Mathematical Meditations

by Thomas Scarborough

I shall call this post an exploration—a survey. Is mathematics, as Galileo Galilei described it, ‘the language in which God has written the universe’? Are the numerical features of the world, in the words of the authors of the Collins Dictionary of Philosophy, Godfrey Vesey and Paul Foulkes, ‘free from the inaccuracies we meet in other fields'? Many would say yes—however, there are things which give us pause for thought.

• Sometimes reality may be too complex for our mathematics to apply. It is impossible to calculate in advance something as simple as the trail of a snail on a wall. Stephen Hawking noted, 'Even if we do achieve a complete unified theory, we shall not be able to make detailed predictions in any but the simplest situations.' If we do try to do so, therefore, we abuse mathematics—or perhaps we should say, in many contexts, mathematics fails. 

• Our measurement of the world may be inadequate to the task—in varying degrees. I take a ruler, and draw a line precisely 100 mm in length. But now I notice the grain of the paper, that my pencil mark is indistinct, and that the ruler's notches are crude. In many cases, mathematics is not the finest fit with the reality we deal with. In some cases, no fit at all. I measure the position of a particle, only to find that theoretical physicist Werner Heisenberg was right: I have lost its velocity. 

• The cosmologist Rodney Holder notes that, with regard to numbers—all numbers—'a finite number of decimal places constitutes an error'. Owners of early Sinclair calculators, such as myself, viewed the propagation of errors in these devices with astonishment. While calculators are now much refined, the problem is still there, and always will be. This error, writes Holder, 'propagates so rapidly that prediction is impossible'. 

• In 1931, the mathematician Kurt Gödel presented his incompleteness theorems. Numbers systems, he showed, have limits of provability. We cannot unite what is provable with what is true—given that what this really means is, in the words of Natalie Wolchover, ‘ill-understood’. A better known consequence of this is that no program can find all the viruses on one’s computer. Consider also that no system in itself can prove one’s own veracity. 

• Then, it is we ourselves who decide what makes up each unit of mathematics. A unit may be one atom, one litre of water, or one summer. But it is not that simple. Albert Einstein noted that a unit 'singles out a complex from nature'. Units may represent clouds with noses, ants which fall off a wall, names which start with a 'J', and so on. How suitable are our units, in each case, for manipulation with mathematics?

• Worst of all, there are always things which lie beyond our equations. Whenever we scope a system, in the words of philosophy professor Simon Blackburn, there is 'the selection of particular facts as the essential ones'. We must first define a system’s boundaries. We must choose what it will include and what not.  This is practically impossible, for the reason that, in the words of Thomas Berry, an Earth historian, 'nothing is completely itself without everything else'. 

• I shall add, myself, a 'post-Gödelian' theorem. Any and every mathematical equation assumes that it represents totality. In the simple equation x + y = z, there is nothing beyond z. As human beings, we can see that many things lie outside z, but if the equation could speak, it would know nothing of it. z revolts against the world, because the equation assumes a unitary result, which treats itself as the whole.
  

Certainly, we can calculate things with such stunning accuracy today that we can send a probe to land on a distant planet’s moon (Titan), to send back moving pictures. We have done even more wonderful things since, with ever increasing precision. Yet still the equations occupy their own totality. Everything else is banished. At what cost?

Staring Statistics in the Face

By Thomas Scarborough

George W. Buck’s dictum has it, ‘Statistics don’t lie.’ Yet the present pandemic should give us reason for pause. The statistics have been grossly at variance with one another.

According to a paper in The Lancet, statistics ‘in the initial period’ estimated a case fatality rate or CFR of 15%. Then, on 3 March, the World Health Organisation announced, ‘Globally, about 3.4% of reported COVID-19 cases have died.’ By 16 June, however, an epidemiologist was quoted in Nature, ‘Studies ... are tending to converge around 0.5–1%’ (now estimating the infection fatality rate, or IFR).

Indeed it is not as simple as all this—but the purpose here is not to side with any particular figures. The purpose is to ask how our statistics could be so wrong. Wrong, rather than, shall we say, slanted. Statistical errors have been of such a magnitude as is hard to believe. A two-fold error should be an enormity, let alone ten-fold, or twenty-fold, or more.

The statistics, in turn, have had major consequences. The Lancet rightly observes, ‘Hard outcomes such as the CFR have a crucial part in forming strategies at national and international levels.’ This was borne out in March, when the World Health Organisation added to its announcement of a 3.4% CFR, ‘It can be contained—which is why we must do everything we can to contain it’. And so we did. At that point, human activity across the globe—sometimes vital human activity—came to a halt.

Over the months, the figures have been adjusted, updated, modified, revised, corrected, and in some cases, deleted. We are at risk of forgetting now. The discrepancies over time could easily slip our attention, where we should be staring them in the face.

The statistical errors are a philosophical problem. Cambridge philosopher Simon Blackburn points out two problems with regard to fact. Fact, he writes, 'may itself involve value judgements, as may the selection of particular facts as the essential ones'. The first of these problems is fairly obvious. For example, ‘Beethoven is overrated’ might seem at first to represent a statement of fact, where it really does not. The second problem is critical. We select facts, yet do so on a doubtful basis.

Facts do not exist in isolation. We typically insert them into equations, algorithms, models (and so on). In fact, we need to form an opinion about the relevance of the facts before we even seek them out—learning algorithms not excepted. In the case of the present pandemic, we began with deaths ÷ cases x 100 = CFR. We may reduce this to the equation a ÷ b x 100 = c. Yet notice now that we have selected variables a, b, and c, to the exclusion of all others. Say, x, y, or z.

What then gave us the authority to select a, b, and c? In fact, before we make any such selection, we need to 'scope the system'. We need to demarcate our enterprise, or we shall easily lose control of it. One cannot introduce any and every variable into the mix. Again, in the words of Simon Blackburn, it is the ‘essential’ facts we need. This in fact requires wisdom—a wisdom we cannot do without. In the words of the statistician William Briggs, we need ‘slow, maturing thought’.

Swiss Policy Research comments on the early phase of the pandemic, ‘Many people with only mild or no symptoms were not taken into account.’ This goes to the selection of facts, and reveals why statistics may be so deceptive. They are facts, indeed, but they are selected facts. For this reason, we have witnessed a sequence of events over recent months, something like this:

• At first we focused on the case fatality rate or CFR
• Then we took the infection fatality rate into account, or IFR
• Then we took social values into account (which led to some crisis of thought)
• Now we take non-viral fatalities into account (which begins to look catastrophic)

This is too simple, yet it illustrates the point. Statistics require the wisdom to tell how we should delineate relevance. Statistics do not select themselves. Subjective humans do it. In fact, I would contend that the selection of facts in the case of the pandemic was largely subconscious and cultural. It stands to reason that, if we have dominant social values, these will tend to come first in our selection process.

In our early response to the pandemic, we quickly developed a mindset—a mental inertia which prevented us from following the most productive steps and the most adaptive reasoning, and every tragic death reinforced this mindset, and distracted us. Time will tell, but today we generally project that far more people will die through our response to the pandemic than died from the pandemic itself—let alone the suffering.

The biggest lesson we should be taking away from it is that we humans are not rational. Knowledge, wrote Confucius, is to know both what one knows, and what one does not know. We do not know how to handle statistics.

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.