Monday, 22 January 2018

The Deceptions of Mathematics

by Thomas Scarborough

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 = 2
the logarithm of 1 = 0
the square root of 1 = 1
and so on.
Yet 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. 

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.


Lina Scarborough said...

"Therefore, various things and various properties may all at once hide inside one and the same single mathematical unit." That reminds me very much of the structure, the morphology, of moss animals, formally called Bryozoa. A sea animal that technically consists of thousands of smaller animals.

docmartincohen said...

Facinating comparison. Which reminds me? We humans too can be seen as a complicated assemblage of different living creatures! Our 'human' cells are outnumbered by those countless alien ones of things like bacteria and viruses.. each being has its own entirely separate aims and functions in life...

Keith said...

An interesting perspective, Thomas, on the place of mathematics.

As for the core consideration of whether mathematics fits reality, even that emblem of genius Einstein seemed at times to have been conflicted . . .

On one occasion, this: “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” On another occasion, this: “How can it be that mathematics . . . is so admirably adapted to the objects of reality.”

(Here, ‘objects of reality’ as in ‘physical/natural reality’ — not, of course, the ‘realities’ of metaphysics, ethics, aesthetics, epistemology, philosophy of the mind, philosophy of religion, Eastern philosophy, political philosophy — and the rest. Nor the ‘realities’ of the other humanities and arts.)

Perhaps Einstein’s eventual confliction-cum-conviction over how close mathematics approaches the reality of natural phenomena was resolved on the heels of his magnificent ‘relativity’ triumph:

He went on to devote some thirty years of his life doggedly trying — alas unsuccessfully — to come up with a so-called Theory of Everything: to reconcile gravity, electromagnetism, and quantum mechanics within a unified theory.

A quintessentially and exquisitely mathematics-dependent enterprise — hand in hand with physics, of course.

An enterprise that — in the mix of vigorously competing solutions like string theory (with its famously taxing calculations) — continues today. In search of an umbrella theoretical framework of physics and mathematics that covers all currently known forces.

The operative words here being ‘all currently known’. One might reasonably argue, without diminishment of this grand and alluring enterprise, there may always be still more of physical reality to peel back for those unending ah-ha moments.

One handy, though overly simplistic, way to describe mathematics is in its comprising a succession of models (theories) that — with multiple revelatory eureka insights along the way that advance humankind’s understanding — come ever closer to ‘natural reality’: a dimension of ‘truth’, perhaps.

Even if, for instance, to figure out the weight-bearing capacity of a newly designed bridge — as well as, say, something grander, such as measuring the mass of the supermassive black hole at the center of the Milky Way galaxy. And much in-between.

From Pythagoras to Kepler to Newton to Bertrand Russell to Mario Livio to Eugene Wigner to [fill in the blank], many notable individuals have given a thumbs-up to the role of mathematics — “the language of mathematics,” “the unreasonable effectiveness of mathematics” — in widening the aperture of our understanding of natural phenomena.

Thomas O. Scarborough said...

Thank you, Lina and Martin. They are good images – as far as they go!

Keith provides much interesting material for discussion. I shall confine myself to two issues in this comment:

With regard to the ‘objects of reality’ – which is a powerful description of a certain view – what can one really mean? It is impossible for an object to reside in the brain, because this is a network. There are no objects there. And objects themselves are never self-contained. Once one has said this much, which objects should qualify as objects and which not? The philosopher Simon Blackburn comments, ‘We should however remember a long tradition in philosophy of calling things ‘fictions’ without disrespect. Matter, force, energy, causes [and so on] – in fact, everything you can think of.’

Then, is this not a rah-rah view of mathematics? Our world is, dare I suggest, in a bit of a strait. I shall put it down to our mathematical thinking (we may exclude pure mathematics for now). The linguist-philosophers Wilhelm Kamlah and Paul Lorenzen, whom I admire for the richness of their thought, wrote that ‘the most plaguing problematic’ of our time is the side-effects of science – which is, a mathematical way of viewing the world. But look at the benefits, you say. It is precisely because of the ability of mathematics to focus – to leave out ever more from those complexes which are its units – that it is the ruination of our world.

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