Putting reality back into the equation

Blind faith in the magic of numbers played a major role in the financial crash. Education must change to ensure that mathematicians have their feet firmly planted on the ground, Chris Ormell argues

一月 20, 2011

Mathematics tends to be both misunderstood and credited with magic powers, especially by those who are intelligent but not mathematically inclined. Arising from this, there is a perennial temptation for mathematicians to play to the gallery and to assume the role of magicians and, even more temptingly, high priests.

The late Bernard Scott, founding professor of mathematics at the University of Sussex, wrote a paper in the 1960s deploring the tendency in some schools to treat mathematics as juju. This, if left unchecked, leads to people acquiring a blind faith in mathematics and mathematical formulae, a faith that bears little relation to their true logical powers.

We know that a major cause of the financial crash of 2008 was the blind faith invested in certain mathematical risk models by managers in the financial institutions. They thought they had covered the risks, but the mathematics and financial reality were, it turned out, miles apart. The problem was exacerbated by the fact that few senior managers in the banks understood the mathematics.

Scott warned against the dangers of letting mathematical educators adopt attitudes that tacitly encouraged their pupils to interpret mathematics the "blind faith" way. It is a warning we need to heed now more than ever.

The great civilisations of the past thrived on blind faith in gods and families of gods. But even when blind faith is directed towards distinctly other-worldy objects, it has a tendency to imply that a particular practical way of life is best. And this, we know, finally led to practical contradictions that contributed to the withering away of once-great civilisations.

Today, faith in gods is rarely as blind as it used to be, because modernity inserts doubts into the minds of even the most devout believers. But blind faith seems to have jumped from theistic objects to the sciences, and in particular mathematics, a subject that puts many people on the defensive and therefore in no position to challenge its single-minded devotees.

That this is a problem for mathematical education hardly needs to be said. But basic mathematical teaching has been stuck in a rut since the practical maths advocated by W.H. Cockcroft in his 1982 report, Mathematics Counts, ran out of steam in schools. It did not engage the divide between maths and reality well enough, or sufficiently interestingly, to carry conviction. The resulting compromise between practical maths and Platonic maths-for-maths'-sake is unconvincing for both teachers and pupils alike. A new paradigm is urgently needed, at all levels - primary, secondary and tertiary - if only to train the financiers of the 2030s.

We have seen blind faith in mathematics in action recently. In addition to the contribution of mathematical models to the great credit crunch of 2008, take physicist Stephen Hawking's claim that philosophy is dead. The reason he gave was that philosophers have stopped bothering trying to understand modern mathematical cosmology. This cosmology is based on current mathematical physics, most of which has been in place for less than 100 years. It is an impressive edifice of concepts and mathematical models, but one that has not yet built up a track record for reliability over a thousand years, let alone a million years.

Around 1900, various theorists wondered whether the velocity of light might be slowly changing. It was a pertinent question. If the velocity of light was changing very slowly, many of our astronomical calculations would have to go into the bin. Perhaps the gravitational constant might be slowly changing: that, too, would throw our calculations out.

Instead it was discovered that light does not travel in absolutely straight lines, but bends slightly due to the Earth's gravitation. It is a minute effect and detectable only with great difficulty, but its consequences are deadly. If this degree of bending occurred in outer space, the light from the nearest star would have completed a circular trajectory on its way from its source to our telescopes. Nothing in the Universe would be where it appears to be.

But, but, but...we all tend to be quite sure that this is only fanciful thinking. Of course light travels in straight lines in outer space! We all have a degree of blind faith in mathematics. We have no reason to believe that merely travelling through space will cause light to bend, though the presence of dark matter in space would have a bending effect if that matter were not absolutely uniformly distributed. (As we know almost nothing about dark matter, it is a material assumption to suppose that it is absolutely uniformly distributed.)

So were the values of c and G - the speed of light and the gravitational constant - the same a million years ago as they are today? It must surely stick in the throat to say that we are quite sure about this. At one time people thought that the magnetic poles were absolutely fixed. Now we know that they are on the move. So let's admit that there is a minute element of doubt here, say 1 per cent. If so, we can be 99 per cent sure that these physical constants were the same 1,000,000 years ago as they are today. If we follow logic - and being blindly sure that the mathematics is right should hardly incline us to reject logic - then we can be only (0.99)2 x 100 per cent sure that these constants were the same 2,000,000 years ago.

And (0.99)100 x 100 per cent sure that they were the same 100,000,000 years ago. This is 36.6 per cent, a reasonable figure perhaps, given that 100,000,000 years ago was a long time ago.

Cosmologists assure us that the Big Bang happened 13.7 billion years ago, that is, 13,700 million years ago.

So what is the figure for the degree to which we can be sure that the physical constants c and G were the same then? Clearly it is (0.99)13,700 x 100 per cent, which comes out as 1.59 x 10-58 per cent.

This may be too minute to reckon with, but it is fully in line with the figure for the same thing one second earlier - namely zero per cent.

When physicists say they are close to a theory of everything that requires 11 additional dimensions, or worse still, an infinity of parallel universes, or a postulated universe before the Big Bang, their blind faith in mathematics is visible for all to see. These 11 new "dimensions" have not been discovered by technologists with instruments capable of actually measuring things in the 11 new dimensions. They are simply parameters in the mathematics that it has been found convenient to use. They are then credited with representing "dimensions".

This is ontology inflated far beyond all previous records, rather like the absurdly inflated figures in Fernando Botero's famous paintings. William of Ockham will be turning in his grave, because any idea of economising with hypotheses seems to have flown out of the window.

But blind faith in mathematics may also be found in pure mathematics itself, that is, uninterpreted mathematics. So we are talking here about mathematical objects defined precisely by means of about 60 symbols, letters and marks. Mathematicians long ago started reifying their symbolic marks, turning /// into 3 and //////////// into 12. These reifications are what pure mathematics is about: this is the basic furniture of the subject.

With 60 symbols, letters and marks one can define an infinity of mathematical expressions. These are then reified and turned into objects. They include numbers, vectors, matrices, tensors, sets, spaces, topoi and so on. Mathematicians are mostly fascinated by infinity, and whether, for example, out of the infinity of even numbers they can all be equated to the sum of two prime numbers. But there is only an infinity of mathematical objects, not a super-infinite (transfinite) totality.

So how has it happened that for a hundred years, the mathematical establishment has swallowed the idea of transfinite sets? Georg Cantor produced an argument that seemed to point to transfinite immensities, but that was before we realised that mathematics was incompletable. In effect Cantor's argument showed that the set of real numbers was incompletable. It did not (could not) show that there were more mathematical objects than an ordinary infinity.

Cantor thought that mathematics was timeless, so his mistaken interpretation looked at the time (the 1890s) quite reasonable. At which point blind faith kicked in and David Hilbert, Bertrand Russell, Alfred North Whitehead and many others who ought to have known better, became intoxicated with the idea of the transfinite. This was amazing! Numbers beyond infinity! It was like discovering that God had an elder brother.

Once installed, blind faith ensured that the transfinite would continue to thrill and amaze generations of students - in spite of the inconvenient fact of its mathematical impossibility.

The lesson from all this is that we need to educate mathematically in a way that is both conceptually interesting and has its feet on the ground. Today the world is far more dependent on mathematical concepts and process (usually executed by microchip) than in any previous era. We need a method of mathematical education that is commensurate with the heavy responsibility this entails.

Neither practical maths nor Platonic maths is up to this task. We have misread the relationship between mathematics and the real world. C.S. Peirce said long ago that "mathematics is the study of hypotheses", which means ideas about new ways to do things, make things, devise explanations and so on. The practicalists treat maths as a language for handling a limited range of fairly humdrum, nitty-gritty, real-life situations. The Platonists treat it as a glorious end-in-itself with no need to think about the real world at all.

Neither approach will prepare the bankers and cosmologists of the future to think carefully about the gap between mathematics and the real world.



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