## Maths and the Case

According to books written by scientists for non-scientists, the normal statement of the physicist is: ‘I propose that such and such is the case in the physical world and have carried out repeatable experiments which, so far, have not falsified this hypothesis.’

The mathematician, on the other hand, says: ‘Within the meanings I have assigned to the symbols and the rules I have laid down for manipulating them, such and such is self-consistent’. His propositions need not relate to the physical world. Unless he claims that they do, he need do no laboratory experiments – indeed it is difficult to see how any could be done. The nearest he will get to falsification is if he or others discover that, within his own terms, his propositions contradict themselves.

That much is neat and tidy. Both physicists and mathematicians know the separate criteria against which their work is to be assessed.

But modern science is not as simple as that. The physicist often claims that much of what he says is validated by maths, and the mathematician often claims that his symbols and equations do say something about the physical world. For us non-scientist onlookers this can lead to confusion.

We can, as regards the physicist, believe his hypothesis to be ‘true’ when, as he predicted, under certain conditions a ship will float or an aerofoil provide lift, even though the maths on which he has relied are beyond our fathoming. We can see that what he said would happen, does happen. We are prepared to accept that his maths must have been sound. But what if he – let’s call him ‘the scientist’ – claims that his mathematical propositions do relate to the physical world, but cannot (yet) be physically tested? Here, us onlookers have neither pragmatic evidence nor the skill to know whether the maths stand up.

The most obvious example is sub-atomic physics. It claims to be about the actual physical world, but at such micro level that its hypotheses cannot presently be verified or contradicted by experiments and may for the indefinite future remain beyond this manner of testing. Its typical statement is of the kind: ‘If we are to make sense of what is within the reach of our microscopes, then such and such might be the sort of thing that is beyond their reach’. (There are those who prefer to put this the other way round, i.e. we construct a model from which the known could be inferred, but I’m doubtful whether this difference in formulation gets us any nearer to actuality.)

Does such a statement deal with the actual physical world, or is it a purely mathematical construct? It might well claim to be both, but to the extent that it is pure maths, its first allegiance is to whatever system of maths it has designed. We, the onlookers, may sense something of desperation on the part of the physicists as more and more strange-sounding entities or qualities are added to the growing community of the non-detectable, but the more they look like maths, the less we can judge.

Not only this, but as part of the package, we are also asked to accept:

that, according to the Uncertainty Principle, what we observe is affected by our act of observing it and that we can therefore never see what, unobserved, it is and

that, at extreme micro level, matter (whether in the form of waves or particles) behaves in ways that are foreign to our experience of the observable world. In this unobservable (but real) world, so we are told, notions such as that events are caused by other events, or that the time-arrow moves only one way, or that one thing can be in only one place at any one time, do not belong.

What are us onlookers to think? (And I do notice that in this territory the overwhelming majority of scientists – who are not involved in sub-atomic physics – are just as much onlookers as us non-scientists.) Are we to trust that the maths must be right? Are we to say that, if it they lead to such absurd conclusions, they must be wrong? Do we give up a quarter of the way through yet another seemingly responsible paperback despite – see back page – the overwhelmingly respectable c.v. and credentials of the author?

Before falling one way or the other, let me look at a less obvious but probably more fundamental example.

What is a cosmologist? It looks as if he is both a physicist and a mathematician, but predominantly the latter. He is aware of what we presently see of the universe, but he is also curious about its future and its past.

As to the past, suppose that, armed with maths, he seeks to determine the condition of the universe half-way back across its expanding life. He uses what we know of its present physical and chemical state, he compares this with what evidence we have of change, he tries to establish a direction and rate of change. He devises equations as measuring instruments and he most certainly uses symbols to represent the values with which he is working.

However complex the result may be, it seems reasonable to say that, faced with an expanding and chemically changing universe, he will arrive at a condition half-way through (in terms of time) the process of expansion.

Suppose now that he seeks to track the process the whole way back. The usual result seems to be that his calculations lead him to a ‘point of infinite density’.

In trying to describe this in non-mathematical terms he stops short of saying that the ‘point’ has no dimension. It could lead to a conclusion with which he might well disagree, namely that something came from nothing. He tends to disappear into formulae which nobody except a fellow mathematician can grasp. The ‘shortest possible’ distance, he might tell you, for example, is ‘the Planck distance’, about 10 to the minus 33 cm. It is linked to ‘the Planck time’, of about 10 to the minus 44 seconds. And how fast does something cross the Planck distance? Well, of course, at the speed of light. It is usually at this point that the layman gives up and lets the mathematician off the hook of ‘Why not something smaller?’ And by now the layman may be too dizzy to recall that, in mathematical terms, a ‘point’ is that which has location but no dimension. If that were not so, then the ‘point’ would be insufficiently precise.

But that leaves the mathematician with what, outside maths, are problems. Physicists seem to agree that density, if not identical with, does have a relationship with mass. And mass, so they seem to say, has, at least in terms of gravitational pull, a relationship with dimension. Against this, what is our cosmologist actually saying?

Then of course there is this word ‘infinitely’. This, so far as I can see, has a useful and necessary function in maths, but what meaning does it have when applied to the actual physical beginning of the universe? It is as if the mathematicians are saying: ‘Look, our retro-active formulae require it to be the case that there was something, almost but not quite nothing, to which we cannot assign any dimension, but which had a density which our greatest imaginable numbers could not describe.’

And of course, as with sub-atomic physics, us non-numerate onlookers must make choices between this and that, neither of which we have any hope of understanding.

This would not matter were it not that, if only because we don’t understand what they are saying, mathematicians have acquired the prestige of the mediaeval priesthood. At great moments their predictions have come true. The eclipse of 1919 proved Einstein to be correct when he said that gravity can bend light. The maths must have been dead right to get a man on the moon. Boolean logic can’t be far wrong if it has given us computers.

(At a lower level, but one which touches most of us more immediately, those acolytes, the actuaries and accountants, govern much of commercial life. He who measures, must be able to evaluate.)

At risk of excommunication, I will say that at both micro and macro levels if the formulae and equations of scientists lead to what reason says are contradictions, then, despite triumphs such as those mentioned above, they are provisionally wrong. The onus is on them to prove themselves right. How they do this, far removed from empirical demonstration, is up to them.