by Thomas Scarborough
On first impressions, it might seem to us then that units of quantity come ready made. Apples come in ones, oranges come in ones – so do people, animals, days, nights, doors, windows, and a great deal more. And if they do not come in ones, then we may make them into ones: one kilogram, one litre, one block, and so on. On this basis, we quantify things and perform various mathematical operations on them.
However, it is not this simple – and even a child might know it. Our ‘ones’ may really be anything at all – say, clouds with noses (‘I saw three of them today’), ants which fall off the wall (dozens), or dogs which wag their tails, and so on to infinity. In each case we are dealing with the mathematical unit ‘one’.
The theoretical physicist Albert Einstein would surely have agreed. He considered that a unit ‘singles out a complex from nature’. This surely seems a contradiction in terms. A complex consists of many different and connected parts – parts (plural) which get ‘singled’, out. That is, one takes a bundle of things or properties, and one defines them as one. Therefore, various things and various properties may all at once hide inside one and the same single mathematical unit.
Now this opens up an obvious question. Who then is to say that our mathematical units – those complexes which we have ‘singled’, out – are precisely the complexes we need for the purpose of our calculations? Supposing that we really ought to have added something to a complex which we call ‘one’ – or that we really should have taken something out – before we began to make use of it?
Besides, does one really find such a thing as a complex which is self-contained and closed? Is not every singled out complex-cum-unit criss-crossed by associations and influences without number?
When we think on it, this is true even of the simplest things in this world. For instance, we might temporarily assume that the complex ‘hamster’ does not include food or water – it merely refers to a rodent, of which there are so and so many millions in the world. Yet this complex breaks down at a certain point, as some children can tragically relate, who forgot the food or water.
Consider a thought experiment – as if it had never been conducted before. Supposing it is true that our complexes might leave things out – or squeeze things in that really ought to be left out. What then would the logical consequences be? Of course, high on the list would be that our mathematics may not fit reality, because our mathematical units are ‘not quite right’. Not only that, but we should easily find examples of this in the world.
And so it is. The mathematics of circular orbits and epicycles had to be replaced with the mathematics of elliptical orbits – the mathematics of scalars, then vectors, had to be replaced by the mathematics of tensors. The mathematics of classical thermodynamics had to be replaced by the mathematics of generalised thermodynamics – and so on. In fact our complexes may contain an entire world-view which needs to be overhauled – for example, Newtonian physics. Yet even with the new, we would do well to remember that we have now carved up our world into four mathematical models.
The nineteenth century American philosopher Charles Sanders Peirce saw that ‘every new concept first comes to the mind in a judgment.’ He was saying, apparently, that our ‘ones’ are simply creations of the mind.
On this basis, we may assume that even the simplest of mathematics is not as straightforward as it seems. In fact mathematics, writes the pioneering statistician William Briggs, requires ‘slow, maturing thought’. It is not just about numbers, but about wisdom and expansive thinking.
The deceptions are, therefore, that mathematics is objective – and that being objective, it makes an excellent fit with our world – perhaps a perfect fit with the cosmos, as Galileo suggested. No. On the contrary, we should see mathematics as a very flawed and very subjective tool – always too simplistic, always in some way violating the totality of the reality in which we live. Mathematics, at the least, should be handled with great humility.
Galileo Galilei, a man of formidable scientific ability, once wrote that ‘the universe cannot be read until we have learned the (mathematical) language.’ Mathematics, he suggested, would reveal the secrets of the entire cosmos. It is a common view – yet it is deceptive. In fact, it may reveal little more than hubris.On the surface of it, mathematics – even more than science – would seem to be thoroughly objective. Here there are no failed experiments, no false interpretations, no paradigm shifts. In mathematics – so it is frequently assumed – there is perfect certainty.
1 + 1 = 2Yet we overlook something, which would seem as simple as one-two-three. We apply mathematics, by and large, to things in the real world (pure mathematics being the exception to the rule) – and in order so to apply it, we identify units of quantity. This identification of units of quantity begins with ‘quantification’ – we map our human sense observations into units of quantity, or simply, quantity.
the logarithm of 1 = 0
the square root of 1 = 1
and so on.
On first impressions, it might seem to us then that units of quantity come ready made. Apples come in ones, oranges come in ones – so do people, animals, days, nights, doors, windows, and a great deal more. And if they do not come in ones, then we may make them into ones: one kilogram, one litre, one block, and so on. On this basis, we quantify things and perform various mathematical operations on them.
However, it is not this simple – and even a child might know it. Our ‘ones’ may really be anything at all – say, clouds with noses (‘I saw three of them today’), ants which fall off the wall (dozens), or dogs which wag their tails, and so on to infinity. In each case we are dealing with the mathematical unit ‘one’.
The theoretical physicist Albert Einstein would surely have agreed. He considered that a unit ‘singles out a complex from nature’. This surely seems a contradiction in terms. A complex consists of many different and connected parts – parts (plural) which get ‘singled’, out. That is, one takes a bundle of things or properties, and one defines them as one. Therefore, various things and various properties may all at once hide inside one and the same single mathematical unit.
Now this opens up an obvious question. Who then is to say that our mathematical units – those complexes which we have ‘singled’, out – are precisely the complexes we need for the purpose of our calculations? Supposing that we really ought to have added something to a complex which we call ‘one’ – or that we really should have taken something out – before we began to make use of it?
Besides, does one really find such a thing as a complex which is self-contained and closed? Is not every singled out complex-cum-unit criss-crossed by associations and influences without number?
When we think on it, this is true even of the simplest things in this world. For instance, we might temporarily assume that the complex ‘hamster’ does not include food or water – it merely refers to a rodent, of which there are so and so many millions in the world. Yet this complex breaks down at a certain point, as some children can tragically relate, who forgot the food or water.
Consider a thought experiment – as if it had never been conducted before. Supposing it is true that our complexes might leave things out – or squeeze things in that really ought to be left out. What then would the logical consequences be? Of course, high on the list would be that our mathematics may not fit reality, because our mathematical units are ‘not quite right’. Not only that, but we should easily find examples of this in the world.
And so it is. The mathematics of circular orbits and epicycles had to be replaced with the mathematics of elliptical orbits – the mathematics of scalars, then vectors, had to be replaced by the mathematics of tensors. The mathematics of classical thermodynamics had to be replaced by the mathematics of generalised thermodynamics – and so on. In fact our complexes may contain an entire world-view which needs to be overhauled – for example, Newtonian physics. Yet even with the new, we would do well to remember that we have now carved up our world into four mathematical models.
The nineteenth century American philosopher Charles Sanders Peirce saw that ‘every new concept first comes to the mind in a judgment.’ He was saying, apparently, that our ‘ones’ are simply creations of the mind.
On this basis, we may assume that even the simplest of mathematics is not as straightforward as it seems. In fact mathematics, writes the pioneering statistician William Briggs, requires ‘slow, maturing thought’. It is not just about numbers, but about wisdom and expansive thinking.
The deceptions are, therefore, that mathematics is objective – and that being objective, it makes an excellent fit with our world – perhaps a perfect fit with the cosmos, as Galileo suggested. No. On the contrary, we should see mathematics as a very flawed and very subjective tool – always too simplistic, always in some way violating the totality of the reality in which we live. Mathematics, at the least, should be handled with great humility.
