, that several differing geometries fit spatial experience equally well. All could not be truths. Apparently mathematical design was not inherent in nature, or if it was, man’s mathematics was not necessarily the account of that design. The key to reality had been lost. This realisation was the first of the calamities to befall mathematics.

The creation of these new geometries and algebras caused mathematicians to experience a shock of another nature. The conviction that they were obtaining truths had entranced them so much that they had rushed impetuously to secure these seeming truths at the cost of sound reasoning. The realisation that mathematics was not a body of truths shook their confidence in what they had created, and they undertook to re-examine their creations. They were dismayed to find that the logic of mathematics was in sad shape.

At the beginning of the 20th century, they set about trying to solve the unsolved problems, remove the contradictions, and elaborate a new and foolproof system of mathematics. As Kline explains:

By 1900 the mathematicians believed they had achieved their goal. Though they had to be content with mathematics as an approximate description of nature and many even abandoned the belief in the mathematical design of nature, they did gloat over their reconstruction of the logical structure of mathematics. But before they had finished toasting their presumed success, contradictions were discovered in the reconstructed mathematics. Commonly these contradictions were referred to as paradoxes, a euphemism that avoids facing the fact that contradictions vitiate the logic of mathematics.

The resolution of the contradictions was undertaken almost immediately by the leading mathematicians and philosophers of the times. In effect four different approaches to mathematics were conceived, formulated, and advanced, each of which gathered many adherents. These foundational schools all attempted not only to resolve the known contradictions but to ensure that no new ones could ever arise, that is, to establish the consistency of mathematics. Other issues arose in the foundational efforts. The acceptability of some axioms and some principles of deductive logic also became bones of contention on which the several schools took differing positions.

The attempt to eliminate contradictions from mathematics only led to new and insoluble contradictions. The final blow was struck in 1930, when Kurt Gödel published his famous theorems, which provoked a crisis, even calling into question the fundamental methods of classical mathematics:

As late as 1930 a mathematician might perhaps have been content with accepting one or another of the several foundations of mathematics and declared that his mathematical proofs were at least in accord with the tenets of that school. But disaster struck again in the form of a famous paper by Kurt Gödel in which he proved, among other significant and disturbing results, that the logical principles accepted by the several schools could not prove the consistency of mathematics. This, Gödel showed, cannot be done without involving logical principles so dubious as to question what is accomplished. Gödel’s theorems produced a debacle. Subsequent developments brought further complications. For example, even the axiomatic-deductive method so highly regarded in the past as the approach to exact knowledge was seen to be flawed. The net effect of these newer developments was to add to the variety of possible approaches to mathematics and to divide mathematicians into an even greater number of differing factions.[10]

The impasse of mathematics has produced a number of different factions and schools, none of which accept the theories of the others. There are the Platonists (yes, that’s right), who regard mathematics as an absolute truth (“God is a mathematician”). There are the Conceptualists, whose conception of mathematics is entirely different to that of the Platonists, but it is merely the difference between objective and subjective idealism. They see mathematics as a series of structures, patterns and symmetries which people have invented for their own purposes – in other words, mathematics has no objective basis, but is purely the product of the human mind! This theory is apparently popular in Britain.

Then we have the Formalist school, which was formed at the beginning of the 20th century, with the specific aim of eliminating contradictions from mathematics. David Hilbert, one of the founders of this school, saw mathematics as nothing more than the manipulation of symbols according to specific rules to produce a system of tautological statements, which have inner consistency, but otherwise no meaning whatsoever. Here mathematics is reduced to an intellectual game, like chess – again a completely subjective approach. The Intuitionist school is equally determined to separate mathematics from objective reality. A mathematical formula, according to these people, is not supposed to represent anything existing independently of the act of computation itself. This has been compared to the attempt of Bohr to use the discoveries of quantum mechanics to introduce new views of physical and mathematical quantities as divorced from objective reality.

All these schools have in common an entirely idealist approach to mathematics. The only difference is that the neo-Platonists are objective idealists, who think that mathematics originated in the mind of God, and the rest – intuitionists, formalists and conceptualists – believes that mathematics is a subjective creation of the human mind, devoid of any objective significance. This, then, is the sorry spectacle presented by the main schools of mathematics in the last decade of the 20th century. But it is not the end of the story.

  1. Quoted in Ferris, T. op. cit., pp. 521-2 and 522-3.