Clive Kilmister's Presidential Address

Clive Kilmister served as President of the Mathematical Association and delivered his Presidential Address to the Association at its Annual Conference in April 1980. His Address was entitled Zeno, Aristotle, Weyl and Shuard: two-and-a-half millenia of worries over number was delivered on 9 April. It was published in The Mathematical Gazette 64 (429) (1980), 149-158. We give a version of Kilmister's Address below.

Zeno, Aristotle, Weyl and Shuard: two-and-a-half millenia of worries over number, by C W Kilmister.

Presidential Address to the Mathematical Association at the Annual Conference, April 1980.

It has been common in recent years, if not entirely obligatory, for your President to address you on this occasion about the difficulties and opportunities he has found in the practice of teaching, and this year will be no exception. But following the excellent example of all past Presidents, as far back as I have investigated, I intend to confine my address to that part of the activity about which I am practically conversant. So while I hope that the general ideas and conclusions of my talk will be enjoyable and possibly useful to everyone here, I shall deal in detail only with the narrow problem of first-year undergraduates. I say simply 'problem' because I believe that the various different questions (the 'interface', the content of first-year pure/applied courses, how should the sixth form teacher best prepare the future university entrant, ...) are all aspects of a single problem. For the sake of precision I shall deal with only one aspect of this problem, as it arises in analysis. Knowing me, you might have expected rather that I would have discussed the difficulties of teaching mechanics, or of persuading young people of 18 plus (although many of them have been brought up since five years of age on a practical approach to mathematics!) that there is any possibility at all of making an application of mathematics in the external world. My depression about that aspect is too strong, and my involvement in it is probably too deep, for me to see it clearly. But my experiences supervising small groups of students who are making their first genuine acquaintance with the real number field suggest to me the possibility of improving matters.

This brings me to the last of the four names in my title and, for those who are still wondering, it is indeed Hilary Shuard who, though she may feel in strange company amongst such eminent historical personages, has prompted my thinking just as much as the other three. She once remarked to me: "Perhaps they do need to know all about the real numbers, but do they need it all in the first year?" And she has developed some kindred points in the Gazette [1] about the teaching difficulties involved in giving definitions of function, limit, and so on. In digging out her Gazette paper I happened to come across the Association's 1962 Analysis report; its general atmosphere is strikingly different from that reigning now. Eighteen years ago we could expect a majority of first-year students to have been exposed to the further topics suggested in the report; many fewer in 1980. However, it isn't only today's young people who find the real numbers difficult, and this brings me to another of my hobby-horses, the importance of the contribution of the history of mathematics to the understanding of mathematics. I do not suppose I need to argue this here; for any subject which has been going on in a recognisably similar form for over two thousand years neglects its own history at its peril. But the real numbers have a history of discussion and paradox which exceeds most mathematical questions. It seems to me almost certain, in such a situation, that the difficulties of the young in confronting such a system must in part echo these historic troubles, and so we can learn from the past something of use today.

The paradoxes of Zeno have come down to us only second-hand, particularly by way of Aristotle. There are of course many possible historical reasons for the loss of Greek texts. (Indeed, when one considers these reasons the wonder is that we have any at all. In the case of the enormous wealth of plays that were written, the vast majority are lost; and we owe the survival of a fair proportion of the best geometry to the good luck of its appreciation by the Arabs.) But at least one scholar (Beth [2]) considers the loss of Zeno's originals as one of simple neglect, because Zeno had attacked and destroyed an earlier theory of actual infinity with such success that the theory and its criticism were not worth preserving. Certainly from Euclid onwards the Greeks handled infinity quite as gingerly as any young person brought up on Hardy's Pure mathematics. Euclid's proof of the infinity of primes is, more precisely, a proof that, given any prime, it is possible to construct a bigger one. He seems to be taking every care to avoid any statement about an actually infinite set of primes. Whether or not this is so, Zeno was considered worth discussion by the most subtle and acute of Greek thinkers. People tend to describe four Zeno paradoxes; they have much in common, but are subtly different, so that to refute one is not to refute the others. Probably Zeno intended to appeal to the overall effect of one after the other; none the less I shall just consider one, the Dichotomy: "Before anything can complete its motion, it must first reach the half-way point. Before it can do this it must first reach half-way to that. And so on. Hence motion is impossible."

As Goodstein has pointed out, mathematicians have for centuries misunderstood Zeno's point, and thought that the paradox showed merely that Zeno did not understand the sum of an infinite series. Well, in the first place, this must be irrelevant, since to say S is the sum of a certain infinite series is precisely to say that, no matter how small an error is prescribed, one can name a number of terms of the series whose sum is nearer to S than the error. But this fact is never called into question by Zeno! His paradox specifically raises the very question left unanswered by the sum to infinity trick: whether the motion can be completed, or, in general, does it always make sense to imagine an infinite set of operations completed? In some of the other paradoxes, time enters, with the question of its infinite divisibility. That is a historical survival of a dispute between Zeno and a follower of Parmenides, Melissus, and by choosing the Dichotomy I have avoided that question. There is a qualitative time reference in the use of "before", it is true; but no reference to a quantitative time.

We now know that some infinite sets have significantly more members than others, in this sense, that they cannot be put in one-one correspondence. The rationals are enumerable (that is, in one-one relation with the natural numbers), as is exhibited by arranging them in groups of ascending sum of numerator and denominator, the members of each group being arranged internally in order of magnitude, But not so the real numbers, defined as sets of infinite decimals. It is useful to look in a little more detail than usual at Cantor's proof that things are different there. Given any enumerable sequence of real numbers, the Cantor construction produces a real number whose first digit is one more than that of the first member of the sequence, whose second digit is one more than that of the second member, and so on; and this number is evidently not in the sequence. However, this statement - that given any sequence of reals we can construct one not in the sequence is not Cantor's theorem on the non-enumerability of the reals, but rather a schema that gives a definite theorem for each given sequence. To derive a single theorem we have to use an indirect argument (that is, reductio ad absurdum) and definitions of a rather queer kind. For we begin by assuming that there is an enumeration of all real numbers, and use this totality to define a real number which cannot occur in the totality. That is, in defining the number we have to refer to the set of all real numbers, of which it is one. Definitions like this, which define an object by reference to a totality to which the object belongs, are impredicative. I shall return to this later; but for the present, back to the Greeks. Some passages in Aristotle suggest that he was beginning to have a glimmer of the distinction between infinities, but no Greek could have posed the following option: "One can consider an enumerable aggregate of steps as a single completed entity." Such an option does not exactly answer the Dichotomy: it challenges it head-on, and says precisely and politely what a more modern speaker might brutally summarise as "So what?". But does this option, if we adopt it, have any unpleasant consequences? You might think of it as changing the furniture of the world, the ontology, by introducing, in addition to 'steps' which might now be called 'steps of order one', the aggregates of them, say 'steps of order two'. Should one go on to consider then aggregates of these, steps of order three, and so on? My instinct is to rule these out on the principle of taking on board one new idea at a time. But even with this narrow formulation, the option is not crystal clear, for we have had to use the word aggregate. Obviously we intended by this to mean something more than the result of adding steps!

Although he had not got a clear idea about enumerable and non-enumerable infinities, Aristotle showed a good deal more gumption than the average child who readily accepts the sum to infinity of a geometric progression. (I make no hypothesis about the ones who won't accept it.) For lack of a better language he distinguishes (in the Physics [3] - not, perhaps, the book of his that comes most readily to mind - Book 3, Chapter 6) between the infinite by addition and the infinite by division, the latter being thought of as produced if we divide a step of the second order into steps of the first. "In a way" he remarks, "the infinite by addition is the same thing as the infinite by division." Yet difficulties do come in. Can a line be composed of points? In Book 6 of the Physics he argues thus. He believes that the relations between the points of the line can be only certain ones, which he begins by explaining:
Now if the terms 'continuous', 'in contact' and 'in succession' are used in this way - things being 'continuous' if their extremities are one, 'in contact' if their extremities are together. and 'in succession' if there is nothing of their own kind intermediate between them - nothing that is continuous can be composed of indivisibles: e.g. a line cannot be composed of points, the line being continuous and the point indivisible. For the extremities of two points can neither be one (since of an indivisible there can be no extremity as distinct from some other part) nor together (since that which has no parts can have no extremity, the extremity and the thing of which it is the extremity being distinct).

Nor, again, can a point be in succession to a point or a moment to a moment in such a way that length can be composed of points or time of moments: for things are in succession if there is nothing of their own kind intermediate between them, whereas that which is intermediate between points is always a line and that which is intermediate between moments is always a period of time.
Here it is the geometric line which is in question. And the question really is: what is the (logical or analytic) model of the geometric continuum? Now there must be something missing in Aristotle's argument, as you can see by taking, instead of the geometric line, a model of it defined as the class of rationals pq\Large\frac p q, for 0pq\Large\frac p q1, with the usual ordering by magnitude. As a model of the line drawn between 0 and 1, we shall see that this has defects. None the less, it has a certain feature which, if we had not been trained to do otherwise, we might call continuity. For between any two points of our model line there lies another point (their average). There is then no doubt that this line is 'composed of points'; moreover, this composition is of an enumerable aggregate of such points, so that we are allowed by our option to consider the aggregate as a single completed entity. Whether these points are or are not continuous, in contact or in succession, is far from clear; what is clear is that if Aristotle's proof were correct, it would apply equally to this model line too. But since the conclusion is then false, for this line does consist of points, he must have omitted to describe the relation which holds between these points; and that such an omission is possible destroys his proof.

Let us use this same model to study another argument of Aristotle in On generation and corruption [4], Book 1, Chapter 2:
To suppose that a body can be divided repeatedly involves a difficulty. What will there be in the body that escapes the division? ... A magnitude? No; that is impossible, since then there will be something not divided ... But if it be admitted that neither a body nor a magnitude will remain ... the constituents of the body will either be points (i.e. without magnitude) or absolutely nothing.
He rejects the latter easily, and goes on: "But if it consists of points, a similar absurdity will result; it will not possess any magnitude." The division here means throwing away the point of division, so we can think of a bisection of the line as removing the point 12\large\frac{1}{2}\normalsize; then further bisections remove 14,34\large\frac{1}{4}\normalsize , \large\frac{3}{4}\normalsize; and so ultimately (after a division-step of the second order) we remove all points pq\Large\frac p q where q=2nq = 2^{n}. Could we say that the line looked any different? Say, a little paler? Certainly it has plenty of points left: even 13\large\frac{1}{3}\normalsize and 23\large\frac{2}{3}\normalsize are still there, and the points left over are in a one-one relation with the original ones.

Of course, you can easily see that 13,23\large\frac{1}{3}\normalsize , \large\frac{2}{3}\normalsize are not particularly inaccessible. Let us modify the division process a little by defining it as bisecting and choosing one half, throwing the other half away. At the first step we can let L, R stand for choosing the left or right half. The four quarters, any one of which will be the one remaining after two bisections are, in order from the 'left-hand end', LL, LR, RL, RR. What about the sequence LRLRLR... (which, according to our option, we are allowed to count as completed)? It is 'a segment of zero length', since after nn steps its length is 12n\large\frac 1 2 ^n. The operation LR gives a left-hand end at 14\large\frac{1}{4}\normalsize. Thus LR repeated nn times gives
14+142+...+14n\large\frac 1 4\normalsize + \large\frac 1 4 ^2\normalsize + ... +\large\frac 1 4 ^n,
and so the completed sequence gives 13\large\frac{1}{3}\normalsize. Notice though that this is still a point of the line. In an unguarded moment, and knowing that this was only a simplified model of the line which must show its faults soon, one might wonder if it is the extreme regularity of the rule for the sequence which ensured that the completed sequence fixed a point of the line. By no means! Here is a rule: For any sequence ss, define a new one Q(s)Q(s) by the rules
(a) For each R in ss write down a new sequence by preceding ss by as many Ls as the order of R, i.e. its number of places from the left-hand end;

(b) Compound the new sequences by starting at the right-hand end and supposing LL → L, LR → R, RL → R, RR → L and also produces an extra R in the next column on the left.

(That must have given the secret away!)
Now the rule for generating a sequence is: begin with R and then proceed thus; if s has been generated already, adjoin R to make sR. Find Q(sR ). If it begins with L, then sR is the next stage in the sequence; if it begins with R, then sL is the next stage. This generates the sequence
and this does not correspond to any point of our model line, as you have no doubt been able to prove mentally.

So the model of the line is a little too pale; for, although between any two points there is always another, yet there seem to be some kind of gaps too. What is wanting in the model is any very natural idea of length along the line, as you can see rather easily by noting that, since the points can be numbered off, we can cover the first one by a small segment of which it is the middle, of length 123=18\large\frac 1 2 ^3 \normalsize = \large\frac 1 8 say, the second one by a similar segment of length 124\large\frac 1 2 ^4, and so on. Using our option, all the points are then covered by segments whose total lengths come to 18112=14\Large\frac {\frac 1 8}{1-\frac 1 2}\normalsize=\large \frac 1 4, which is devastating, quite apart from the fact that the segments do this with a certain amount of overlapping. Now it is precisely at this point that the usual approach introduces the real number field, or continuum, by supposing that, by magic, all the limits of sequences have been adjoined. Since we can then no longer count off all the points, there is no very serious conflict with our original option, except of course that by its use we shall not be able to produce the line from points. There is a somewhat similar argument of L E J Brouwer seeking to prove that classical mathematics cannot construct the continuum; it goes like this. Consider all numbers constructed up till 9 April 1980. This number is evidently finite. Now let mathematicians continue to construct numbers, one after the other; even if we put no limit on the future duration of the world, they will construct only an enumerable set. So, even if we might want to pick holes in Brouwer's uncompromising approach, it seems clear that jumping in a bold but non-constructive way all the way to the real numbers is going too fast. It is interesting here to see how this problem troubled one of the deepest of modern thinkers. We are able to do this by means of his posthumous writings [5], just published. In a fragment written towards the end of his life, in 1924-5, Frege says:
When I first set out to answer for myself the question of what is to be understood by numbers and arithmetic, I encountered - in an apparently predominant position - what was called formal arithmetic. The hallmark of formal arithmetic was the thesis 'Numbers are numerals'. ... Quite a dodge, a degree of cunning amounting, one might say, almost to genius; it's only a shame that it makes the numerals, and so the numbers themselves, completely devoid of content and quite useless.
For if the numbers are just numerals, the numerals denote nothing and so are simply shapes on the paper, without content. So numerals must denote some kind of objects; and Frege had himself tried to answer this question, of what kind, by a logical analysis of the natural numbers. But by 1924 he finds this still deficient; the natural numbers form a discontinuous series, with always a jump from one member to the next. Even the introduction of the rationals does not help, for the series still has gaps (though not such obvious ones). "Anything resembling a continuum remains as impossible as ever." There is no bridge to get you to the real numbers; and empirical facts (sense perceptions) cannot be brought in, since then one would also have to accept as a possible fact that the series of natural numbers might one day come to an end, just as one might have to accept one day that there were no stars larger than a certain size. "But surely here the position is different: that the series of whole numbers should eventually come to an end is not just false; we find the idea absurd." Frege proposes a rather desperate way out: to base arithmetic on geometry. His detailed plan is to start at the opposite end from usual, with the complex numbers, defined in terms of a suitable fragment of plane geometry based on the notion of two points being symmetric with respect to a line (mirror images). The path to the real numbers is then obvious; unfortunately Frege did not live to show how to isolate the integers and rationals from them. But this does not help at present. Would not some intermediate path be more acceptable? As far as I know, an investigation in this spirit (perhaps with details a little different) has been carried out only once, by the third of my four heroes, Hermann Weyl in Das Kontinuum in 1917 [6].

It cannot be said that this is an easy book; the language is hard and is allied to a unique mathematical notation which is ingenious but quite unlike that now used. (Essentially, a first-order logic is formulated without quantifiers - because an existential quantifier is avoided by the device of leaving a space instead of the bound variable. Of course, there are some difficulties over successive quantifications!) The notation plays no essential part. The idea of the book is to take the natural numbers as given; then to construct from them, by the construction originally given by Hamilton in 1833 (see [7]), the integers as pairs (p,q)(p, q), the operations between such pairs being defined, entirely in terms of addition of natural numbers, so that (p,q)(p, q) has always the formal properties that pqp - q would have, if it existed. Similarly, following Hamilton again, for the rationals. Let us leave the question of the reals for the moment. Weyl insists, though, that any statement like "there exists an integer such that ..." has always to be translated back into "there exist a pair of natural numbers such that ...". This distinction becomes important for him because his view on set-theory is very restrictive. A property of the natural numbers defines what he calls a one-dimensional set, a two-place relation (which might be a similar set of integers, since these are pairs of natural numbers) is a two-dimensional set, and so on. Moreover these new objects, the sets, are quite different from the original ones: "sie gehören einer ganz anderen Existenzsphäre an." There is to be no general set-theory, but this very restricted one is in fact sufficient for elementary algebra and geometry, and so for everyday life. Not so for the whole of analysis, though. There is no time, and this would not be the occasion, to try to go through Weyl's discussion in detail. But I would like to try to give you the flavour of his trenchant approach. He says:
In this way I can give here an exact formulation of the concept of function, in place of the completely vague idea which has, since Dirichlet, become standard in analysis. ... The nineteenth-century critique of classical analysis was right ... and has produced an enormous step forward in precise thought. But what is put in place of the old reasoning, if one looks at the basic principles, is just as unclear and contestable. ... The great problem posed by the Pythagorean discovery of the irrationals - which we try to solve in connexion with the flow of time and motion by a concept of continuity - is, despite Dedekind, Cantor and Weierstrass, just as unsolved as ever.
How does he go about this? He begins, as he says, with the primitive objects, the natural numbers, and their primitive relation of succession. But at this point temptation enters:
From the primitive properties and relations others are derived and such primitive or derived relations form, in the mathematical process, one or more dimensional sets. The category which belongs to such a set is determined by the number of empty places of the relations from which it comes and by the basic categories from which these empty places can be filled. For the moment we denote all these properties and relations, sets and functional relations as of the first order.

If a, b ... are elements of a set M, we denote the relation between them and M by e. The two empty places of this relation correspond (a) to a category of sets of the first order, (b) to the basic category. ... In this way there corresponds an extended domain of operations to which we have been led by the 'mathematical process'. We refer to (one or more dimensional) 'sets of the second order' whose empty places are filled in part from the basic category, in part from categories of the first order ... [and so on] ... . In the case of an existential statement particularly, e.g. for a relation R with an empty place for a set, we would have to say 'there exists a relation of the first kind with the property that ...'. If at this point we ignore the distinction between first, second, ... orders, we are led into senselessness and contradiction fully analogous to the Russell paradox of the set of all sets that do not belong to themselves. (I believe, and will shortly show, that modern analysis wanders 'auf Schritt und Tritt' in such circles.)
The way in which troubles can arise is precisely when we want to extend the construction of rationals one step further. Weyl goes on, on the same lines as Dedekind, to define a real number as the one-dimensional set of all rationals smaller than it, that is, as a set of rationals such that, if r is a member, then, firstly, so is any smaller rational, but, secondly, there is always a larger rational than r in the set. But then he asks:
How is 'set' here to be understood? No mathematical process is to be used once only. Analysis considers real numbers, sets of real numbers. ... Should we follow the strict interpretation in this iteration or not? If not, we derive 'analysis with orders', with real numbers of order 1, 2, 3, ... and so functions likewise. All the existential statements of analysis have to be augmented with qualifications ' ... of order 2' and so on. But this leads straight to definitions and proofs of vicious circle form. For example, suppose M to be a bounded set of real numbers of the first order. Its upper bound is a real number, that is, a set of rationals, with the property that every rational in the set is less than some member of M. But this set is now a real number of order 2.

This vicious circle, which has crept into analysis through the foggy nature of the usual set and function concepts, is not a minor, easily avoided form of error in analysis. ...

An analysis 'with orders' is artificial and useless. It is clear that we must proceed otherwise - namely apply the existence concept only at the level of the primitive categories, not to the system of properties and relations, i.e. it is uniquely natural to follow the strict interpretation. Only by this means can we guarantee what with hindsight from applications is seen as of decisive importance - that all concepts, magnitudes, operations of such a precise analysis are introduced, as idealisations of analogous things in an approximate mathematics operating with rough numbers. If this should mean that, to avoid error, we have to give up such a proposition as that a bounded set of real numbers has a least upper bound, then so be it.
Actually somewhat more, it turns out, has to be given up. The standard texts on analysis list four statements which are equivalent:
(i) a sequence of enfolded segments of the line, decreasing to zero, defines a point;

(ii) a monotonic increasing bounded sequence has a limit;

(iii) a bounded infinite set has a least upper bound;

(iv) a bounded infinite set has a limit point (the Bolzano-Weierstrass theorem).
For Weyl's strict interpretation, only the first two of these hold. The other
two, it is interesting to remark, involve impredicative definitions. This fact
has led those people, regrettably few, who have read Das Kontinuum to see
it with hindsight as concerned only with this. For example Kleene [8] says:
The impredicative character of some of the definitions in analysis has been especially emphasised by Weyl, who in his book undertook to find out how much of analysis could be reconstructed without impredicative definitions.
But I think Weyl's argument is more subtle than this, and, in any case, it is not too hard to avoid the difficulty of the construction described by Weyl in the 'analysis with orders' - one simply considers, not all real numbers, but all those of finite order.

On 9 February 1918 Polya and Weyl made a bet in Zurich, with twelve witnesses (all mathematicians) (see [9]). About (iii), above, Weyl prophesied:

  1. Within twenty years Polya, or a majority of leading mathematicians, will admit that the concepts of number, set and countability involved are completely vague; and that there is no more point in asking about the truth of (iii) than of the main assertions of Hegel's physics.

  2. It will be recognised by Polya, or a majority of leading mathematicians, that in any wording (iii) is false ...
The bet was called in 1940. With one exception, everyone agreed that Polya had won; but the exception was Kurt Gödel, This must give us some food for thought. Let me make it quite clear that I am doing no more than to press the claims of Das Kontinuum as a fascinating historical document, with the inference that, if the construction of the real numbers contains subtleties that troubled such an acute intellect as Weyl's as recently as 1917, and still worried Gödel in 1940, it is not to be wondered at that some of our first-year undergraduates find it hard to stomach. Perhaps they are wiser than we are.

I think there is a moral in this for all our teaching, and not only in analysis for first-year undergraduates, It is this: if, year after year, they seem to find particular difficulty with something, it may be because the difficulty is really there. Our modern axiomatic method is very powerful; but it has a power which can sometimes be used to sweep difficulties under the carpet. And a study of history can help by indicating when the danger of this is present.

I am very grateful to Richard Sorabji for his painstaking introduction of me to Aristotle.

C W Kilmister

King's College, Strand, London WC2R 2LS

References (show)

  1. H B Shuard, Does it matter?, The Mathematical Gazette 59 (407) (1975), 7-15.
  2. E W Beth, The foundations of mathematics (North-Holland, 1959).
  3. Aristotle, Physics.
  4. Aristotle, On generation and corruption.
  5. H Hermes, F Kambartel and F Kaulbach (eds.) (trans. P Long and F White), G Frege, Posthumous writings (Blackwell, 1979).
  6. H Weyl, Das Kontinuum (Zurich, 1917).
  7. W R Hamilton, Trans. Roy. Irish Acad. 17 (1837), 293-422.
  8. S C Kleene, Introduction to metamathematics (North-Holland, 1962).
  9. Hao Wang, From mathematics to philosophy (Routledge and Kegan Paul, 1974).

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