The Shaky Foundations of Mathematics

The claim is often made that mathematical results are immutable. Once proven, they remain forever valid. But things are not so simple. There are problems at the very core of mathematics that cast a shadow of uncertainty. We can never be absolutely sure that the foundations of our subject are rock-solid [TM104 or search for “thatsmaths” at].


Left: Plato and Aristotle. Centre: Pythagoras. Right: Euclid [Raphael, The School of Athens]

The ancient Greeks put geometry on a firm footing. Euclid set down a list of axioms, or basic intuitive assumptions. Upon these, the entire edifice of Euclidean geometry is constructed. This axiomatic approach has been the model for mathematics ever since.

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Taylor Expansions from India


NPG 1920; Brook Taylor probably by Louis Goupy

FIg. 1: Brook Taylor (1685-1731). Image from NPG.

The English mathematician Brook Taylor (1685-1731) introduced the calculus of finite differences in his Methodus Incrementorum Directa et Inversa, published in 1715. This work contained the famous formula known today as Taylor’s formula. In 1772, Lagrange described it as “the main foundation of differential calculus” (Wikipedia: Brook Taylor). Taylor also wrote a treatise on linear perspective (see Fig. 1).

It is noteworthy that the series for {\sin x}, {\cos x} and {\arctan x} were known to mathematicians in India about 400 years before Taylor’s time.
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Marvellous Merchiston’s Logarithms

Log tables, invaluable in science, industry and commerce for 350 years, have been consigned to the scrap heap. But logarithms remain at the core of science, as a wide range of physical phenomena follow logarithmic laws  [TM103 or search for “thatsmaths” at].


Android app RealCalc with natural and common log buttons indicated.

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Which is larger, e^pi or pi^e?

Which is greater, {x^y} or {y^x}? Of course, it depends on the values of x and y. We might consider a particular case: Is {e^\pi > \pi^e} or {\pi^e > e^\pi}?


Contour plot of x^y – y^x, positive in the yellow regions, negative in the blue ones.

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A New Window on the World

The motto of the Pythagoreans was “All is Number” and Pythagoras may have been the first person to imagine that the workings of the world might be understood in mathematical terms. This idea has now brought us to the point where, at a fundamental level, mathematics is the primary means of describing the physical world. Galileo put it this way: the book of nature is written in the language of mathematics [TM102, or search for “thatsmaths” at].


Visualization of gravitational waves. Image credit MPI/Gravitational Physics/ITP Frankfurt/ZI Berlin.

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That’s Maths Book Published

A book of mathematical articles, That’s Maths, has just been published. The collection of 100 articles includes pieces that have appeared in The Irish Times over the past few years, blog posts from this website and a number of articles that have not appeared before.


The book has been published by Gill Books and copies are available through all good booksellers in Ireland, and from major online booksellers. An E-Book is also available online.

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Thank Heaven for Turbulence

The chaotic flow of water cascading down a mountainside is known as turbulence. It is complex, irregular and unpredictable, but we should count our blessings that it exists. Without turbulence, we would gasp for breath, struggling to absorb oxygen or be asphyxiated by the noxious fumes belching from motorcars, since pollutants would not be dispersed through the atmosphere [TM101, or search for “thatsmaths” at].


Turbulent flow behind a cylindrical obstacle [image from “An Album of Fluid Motion”, Milton Van Dyke, 1982].

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