Sebastian Hayes

Arithmetic and the Natural Numbers

In principle the whole of contemporary mathematics can be deduced from the six or seven basic axioms of Zermelo-Fraenkel Set Theory.  No one, of course, ever learned mathematics that way (including Zermelo and Fraenkel) and doubtless no one ever will.

As far as we can tell, mathematics did not evolve as the result of philosophic speculation or as a formal exercise in symbol manipulation. It was the large, centrally controlled societies of the Middle East, Sumeria, Assyria and Babylon in particular, who developed both writing and numbering (2). Why? Their reasons are pretty obvious: a hunter/gatherer, goatherd or small farmer who is in constant contact with his small store of worldly wealth does not need much of a number system, but a state official put in charge of a vast area with varied resources does (3). Arithmetic was invented and rapidly brought to quite an advanced level for mundane and very unromantic reasons : it was needed for stock-taking, censuses and above all taxation. Geometria, literally ‘land measurement’, was developed by the Egyptians for similar reasons : it was found necessary to assess accurately the surface area of very dissimilar plots of land bordering the Nile so that the peasants working these plots could be taxed more or less fairly. It was only much later that the Greeks turned geometry into a recondite and stylish branch of higher mathematics.

J.S. Mill, almost alone amongst ‘modern’ writers on logic and mathematics, took a pragmatic view of arithmetic. “’2 + 2 = 4’ is a physical fact”, Mill dared to write in his Logic for which he has endlessly been ridiculed since by the likes of Frege, Russell and countless others. Strictly speaking, Mill is wrong. ‘2 + 2 = 4’ is not a physical fact but the symbolic representation of a  physical fact — but this is splitting hairs. What Mill meant is undoubtedly correct, namely that ‘2 + 2 = 4’ is a faithful representation of what happens when you take //, or ‘two’ objects and bring them together with another // objects, making up a group of //// or ‘four’ objects. Does anyone seriously doubt that this is what happens?

‘1 + 1 = 2’ is untrue if we are dealing with entities which merge when they are brought into close proximity. For droplets of water ‘1 + 1 = 1’. Droplets of oil are a little more complicated since I have it from a physics textbook that, if you keep on adding oil, drop by drop, to a blob on a sheet of water, the original blob eventually separates into two blobs. There is thus an upper limit on n in oil-droplet arithmetic. For the limiting value N, when     n < N ‘1 + n = 1’, but if n ≥ N, ‘1 + n = 2’.

I cannot for the life of me see that ‘1 + 1 = 2’ is a ‘truth of logic’ as Russell and Whitehead consider it to be. If it were to be so considered, then we would have the undesirable situation where two incompatible statements were both ‘logical truths’ — since ‘1 + 1 = 1’ is  just as valid, merely less interesting and useful to such creatures as ourselves. The fact of the matter is that each statement is true in the appropriate context, voilà tout.

However, this does not mean that our elementary mathematics is a ‘free creation’ or that the rules of arithmetic we have are completely arbitrary. They apply very well to the manipulation of objects that can be combined without merging : if they did not so apply, we would not bother with them and use other ones. This has nothing to do with whether or not our rules of arithmetic can be deduced from the Peano Axioms : Nature did not consult Peano in the matter, nor did it need to. We need mathematics to work out certain things but Nature works things out the hard and long way, namely by trial and error which is basically what natural selection is. There are indeed certain ‘givens’ but one does not need to know what they are : any animal which tried to get as large as it possibly could would eventually collapse under its own weight and the species would soon go extinct. But that doesn’t mean animals or trees or anything else know the ‘laws of mechanics’ : of course they don’t. Mathematicians labour under the  delusion that everyone and everything depends on their say-so.

As Mill correctly said, it is a matter of fact, and not of logic, that if you have a collection of stones, say  ΟΟΟΟΟΟΟΟΟΟΟΟ and you are told to put them into containers  that have room for ΟΟΟΟ only, you will need ‘three’ containers no more, no less. In our rather muddled terminology, ’12 divided by 4 gives 3’ (better would be to say ’12 divided into 4 gives 3’).

Theorems of so-called elementary Number Theory are not only ‘provable’ in the pure-mathematic sense, but in the majority of cases actually testable (though I am not suggesting that we should do this). The point is that they pass the Popperian criterion, that these assertions can be shown not to be the case. For example, if I read in a book that the number 4900 is at once a so-called pyramidal number and a square one, I can see whether this is in fact the case. Given enough ball bearings I can build up a pyramid on a base 24 × 24, reducing each level by one bearing on each side, so that the next layer is 23² and so on down to a single ball bearing. Flattening all this out onto the table I find I can form a square which turns out to have side 70.

Obviously, I am not going to test such statements most of the time since I have confidence that the normal rules of arithmetic are soundly based, but at least I know I have this possibility. It will be objected that, when dealing with general statements which apply to an unlimited number of cases, I cannot test them all. This is indeed so but what I can do is examine a particular case and then convince myself that what makes the proposition true in this case is not something specific to the particular case, but which will extend to all other cases of this type. Such a procedure does not cover non-constructive proofs of theorems which provide for the ‘existence’ of such and such a number without giving any indication of how such a number can be produced. However, such proofs do not have the persuasive power of constructive proofs and have rightly been treated with suspicion by many mathematicians. The proofs given in Euclid Books VII, VIII and IX, which are devoted to Number Theory, on the other hand are strictly constructive.

Moreover, theorems about the so-called ‘natural numbers’ are, in general, not just ‘roughly true’, ‘true in the limiting case’, ‘statistically true’  and so on, but are either completely true or wrong. Such a situation can only make practitioners of other sciences gasp with envy. Aristotle’s physics, in its day no mean achievement, had to give way to Newton and classical mechanics has had to give way to Quantum Mechanics. But the substance of Greek Number Theory has, apart from a greatly improved notation, scarcely changed in twenty-three centuries. It is in this sense that we should interpret the oft-quoted statement of Gauss to the effect that “Mathematics is the Queen of the Sciences and Number Theory the Queen of Mathematics”.

And the reason for the much greater sureness of results in Number Theory is that numbers (whole numbers) are far more basic than everything else. The distribution of the prime numbers is a fait accompli which does not depend on a formula, even if one could be found, it is ‘what it is’ and  follows ineluctably as soon as we have something which is repeatedly divided up into little bits. Physicists have imagined all sorts of universes where not only the basic constants but many of the ‘laws’ themselves would be different, but it is impossible to imagine a physical world where, for example, Unique Prime Factorisation does not exist : if you don’t agree try to imagine one. The divisibility properties of numbers are ‘given’ and no intelligence is necessarily involved : Nature does not know and does not need to ‘know’ what quantities can be divided up in such and such ways. Perhaps, the same goes for physical laws but this is harder to believe : even though scientists have long since dispensed with an intelligent Creator God, they still need to appeal to certain ‘physical laws’ which are conceived somehow to have been there before even the universe existed.

Calculus and Infinitesimals

It is distressing in the extreme that practically everyone assumes that because Calculus is more difficult than ordinary arithmetic, it is in some sense ‘truer’. The exact opposite is the case. Except in very simple examples where it is not needed, Calculus always involves ‘rounding off’ whilst elementary arithmetic doesn’t. If I amalgamate two flocks comprising 100 and 200 sheep respectively, the resulting flock will have 300 sheep, not approximately but exactly. In such cases the mathematical model is 100% accurate.

In the Differential Calculus, and representing the increments in the independent and dependent variables respectively, can always be arbitrarily decreased, at any rate in ‘continuous functions’. This means, amongst other things, that time cab be chopped up into ‘infinitely small’ segments –  can one really believe this? Even if one can, the assumptions on which Calculus is based are obviously wrong if we are dealing with phenomena that are known to be discrete. Also, in Calculus the roles of the dependent and independent variables, x and y, can be, and very frequently are, inverted at will : this means, in realistic terms that effects can cause causes which is fatuous.

Suppose we have a machine powered by steam or diesel and we set it to work. Can the input we give to it be arbitrarily decreased? Obviously not. Any energy input beyond a certain level will not be sufficient to overcome internal friction and so no work will ne done at all. (To think otherwise is to quarrel with the 2^nd Law of Thermo-dynamics.)

Are the roles of energy in put and work done interchangeable? No, they are not : output depends on input but input does not depend on output except in sophisticated machines which have feedback devices, and even then only to a small degree. Also, Calculus is blithely used in molecular thermo-dynamics even though (dn) can, in reality, never be less than 1, i.e. a single molecule. The same goes for population studies.

So how on earth does it come about that such an inaccurate mathematical model somehow ‘gives the right result’? “By virtue of a twofold error, you arrive, though not at science, yet at the truth!” as Bishop Berkeley exclaimed in wonderment. The good Bishop’s objections were more philosophical than technical though for all that unanswerable at the time. During the nineteenth century when the Queen of the Sciences parted company from her husband Natural Philosophy, the mathematical inconsistencies were sorted out and the conceptual problems swept under the carpet where they have remained ever since.

How does an increment in the dependent variable, call it (δy), change with respect to a small increment in the independent variable (δx) : this is essentially the issue which gave rise to the Infinitesimal Calculus. Newton needed to solve it to determine orbits amongst other things. Now, if y is strictly proportionate to x, y = Ax + C, with A, C, constants, then the rate of change will be the same no matter how large or small we make (δx) and the graph of the ‘derivative’ (giving the rate of change) will be a flat straight line (or nothing at all if f(x) is a constant function). In such cases we do not need Calculus. In every other case – and this will come as a shock to most readers ¾ the so-called derivative can only be determined by discarding non-zero quantities and so does not give the exact rate of change of any actual physical process.

The eventual mathematical solution was to view the ‘derivative’ not as a ‘final ratio’ as Leibnitz and Newton did, but as a ‘limit’, where ‘limit’ has a very precise mathematical sense. The way in which a limit is defined in Analysis neatly sidesteps the issue of whether the limiting value is actually attained, or not. Roughly, the idea is that if (and only if) we can make the difference between a proposed value and the actual value of a function as it approaches a certain point less than any given quantity, then the function  ‘goes to the limit’. For example, in the above example, we are allowed to consider the limit of the derived function to be 2x since, given any assigned quantity, we are free to make (dx) even smaller.

So how did it come about that geniuses like Newton and Leibnitz were incapable of grasping what most sixth formers today absorb in a single lesson ? The reason is rather an odd one. Newton and Leibnitz were mathematical realists and not Formalists : they believed that mathematics should, and could, provide a model for what actually went on in the real world. The analytic assumption does the trick but it involves the wholly unrealistic assumption that (δx) and (δy) can be made arbitrarily small.

Leibnitz, in his version of Calculus, always dealt in definite ratios between definite quantities however small, and could not rid himself of the conviction that there must be a final ratio between two quantities in tandem where one changes continually with respect to the other. Against all the odds, he has been proved right. For, as stated earlier, in all working machines there is a lower limit to the energy input that can overcome internal friction and produce useful work. We also know now for a fact that all energy, light, heat, chemical bonding and so on, is quantized so there is always a lower limit to all energy transfers. The quantities involved are, of course, very small by our standards but there is no longer any need to call them ‘infinitesimal’, a vague and meaningless word : in many cases the lower limits can actually be given a number. It is the nineteenth century analytic mathematical model which has been shown to be unrealistic in its assumptions ¾ not that this bothers the pure mathematicians. All that remains is Space and Time which today are still usually considered to be ‘continuous’ ¾ though this has not been proved and probably never will be. Some physicists are already suggesting that Space/Time may be ‘grainy’ at a certain level and Wolfram in A New Kind of Science writes,

“The only thing that ultimately makes sense is to measure space and time taking each connection in the causal network to correspond to an identical elementary distance in space and elementary interval in time” (my italics). (p. 520)

He even puts a number to these Space/Time infinitesimals, guessing that the “elementary distance is around 10^-35 metres, and the elementary time interval around 10^-43 seconds” (5). Whitrow before him launched the concept of the ‘chronon’, an ‘atom of time’, evaluating it as the diameter of the smallest elementary particle divided by c, the speed of light.

The Integral Calculus is perhaps a better model of real life conditions than the Differential –  which is why it was developed first — but, nonetheless, in its modern form it involves treating a whole host of quantities, speed, momentum, force &c. as continuous when in most cases we know perfectly well that they are not. And the Integral Calculus works for much the same reasons as the Differential does : if the quantities under consideration are small enough, the unreal mathematical assumptions don’t make much odds.

Calculus was developed to deal with a situation where, typically, we have two widely different scales of values, macroscopic and microscopic. The macroscopic values correspond to things we can actually observe as human beings but we usually assume, and in many cases know for a fact, that these quantities are built up, or broken down, bit  by bit in very small stages, too small to be recorded other than with high precision instruments. The growth of a bacterial disease can be modelled by Calculus, but it depends on the number of bacilli or viruses within the human body, and  this number, though large, is certainly not infinite. Likewise, we may suppose that the growth of a bacterial colony is ‘continuous’, is ‘going on all the time’, but this is not the case since even the fastest growing bacteria require about twenty minutes to reproduce themselves.

Practically speaking, the microscopic changes can usually be safely neglected beyond a certain point : this is why throwing away all these little δxs doesn’t make a lot of odds. But how do we know where to draw the line? This is a matter to be decided by the practising physicist or engineer: the business of the mathematician is to provide a coherent model which can be adapted to varying circumstances. We, as human beings, consider that if δn, a small increment, or reduction, in the human population variable, is a single person and it is someone we know, then this increment is not negligible. But if we are dealing in billions, as in world population studies, such a quantity really is negligible and Calculus is perfectly acceptable as a model even though we know that the quantity we are concerned with is always discrete. In molecular thermo-dynamics, δn cannot be smaller than a single molecule and is usually negligible : nonetheless, the more reputable texts warn the student about the dangers of using the Integral Calculus at the quantum level — it can quite simply give the wrong answers.

The centuries old ‘mystery’ of Calculus turns out to be nothing more than a failure to carefully distinguish between the very different requirements of pure and applied mathematics. The pure mathematician seeks consistency, generality and elegance, the applied mathematician wants fidelity to the facts of the matter. In the pure mathematical model δx is quite properly left as a free variable without a lower limit even though in most (all?) applications it will have a precise non-zero value. In pure mathematics the ‘rate of change’ of one quantity with respect to another ‘converges to a limit’ and it is immaterial to the pure mathematician whether it actually attains the limit (generally it doesn’t). But in the real world there is always a final ratio between causes and effects as Leibnitz stressed. Instead of being ‘God’s shorthand’, Calculus simply turns out to be an ingenious method of getting approximately true results when we do not know the exact values of certain small quantities. Today, in the computer age, the tendency is, increasingly, to dispense with Calculus and to slog it out numerically. Sic transit gloria mundi.

To be continued