What then is Mathematics?

Kurt Gödel’s Proof of the Existence of God

Anyone who thinks there is a simple answer to that question will be seriously disappointed. What have famous mathematicians said about it?

Mathematics is the art of giving the same name to different things (Henri Poincaré)

Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true (Bertrand Russell)

Is mathematics something ‘out there’ waiting to be discovered, or is it a product of the human mind? Do we discover mathematical laws and relationships in nature, or do we impose mathematical descriptions on nature as a way of making sense of it, and harnessing it? Do we even know what number and arithmetic are? Yes, we all know what five loaves and three fishes are because those numbers have referents, but what of the number ‘three’ apart from its being a word or a symbol – does it exist as an immaterial entity? If so, where?

Questions such as these have interested philosophers and mathematicians for centuries. There are at least three fundamental questions to be addressed, on ontology, epistemology, and truth:

• What is the nature of mathematical objects?
• How do we obtain knowledge of them?
• How do we account for certitude in mathematics?

We could add another question on effectiveness:

• How do we account for the utility of mathematics in physics?

The various views on these questions correspond to the two camps of the medieval philosophers, the Realists and the Nominalists (or Anti-Realists), though Realism goes back much further to ancient Greece. Essentially, Realists believe that abstract entities or universals exist in their own right independently of the mind that thinks them, whereas Nominalists deny the extra-mental reality of universals and abstract ideas.

The Du Sautoy Code

Professor Marcus du Sautoy

Marcus du Sautoy, Simonyi Professor for the Public Understanding of Science (having succeeded Richard Dawkins in the Chair), is currently presenting a series of TV programmes about mathematics and nature entitled ‘The Code’. Viewers could be forgiven for believing that what he is presenting is a mainstream view of mathematics rather than peddling his own peculiar brand of atheistic metaphysics. Since no appropriate caveats have been employed by the BBC, we feel it necessary to make a few of our own.

Firstly, Du Sautoy’s view that, as the Pythagoreans expressed it, ‘Number is everything’ is of very ancient pedigree; but, nothwithstanding, it is undemonstrable (which should be anathema to a mathematician) and a faith-based religious concept. Secondly, philosophers of mathematics and informed students of mathematics know that there is, to date, no satisfactory understanding of the relationship, if any, between mathematics and reality; to suggest that there is a relationship, and what such a relationship might be, is an act of faith. And thirdly, it is very unfortunate for scientists to be working with mathematics as though mathematics itself is the original reality to which the physical world ‘must’ conform through such things as ‘laws’; science has been hideously corrupted in the last 80 years because of this.

Some Christians might be heartened to see and hear Du Sautoy suggesting that numbers are at the root of all reality, that this is in some way all grist to the mill of Intelligent Design. Not so fast: Du Sautoy is an avowed atheist (who not very wittily gives his religion as ‘Arsenal’) who by his own admission is trying a more ‘softly softly’ approach than Richard ‘The Rottweiler’ Dawkins (whom all can see is a bigoted fanatic) and is not appealing to design, or even apparent design, but to some mysterious entity he calls ‘The Code’. A code at the very least implies information content, but The Code (as a proper noun and with the definite article) suggests something unique and powerful. Thus Du Sautoy:

…underlying everything that surrounds us, from the natural world to the cities we live in, there is a hidden code that explains why things look and behave they way they do.

[This hidden code (‘The Code’)] has the power to unlock the laws that govern the universe.

The Code is the truth of the universe, and its numbers dictate the way the world must be.

So, this hidden code, this entity that Du Sautoy calls ‘The Code’, has total and complete explanatory power, is identical to Absolute Truth, can lead us into All Truth, and is completely deterministic. This is unquestionably a religious position. And it is none other than the old heresy of Pythagoras, the pagan Greek philosopher, re-worked by gnostics, Kabbalists, Freemasons, Rosicrucians, Illuminists, and now, it appears, New Atheists. What a wheeze if they can pull this one off!

Arabic Myths

It was once mistakenly thought that the Arabs invented ‘Arabic’ numerals, including the number ‘0’ (zero), a great improvement over Roman numerals for algebra, another supposed Arabic invention. That’s complete nonsense. For example, Cajori’s A History of Mathematics (Fifth edition, 1991) informs us:

The grandest achievement of the Hindus, and the one which, of all mathematical inventions, has contributed most to the progress of intelligence, is the perfecting of the so-called “Arabic Notation.” That this notation did not originate with the Arabs is now admitted by everyone.

What we call ‘Arabic’ numerals are not the characters that appear in Arabic script. Even the Arabs themselves don’t call these signs Arabic numerals but Western numerals, or Hindu numerals, which was their actual source around 200 BC. By the fifth century AD the Indians were using decimal notation and had developed the use of zero from an earlier concept of it in Babylonian mathematics.

Through the writings of Gerbert D’Aurillac (c.946-1003) these numerals became known in Western ‘Latin’ Europe. In 1202, the mathematician Fibonacci in his Liber Abaci [The Book of the Abacus] describes the notation and calculation method that he learned from his youth thus:

…following my introduction…to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others…Therefore, embracing more stringently that method of the Hindus [Modus Indorum], and taking stricter pains in its study…I have striven to compose this book. The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0… any number may be written…