Truth in Mathematics

 

Though mathematics might seem the clearest and most certain kind of knowledge we possess, there are problems just as serious as those in any other branch of philosophy. What is the nature of mathematics? In what sense do its propositions have meaning?

Plato believed in Forms or Ideas that were eternal, capable of precise definition and independent of perception. Among such entities he included numbers and the objects of geometry — lines, points, circles — which were therefore apprehended not with the senses but with reason. "Mathematicals" — the objects mathematics deals with — were specific instances of ideal Forms. Since the true propositions of mathematics were true of the unchangeable relations between unchangeable objects, they were inevitably true, which means that mathematics discovered pre-existing truths "out there" rather than created something from our mental predispositions.

 

Aristotle disagreed. Forms were not entities remote from appearance but something which entered into objects of the world. That we can abstract oneness or circularity does not mean that these abstractions represent something remote and eternal. Mathematics was simply reasoning about idealizations.

 

Leibniz brought together logic and mathematics. Mathematical propositions are not true because they deal in eternal or idealized entities, but because their denial is logically impossible. They are true not only of this world, or the world of eternal Forms, but of all possible worlds.

 

Gottlob Frege , Bertrand Russell and their followers developed Leibniz's idea that mathematics was something logically undeniable. Frege used general laws of logic plus definitions, formulating a symbolic notation for the reasoning required. Inevitably, through the long chains of reasoning, these symbols became less intuitively obvious, the transition being mediated by definitions. What were these definitions? Russell saw them as notational conveniences, mere steps in the argument. Frege saw them as implying something worthy of careful thought, often presenting key mathematical concepts from new angles. Russell carried on, however, resolving and side-stepping many logical paradoxes, to create with Whitehead the monumental system of description and notation of the Principia Mathematica (1910-13).

 

But then there was the logical paradox of a set both belonging and not belonging to itself. Ad-hoc solutions could be found, but something more substantial was wanted. David Hilbert (1862-1943) and his school tried to reach the same ends as Russell, but abandoned some of the larger claims of mathematics. Mathematics was simply the manipulation of symbols according to specified rules. The focus of interest was the entities themselves and the rules governing their manipulation, not the references they might or might not have to logic or to the physical world. Hilbert's continued promisingly until 1931, when Gödel's second incompleteness theorem brought the programme to an end. Gödel showed, fairly simply and quite conclusively, that such formalisms could not formalize arithmetic completely.

 

For intuitionists like L.E.J. Brouwer the subject matter of mathematics is intuited non-perceptual objects and constructions, these being introspectively self-evident. Indeed, mathematics begins with a languageless activity of the mind which moves on from one thing to another but keeps a memory of the first as the empty form of a common substratum of all such moves. Subsequently, such constructions have to be communicated so that they can be repeated — i.e. clearly, succinctly and honestly, as there is always the danger of mathematical language outrunning its content.

 

Social constructivists took a very different line. Mathematics is simply what mathematicians do. Mathematics arises out of its practice, and must ultimately be a free creation of the human mind, not an exercise in logic or a discovery of preexisting fundamentals. True, mathematics does tell us something about the physical world, but it is a physical world sensed and understood by human beings, as Kant pointed out long ago.

 

Jungian psychiatrists regarded numbers as archetypes, autonomous and self-organizing entities buried deep in the collective unconscious. Scientists and mathematician have found that approach much too shadowy, lacking real evidence or explanatory power. But numbers as predispositions of inner body processes have reappeared in metaphor theory, this time supported by clinical study.

As the embodied mind theory has yet to be widely accepted, there flourish today the four interpretations of mathematics: Platonism, formalism, intuitionism and social constructivism. All have their advocates, but practising mathematicians often have mixed views. A mathematician may be fortified by the Platonist view, for example, but also regard mathematics as an communal activity, one which generates deep relationships that are sometimes applicable to the "real world", a view that brings him close to social constructivism.

But most mathematicians do not fish these nebulous waters. The theoretical basis of mathematics is one aspect of the subject, but not the most interesting, nor the most important. Like their scientist colleagues, they assert simply that their discipline "works". They accept that mathematics cannot entirely know or describe itself, that it may not be a seamless activity, and that contradictions may arise from unexpected quarters. Mathematics is an intellectual adventure, and it would be disappointing if its insights could be explained away in concepts or procedures we could fully circumscribe.

What is the relevance to poetry? Only that both mathematics and poetry seem partly creations and partly discoveries of something fundamental about ourselves and the world around. Elegance, fertility and depth are important qualities in both disciplines, and behind them both lurks incompleteness and unfathomable strangeness.

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Truth in Mathematics

 

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