Arithmetic and the Natural Numbers

In principle the whole of contemporary mathematics can de 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 ¾ 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 the alleged fact but the symbolic representation of the alleged 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 ‘2’ 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 = 2’. 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’.

In 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’ seems to me just as valid, merely less interesting. The fact of the matter is that each statement is true in the appropriate context, that is all there is to it.

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 exactly to objects that can be combined without merging : if they did not so apply, we would disregard 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.

As Mill correctly said, it is a matter of fact, and not of logic, that if you have, say, a collection of stones, say  ΟΟΟΟΟΟΟΟΟΟΟΟΟ   and you are told to put them into containers Δ that have room for  ΟΟΟΟ only, you will need  ΔΔΔ  containers, no more, no less. In our rather muddled terminology, ’12 divided by 4 gives 3’ (it would be better 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 many instances actually testable, i.e. they pass the Popperian test for empirical disqualification. For example, if I read in a textbook that a pyramidal number with base 24  24 is also a square number I can check whether this is the case by building up a pyramid on this base and then flattening the whole lot and making them into 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 ¾ 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 ¾ 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 2nd 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 (dy), change with respect to a small increment in the independent variable (dx) : 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 (dx) 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 ¾ and this will come as a shock to most readers — 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 (dx) and (dy) 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 —  not that this bothers the pure mathematicians. All that remains is Space and Time which today are still usually considered to be ‘continuous’ —  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 –Ο 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 dxs 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 dn, 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, dn 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 ¾ 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 dis 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.

Definite and Indefinite Totalities

it would be wearisome to give here an empirical underpinning to all branches of classical mathematics, though I believe it can and should be done. Pending a through reform of the subject which is long overdue, mathematical sceptics such as myself are willy-nilly obliged to attend divine service though we do not have to take at face value everything the minister says. To judge by the frequent occurrence of the infinity sign, ∞, in mathematical textbooks, you would imagine that we are surrounded by double infinities at every moment of our lives, but such is not my impression. Expressions like ‘n  goes to ∞’  are basically directives, not formulae : they tell the reader to allow n to increase indefinitely and one could quite easily dispense with the infinity sign and just write n followed by an arrow when we are letting n increase without bound and have the arrow going the other way when n  is decreased indefinitely while remaining positive. This does away with the temptation to view ‘infinity’ as a specific quantity which it certainly is not.

Similarly, I don’t as a matter of fact believe in the reality of n spatial dimensions with n > 3 but I don’t have any special problem with ‘n-dimensional phase space’ so-called because the mathematical treatment has in reality nothing whatsoever to do with spatial dimensions in the normal sense. Anything that can be quantified (stick a number to), e.g. speed, mass, pressure &c. can be treated as a ‘dimension’ and, since there are certainly more than three types of quantifiable entities, it is up to a point legitimate to  talk of ‘Hilbert Space’, or of ‘n-dimensional space’ — though I still dislike the term because it thoroughly confuses the layman as it is intended to. (In string theory there are supposed to be 8 or 11 spatial dimensions but one is never sure how literally one is to take this and anyway there is to date no evidence at all that they exist.)

Infinity, if it is a meaningful term at all which I doubt, lies outside our normal sense experience and is hardly something that falls within the remit of the natural sciences. The Greeks very sensibly would have nothing to do with it : Euclid is careful not to prove that the set of prime numbers is ‘infinite’, only that there is no greatest prime number. As late as the early nineteenth century the greatest mathematician of his time, Gauss, stated categorically, “There is and can be no actual infinity”. But Cantor, the mentally unbalanced founder of Set Theory, really believed in actual infinity ¾ he specifically rejected the concept of ‘potential infinity’ (which is all in practice we need) as inadequate. And hot on his heels Russell, despite being an agnostic and a positivist, introduced into his and Whitehead’s Principia Mathematicae the ridiculous Axiom, “That infinities exist” (RW, *120.03).

The belief in ‘infinite sets’ only comes about  because of a failure to distinguish between ‘definite’ sets  and what I call ‘indefinitely extendable’ sets. A ‘definite’ set is fully constituted once and for all, and its members are, at least in theory, listable, one, two, three…… There is also a last member (though the actual choice of which member comes at the bottom of the list is usually arbitrary). An ‘indefinitely extendable set’ cannot be listed ‘once and for all’ because it is capable of being extended. The following anecdote will perhaps make clearer what I am driving at.

Two schoolboy philosopher meet up during the lunch break. Sceptic  exclaims, “You know, there’s not a single thing I’m sure about!” His companion rejoins, “Ah! but there is one thing at least you’re sure about, and that is that you aren’t sure about anything!” This rather floors Sceptic —  for the moment.

But in fact there is no contradiction. Sceptic’s first statement only referred to the fully definite set of all beliefs he had actually considered up to that moment, and a standpoint of all-round scepticism was  not one of them. It would be quite perverse to consider his first statement as referring to the collection of all possible beliefs he as a human individual might conceivably entertain. The belief  “I don’t believe in anything” was not, at the beginning of the discussion, a member of the Set of All Beliefs Sceptic Had Considered So Far (a definite set) but after the end of the conversation it was. His first statement was time and context dependent : it was not an intemporal assertion.

All this hardly seems worth dwelling on. So why the fuss ? Because, when it comes to mathematics, the situation is very, very different. Mathematical assertions are not generally considered to be time and context dependent, they are in some sense held to be ‘eternally true’, true even before human beings or the universe we live in existed. Once true, always true, when it comes to mathematics.

A ridiculous amount of printing ink has been devoted to such earth-shaking issues as whether the Set of All Sets contains itself or what exactly is the status of the Ordinal of the set of All Ordinal Numbers. In common speech we don’t tie ourselves up in mathematical knots because we usually restrict ourselves to statements which have a meaningful context, and above all we do not confuse finite definite Sets with what one might call ‘partly definite Sets that are capable of being extended’. Most of the meaningful categories such as ‘the human race’, ‘members of the EU’, ‘species of insects’, and so on, are collections which are particle definite in the sense that we can list bona fide members but which are continually being extended as mew human beings are born, Eastern European countries are allowed into the EU, and the procedures of natural selection produce new species. This does not make these Sets ‘infinite’, only indefinitely extendable and no one except mathematicians ever have a problem with this.

Viewed in this light, many of the ‘paradoxical’ features of so-called infinite sets vanish like morning dew. According to Dedekind and Cantor, the Set of All Positive Integers, itself an ‘infinite’ set, can be put in one-one correspondence (paired off) with the Set of All Even Numbers, since for any positive number you like to mention I can match this with its double. What of it? This is only astonishing and mystical and extraordinary if we consider ‘all’ the natural numbers laid out on the grass in some Platonic Never-never Land and ‘all’ the even numbers lying on top of them. Even though there are ‘more’ natural numbers than evens ¾ since 3 and 5 and 151 are not even ¾  the two sets can be paired off ! Amazing! How does this fellow Cantor do it?

If, however, N, the Set of All Natural Numbers, is viewed simply as an indefinitely extendable set starting 2, 4, 6, then the paired set {(1,2), (2,4), (3, 6)…..} is also an indefinitely extendable set. So what? But for any given definite set of integers, a subset comprising the evens within this set cannot be paired off with the whole set.

As a matter of fact most of the mathematical sets we are interested in turn out to be ‘partly definite indefinitely extendable sets’, i.e. we can list a few members but are free to extend the list ‘indefinitely’. What above all we must not do is, however, to confuse or compare an open-ended indefinitely extendable set with a fully constituted one.

In real life, not only do most sets get extended all the time, some members drop out (through death for example) and certain individuals oscillate endlessly between different sets. This sort of situation wouldn’t make for good mathematics, since mathematics needs things to be clearcut, but the real world is actually not that mathematical.

A typical example of ‘oscillating membership’  is provided by Russell’s Village Barber Paradox though Russell did not realize this. Russell invites us to consider a Village Barber who claims he shaves everyone in the village who does not shave himself and only such persons. The big question is : Does he shave himself? If he does shave himself, he shouldn’t be doing so — since, as a barber, he shouldn’t be shaving self-shavers. On the other hand, if he doesn’t shave himself, that is exactly what he ought to be doing.

The contradiction only arises because Russell, like practically all modern mathematicians, insists on viewing sets as being constituted once and for all in the usual Platonic manner. Let us see what would actually happen in real life.  It is first of all necessary to define what we mean by being a self-shaver : how many days do you have to shave yourself consecutively to qualify ? Ten? Four? One? It doesn’t really matter as long as everyone agrees on a fixed length of time, otherwise the question is completely meaningless. Secondly, it is important to realize that the Barber has not always been the Village Barber : there was a time when he was a boy or perhaps inhabited a different village. On some day d he took up his functions as Village Barber in the village in question. Suppose our man has been shaving himself for the last four days prior to taking on the job, so, if four days is the length of time needed to qualify as a self-shaver, he  classes himself on day d as a self-shaver. He does not get a shave that day since he belongs to the Self-shaving set and the Village Barber does not shave such invididuals.

The next day he reviews the situation and decides he is no longer in the Self-shaving category — he didn’t get a shave the previous day — so he shaves himself on Day 2. On Day 3 he carries on shaving himself — since he has not yet got a run of four successive self-shaving days behind him. This goes on until Day 6 when he doesn’t shave himself. The Barber spends his entire adult active life oscillating between the Self-shaving and the Non-self-shaving sets. There is nothing especially strange about this : most people except strict teetotallers and alcoholics oscillate between being members of the Set of Drinkers and Non-drinkers — depending of course on how much and how often you have to drink to be classed as a ‘drinker’.

This example was originally chosen by Russell to show that the self-referential issue has nothing necessarily to do with infinity. Nor does it, but it does depend on the question of whether sets or collections are time and context dependent.

Important Numbers and Formulae

Specific numbers, sequences and formulae can be important in three rather different senses :

1. 1.   they are significant in Nature and the physical world;
2. 2.   they are key items in the construction of a workable symbolic system;
3. 3.   they are important for aesthetic reasons.

Ideally, that is to say Platonically, the really significant numbers and formulae ought to be important in all three ways at once, and certainly this is how most of us would like them to be. But usually they are not. The constant of gravitation and the fine structure constant are extremely important in sense (1) but not at all in senses (2) and (3). (This is in itself a strong argument against Platonism.) To date all attempts to deduce the basic constants found in Nature, g, c and so on, from logical or philosophic principles have failed dismally. But if mathematics really were a window on a timeless world which lies within and behind this one, it should theoretically be possible to do precisely this ¾ as Eddington for one tried to do. Plato and Hegel thought it was possible to deduce astronomy from mathematics : Plato decided that all planetary orbits must be circular because the circle was a perfect shape, Hegel proved from first principles that there could be no more than seven planets in the solar system.

The great Platonic argument from the alleged ‘simplicity’ of Nature has always struck me as extremely weak. The motions of the planets round the Sun are so complicated that it took humanity thousands of years to painfully work out a decent calendar. Had a mathematician been put in charge of the solar system, he would undoubtedly have arranged things very differently, perhaps making the orbits fit into the shapes of the Platonic solids as Kepler at first supposed was the case. More recently, Solomon Golomb has invented a genetic code far more elegant than the messy UAGT  one used in our bodies since it is not only ‘comma-free’ (cannot be read in two or more incompatible ways) but “can correct any two simultaneous errors in translation, and detect a third error”. As someone put it, “Life would be more reliable if Golomb had been put in charge”. Most of the really  elegant pieces of pure mathematics are of little utility while the mathematics of Relativity and Quantum Theory, though undoubtedly ingenious, could not remotely be called either simple or beautiful. A pure mathematician is supposed to have said about partial differential equations (which are useful), “This is not mathematics but stamp collecting” and I am inclined to agree with him.

There are a few numbers and functions ¾ but not that many ¾ which are equally significant to the pure mathematician and the scientist, e being a case in point. There is nothing surprising about the omnipresence of e, or rather the importance of exponential and logarithmic functions in Nature since they are central to biological processes. The more bacteria or humans there are, the more there are going to be in the near future when they reproduce, so we would expect population growth to be ‘exponential’ (proportional to size) ¾ until the food supply gets used up, or other limiting factors kick in.

The numbers we use the most are those which serve as bases, notably 10, the Babylonian base 60 (still used for seconds and minutes), also 2 which has had a new lease of life because of computers. Apart from 2, none of these numbers are especially important in Nature which, though it has a certain tendency to favour bilateral symmetry (right-left) does not use numerical bases. 10 is a dull number and its quasi-universality as a numerical base is the result of a historical accident ¾ primates have ten fingers and thumbs.

Conversely, many of the theorems and numerical conjectures that have delighted and obsessed  mathematicians are of no importance in physics. I doubt if it makes any great odds scientifically speaking if the Riemann Conjecture, the leading unsolved problem of pure mathematics, is true or not, and the same applies to Euler’s surprising discovery that the inverse squares 1 + ½2 + 1/32 + ….  sum to p2/6. Primes have limited use since they are currently used for encryption but otherwise there is no particular significance in their distribution amongst the natural numbers. Indeed, the most prized mathematical theorems are like emeralds : they delight humans but are of no great significance to the physical world.

The Unholy Alliance of Platonism and Formalism

Mathematical Platonism, apart from being a hard sell in a godless world, is difficult to maintain across the board. In practice most pure mathematicians sign up to some sort of amalgam of Platonism and Formalism : the best bits partake of the eternal but the more mundane parts are human invention. As one eminent contemporary mathematician puts it

“There are things in mathematics for which the term ‘discovery’ is much more appropriate than ‘invention’.(…) One may take the view that in such cases the mathematicians have stumbled upon ‘works of God’ ” (Penrose, The Road to Reality).

This is all very well up to a point and I am myself mathematician enough to have felt this frisson when confronted with certain theorems which, as Hardy said when being confronted by certain theorems of  Ramanujan, “must be true because no one could possibly have conceivably have made them up”. One is reminded of Coleridge’s distinction between Fancy and Imagination. But the lack of any clear guidelines as to how to distinguish the dross from the gold, the invented part from the divine, is embarrassing to say the least especially since we are dealing with a discipline that prides itself on its rigour. The criteria are vague : ‘simplicity’, ‘generality’, ‘elegance’, ‘profundity’…… On top of that, these criteria, for what they are worth ¾ and they are worth something ¾ are rarely met at the same time. I wouldn’t call the formula for solving a quadratic ‘beautiful’ (x = (b ±  Ö(b2 4ac))/2a  but it certainly has generality. Leibnitz’s formula

p/4 = 1 1/3 + 1/5 1/7 ……

is delightful but more a curiosity than anything else and useless for calculation purposes (because you have to add  hundreds of terms to get p even to three or four places after the decimal point). Indeed, I would go so far as to say that the elegance of pure mathematical theorems is inversely proportional to their usefulness (even in  strictly pure mathematical terms). But if a theorem of formula really were ‘universal’, it would crop up all over the place, both in nature and in man-made systems.

The extravagance of the claims made for mathematics (by Platonists) is sometimes really alarming : certainly no practitioner of any other art or science would dare to speak in such terms without fear of ridicule. Anyone passionately involved in something tends to envisage the whole universe in terms of it : the ballet dancer sees life as a dance and countless people from Empedocles onwards have envisaged the whole of life as a kind of erotic play. For the musician, in the beginning was the sound, for the calligraphist, all started with a brushstroke. But Penrose typically discounts all such claims except those made for mathematics :

“I cannot help thinking that, with mathematics, the case for believing in some kind of ethereal, eternal existence, at least for the more profound mathematical concepts, is a good deal stronger than in those other cases. There is a compelling uniqueness and universality in such mathematical ideas which seems to be of quite a different order from that which one could expect in the arts or engineering.”

Penrose, (1989)

But why on earth should we believe that complex analysis is more ‘eternally true’ than Bach’s fugues or Highland Dancing? Certainly, of all symbolic systems yet invented, mathematics is by far the most efficient as a prediction system for physical phenomena ¾ but this is not what is being claimed.
Wonderful though it is (or can be), in recent years mathematics has become rather too big for its boots and needs to be brought down a peg or two. Mathematics arises in the main from our human concern to understand the workings of Nature, so at any rate I would like to believe. But this does not mean that Nature is consciously mathematical ¾ it neither is nor needs to be. In mathematical terms, every time a young child, or a cat, reaches out to snatch a dangling object it is, as someone put it in a popular maths book, ‘solving a succession of differential equations’. But is it? Of course not. Snatching a moving object is just something you have to learn to do if you want to survive. And if biologists are to be believed ¾ and no one these days wants to take them on ¾ evolution is blind and devoid of intelligence as we understand the term : the rather disappointing message is that if you’ve got the time, trial and error suffices for most things, for that is basically all natural selection is.

A final point that is worth making is that the much vaunted ‘objectivity’ of mathematics is self-limiting : it means that mathematics, has precisely nothing to say about the human condition, nothing about the joyful/painful sense of the ephemerality of life, the perennial theme of most  art, especially painting and lyrical poetry. We may reasonably doubt that a Platonic world of eternal Forms and/or mathematical formulae exists, but it is not possible to doubt that the passing world of ephemeral sensations exists, the ‘Floating World’ of the Japanese printmakers who in turn gave rise to French Impressionism. If everything changeless and timeless turns out in the last resort to be an invention (= illusion), as Zen Buddhists believe, then this ‘Floating World’ of sensations is ‘more real’ than the Platonic ¾ and indeed certain Indian Buddhists such as Vasubandhu and certain Chinese Taoists sais precisely this. Our mathematics, manufactured to deal with the changeless, is a clumsy and inadequate instrument for approaching the ephemeral.

Conclusion

The belief that mathematics, in its entirety, is ‘a free invention of the human mind’ is completely unacceptable 1. because this is not, as far as we know, how arithmetic and geometry were first conceived and why they were developed, and 2. because it would make mathematics’ success as a predictive system and model of physical behaviour a pure fluke. Mathematics is indeed not the physical reality itself but a product of human effort. However, that part of it which is the most useful and the most important is a constrained invention, constrained by the way things are even though the ‘things’ do not know how and what they are.
So much for Formalism. The rather different belief that mathematics ¾ those bits of it that we find ‘profound’ or beautiful ¾ is equally hard to sustain : certainly such a belief is, in the Popperian sense, unscientific since there is apparently no way this side of the grave of disproving (or proving) such an extravagant claim. The argument from simplicity is weak because, although the universe may have looked reasonably simple to Archimedes and Newton, it certainly does not look simple today. The alleged ‘simplicity’ of mathematics is perhaps merely a human convenience. The argument from design is no longer accepted in biology ¾ so why should it pass water in mathematics? The trend in evolution is rather from simplicity to greater and greater complexity. As to beauty I am responsive enough to some areas of mathematics to be sympathetic to such a line of argument but it is really not much more than the ‘argument’ of a nineteenth century vicar who cites Beethoven’s Ninth Symphony as a ‘proof’ that there must be a God. I have myself witnessed one of Europe’s most eminent mathematicians show the audience the Mandelbrot Set on a screen as evidence of…well, evidence that mathematics could not be entirely a human invention because ‘who could have invented this’? Yes, the world is full of strange and wonderful things, the Mandelbrot Set perhaps included (though I don’t find it beautiful) ¾ but why shouldn’t it be? One could equally argue that there must be a Devil by exhibiting many of the horrible and monstrous things also to be found in Nature.

The commonsense view on mathematics is that it is a mental construction intended to represent those parts of the natural world in which we humans, or some of us humans, were most interested. This at any rate is how mathematics started. From the empirical/constructionist point of view which is mine the success of (basic) mathematics is no puzzle at all : what has been extracted from the physical world can be returned to it. I believe that science’s main concern should be with truth and that scientific assertions should be if humanly possible empirically testable or, if not, have ‘explanatory power’. Now on these counts elementary mathematics up to and including early calculus is a science, indeed ‘Elementary’ Number Theory is probably the  most successful science that will ever be invented since the properties of ‘numbers’, their divisibility or not in particular, are very deeply rooted in the natural world.

As to the more fanciful branches of mathematics, especially all those involving the transfinite for whose ‘existence’ (other than mathematical) there is not a scrap of evidence, they do not in general have much physical content at all though occasionally they contain information about the world in a veiled form, usually unbeknown to their inventors. All this area of mathematics is not a science at all but at best an abstract art. Certainly, it is sometimes possible for science fiction to become science, and one example is Riemann’s hyperbolic geometry against all odds turning out to be invaluable as a tool in astronomy, but this sort of thing happens far less than mathematicians would have you believe. The typical modern pure mathematician has a distaste for physical reality and during the last hundred years immense effort has been expended to move mathematics away from contamination with the real. In return, society as a whole would do well to treat the more outlandish claims of mathematicians concerning the importance of their pet subject with distrust and scepticism.

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