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Is Mathematics a free Creation of the human mind? : Arithmetic and the Natural Numbers

October 28, 2012

 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 ¾ 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 ³ 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’ ¾ since ‘1 + 1 = 1’ is just as valid, merely less interesting and fruitful.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  ¢¢¢¢¢¢¢¢¢¢¢   and you are told to put them into containers 5 that have room for ¢¢¢¢ only, you will need 555 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  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  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 so-called physical laws

To be continued      S.H.  28/10/12

 

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