The Many Schools of Mathematical Thought

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 … Continue reading The Many Schools of Mathematical Thought →

The Axiom of Choice: Shoes & Socks and Non-constructive Proofs

Recall Euclid's proof that there is no limit to the list of prime numbers. One way to show this is that, by assuming that some number $latex {p}&fg=000000$ is the largest prime, we arrive at a contradiction. The idea is simple yet powerful. A Non-constructive Proof Suppose $latex {p}&fg=000000$ is prime and there are no … Continue reading The Axiom of Choice: Shoes & Socks and Non-constructive Proofs →

Amusical Permutations and Unsettleable Problems

In a memorial tribute in the Notices of the American Mathematical Society (Ryba, et al, 2022), Dierk Schleicher wrote of how he convinced John Conway to publish a paper, ``On unsettleable arithmetical problems'', which included a discussion of the Amusical Permutations. This paper, which discusses arithmetical statements that are almost certainly true but likely unprovable, … Continue reading Amusical Permutations and Unsettleable Problems →

Sets that are Elements of Themselves: Verboten

Can a set be an element of itself? A simple example will provide an answer to this question. Let us define a set to be small if it has less than 100 elements. There are clearly an enormous number of small sets. For example, The set of continents. The set of Platonic solids. The set … Continue reading Sets that are Elements of Themselves: Verboten →

Goldbach’s Conjecture and Goldbach’s Variation

Goldbach's Conjecture is one of the great unresolved problems of number theory. It simply states that every even natural number greater than two is the sum of two prime numbers. It is easily confirmed for even numbers of small magnitude. The conjecture first appeared in a letter dated 1742 from German mathematician Christian Goldbach to … Continue reading Goldbach’s Conjecture and Goldbach’s Variation →

Hyperreals and Nonstandard Analysis

Following the invention of calculus, serious concerns persisted about the mathematical integrity of the method of infinitesimals. Leibniz made liberal use of infinitesimals, with great effect, but his reasoning was felt to lack rigour. The Irish bishop George Berkeley criticised the assumptions underlying calculus, and his objections were not properly addressed for several centuries. In … Continue reading Hyperreals and Nonstandard Analysis →

Topsy-turvy Maths: Proving Axioms from Theorems

Mathematics is distinguished from the sciences by the freedom it enjoys in choosing basic assumptions from which consequences can be deduced by applying the laws of logic. We call the basic assumptions axioms and the consequent results theorems. But can things be done the other way around, using theorems to prove axioms? This is a … Continue reading Topsy-turvy Maths: Proving Axioms from Theorems →

Goldbach’s Conjecture: if it’s Unprovable, it must be True

The starting point for rigorous reasoning in maths is a system of axioms. An axiom is a statement that is assumed, without demonstration, to be true. The Greek mathematician Thales is credited with introducing the axiomatic method, in which each statement is deduced either from axioms or from previously proven statements, using the laws of … Continue reading Goldbach’s Conjecture: if it’s Unprovable, it must be True →

Laczkovich Squares the Circle

The phrase `squaring the circle' generally denotes an impossible task. The original problem was one of three unsolved challenges in Greek geometry, along with trisecting an angle and duplicating a cube. The problem was to construct a square with area equal to that of a given circle, using only straightedge and compass. That's Maths … Continue reading Laczkovich Squares the Circle →

Aleph, Beth, Continuum

Georg Cantor developed a remarkable theory of infinite sets. He was the first person to show that not all infinite sets are created equal. The number of elements in a set is indicated by its cardinality. Two sets with the same cardinal number are ``the same size''. For two finite sets, if there is a … Continue reading Aleph, Beth, Continuum →

Mathematics and the Nature of Physical Reality

Applied mathematics is the use of maths to address questions and solve problems outside maths itself. Counting money, designing rockets and vaccines, analysing internet traffic and predicting the weather all involve maths. But why does this work? Why is maths so successful in describing physical reality? How is it that the world can be understood … Continue reading Mathematics and the Nature of Physical Reality →

Berry’s Paradox and Gödel’s Incompleteness Theorem

A young librarian at the Bodleian Library in Oxford devised an intriguing paradox. He defined a number by means of a statement of the form THE SMALLEST NATURAL NUMBER THAT CANNOT BE DEFINED IN FEWER THAN TWENTY WORDS. This appears to indicate a specific positive integer, which we denote $latex {\mathcal{B}}&fg=000000$. But there is … Continue reading Berry’s Paradox and Gödel’s Incompleteness Theorem →

Buridan’s Ass

``Buridan's Ass'' is a paradox in philosophy, in which a hungry donkey, located at the mid-point between two bales of hay, is frozen in indecision about which way to go and faces starvation --- he is unable to move one way or the other. Jean Buridan was a French philosopher who lived in the fourteenth … Continue reading Buridan’s Ass →

The Ross-Littlewood Paradox

A most perplexing paradox appeared in Littlewood's book A Mathematician's Miscellany. It was later analysed in detail by Sheldon Ross in his 1988 book A First Course in Probability. Littlewood wrote: Balls numbered 1, 2, ... (or for a mathematician the numbers themselves) are put into a box as follows. At 1 minute to noon … Continue reading The Ross-Littlewood Paradox →

The “Napoleon of Crime” and The Laws of Thought

A fascinating parallel between a brilliant mathematician and an arch-villain of crime fiction is drawn in a forthcoming book – New Light on George Boole – by Des MacHale and Yvonne Cohen. Professor James Moriarty, master criminal and nemesis of Sherlock Holmes, was described by the detective as “the Napoleon of crime”. The book presents … Continue reading The “Napoleon of Crime” and The Laws of Thought →

“Dividends and Divisors Ever Diminishing”

Next Saturday is Bloomsday, the anniversary of the date on which the action of Ulysses took place. Mathematical themes occur occasionally throughout Ulysses, most notably in the penultimate episode, Ithaca, where the exchanges between Leopold Bloom and Stephen Dedalus frequently touch on weighty scientific matters. [Last week's ThatsMaths post] In Ithaca, the narrator takes every … Continue reading “Dividends and Divisors Ever Diminishing” →

Enigmas of Infinity

Children sometimes amuse themselves searching for the biggest number. After trying millions, billions and trillions, they realize that there is no end to the game: however big a number may be, we can always add 1 to produce a bigger number: the set of counting numbers is infinite. The concept of infinity has intrigued philosophers … Continue reading Enigmas of Infinity →

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 … Continue reading The Shaky Foundations of Mathematics →

Peano Music

The links between mathematics and music are manifold. Mathematics can be set to music in a simple but surprising manner. For the award ceremony of the Gödel Medal in 2014, a musical interpretation of Gödel's incompleteness Theorems was written by Danish composer Niels Marthinsen. It encodes the basic axioms of number theory that form the … Continue reading Peano Music →

The Year of George Boole

This week’s That’s Maths column in The Irish Times (TM058, or search for “thatsmaths” at irishtimes.com) is about George Boole, the first Professor of Mathematics at Queen's College Cork. Mathematician and logician George Boole died just 150 years ago, on 8 December 1864, following a drenching as he was walking between his home and Queen's … Continue reading The Year of George Boole →

Degrees of Infinity

Many of us recall the sense of wonder we felt upon learning that there is no biggest number; for some of us, that wonder has never quite gone away. It is obvious that, given any counting number, one can be added to it to give a larger number. But the implication that there is no … Continue reading Degrees of Infinity →

The Loaves and the Fishes

One of the most amazing and counter-intuitive results in mathematics was proved in 1924 by two Polish mathematicians, Stefan Banach and Alfred Tarski. Banach was a mathematical prodigy, and was the founder of modern functional analysis. Tarski was a logician, educated at the University of Warsaw who, according to his biographer, “changed the face of … Continue reading The Loaves and the Fishes →