Science

The many schools of mathematical thought

The Formalist School regards mathematics as a kind of game where symbols are manipulated according to clear rules but have no inherent or underlying meaning

A statue of Plato in Greece. Platonism does not explain how ideas change over time: the underlying axioms and definitions of mathematics have evolved over the centuries and do not fit the unchanging model of Plato

Peter Lynch
Thu Mar 21 2024 - 06:00

Mathematics is used widely, playing a central role in science and engineering and increasingly in the social and biological sciences. But users seldom consider the fundamental nature of mathematics.

Many cannot improve on the vacuous definition: mathematics is what is done by mathematicians. We could try harder, with something like “mathematics is the language of clear thinking”, although brilliant but innumerate philosophers might cavil at that.

There are several schools of thought on the fundamental nature of maths. Plato claimed that the world of ideas is more real than the physical world: the idea of a rock is more fundamental than an actual rock. The theorems of mathematics belong in the world of ideas. The Platonic School holds that mathematical concepts are eternal, ethereal, pure. We see circles everywhere in nature, but they are only shadows of ideal circles, which are perfect in every detail, eternal and immutable. Platonists claim to “discover” mathematical results, not to invent them. The results have always been there awaiting discovery.

However, Platonism does not explain how ideas change over time: the underlying axioms and definitions of mathematics have evolved over the centuries, and do not fit the unchanging model of Plato. Euclid set down a system of axioms, or basic assumptions, underlying geometry. But centuries later mathematicians “tweaked” one of these axioms – the parallel postulate – and a whole universe of non-Euclidean geometries emerged.

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Contrasting with Plato, the Formalist School regards mathematics as a kind of game where symbols are manipulated according to clear rules but have no inherent or underlying meaning. The great 20th century German mathematician David Hilbert was a proponent of this view, and formalism is attractive to people developing artificial intelligence today.

The Logical School, closely related to the Formalists, aims to reduce mathematics to a branch of logic. A notorious example of this was the monumental work of philosophers Alfred North Whitehead and Bertrand Russell: in their three-volume magnum opus, Principia Mathematica, on the foundations of maths, Whitehead and Russell famously wrote on page 360 “From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2″. Some of the ideas in this work are still relevant, but the grand aims of the project have largely been abandoned.

Several important results called “existence theorems” show that objects or solutions exist, but without producing concrete examples. Mathematicians of the Intuitionist School insist that there can be no existence without construction. Dutch mathematician L E J Brouwer championed this view, which conflicted directly with Hilbert’s perspective, leading to bitter controversy that continued for decades.

During coffee-break discussions most mathematicians lean towards a formalist view, where maths may be a meaningless game, justified by its inherent elegance and beauty or – if they are applied mathematicians – by its manifold applications and its “unreasonable effectiveness” in describing the physical world.

However, in their day-to-day work they probably tend towards Platonism. The idea that stark logic is not enough was expressed cogently by Henri Poincaré, who wrote “it is by logic that we prove, but by intuition that we discover”.

Peter Lynch is emeritus professor at the School of Mathematics & Statistics, University College Dublin – he blogs at thatsmaths.com

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