Time for another guest post from my old chum Anton, this time on the topic of mathematics. I’m not sure any mathematicians reading this piece will be too happy, but if that applies to you then blame him not me. As usual, comments are welcome through the proper channel at the bottom of the page..

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Nowadays a page of mathematics looks to a physicist or engineer like gobbledygook. This was not always so: a century ago a physicist might hope to understand everything in journals of mathematics, and even contribute to them. Fifty years ago a physicist might not be able to understand everything written there, but the mathematics would appear comprehensible in principle. A qualitative change has since taken place.

This change has coincided, roughly, with acceptance of the distinction between ‘pure’ and ‘applied’ (impure??) mathematics, and with the consequent, deliberate, emancipation of ‘pure’ mathematics. This is a new departure: for centuries mathematics evolved side by side with physics, and the mathematics that was studied was the mathematics used in tackling physical problems. Galileo had said (in his work *Il Saggiatore*, The Assayer) that

…the universe… cannot be understood unless one first learns to comprehend the language… in which it is written. It is written in the language of mathematics.

So the change is recent, and it is huge. I suggest that it is a change for the worse; that in divorcing themselves from physical science pure mathematicians have cut off their air supply; and that the suffocating style of modern pure mathematics is a result. Mathematics was not born in a vacuum, and it will not ultimately flourish in one.

A pure mathematician might respond that I would say that, since I am a physicist. But perhaps an outsider is needed to see the problem; insiders generally adopt the party line. The justification for my stance is this. Mathematicians acknowledge that their subject is the formal study of patterns. And mathematicians think in patterns, not formulae – which are really a highly efficient way to express their thoughts. Crucially, the patterns arising in the natural world are far richer and more diverse than the patterns that even the best pure mathematicians can pull out of their heads by introspection. Even number theory is not an exception, for the positive integers are abstractions – ideals – of the physical realisations of one, two, three *etc* sheep in a field, or boats on a lake.

The role of pattern explains the “unreasonable effectiveness” of mathematics in physical science (as Eugene Wigner put it), since physics is concerned with relations – correlations – between variables in space and time, and correlation is synonymous with pattern. The theoretical physicist Lev Landau vehemently believed that the best mathematics is the mathematics used in physics. An opposing point of view was taken by the pure mathematician Paul Halmos, in an essay titled *Applied Mathematics is Bad Mathematics*. Not all mathematicians share Halmos’ view, however. The mathematician Morris Kline was the author of many books about mathematics and its embedding in the cultures which nurtured it. In his book *Mathematics: The Loss of Certainty*, Kline demonstrated that the history of mathematics in the 20th century has not been the smooth progression that it appears to the outsider; and that arguments about the foundations have led not to resolution, but to schism into differing schools – based on different foundations – that do not talk to each other. Mathematics is not in fact a one-way road running from self-evident axioms to consequences, but is open at both top and bottom.

Already in the 19th century a formal style was developing in the mathematical study of logic, and such distinguished noses as Henri Poincaré (in *Science et Methodes*, part II) protested as early as 1909 that this tended to hide misleading or negligible content. To no avail: the dominance of the formalistic logical viewpoint led to the adoption of its house style across the whole of mathematics. Below university level, mathematics is still taught today as it used to be, with the emphasis on the understanding of ideas rather than their formal presentation. Freshmen are often shocked when they first meet the new way of doing things, in university lectures given by professional mathematicians. I doubt that the form of modern mathematical writing is governed by its content, for whenever my research has demanded I read some contemporary mathematics, and I have had to translate a piece of modern mathematical writing into something comprehensible to scientists, I have found it difficult to distinguish substantial points from trivia. When, for instance, four axioms are needed to establish a result, they will typically be presented as having equal weight, even if one is the crucial axiom that allows most of the proof to be constructed, and another is used only in closing loopholes. Acknowledging the quality of axioms, as well as the quantity, does not compromise rigour.

When I think of the work of Andrew Wiles and Grigori Perelman, I realise that magnificent work is done today by mathematicians far beyond my own competence. But might mathematicians question whether what they regard as the only way to write mathematics is actually a convention, and not necessarily a good one? If they wrote mathematics as they did fifty years ago, others might be able to see for themselves. More fundamentally, might they also realise what their predecessors understood, that by its abstraction mathematics is given an autonomy of its own, and that to look to the physical world for inspiration is not to make mathematics a slave of physics? The present divorce between mathematics and physics impoverishes everyone.