'Who would believe it-one and the same concept can be posited
repeatedly!
Gottlob-Frege-Posthumous-Writings
Such definitions are to be compared with thosf stucco-embellishments on buildings which look as though they supported something whereas in reality they could be removed without the slightest detriment to the building.
We can recognize such definitions by the fact that no use is made of them, that no proof ever draws upon them.
But if a wore or sign which has been introduced by definition is used in a theorem, thf only way in which it can make its appearance there is by applying thf definition or the identity which follows immediately from it.
If such an application is never made, then there must be a mistake somewhere.
Of course the application may be tacit.
That is why it is so important, if we are to have a clear insight into what is going on, for us to be able to recognize the premises of every inference which occurs in a proof and the law of inference in accordance with which it takes place.
So long as proofs are drawn up in conformity with the practice which is everywhere current at the present time, we cannot be certain what is really used in the proof, what it rests on.
And so we cannot tell either whether a definition is a mere stucco- , definition which serves only as an ornament, and is only included because it is in fact usual to do so, or whether it has a deeper justification.
That is why it is so important that proofs should be drawn up in accordance with the requirements we have laid down.
We can characterize another kind of inadmissible definition by a metaphor from algebra. Let us assume that three unknowns x, y, z occur in three equations. Then they can be determined by means of these equations. , Strictly speaking, however, they are determined only for the case where there is only one solution. In a similar way the words 'point', 'straight line',, 'surface' may occur in several sentences. Let us assume that these words have as yet no sense. It may be required to find a sense for each of these words such that the sentences in question express true thoughts. But have, we here provided a means for determining the sense uniquely? At any rate not in general; and in most cases it must remain undecided how many solutions are possible. But if it can be proved that only one solution is' possible, then this is given by assigning, via a constructive definition, a sense in turn to each of the words that needs defining. But we cannot regard as a definition the system of sentences in each of which there occur several of the expressions that need defining.
A special case of this is where only one sign, which has as yet no sense, occurs in one or more sentences. Let us assume that the other constituents of the sentences are known. The question is now what sense has to be given to this sign for the sentences to have a sense such that the thoughts expressed in them are true. This case is to be compared to that in which the letter x occ. urs in one or more equations whose other constituents are known, where the problem is: what meaning do we have to give the letter Jt for the equations to express true thoughts? If there are several equations,j
? Logic in Mathematics 213
this problem will usually be insoluble. It is obvious that in general no number whatsoever is determined in this way. And it is like this with the case in hand. No sense accrues to a sign by the mere fact that it is used in one or more sentences, the other constituents of which are known. In algebra we have the advantage that we can say something about the possible solutions and how many there are-an advantage one does not have in the general case. But a sign must not be ambiguous. Freedom from ambiguity is the most important requirement for a system of signs which is to be used for scientific purposes. One surely needs to know what one is talking about and the statements one is making, what thoughts one is expressing.
Now it is true that there have even been people, who have fancied themselves logicians, who have held that concept-words (nomina appel- /ativa) are distinguished from proper names by the fact that they are
ambiguous. The word 'man', for example, means Plato as well as Socrates and Charlemagne. The word 'number' designates the number 1 as well as the number 2, and so on. Nothing is more wrong-headed. Of course I can use the words 'this man' to designate now this man, now that man. But still on each single occasion I mean them to designate just one man. The sentences of our everyday language leave a good deal to guesswork. It is the surrounding circumstances that enable us to make the right guess. The sentence I utter does not always contain everything that is necessary; a great deal has to be supplied by the context, by the gestures I make and the direction of my eyes. But a language that is intended for scientific employment must not leave anything to guesswork. A concept-word combined with the demonstrative pronoun or definite article often has in this way the logical status of a proper name in that it serves to designate a single determinate object. But then it is not the concept-word alone, but the whole consisting of the concept-word together with the demonstrative pronoun and accompanying circumstances which has to be understood as a proper name. We have an actual concept-word when it is not accompanied by the definite article or demonstrative pronoun and is accompanied either by no article or by the indefinite article, or when it is combined with 'all', 'no' and 'some'. We must not think that I mean to assert something about an African chieftain from darkest Africa who is wholly unknown to me, when I say 'All men are mortal'. I am not saying anything about either this man or that man, but I am subordinating the concept man to the concept of what is mortal. In the sentence 'Plato is mortal' we have an instance of subsumption, in the sentence 'All men are mortal' one of subordination. What is being spoken about here is a concept, not an individual thing. We must not think either that the sense of the sentence 'Cato is mortal' is contained in that of the sentence 'All men are mortal', so that by uttering the latter sentence I should at the same time have expressed the thought contained in the former sentence. The matter is rather as follows. By the sentence 'All men are mortal' I say 'If anything is a man, it is mortal'. By an inference from the general to the particular, I obtain from this the sentence 'If Cato is a man,
? 214 Logic in Mathematics
then Cato is mortal'. Now I still need a second premise, namely 'Cato is a man'. From these two premises I infer 'Cato is mortal'.
Since therefore we need inferences and a second premise, the thought that Cato is mortal is not included in what is expressed by the sentence 'All men are mortal', and so 'man' is not an ambiguous word which amongst its many meanings has that which we designate by the proper name 'Plato'. On the contrary, a concept-word simply serves to designate a concept. And a concept is quite different from an individual. If I say 'Plato is a man', I am not as it were giving Plato a new name-the name 'man'-but I am saying that Plato falls under the concept man. Likewise we have two quite different cases when I give the definition '2 + 1 = 3' and when I say '2 + 1 is a prime number'. In the first case I confer on the sign '3', which is so far empty, a sense and a meaning by saying that it is to mean the same as the combination of signs '2 + 1'. In the second case I am subsuming the meaning of '2 + 1' under the concept prime number. I do not give it a new name by doing that. The fact therefore that I subsume different objects under the same concept does not make the concept-word ambiguous. So in the sentences
'2 is a prime number' '3 is a prime number' '5 is a prime number'
the word 'prime number' is not somehow ambiguous because 2, 3, 5 are different numbers; for 'prime number' is not a name which is given to these numbers.
It is of the essence of a concept to be predicative. If an empty proper name occurs in a sentence, the other parts of which are known, so that the sentence has a sense once a sense is given to that proper name, then, so long as the proper name remains empty, the sentence contains the possibility of a statement, but we do not have an object about which anything is being said. So the sentence 'x is a prime number', does indeed contain the possibility of a statement, but so long as no meaning is given to the letter 'x', we do not have an object about which anything is being said. Another way of putting this would be to say: we have a concept but we have no object subsumed under it. If we take as a further instance the sentence 'x increased by 2 is divisible by 4' then we have a concept again. We can take these two concepts as characteristic marks of a new concept by putting together the sentences 'x is a prime number' and 'x increased by 2 is divisible by 4'. Under this concept there falls only one object-the number 2. But a concept under which only one object falls is still a concept; this does not make the expression for it into a proper name.
Our position is this: we cannot recognize sentences containing an empty sign, the otrn:r constituents of which are known, as definitions. But such sentences can have an explanatory role by providing a clue to what is to be understood by the sign or word in question.
? ? Logic in Mathematics 215
I have read that verbal definitions are considered faulty, and 1t ts argued that we should really have no further truck with such definitions. By way of example reference was made to a definition given by me, but it was not said what a verbal definition was. 1 Of course every definition makes use of words or signs. Perhaps what is meant by a verbal definition is one in which the definiens contains a word which is a mere word as such, having no sense. Certainly this should not be allowed, but from the fact that the reader attaches no sense to a word it does not follow that the author of the definition has attached no sense to it. The insistence on sense is absolutely justified, and all the more so since many mathematicians seem to prove what are merely sentences without bothering whether they have a sense and what sense it is they have.
How little value is commonly placed on sense and definitions can be seen from the sharply conflicting accounts that mathematicians give of what number is. (We are speaking here of the natural numbers. ) Weierstrass says 'Number is a series of things of the same kind'. Another says that certain conventional shapes produced by writing, such as 2 and 3, are numbers. 2 A third is of the opinion: if I hear the clock strike three I see nothing in this of what three is. Therefore it cannot be anything visible. If I see three lines, then I hear nothing in this of what three is. Therefore it can be nothing audible either. An axiom is not a visible thing and so if we speak of three axioms, the three here is nothing visible either. Number cannot be anything whatever which can be perceived by the senses. 3
Obviously each of these attaches a different sense to the word 'number'. So the arithmetics of these three mathematicians must be quite different. A sentence from the first mathematician must have a quite different sense from the equivalent-sounding sentence of the second mathematician. This resembles what it would be like if botanists were not agreed about what they wished to understand by a plant, so that for one botanist a plant was, say, an organically developing structure, for another a human artefact, and for a third something that was not perceptible by the senses at all. Such a situation would certainly not give rise to a common science of botany.
But why should it not be possible to lay it down that by a number is to be meant a series of things of the same kind? Admittedly we can raise objections to such a course. For one thing, it may be thought that the sense of the word 'series' is not firmly enough established. Are we to think in this connection of a spatial ordering, or of a temporal ordering or of a spatia- temporal one perhaps? Further it is not clear what we are to understand by 'of the same kind'. For example, are the notes of a scale of the same kind qua
1 Frege is here referring to the article Ober die Stellung der Definition in der Axiomatik by A. Schoenjliess in Jahresbericht der deutschen Mathematiker- Vereinlgung XX (1911), pp. 222-255 (ed. ).
2 The reference is to the so-called 'formal' theory of arithmetic (arithmetic as a game with signs) held by Frege's contemporaries E. He/ne and J. Thomae (ed. ).
3 The reference is probably to G. Cantor (ed. ).
? 216 Logic in Mathematics
notes, or are they of the same kind only if they have the same pitch? But let us assume that explanations were given which cleared these matters up. A train is a series of objects of the same kind "'hich moves along rails on wheels. It may be thought that the engine is nevertheless something of a different kind. Still that makes no essential difference. And so such a number comes steaming here from Berlin. Let us assume that the science of these numbers has been set up. There is no doubt that it must be entirely different from the science in which certain shapes that one makes on a writing surface with a writing instrument are called numbers. Even if the form of words is the same, the thought expressed must be quite different. Now it is striking that the sentences of these fundamentally different sciences, each of which is called arithmetic, are constituted by precisely the same words. And it is even more striking that the practitioners of these sciences have no inkling that their sciences are fundamentally different. They all believe that they are doing arithmetic, and the same arithmetic at that, the same number theory, although what one of them is calling a number has no resemblance at all to what another is calling a number.
How is this possible? One would almost think that mathematicians regard the words used, the form of an expression, as the essential thing, and the thought expressed as quite inessential. Perhaps they think 'The thoughts contained in sentences are really no concern of mathematicians-they are a matter for philosophers; and everything to do with philosophy is of course extremely imprecise, uncertain and essentially unscientific. A mathematician who remains true to his scientific calling will have nothing to do with it. True, it can happen to even the best of them, in a moment of weakness, to let a definition slip, or something which looks like one, but we should not accord any significance to that. It is all one with a man sneezing. Really the only thing that matters is that they should all agree on the words and formulae they use. That is enough for a mathematician who has not been infected by philosophy. '
But is that then a science which proves sentences without knowing what it proves? But is it the case in actual fact that scientists do agree in the words they use? Are not mathematical works written in different languages, and are they not translated into other languages? In which case, of course, we no longer have the same form of words. But there must be something else which is preserved. And what can this be but the sense? So the thought, the sense of a sentence, cannot after all be wholly irrelevant. And does one not feel in the depth of one's being that the thoughts are the essential thing-that it is in fact these alone that we are concerned about?
But how do they come to be treated as irrelevant? How can one possibly imagine that two quite different sciences should really have the same content? Is it only because they are both called arithmetic and both treat of numbers, althpugh what is called a number in the one is quite different from what is called a number in the other? Or is not the explanation rather that we have really to do with the same science; that this man does attach the
? ? Logic in Mathematics 217
same sense to the word number as that man, only he doesn't manage to get hold of it properly? Perhaps the sense appears to both through such a haze that when they make to get hold of it, they miss it. One of them makes a grasp to the right perhaps and the other to the left, and so although they mean to get hold of the same thing, they fail to do so. How thick the fog must be for this to be possible! But it must surely show up in the proofs that they have not got hold of the same sense. Yes, it could not fail to, if the proofs were drawn up in a logically rigorous way, with no gaps in the chains of inference. But this is just what we shall fail to find. If no use is ever made of a definition, there might as well not be one. However wide it may be of the actual target, no-one will notice. Another mathematician's shot may miscarry on the other side; but since he makes no use of his definition, this too might as well not be there. We can see in this way how definitions, which seem to be utterly irreconcilable, lie peacefully alongside one another like the animals in paradise. I f only things were really like that!
Really the question that must surely exercise one is how multiplication takes place with Weierstrass's numbers. Next to my window there is a bookcase; on its top shelf there is a series of things of the same kind, a number. This afternoon at approximately 5. 15 an express train, which is likewise a number, arrives at Saal station from Berlin. It is a widely held opinion that if one number is multiplied by another, the result is again a number. Accordingly the result of multiplying our series of books by the Berlin express would again have to be a series of things of the same kind. Now how are we to do that? I read in a set of lecture notes which contains a lecture by Weierstrass1 'According to the definition a numerical magnitude is formed by the repeated positing of elements of the same kind. ' So this, apparently, is meant to be an application of the definition. How does the definition itself go? 'We can imagine a series of things of the same kind if by things of the same kind we understand things which have a complex of determinate characteristics in common. We shall understand by the concept numerical magnitude such a series. '
Mention is made here of numerical magnitude instead of number, but this is immaterial. To begin with, we have an assertion 'We can imagine a series of things of the same kind. ' This is a psychological truth, which is really of no concern to us in the present context. But now does it follow from the definition that a numerical magnitude is formed by the repeated positing of elements of the same kind? There is no doubt that on the definition our express is a numerical magnitude, for it is a series of things which have in common a complex of determinate characteristics. Well, is a train formed by the repeated positing of carriages? Do I have to posit repeatedly one and the same carriage? And how do I do that? Or do I have to put one carriage
1 Presumably the lecture notes entitled 'Analytische Funktionen' which are referred to in Der wissenschqftliche Nachlass von Gottlob Frege, by H. Scholz and F. Bachmann. See footnote 2, p. 29 of Actes du congres international de philosophie scient(fique, Paris 1935 VIII, Paris 1936 (ed. ).
? 218 Logic in Mathematics
next to another? In that case it would be better to say 'A train is formed by putting one carriage next to another. ' I do not believe that railwaymen are yet acquainted with this method of forming a train. So I should like to put it in question that a numerical magnitude can thus be formed by the repeated positing of things of the same kind. Certainly nothing can be gathered from the definition about how this is to be done.
The lecture notes go on to say, 'Now the concept "magnitude" can itself be viewed as a unit and posited repeatedly e. g. b, b, b . . .
'Who would believe it-one and the same concept can be posited repeatedly! Here b seems to be the sign for the concept magnitude. Is this concept posited repeatedly by writing down the sign for it over and over again? Incidentally there seems to be a mistake here. At least it seems to me as if it is not the concept magnitude, but a particular magnitude that is meant to be repeated. In that case we should presumably have to see b as a sign for this magnitude-for this express train, for example. But what has writing this sign down over and over again got to do with the repeated positing of the express? Or is it perhaps not the express itself, but an idea which I have of it, that we are to take as the numerical magnitude? This would turn the issue into a psychological and subjective one, without its becoming any clearer. Numerical magnitudes would be psychological structures and arithmetic a branch of psychology. But, to return to the point, how do we arrive at multiplication? The lecture notes continue 'Now there is a magnitude which contains all these b. If b occurs a-times we designate the sum consisting of a additions of b by a x b. '
This account is open to the objection that the sign 'b' has suddenly turned into a concept-word. At first it was a proper name of a numerical magnitude, of an express train, for instance; now all of a sudden it is a question of all these b. Let us make this plain by an example. We posit, say, the President of the United States repeatedly and thus get a series of President Wilsons, and the proper name we began with turns into a nomen appellativum, and each single one of the specimens we obtain by the repeated positing is a President Wilson. So as a result of the repeated positing of President Wilson we have got a series of President Wilsons, and in this series the man President Wilson (now we have a proper name again, as the definite article shows) occurs, and so in this series of President Wilsons the man President Wilson occurs more than once.
And this is how we have to think of the matter here. We designate the express which arrives here at approximately 5. 15 this afternoon from Berlin by b. b is a numerical magnitude. We posit this numerical magnitude repeatedly, and thus obtain a series of expresses b. We now have a numerical magnitude which contains all these expresses b. Really? This will presumably be an express in turn; but where does it stop? Now the express b occurs more than once in this series. If it occurs a-times, we designate by a x b the sum consisting of a additions of b. So far not a word has been said about this sum. Probably the numerical magnitude is what contains all the
? Logic in Mathematics 219
expresses won by positing; and this numerical magnitude is, I suppose, itself a train. Do we now know what a x b is? a of course is a numerical magnitude too and we were eager to learn how to multiply the express with the series of books on the top of my bookcase by the window. So we want to call this series of books a. But what then are we to understand by a-times? An infernally difficult matter, multiplication of this sort! But according to the lecture notes, we can obtain the numerical magnitude both by positing b a- times and by positing a b-times. So we have a choice. Is it perhaps easier to take the series of books a b-times? It seems just as difficult. Now does the numerical magnitude, which we designate by a x b, actually consist of books or of trains? Who would have thought that multiplying was so diffi- cult! And we expect nine year olds to master it. But just consider the diffi- culty of positing an express repeatedly. There is nothing wonderful in speed but the aplomb with which numerical magnitudes are made to vanish and what is normally called number to appear in their place is really staggering.
There is yet another way in which number is introduced surreptitiously. We read at ? 2 'Since what matters here, however, is not the order of the elements, but only the set of them, it follows that
a+ b= b +a. '
If a numerical magnitude were really a series of things of the same kind, the order of the elements would be relevant; for if you alter the order of the elements, you have a different series. And what is here being called the set of the elements-is not this really what is called the number of the elements? So it is not a question of a series of things of the same kind but of a number, and this shows that a series of similar things and a number are different.
As number proper gets smuggled in here under the guise of a set, in other places it gets smuggled in under the guise of a value. We have seen that in the equation
a, b, and c are meant to be numerical magnitudes, and we read now 'If we have two equations
a+b=c a? b=c,
then the value of c can be determined by an addition and multiplication if we are given the value of a and b. ' Here the value of a numerical magnitude is distinguished from the numerical magnitude itself. And what else can this value be but a number? Now, on Weierstrass's account, is a value really determined by addition? Let us assume that we have a train a and a train b. We uncouple the carriages of band couple them to a. We thus obtain a train c, and Weierstrass says that it is a result of adding b to a. Here all that has taken place is that a new series c has been formed from a series a and a series b; but there is nothing about how to determine the value of c. It is not
? 220 Logic in Mathematics
that we have fixed what value c has-we have constructed c. So there is apparent throughout a conflict between the definition Weierstrass gives and the things he goes on to say. What Weierstrass is here calling a value can hardly be anything other that is normally called a number.
We read further in the lecture notes 'A numerical magnitude is determined once we are given the elements and how often each is contained in it'.
Now it is surely the carriages that have to be taken as the elements of a train. Thus a train is determined when we are given its carriages and how often each is contained in it.
One of my university teachers once told of an inventor of a perpetuum mobile who exclaimed 'Now I have,it; the only thing I lack is a little device which keeps doing this', illustrating the movement with his index finger. This 'how often' strikes me as such a little device which keeps doing this. Does it not in fact conceal the whole difficulty? If we have the little device, then we have a perpetuum mobile; and if we can define the words 'how often', we can also define number.
However there is something I have passed over here. We have earlier the statement that the concept of a numerical magnitude has to be extended. 'To this end numerical magnitudes are now to be formed out of different units, whereas the numerical magnitudes considered previously all came from one unit'.
Really? Before this we had the statement 'Each single one of the elements which recur in the series is called the unit of the numerical magnitude'.
The unit? 'Each single element is a unit' is all right, but 'each element is the unit' is nonsense. I f the word 'unit' is meant to have the same meaning as 'element', then we have units if we have elements, but not the unit. Several things can indeed be subsumed under one concept: we do this when we call each of them a unit; but we are not entitled to call each of them by the same proper name. And 'the unit' is to be regarded as a proper name, since the form of this expression is such that it designates one determinate object. If we call each of several objects 'the unit', we are making a mistake. It leads to a curious interplay between singular and plural. A numerical magnitude consists of several elements, and yet of only one unit, because each element is the unit. How is this to be imagined? Well, we take a railway wagon, say
goods wagon no. 1061 from the Erfurt region. We posit this repeatedly and construct a goods train out of it. The goods train consists of several elements, namely goods wagons, but of only one unit, for each of these goods wagons is the unit-namely, the goods wagon no. 1061 that we began with. This occurs repeatedly. It is true that I have not yet seen a train in which one and the same goods wagon occurs repeatedly, but according to Weierstrass there is no doubt that such a thing must be possible. Thus jt is possible for a numerical magnitude or series of things of the same kind to consist of several elements and yet of only one unit.
However let us go back to the sentence 'A numerical magnitude is
? Logic in Mathematics 221
determined once we are given the elements and how often each is contained in it'.
We have just made every effort to distinguish between element and unit, and now everything is confounded again. The layman will say 'But with a train the question of ordering comes in'. Not at all! We have only a single wagon which occurs repeatedly. In such a case there can be no talk of an ordering. Ordering comes in only when we have different things, not when we have a single thing which occurs repeatedly.
But Weierstrass says 'Numerical magnitudes are now to be formed out of different units'. Because of course there has to be an ordering! And so we get more and more snarled up.
His project being obviously a complete failure, Weierstrass felt himself obliged to bring number proper back in again by the back door. Again and again he comes into conflict with his own statements. Ifon his definition a is a numerical magnitude, then a-times has no sense. Number proper is dressed up as a set or value or introduced by the phrase 'how often'. In this way we have a curious interplay between singular and plural and correspondingly between proper names and concept-words. lf someone who had given the matter no thought were roused from sleep by the question 'What is number? ', he would probably come out with an answer not far removed from Weierstrass's. And yet here's a man who, one would have thought, had already reflected on the question.
How, we may then ask, is it possible for so distinguished a mathematician to go so badly astray over this issue? If only he had given it some thought, he could not have failed to get clearer about it. But that is just what he has not done-given it any thought at all. And why not? He obviously believed that none at all was necessary. He was lacking in the first re- quirement-knowledge of his own ignorance. He saw no difficulties at all, everything seemed clear to him, and he didn't notice that he was constantly deluding himself. He did not possess the ideal of a system of mathematics. We do not come across any proofs; no axioms are laid down: we have nothing but assertions which contradict one another. And when on occasion an inference does seem to be drawn from his definition, it is fallacious. If he had but made the attempt to construct a system from the foundation upwards, he could not have failed straightway to see the uselessness of his definition. He had a notion of what number is, but a very hazy one; and working from this he kept on revising and adding to what should really have been inferred from his definition. Thus he asserts that ordering does not come in, and yet ordering is essential for a series. And so he quite fails to see that what he asserted does not flow from his definition, but from his inkling of what number is.
We may add the following. We cannot insist on complete scientific rigour in the classroom because the pupils do not have the intellectual maturity to feel so much as the need for it. It will probably be impossible, in the third or fourth forms, to handle irrational fractions in the way Euclid does--indeed
? 222 Logic in Mathematics
it may scarcely be possible in the fifth form. In all likelihood such matters will, for the most part, be treated very superficially. For didactic reasons difficulties are ironed out, sharp logical edges are rounded off. And no doubt it has to be like that to begin with, but it should not continue so. Later on we should bring up the question of a rigorous deployment of proofs by awakening the need for it and then satisfying it. But it happens all too easily that the teachers, in their efforts to make everything palatable to the pupils, forget this second part of their task altogether. Mathematics can attain its full educational value only if it is pursued with the utmost logical rigour. And if there has to be some slackening of rigour in the early stages, we ought to make up for this later on. If we can only give a thorough logical grounding at the cost of sacrificing some of the material, then we should do this. But such a grounding will often be lacking. In later life people look back on these school topics as something that was mastered a long time ago, which it would not befit a serious thinker to devote any attention to. We are so prone to regarding these things as matters only for the schoolroom, that they seem to be too elementary to be worth reflecting on.
But how, it may be asked, can a man do effective work in a science when he is completely unclear about one of its basic concepts? The concept of a positive integer is indeed fundamental for the whole arithmetical part of mathematics. And any unclarity about this must spread throughout the whole of arithmetic. This is obviously a serious defect and one would imagine that it could not but prevent a man from doing any effective work whatsoever in this science. Surely no arithmetical sentence can have a completely clear sense to someone who is in the dark about what a number is? This question is not an arithmetical one, nor a logical one, but a psychological one. We simply do not have the mental capacity to hold before our minds a very complex logical structure so that it is equally clear to us in every detail. For instance, what man, when he uses the word 'integral' in a proof, ever has clearly before him everything which appertains to the sense of this word! And yet we can still draw correct inferences, even though in doing so there is always a part of the sense in penumbra. Weierstrass has a sound intuition of what number is and working from this he constantly revises and adds to what should really follow from his official definitions. In so doing he involves himself in contradictions and yet arrives at true thoughts, which, one must admit, come into his mind in a purely haphazard way. His sentences express true thoughts, if they are rightly understood. But if one tried to understand them in accordance with his own definitions, one would go astray.
We may look at a few more points in Weierstrass's theory (? 2): '(. . . )and defines it by the equation c = a + b'. What is being defined here? For neither the plus sign nor the equals sign has occurred previously. A definition must not have the fqrm of an equation in several unknowns. What construction should be placed on the equals sign? The words might lead one to think that '='and'+' are not to be understood as independent signs at all, each having
? Logic in Mathematics 223
a sense in its own right, but only that the sentence, as a whole, was meant to say that the series c had arisen in the way described out of the series a and b. This would be perfectly alright in itself; only it does not agree with normal usage; for both'=' and'+' occur in other combinations. And Weierstrass himself immediately afterwards uses the combinations of signs
'b +a =a+ b'
with the observation that this is an instance of the general law that two things which are not identical may be equal to one another according to a particular definition. And it is true that he has not defined the sign '=' as between numerical magnitudes, but the word 'equals'.
Accordingly the word 'equals' does not have the sense of 'the same as'. If we understand the sign '=' according to the definition given of the word 'equals', then we must expect that what stands to the left of the sign, as well as what stands to the right, designates a series of things of the same kind. But we still do not know what 'a + b' is meant to designate. When in the ordinary way we write down '5 = 3 + 2' we are not designating a series, a numerical magnitude, by '5' or by '3 + 2', as Weierstrass says we are. For what series would it be? What members could it consist of? It is clear that on Weierstrass's definition his numerical magnitudes can be equal to one another without agreeing in every respect; e. g. one might consist of railway wagons, the other of books. Hence a numerical magnitude would not just have one successor, but very many, perhaps infinitely many, all indeed equal to one another, but nevertheless different. But this is a departure from arithmetical usage. What we designate by the numerals are not numerical magnitudes in Weierstrass's sense.
The question now arises whether in arithmetic, according to our usual way of speaking and writing, numbers which are equal to one another may yet be distinguished from one another in any way. Most mathematicians are inclined to say they can; but what they give out as their opinion, though it is quite sincere, does not always agree with what, at rock bottom, their real opinion is. We have seen this from the case of Weierstrass; we had to assume that, contrary to his own words, he had an inkling of the true state of affairs.
Most mathematicians don't express any view at all about the equals sign, but rather take its sense for granted. But we cannot without more ado take it as certain that its sense is quite clear to them.
What are we really doing when we write down '3 + 2'? Are we presenting a problem for solution? When we write down '7 - 3', is it as if we were saying 'look for a number which gives 7 when 3 is added? It might perhaps look to be so, if this combination of signs occurred only on its own. But we also write '(3 + 2) + 4'. Are we meant here to add the number 4 to a problem? No, to the number which is the solution to this problem. On the normal reading what comes before the sign '+'designates a number. And likewise what occurs to the right of'+' designates a number.
? 224 Logic in Mathematics
It follows that the '(3 + 2)' in '4 + (3 + 2)' must also be regarded as a sign for a number, for that number in fact which is also designated by '5'. So in '3 + 2' and '5' we have signs for the same number. And when we write down '5 = 3 + 2' the meanings of the signs to the left and right of the equals sign don't just agree in such and such properties, or in this or that respect, but agree completely and in every respect. What is designated on the left is the same as what is designated on the right.
But surely the two signs are different; one can see at the first glance that they are different!
We can characterize another kind of inadmissible definition by a metaphor from algebra. Let us assume that three unknowns x, y, z occur in three equations. Then they can be determined by means of these equations. , Strictly speaking, however, they are determined only for the case where there is only one solution. In a similar way the words 'point', 'straight line',, 'surface' may occur in several sentences. Let us assume that these words have as yet no sense. It may be required to find a sense for each of these words such that the sentences in question express true thoughts. But have, we here provided a means for determining the sense uniquely? At any rate not in general; and in most cases it must remain undecided how many solutions are possible. But if it can be proved that only one solution is' possible, then this is given by assigning, via a constructive definition, a sense in turn to each of the words that needs defining. But we cannot regard as a definition the system of sentences in each of which there occur several of the expressions that need defining.
A special case of this is where only one sign, which has as yet no sense, occurs in one or more sentences. Let us assume that the other constituents of the sentences are known. The question is now what sense has to be given to this sign for the sentences to have a sense such that the thoughts expressed in them are true. This case is to be compared to that in which the letter x occ. urs in one or more equations whose other constituents are known, where the problem is: what meaning do we have to give the letter Jt for the equations to express true thoughts? If there are several equations,j
? Logic in Mathematics 213
this problem will usually be insoluble. It is obvious that in general no number whatsoever is determined in this way. And it is like this with the case in hand. No sense accrues to a sign by the mere fact that it is used in one or more sentences, the other constituents of which are known. In algebra we have the advantage that we can say something about the possible solutions and how many there are-an advantage one does not have in the general case. But a sign must not be ambiguous. Freedom from ambiguity is the most important requirement for a system of signs which is to be used for scientific purposes. One surely needs to know what one is talking about and the statements one is making, what thoughts one is expressing.
Now it is true that there have even been people, who have fancied themselves logicians, who have held that concept-words (nomina appel- /ativa) are distinguished from proper names by the fact that they are
ambiguous. The word 'man', for example, means Plato as well as Socrates and Charlemagne. The word 'number' designates the number 1 as well as the number 2, and so on. Nothing is more wrong-headed. Of course I can use the words 'this man' to designate now this man, now that man. But still on each single occasion I mean them to designate just one man. The sentences of our everyday language leave a good deal to guesswork. It is the surrounding circumstances that enable us to make the right guess. The sentence I utter does not always contain everything that is necessary; a great deal has to be supplied by the context, by the gestures I make and the direction of my eyes. But a language that is intended for scientific employment must not leave anything to guesswork. A concept-word combined with the demonstrative pronoun or definite article often has in this way the logical status of a proper name in that it serves to designate a single determinate object. But then it is not the concept-word alone, but the whole consisting of the concept-word together with the demonstrative pronoun and accompanying circumstances which has to be understood as a proper name. We have an actual concept-word when it is not accompanied by the definite article or demonstrative pronoun and is accompanied either by no article or by the indefinite article, or when it is combined with 'all', 'no' and 'some'. We must not think that I mean to assert something about an African chieftain from darkest Africa who is wholly unknown to me, when I say 'All men are mortal'. I am not saying anything about either this man or that man, but I am subordinating the concept man to the concept of what is mortal. In the sentence 'Plato is mortal' we have an instance of subsumption, in the sentence 'All men are mortal' one of subordination. What is being spoken about here is a concept, not an individual thing. We must not think either that the sense of the sentence 'Cato is mortal' is contained in that of the sentence 'All men are mortal', so that by uttering the latter sentence I should at the same time have expressed the thought contained in the former sentence. The matter is rather as follows. By the sentence 'All men are mortal' I say 'If anything is a man, it is mortal'. By an inference from the general to the particular, I obtain from this the sentence 'If Cato is a man,
? 214 Logic in Mathematics
then Cato is mortal'. Now I still need a second premise, namely 'Cato is a man'. From these two premises I infer 'Cato is mortal'.
Since therefore we need inferences and a second premise, the thought that Cato is mortal is not included in what is expressed by the sentence 'All men are mortal', and so 'man' is not an ambiguous word which amongst its many meanings has that which we designate by the proper name 'Plato'. On the contrary, a concept-word simply serves to designate a concept. And a concept is quite different from an individual. If I say 'Plato is a man', I am not as it were giving Plato a new name-the name 'man'-but I am saying that Plato falls under the concept man. Likewise we have two quite different cases when I give the definition '2 + 1 = 3' and when I say '2 + 1 is a prime number'. In the first case I confer on the sign '3', which is so far empty, a sense and a meaning by saying that it is to mean the same as the combination of signs '2 + 1'. In the second case I am subsuming the meaning of '2 + 1' under the concept prime number. I do not give it a new name by doing that. The fact therefore that I subsume different objects under the same concept does not make the concept-word ambiguous. So in the sentences
'2 is a prime number' '3 is a prime number' '5 is a prime number'
the word 'prime number' is not somehow ambiguous because 2, 3, 5 are different numbers; for 'prime number' is not a name which is given to these numbers.
It is of the essence of a concept to be predicative. If an empty proper name occurs in a sentence, the other parts of which are known, so that the sentence has a sense once a sense is given to that proper name, then, so long as the proper name remains empty, the sentence contains the possibility of a statement, but we do not have an object about which anything is being said. So the sentence 'x is a prime number', does indeed contain the possibility of a statement, but so long as no meaning is given to the letter 'x', we do not have an object about which anything is being said. Another way of putting this would be to say: we have a concept but we have no object subsumed under it. If we take as a further instance the sentence 'x increased by 2 is divisible by 4' then we have a concept again. We can take these two concepts as characteristic marks of a new concept by putting together the sentences 'x is a prime number' and 'x increased by 2 is divisible by 4'. Under this concept there falls only one object-the number 2. But a concept under which only one object falls is still a concept; this does not make the expression for it into a proper name.
Our position is this: we cannot recognize sentences containing an empty sign, the otrn:r constituents of which are known, as definitions. But such sentences can have an explanatory role by providing a clue to what is to be understood by the sign or word in question.
? ? Logic in Mathematics 215
I have read that verbal definitions are considered faulty, and 1t ts argued that we should really have no further truck with such definitions. By way of example reference was made to a definition given by me, but it was not said what a verbal definition was. 1 Of course every definition makes use of words or signs. Perhaps what is meant by a verbal definition is one in which the definiens contains a word which is a mere word as such, having no sense. Certainly this should not be allowed, but from the fact that the reader attaches no sense to a word it does not follow that the author of the definition has attached no sense to it. The insistence on sense is absolutely justified, and all the more so since many mathematicians seem to prove what are merely sentences without bothering whether they have a sense and what sense it is they have.
How little value is commonly placed on sense and definitions can be seen from the sharply conflicting accounts that mathematicians give of what number is. (We are speaking here of the natural numbers. ) Weierstrass says 'Number is a series of things of the same kind'. Another says that certain conventional shapes produced by writing, such as 2 and 3, are numbers. 2 A third is of the opinion: if I hear the clock strike three I see nothing in this of what three is. Therefore it cannot be anything visible. If I see three lines, then I hear nothing in this of what three is. Therefore it can be nothing audible either. An axiom is not a visible thing and so if we speak of three axioms, the three here is nothing visible either. Number cannot be anything whatever which can be perceived by the senses. 3
Obviously each of these attaches a different sense to the word 'number'. So the arithmetics of these three mathematicians must be quite different. A sentence from the first mathematician must have a quite different sense from the equivalent-sounding sentence of the second mathematician. This resembles what it would be like if botanists were not agreed about what they wished to understand by a plant, so that for one botanist a plant was, say, an organically developing structure, for another a human artefact, and for a third something that was not perceptible by the senses at all. Such a situation would certainly not give rise to a common science of botany.
But why should it not be possible to lay it down that by a number is to be meant a series of things of the same kind? Admittedly we can raise objections to such a course. For one thing, it may be thought that the sense of the word 'series' is not firmly enough established. Are we to think in this connection of a spatial ordering, or of a temporal ordering or of a spatia- temporal one perhaps? Further it is not clear what we are to understand by 'of the same kind'. For example, are the notes of a scale of the same kind qua
1 Frege is here referring to the article Ober die Stellung der Definition in der Axiomatik by A. Schoenjliess in Jahresbericht der deutschen Mathematiker- Vereinlgung XX (1911), pp. 222-255 (ed. ).
2 The reference is to the so-called 'formal' theory of arithmetic (arithmetic as a game with signs) held by Frege's contemporaries E. He/ne and J. Thomae (ed. ).
3 The reference is probably to G. Cantor (ed. ).
? 216 Logic in Mathematics
notes, or are they of the same kind only if they have the same pitch? But let us assume that explanations were given which cleared these matters up. A train is a series of objects of the same kind "'hich moves along rails on wheels. It may be thought that the engine is nevertheless something of a different kind. Still that makes no essential difference. And so such a number comes steaming here from Berlin. Let us assume that the science of these numbers has been set up. There is no doubt that it must be entirely different from the science in which certain shapes that one makes on a writing surface with a writing instrument are called numbers. Even if the form of words is the same, the thought expressed must be quite different. Now it is striking that the sentences of these fundamentally different sciences, each of which is called arithmetic, are constituted by precisely the same words. And it is even more striking that the practitioners of these sciences have no inkling that their sciences are fundamentally different. They all believe that they are doing arithmetic, and the same arithmetic at that, the same number theory, although what one of them is calling a number has no resemblance at all to what another is calling a number.
How is this possible? One would almost think that mathematicians regard the words used, the form of an expression, as the essential thing, and the thought expressed as quite inessential. Perhaps they think 'The thoughts contained in sentences are really no concern of mathematicians-they are a matter for philosophers; and everything to do with philosophy is of course extremely imprecise, uncertain and essentially unscientific. A mathematician who remains true to his scientific calling will have nothing to do with it. True, it can happen to even the best of them, in a moment of weakness, to let a definition slip, or something which looks like one, but we should not accord any significance to that. It is all one with a man sneezing. Really the only thing that matters is that they should all agree on the words and formulae they use. That is enough for a mathematician who has not been infected by philosophy. '
But is that then a science which proves sentences without knowing what it proves? But is it the case in actual fact that scientists do agree in the words they use? Are not mathematical works written in different languages, and are they not translated into other languages? In which case, of course, we no longer have the same form of words. But there must be something else which is preserved. And what can this be but the sense? So the thought, the sense of a sentence, cannot after all be wholly irrelevant. And does one not feel in the depth of one's being that the thoughts are the essential thing-that it is in fact these alone that we are concerned about?
But how do they come to be treated as irrelevant? How can one possibly imagine that two quite different sciences should really have the same content? Is it only because they are both called arithmetic and both treat of numbers, althpugh what is called a number in the one is quite different from what is called a number in the other? Or is not the explanation rather that we have really to do with the same science; that this man does attach the
? ? Logic in Mathematics 217
same sense to the word number as that man, only he doesn't manage to get hold of it properly? Perhaps the sense appears to both through such a haze that when they make to get hold of it, they miss it. One of them makes a grasp to the right perhaps and the other to the left, and so although they mean to get hold of the same thing, they fail to do so. How thick the fog must be for this to be possible! But it must surely show up in the proofs that they have not got hold of the same sense. Yes, it could not fail to, if the proofs were drawn up in a logically rigorous way, with no gaps in the chains of inference. But this is just what we shall fail to find. If no use is ever made of a definition, there might as well not be one. However wide it may be of the actual target, no-one will notice. Another mathematician's shot may miscarry on the other side; but since he makes no use of his definition, this too might as well not be there. We can see in this way how definitions, which seem to be utterly irreconcilable, lie peacefully alongside one another like the animals in paradise. I f only things were really like that!
Really the question that must surely exercise one is how multiplication takes place with Weierstrass's numbers. Next to my window there is a bookcase; on its top shelf there is a series of things of the same kind, a number. This afternoon at approximately 5. 15 an express train, which is likewise a number, arrives at Saal station from Berlin. It is a widely held opinion that if one number is multiplied by another, the result is again a number. Accordingly the result of multiplying our series of books by the Berlin express would again have to be a series of things of the same kind. Now how are we to do that? I read in a set of lecture notes which contains a lecture by Weierstrass1 'According to the definition a numerical magnitude is formed by the repeated positing of elements of the same kind. ' So this, apparently, is meant to be an application of the definition. How does the definition itself go? 'We can imagine a series of things of the same kind if by things of the same kind we understand things which have a complex of determinate characteristics in common. We shall understand by the concept numerical magnitude such a series. '
Mention is made here of numerical magnitude instead of number, but this is immaterial. To begin with, we have an assertion 'We can imagine a series of things of the same kind. ' This is a psychological truth, which is really of no concern to us in the present context. But now does it follow from the definition that a numerical magnitude is formed by the repeated positing of elements of the same kind? There is no doubt that on the definition our express is a numerical magnitude, for it is a series of things which have in common a complex of determinate characteristics. Well, is a train formed by the repeated positing of carriages? Do I have to posit repeatedly one and the same carriage? And how do I do that? Or do I have to put one carriage
1 Presumably the lecture notes entitled 'Analytische Funktionen' which are referred to in Der wissenschqftliche Nachlass von Gottlob Frege, by H. Scholz and F. Bachmann. See footnote 2, p. 29 of Actes du congres international de philosophie scient(fique, Paris 1935 VIII, Paris 1936 (ed. ).
? 218 Logic in Mathematics
next to another? In that case it would be better to say 'A train is formed by putting one carriage next to another. ' I do not believe that railwaymen are yet acquainted with this method of forming a train. So I should like to put it in question that a numerical magnitude can thus be formed by the repeated positing of things of the same kind. Certainly nothing can be gathered from the definition about how this is to be done.
The lecture notes go on to say, 'Now the concept "magnitude" can itself be viewed as a unit and posited repeatedly e. g. b, b, b . . .
'Who would believe it-one and the same concept can be posited repeatedly! Here b seems to be the sign for the concept magnitude. Is this concept posited repeatedly by writing down the sign for it over and over again? Incidentally there seems to be a mistake here. At least it seems to me as if it is not the concept magnitude, but a particular magnitude that is meant to be repeated. In that case we should presumably have to see b as a sign for this magnitude-for this express train, for example. But what has writing this sign down over and over again got to do with the repeated positing of the express? Or is it perhaps not the express itself, but an idea which I have of it, that we are to take as the numerical magnitude? This would turn the issue into a psychological and subjective one, without its becoming any clearer. Numerical magnitudes would be psychological structures and arithmetic a branch of psychology. But, to return to the point, how do we arrive at multiplication? The lecture notes continue 'Now there is a magnitude which contains all these b. If b occurs a-times we designate the sum consisting of a additions of b by a x b. '
This account is open to the objection that the sign 'b' has suddenly turned into a concept-word. At first it was a proper name of a numerical magnitude, of an express train, for instance; now all of a sudden it is a question of all these b. Let us make this plain by an example. We posit, say, the President of the United States repeatedly and thus get a series of President Wilsons, and the proper name we began with turns into a nomen appellativum, and each single one of the specimens we obtain by the repeated positing is a President Wilson. So as a result of the repeated positing of President Wilson we have got a series of President Wilsons, and in this series the man President Wilson (now we have a proper name again, as the definite article shows) occurs, and so in this series of President Wilsons the man President Wilson occurs more than once.
And this is how we have to think of the matter here. We designate the express which arrives here at approximately 5. 15 this afternoon from Berlin by b. b is a numerical magnitude. We posit this numerical magnitude repeatedly, and thus obtain a series of expresses b. We now have a numerical magnitude which contains all these expresses b. Really? This will presumably be an express in turn; but where does it stop? Now the express b occurs more than once in this series. If it occurs a-times, we designate by a x b the sum consisting of a additions of b. So far not a word has been said about this sum. Probably the numerical magnitude is what contains all the
? Logic in Mathematics 219
expresses won by positing; and this numerical magnitude is, I suppose, itself a train. Do we now know what a x b is? a of course is a numerical magnitude too and we were eager to learn how to multiply the express with the series of books on the top of my bookcase by the window. So we want to call this series of books a. But what then are we to understand by a-times? An infernally difficult matter, multiplication of this sort! But according to the lecture notes, we can obtain the numerical magnitude both by positing b a- times and by positing a b-times. So we have a choice. Is it perhaps easier to take the series of books a b-times? It seems just as difficult. Now does the numerical magnitude, which we designate by a x b, actually consist of books or of trains? Who would have thought that multiplying was so diffi- cult! And we expect nine year olds to master it. But just consider the diffi- culty of positing an express repeatedly. There is nothing wonderful in speed but the aplomb with which numerical magnitudes are made to vanish and what is normally called number to appear in their place is really staggering.
There is yet another way in which number is introduced surreptitiously. We read at ? 2 'Since what matters here, however, is not the order of the elements, but only the set of them, it follows that
a+ b= b +a. '
If a numerical magnitude were really a series of things of the same kind, the order of the elements would be relevant; for if you alter the order of the elements, you have a different series. And what is here being called the set of the elements-is not this really what is called the number of the elements? So it is not a question of a series of things of the same kind but of a number, and this shows that a series of similar things and a number are different.
As number proper gets smuggled in here under the guise of a set, in other places it gets smuggled in under the guise of a value. We have seen that in the equation
a, b, and c are meant to be numerical magnitudes, and we read now 'If we have two equations
a+b=c a? b=c,
then the value of c can be determined by an addition and multiplication if we are given the value of a and b. ' Here the value of a numerical magnitude is distinguished from the numerical magnitude itself. And what else can this value be but a number? Now, on Weierstrass's account, is a value really determined by addition? Let us assume that we have a train a and a train b. We uncouple the carriages of band couple them to a. We thus obtain a train c, and Weierstrass says that it is a result of adding b to a. Here all that has taken place is that a new series c has been formed from a series a and a series b; but there is nothing about how to determine the value of c. It is not
? 220 Logic in Mathematics
that we have fixed what value c has-we have constructed c. So there is apparent throughout a conflict between the definition Weierstrass gives and the things he goes on to say. What Weierstrass is here calling a value can hardly be anything other that is normally called a number.
We read further in the lecture notes 'A numerical magnitude is determined once we are given the elements and how often each is contained in it'.
Now it is surely the carriages that have to be taken as the elements of a train. Thus a train is determined when we are given its carriages and how often each is contained in it.
One of my university teachers once told of an inventor of a perpetuum mobile who exclaimed 'Now I have,it; the only thing I lack is a little device which keeps doing this', illustrating the movement with his index finger. This 'how often' strikes me as such a little device which keeps doing this. Does it not in fact conceal the whole difficulty? If we have the little device, then we have a perpetuum mobile; and if we can define the words 'how often', we can also define number.
However there is something I have passed over here. We have earlier the statement that the concept of a numerical magnitude has to be extended. 'To this end numerical magnitudes are now to be formed out of different units, whereas the numerical magnitudes considered previously all came from one unit'.
Really? Before this we had the statement 'Each single one of the elements which recur in the series is called the unit of the numerical magnitude'.
The unit? 'Each single element is a unit' is all right, but 'each element is the unit' is nonsense. I f the word 'unit' is meant to have the same meaning as 'element', then we have units if we have elements, but not the unit. Several things can indeed be subsumed under one concept: we do this when we call each of them a unit; but we are not entitled to call each of them by the same proper name. And 'the unit' is to be regarded as a proper name, since the form of this expression is such that it designates one determinate object. If we call each of several objects 'the unit', we are making a mistake. It leads to a curious interplay between singular and plural. A numerical magnitude consists of several elements, and yet of only one unit, because each element is the unit. How is this to be imagined? Well, we take a railway wagon, say
goods wagon no. 1061 from the Erfurt region. We posit this repeatedly and construct a goods train out of it. The goods train consists of several elements, namely goods wagons, but of only one unit, for each of these goods wagons is the unit-namely, the goods wagon no. 1061 that we began with. This occurs repeatedly. It is true that I have not yet seen a train in which one and the same goods wagon occurs repeatedly, but according to Weierstrass there is no doubt that such a thing must be possible. Thus jt is possible for a numerical magnitude or series of things of the same kind to consist of several elements and yet of only one unit.
However let us go back to the sentence 'A numerical magnitude is
? Logic in Mathematics 221
determined once we are given the elements and how often each is contained in it'.
We have just made every effort to distinguish between element and unit, and now everything is confounded again. The layman will say 'But with a train the question of ordering comes in'. Not at all! We have only a single wagon which occurs repeatedly. In such a case there can be no talk of an ordering. Ordering comes in only when we have different things, not when we have a single thing which occurs repeatedly.
But Weierstrass says 'Numerical magnitudes are now to be formed out of different units'. Because of course there has to be an ordering! And so we get more and more snarled up.
His project being obviously a complete failure, Weierstrass felt himself obliged to bring number proper back in again by the back door. Again and again he comes into conflict with his own statements. Ifon his definition a is a numerical magnitude, then a-times has no sense. Number proper is dressed up as a set or value or introduced by the phrase 'how often'. In this way we have a curious interplay between singular and plural and correspondingly between proper names and concept-words. lf someone who had given the matter no thought were roused from sleep by the question 'What is number? ', he would probably come out with an answer not far removed from Weierstrass's. And yet here's a man who, one would have thought, had already reflected on the question.
How, we may then ask, is it possible for so distinguished a mathematician to go so badly astray over this issue? If only he had given it some thought, he could not have failed to get clearer about it. But that is just what he has not done-given it any thought at all. And why not? He obviously believed that none at all was necessary. He was lacking in the first re- quirement-knowledge of his own ignorance. He saw no difficulties at all, everything seemed clear to him, and he didn't notice that he was constantly deluding himself. He did not possess the ideal of a system of mathematics. We do not come across any proofs; no axioms are laid down: we have nothing but assertions which contradict one another. And when on occasion an inference does seem to be drawn from his definition, it is fallacious. If he had but made the attempt to construct a system from the foundation upwards, he could not have failed straightway to see the uselessness of his definition. He had a notion of what number is, but a very hazy one; and working from this he kept on revising and adding to what should really have been inferred from his definition. Thus he asserts that ordering does not come in, and yet ordering is essential for a series. And so he quite fails to see that what he asserted does not flow from his definition, but from his inkling of what number is.
We may add the following. We cannot insist on complete scientific rigour in the classroom because the pupils do not have the intellectual maturity to feel so much as the need for it. It will probably be impossible, in the third or fourth forms, to handle irrational fractions in the way Euclid does--indeed
? 222 Logic in Mathematics
it may scarcely be possible in the fifth form. In all likelihood such matters will, for the most part, be treated very superficially. For didactic reasons difficulties are ironed out, sharp logical edges are rounded off. And no doubt it has to be like that to begin with, but it should not continue so. Later on we should bring up the question of a rigorous deployment of proofs by awakening the need for it and then satisfying it. But it happens all too easily that the teachers, in their efforts to make everything palatable to the pupils, forget this second part of their task altogether. Mathematics can attain its full educational value only if it is pursued with the utmost logical rigour. And if there has to be some slackening of rigour in the early stages, we ought to make up for this later on. If we can only give a thorough logical grounding at the cost of sacrificing some of the material, then we should do this. But such a grounding will often be lacking. In later life people look back on these school topics as something that was mastered a long time ago, which it would not befit a serious thinker to devote any attention to. We are so prone to regarding these things as matters only for the schoolroom, that they seem to be too elementary to be worth reflecting on.
But how, it may be asked, can a man do effective work in a science when he is completely unclear about one of its basic concepts? The concept of a positive integer is indeed fundamental for the whole arithmetical part of mathematics. And any unclarity about this must spread throughout the whole of arithmetic. This is obviously a serious defect and one would imagine that it could not but prevent a man from doing any effective work whatsoever in this science. Surely no arithmetical sentence can have a completely clear sense to someone who is in the dark about what a number is? This question is not an arithmetical one, nor a logical one, but a psychological one. We simply do not have the mental capacity to hold before our minds a very complex logical structure so that it is equally clear to us in every detail. For instance, what man, when he uses the word 'integral' in a proof, ever has clearly before him everything which appertains to the sense of this word! And yet we can still draw correct inferences, even though in doing so there is always a part of the sense in penumbra. Weierstrass has a sound intuition of what number is and working from this he constantly revises and adds to what should really follow from his official definitions. In so doing he involves himself in contradictions and yet arrives at true thoughts, which, one must admit, come into his mind in a purely haphazard way. His sentences express true thoughts, if they are rightly understood. But if one tried to understand them in accordance with his own definitions, one would go astray.
We may look at a few more points in Weierstrass's theory (? 2): '(. . . )and defines it by the equation c = a + b'. What is being defined here? For neither the plus sign nor the equals sign has occurred previously. A definition must not have the fqrm of an equation in several unknowns. What construction should be placed on the equals sign? The words might lead one to think that '='and'+' are not to be understood as independent signs at all, each having
? Logic in Mathematics 223
a sense in its own right, but only that the sentence, as a whole, was meant to say that the series c had arisen in the way described out of the series a and b. This would be perfectly alright in itself; only it does not agree with normal usage; for both'=' and'+' occur in other combinations. And Weierstrass himself immediately afterwards uses the combinations of signs
'b +a =a+ b'
with the observation that this is an instance of the general law that two things which are not identical may be equal to one another according to a particular definition. And it is true that he has not defined the sign '=' as between numerical magnitudes, but the word 'equals'.
Accordingly the word 'equals' does not have the sense of 'the same as'. If we understand the sign '=' according to the definition given of the word 'equals', then we must expect that what stands to the left of the sign, as well as what stands to the right, designates a series of things of the same kind. But we still do not know what 'a + b' is meant to designate. When in the ordinary way we write down '5 = 3 + 2' we are not designating a series, a numerical magnitude, by '5' or by '3 + 2', as Weierstrass says we are. For what series would it be? What members could it consist of? It is clear that on Weierstrass's definition his numerical magnitudes can be equal to one another without agreeing in every respect; e. g. one might consist of railway wagons, the other of books. Hence a numerical magnitude would not just have one successor, but very many, perhaps infinitely many, all indeed equal to one another, but nevertheless different. But this is a departure from arithmetical usage. What we designate by the numerals are not numerical magnitudes in Weierstrass's sense.
The question now arises whether in arithmetic, according to our usual way of speaking and writing, numbers which are equal to one another may yet be distinguished from one another in any way. Most mathematicians are inclined to say they can; but what they give out as their opinion, though it is quite sincere, does not always agree with what, at rock bottom, their real opinion is. We have seen this from the case of Weierstrass; we had to assume that, contrary to his own words, he had an inkling of the true state of affairs.
Most mathematicians don't express any view at all about the equals sign, but rather take its sense for granted. But we cannot without more ado take it as certain that its sense is quite clear to them.
What are we really doing when we write down '3 + 2'? Are we presenting a problem for solution? When we write down '7 - 3', is it as if we were saying 'look for a number which gives 7 when 3 is added? It might perhaps look to be so, if this combination of signs occurred only on its own. But we also write '(3 + 2) + 4'. Are we meant here to add the number 4 to a problem? No, to the number which is the solution to this problem. On the normal reading what comes before the sign '+'designates a number. And likewise what occurs to the right of'+' designates a number.
? 224 Logic in Mathematics
It follows that the '(3 + 2)' in '4 + (3 + 2)' must also be regarded as a sign for a number, for that number in fact which is also designated by '5'. So in '3 + 2' and '5' we have signs for the same number. And when we write down '5 = 3 + 2' the meanings of the signs to the left and right of the equals sign don't just agree in such and such properties, or in this or that respect, but agree completely and in every respect. What is designated on the left is the same as what is designated on the right.
But surely the two signs are different; one can see at the first glance that they are different!
