What would be the result of taking 2 + 3 to be
different
from 5?
Gottlob-Frege-Posthumous-Writings
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! Here we come up against a disease endemic amongst mathematicians, which I should like to call 'morbus mathematicorum recens'. Its chief symptom is the incapacity to distinguish between a sign and what it designates. Is it really quite impossible to designate the same thing by different signs? Can the mere fact of a difference in signs be of itself a sufficient ground for assuming that what is designated is also different?
What would be the result of taking 2 + 3 to be different from 5? To the question 'Which number follows immediately after 4 in the series of whole numbers? ', we should have to answer 'There are infinitely many. Some of themare5,1+4,2+3,7- 2,(32- 22). 'Weshouldnothaveasimple series of whole numbers at all, but a chaos. The whole numbers which follow immediately after 4 would not follow immediately after 4 alone, but immediately after 22, and 2 ? 2 as well. It is true that these numbers would also be equal to one another, but they would be different nonetheless. Surely we cannot accept this. We hold that the signs '2 + 3', '3 + 2', '1 + 4', '5' do designate the same number. Still an objection may be urged against this, for don't the sentences '5 = 5' and '5 = 2 + 3' have a different content? The former is an immediate consequence of the general principal of identity; but is the latter?
We might say: if we designated the same number by '2 + 3' as by '5' then we should surely have to know that 5 = 2 +3 straightoff, and not need first to work it out. It is clearer if we take the case of larger numbers. It is surely not self-evident that 137 + 469 = 606; on the contrary we only come to see this as the result of first working it out. This sentence says much more than the sentence '606 = 606'; the former increases our knowledge, not so the latter. So the thoughts contained in the two sentences must be different too. Is it possible to designate the same thing by two different names or signs without knowing that it is the same thing one has designated? Of course it is, and this also happens in other contexts. For instance, we have observed a small planet and given it a provisional designation. After a long period of observation we are able to work out the same planet had been observed at an earlier time and had already received a name. Now it can easily happen that the same astronomer has used both names without knowing that they designate the same planet. Again, in exploring a new country, it may happen that two explor,ers, who have seen the same mountain from different sides, have given it different names, and that it is only subsequently, when they compare maps, that it comes out that they have seen the same mountain and
? Logic in Mathematics 225
named it differently. It must certainly be conceded therefore that we can name the same object by different names without knowing that it is the same.
On the other hand, one cannot fail to recognize that the thought expressed by '5 = 2 + 3' is different from that expressed by the sentence '5 = 5', although the difference only consists in the fact that in the second sentence '5', which designates the same number as '2 + 3', takes the place of '2 + 3'. So the two signs are not equivalent from the point of view of the thought expressed, although they designate the very same number. Hence I say that the signs '5' and '2 + 3' do indeed designate the same thing, but do not express the same sense. In the same way 'Copernicus' and 'the author of the heliocentric view of the planetary system' designate the same man, but have different senses; for the sentence 'Copernicus is Copernicus' and 'Copernicus is the author of the heliocentric view of the planetary system' do not express the same thought.
It is remarkable what language can achieve. With a few sounds and combinations of sounds it is capable of expressing a huge number of thoughts, and, in particular, thoughts which have not hitherto been grasped or expressed by any man. How can it achieve so much? By virtue of the fact that thoughts have parts out of which they are built up. And these parts, these building blocks, correspond to groups of sounds, out of which the sentence expressing the thought is built up, so that the construction of the sentence out of parts of a sentence corresponds to the construction of a thought out of parts of a thought. And as we take a thought to be the sense of a sentence, so we may call a part of a thought the sense of that part of the sentence which corresponds to it.
Let us now look at the sentence 'Etna is larger than Vesuvius'. A part of a thought corresponds to the word 'Etna', namely the sense of this word. But is the mountain itself with its rocks and lava part of the thought? Obviously not, for one can see Etna, but one cannot see the thought that Etna is higher than Vesuvius. But what are we making a statement about? Obviously about Etna itself. And when we say 'Scylla has 6 heads', what are we making a statement about? In this case, nothing whatsoever; for the word 'Scylla' designates nothing. Nevertheless we can find a thought expressed by the sentence, and concede a sense to the word 'Scylla'. This thought, however, does not belong to the realm of truth and science but to that of myth and fiction. Such a case apart, a proper name must designate something and in a sentence containing a proper name, we are making a statement about that which it designates, about its meaning. But a proper name must have a sense as well, and this will be part of the thought of the sentence in which it occurs. From this we can see that it is possible for two signs to designate the same thing and yet, because they have different senses, not to be interchangeable as far as the thought-content of sentences in which they occur is concerned. But the fact that they are not interchangeable may be the reason why some people have refused to
? 226 Logic in Mathematics
acknowledge that they designate the same number. But we have now seen this reason won't hold water; and we maintain our position that the equals sign in arithmetic is to be construed as a sign of identity.
Moreover we can find confirmation for this in the lecture notes. One of the concerns there is to investigate how the number domain must be extended for subtraction to be always possible. In this connection we read 'In that case (a -a) must also have a meaning: it has the meaning that it leaves unchanged the value of whatever number it is added to. '
Here the value of a numerical magnitude is distinguished from the magnitude itself: this value is meant to be the same after the addition of (a- a) as before. We must now take it that what Weierstrass is calling the value of a numerical magnitude is really a number. The number has there- fore remained the same. So we arrive at the view that according to Weier- strass equal numerical magnitudes have the same value. Therefore in making the transition from Weierstrass's numerical magnitudes to their values, we are at the same time making the transition from Weierstrass's equality to identity. If now, as is probable, Weierstrass means by 'value of a numerical magnitude' what is ordinarily called number, then with these numbers we also arrive at identity.
So the situation is as follows. First for Weierstrass the distinction between what he calls a numerical magnitude and a number in arithmetic is blurred. But he still cannot avoid introducing numbers themselves under the guise of values of numerical magnitudes and thus distinguishing between a numerical magnitude and its value; at the same time it incidentally emerges that numerical magnitudes have the same value according as they are, in Weierstrass's terms, equal to one another. But, on close inspection, the equals sign does not occur in arithmetic between names of Weierstrass's numerical magnitudes, but between names of numbers proper, which Weierstrass introduces, albeit covertly, under the name of values of numerical magnitudes.
In this way therefore the conception of number as a series of things of the same kind, as a mass, heap or whole consisting of parts of the same kind, is very closely bound up with the view that the equals sign is not used only as a sign for identity. But then as soon as, by some logical sleight of hand, we get numbers in the sense of arithmetic, as it is inevitable we should, the equals sign is at the same time transmuted into an identity sign. So it is not to be wondered at if there is a constant fluctuation from one conception to another.
We have something analogous in the case of the plus sign. This first made its appearance when addition was defined. According to this definition a + b is to designate the numerical magnitude which results by adding one after another all the units of b to the units of a. So the plus sign here occurs between signs of numerical magnitudes. But in the case of multiplication we read 'If one designates the units of b by a and forms
b tipes
I ~--~, ? ? ? ,
a+ a+ a+ . . . + a
? ? Logic in Mathematics 227
Here the plus sign occurs between signs of units, and by a unit must be understood a member of a series of things of the same kind. So we must assume that it is Weierstrass's view that even a single thing is to be regarded as a series of things of the same kind, as a numerical magnitude, that is, as a series consisting of a single member. So in Weierstrass's sense even a bean is to be regarded as a numerical magnitude. Let us now take a bean and designate it by 'cf. Let us take another bean and designate it by 'P'? If we now put bean Pnext to bean a; we obtain a new series of things of the same kind, and this Weierstrass will no doubt designate by 'a+ p? . If we now take a further bean, designate it by 'y' and put it next to the numerical magnitude a + p, consisting of beans a and p we obtain by addition a new numerical magnitude which, following Weierstrass, we designate by '(a+ {J) + y'. In this way we shall, according to Weierstrass, be able to form the name of a numerical magnitude from the names of its elements or units by means of the plus sign. But what can 'a+ et designate on this account? We can of course put bean Pnext to bean a to form a series of things of the same kind, but how do we manage to put bean a alongside itself? Presumably a will have to be so good as to occur more than once.
Let us now take the planets Jupiter, Saturn, Uranus, and Neptune as an example of a series of things of the same kind. Let us suppose this series to be designated by
? ~ + Q + 0 + ~ '.
Let us call this numerical magnitude b. ~ is therefore a unit of b and so is Q and so also is 0, and so finally is~ .
Fortunately none of these units is repeated. Can we now say 'If we designate the units of b by " 2j. " and form
"~ + ~ + ~ + ~"'? Does this really designate the same as
'~ + Q + 0 + ~ ' ?
It is not permissible to designate different things by the same sign, for the first thing that we must require of our signs is that they should be unambiguous. Obviously the plus sign cannot be employed here, if one wishes to use it as it is normally used in arithmetic. We do write' 1 + 1 + 1 + 1'; but here the first unit-sign means the same as the second, the same as the third, and the same as the fourth. We do not have here different things which form a series, a group or a heap: we just have the number one. It is clear from this that the plus sign cannot correspond to the 'and' of speech. If we say 'Schiller and Goethe are poets', we are not really connecting the proper names by 'and', but the sentences 'Schiller is a poet' and 'Goethe is a poet', which have been telescoped into one. It is different with the sentence 'Siemens and Halske have built the first major telegraph network'. ' Here we
1 In Frege's time 'Siemens and Halske' formed the name of a company, now known as 'Siemens A. G. ' It seems clear, however, that Frege does not here intend 'Siemens and Halske' to be undentood as the name of the company. The example is
? ? 228 Logic in Mathematics
don't have a telescoped form of two sentences, but 'Siemens and Halske' designates a compound object about which a statement is being made, and the word 'and' is used to help form the sign for this object. It is only this 'and' which is to be compared with the plus sign. But this comparison also shows at once that the cases 'Siemens and Halske' or 'Earth and Moon' and '1 + 1' are quite different.
Here we see throughout a crude and mathematically unworkable conception in conflict with the only viable one.
The conception of number as a series of things of the same kind, as a group, heap, etc. is very closely connected with the view which takes the sign of equality to designate partial agreement only, and with the view of the plus sign as synonymous with 'and'. But these views collapse at the first serious attempt to use them as a foundation for arithmetic. If we make such an attempt, we are always obliged to smuggle in something which is in conflict with these views. They provide us with examples of what a definition should not be. It is only if we do not go so far as to try and construct a system with the help of these definitions that we can deceive ourselves as to how utterly unworkable they are. And from this we may derive the principle that a definition must prove itself in the construction of a scientific system.
We shall now go into the definition ofconcepts more closely. The simplest case of the occurrence of a concept is that of a sentence whose grammatical subject is a proper name. We may say that in such a sentence an object is subsumed under a concept, namely that object of which the grammatical subject is the proper name. The remaining, predicative, part of the sentence means a concept. I say therefore: the concept is predicative in character, it is in need of supplementation, just as the predicative part of the sentence always demands a grammatical subject, being manifestly incomplete without it. Because it is thus incomplete, or in need of supplementation, we cannot have the predicative part occurring by itself on one side of a definition, but we have to supplement the predicative part with something which takes the place of the grammatical subject. For this purpose we may take the letter 'a'. On the right and left sides of the definition we now have an expression which contains the letter 'a', and we wish to stipulate by its means that both sides of the definition are to have the same sense, no matter what meaningful proper name be put in the place of a. We make use of this letter in order to give generality to our stipulation.
Now the simplest case of the definition of a concept is where one concept is combined with another to form a new concept in the following way. We begin by connecting two sentences with 'and'; the compound sentence, like each of its clauses, can be regarded as the expression of a thought-the one we have compounded-just because we can affirm or deny it as a whole. If
rather to be compared with 'Bunsen and KirchhotT laid the foundations of spectrum analysis', which occurs in one of Frege's letters to Russell. See p. 222 of Gottlob Frege Wissenschqftlicher Briefwechsel (Felix Meiner 1976) (trans. ).
? ? ? a 1s pos1tive
If a is a multiple of an integer greater than 1, then a is that integer
and
a is an integer
and
a is greater that 1
=a is a prime number.
Logic in Mathematics 229
we affirm it, we thereby affirm each of the clauses; if we deny it, we leave it open whether the first or second clause is false or whether both are.
Let us take the case '8 is a perfect cube and 8 is positive'.
In this compound sentence something is asserted of the number 8. We can construe what is present in the sentence over and above the numeral '8' as a sign for a concept, and introduce by definition a new sign for it, as in
a is a cube}
. and. . = a is a positive cube.
{
Here the letter 'a' occurs twice on the left. There is no objection to this; but on the right-hand side, where the dejiniendum occurs, the letter 'a' should occur only once. For if 'a' occurred in different places, it would be possible to fill them with different proper names, and thus obtain an undefined expression; but we must make it impossible for an expression to occur which has no sense.
Here we have a concept (that of a positive cube) that is formed by putting together the component concepts (perfect cube and positive). These we call the characteristic marks of this compound concept.
The case where a proper name occurs as the grammatical subject of a sentence, the predicate of which designates a concept, is the simplest language affords, but is not the only possible one. Whenever a proper name occurs in an assertoric sentence, we can regard the remaining part as a concept-sign. A concept-sign always stands in need of completion by a proper name or a sign standing in for a proper name, such as a letter.
The definition ofprime number will serve as an example. We stipulate
Here on the left we have the dejiniens, on the right the definiendum. We are here saying: No matter what meaningful proper name we put in place of the letter 'a', the expression on the right is always to have the same sense as the expression on the left.
It is an imperative requirement that a concept have sharp boundaries. What this means is that it must hold of every object either that it falls under the concept or that it does not. We cannot allow a third case in which it is somehow undecided or indeterminate whether an object falls under the concept.
This yields a requirement for concept-signs. Such a sign, when completed by a proper name, must always result in a sentence. And this sentence must always express a thouaht which is either true or false. If therefore a new
? 230 Logic in Mathematics
concept-sign is introduced by definition, this requirement has to be satisfied. This happens automatically if the defining side of the definition consists of a properly constructed concept-sign that is completed by a letter which stands in for a proper name and serves to give the necessary generality to the definition. Of course a definition may not be conditional. The stipulation that the dejiniendum is to have in every case the same sense as the dejiniens may not be made to depend on a condition; for then nothing is determined in the case where the condition is not satisfied. The dejiniendum is then without sense. Thus if we replace the indicating letter by the name of an object which fails to satisfy the condition, we shall not have, on the right of the definition, a sentence which expresses a thought that is either true or false, but a senseless combination of signs.
In general, therefore, we must have no truck with conditional definitions. In any case such a definition, if it should be employed, would require a special kind ofjustification.
We move on to the definitions o f relations. If two proper names occur in a sentence, the remaining part of the sentence can be regarded as a sign for a relation, as for example in
'3 is greater than 2',
where the numerals '3' and '2' are to be regarded as proper names. When we define a relation we need two indicating signs to stand in for the names of the objects which stand in the relation. For the same reason that we have seen to hold for the definition of concepts each of these letters may occur only once on the side that is defined. Let us take as an example the definition of congruence between numbers.
(a-b)isamultipleof7 ) and
1aisaninteger and
b is an integer
= aiscongruenttobmodulo7.
The purpose of the letters a and b is to make the stipulation general. We say by this that no matter what meaningful proper names are substituted for 'a' and 'b', the expression on the right-hand side is always to have the same sense as the expression on the left. So if e. g. we put' 16' for 'a' and 2 for 'b' we get
(16- 2) is a multiple of 7 ) and
16 is ~n integer and
2 is an integer
= 16 is congruent to 2 modulo 7.
? Logic in Mathematics 231
So long as the indefinitely indicating letters are not replaced by proper names, the left-hand side has no sense on its own and neither does the right- hand side. But it is necessary that everything on the left-hand side, apart from the letters, should be understood, so that the left-hand side always has a sense when meaningful proper names are substituted for 'a' and 'b'. From this it follows that the minus sign must not only be defined for the case where it stands between numerals, because otherwise the sentence ' ( a - b) is a multiple of 7' would not always have a sense when we substitute meaningful proper names for 'a' and 'b'. In that case it would follow that the expression
'a is congruent to b modulo 7'
would not always have a sense either.
The combination of signs '(16 - 2)' is a proper name of a number.
Accordingly the sign '(a - b)' stands in for a proper name. We obtain a proper name from it by replacing 'a' by a proper name and 'b' likewise. Therefore in the sentence
'(16- 2) is a multiple of 7'
the sign '(16 - 2)' has a meaning: it means the number 14. Therefore the sign '(16 - 2)' has a sense as well, which is a part of the sense of the above sentence, and this sense is the contribution which the sign '(16 - 2)' makes to the expression of the thought. So we may say that the sign 'a - b' acquires a sense by our replacing each of the two letters by a meaningful proper name. In this respect the sign ' ( a - b)' is on all fours with
'(a- b) is a multiple of 7'
By replacing 'a' here by '16' and 'b' by '2' we obtain a sense, namely the
sense of the sentence
'(16- 2)isamultipleof7'
and this sense is a thought. '(16 - 2)' has a sense too; but this is not a
thought: it is only part of the thought. Still the compound sign '(a- b)' and '(a- b) is a multiple of 7'
agree in both acquiring a sense if the letters are replaced by meaningful proper names. The sign '(a - b)' at the same time acquires a meaning by such a replacement.
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
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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
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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
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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.
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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! Here we come up against a disease endemic amongst mathematicians, which I should like to call 'morbus mathematicorum recens'. Its chief symptom is the incapacity to distinguish between a sign and what it designates. Is it really quite impossible to designate the same thing by different signs? Can the mere fact of a difference in signs be of itself a sufficient ground for assuming that what is designated is also different?
What would be the result of taking 2 + 3 to be different from 5? To the question 'Which number follows immediately after 4 in the series of whole numbers? ', we should have to answer 'There are infinitely many. Some of themare5,1+4,2+3,7- 2,(32- 22). 'Weshouldnothaveasimple series of whole numbers at all, but a chaos. The whole numbers which follow immediately after 4 would not follow immediately after 4 alone, but immediately after 22, and 2 ? 2 as well. It is true that these numbers would also be equal to one another, but they would be different nonetheless. Surely we cannot accept this. We hold that the signs '2 + 3', '3 + 2', '1 + 4', '5' do designate the same number. Still an objection may be urged against this, for don't the sentences '5 = 5' and '5 = 2 + 3' have a different content? The former is an immediate consequence of the general principal of identity; but is the latter?
We might say: if we designated the same number by '2 + 3' as by '5' then we should surely have to know that 5 = 2 +3 straightoff, and not need first to work it out. It is clearer if we take the case of larger numbers. It is surely not self-evident that 137 + 469 = 606; on the contrary we only come to see this as the result of first working it out. This sentence says much more than the sentence '606 = 606'; the former increases our knowledge, not so the latter. So the thoughts contained in the two sentences must be different too. Is it possible to designate the same thing by two different names or signs without knowing that it is the same thing one has designated? Of course it is, and this also happens in other contexts. For instance, we have observed a small planet and given it a provisional designation. After a long period of observation we are able to work out the same planet had been observed at an earlier time and had already received a name. Now it can easily happen that the same astronomer has used both names without knowing that they designate the same planet. Again, in exploring a new country, it may happen that two explor,ers, who have seen the same mountain from different sides, have given it different names, and that it is only subsequently, when they compare maps, that it comes out that they have seen the same mountain and
? Logic in Mathematics 225
named it differently. It must certainly be conceded therefore that we can name the same object by different names without knowing that it is the same.
On the other hand, one cannot fail to recognize that the thought expressed by '5 = 2 + 3' is different from that expressed by the sentence '5 = 5', although the difference only consists in the fact that in the second sentence '5', which designates the same number as '2 + 3', takes the place of '2 + 3'. So the two signs are not equivalent from the point of view of the thought expressed, although they designate the very same number. Hence I say that the signs '5' and '2 + 3' do indeed designate the same thing, but do not express the same sense. In the same way 'Copernicus' and 'the author of the heliocentric view of the planetary system' designate the same man, but have different senses; for the sentence 'Copernicus is Copernicus' and 'Copernicus is the author of the heliocentric view of the planetary system' do not express the same thought.
It is remarkable what language can achieve. With a few sounds and combinations of sounds it is capable of expressing a huge number of thoughts, and, in particular, thoughts which have not hitherto been grasped or expressed by any man. How can it achieve so much? By virtue of the fact that thoughts have parts out of which they are built up. And these parts, these building blocks, correspond to groups of sounds, out of which the sentence expressing the thought is built up, so that the construction of the sentence out of parts of a sentence corresponds to the construction of a thought out of parts of a thought. And as we take a thought to be the sense of a sentence, so we may call a part of a thought the sense of that part of the sentence which corresponds to it.
Let us now look at the sentence 'Etna is larger than Vesuvius'. A part of a thought corresponds to the word 'Etna', namely the sense of this word. But is the mountain itself with its rocks and lava part of the thought? Obviously not, for one can see Etna, but one cannot see the thought that Etna is higher than Vesuvius. But what are we making a statement about? Obviously about Etna itself. And when we say 'Scylla has 6 heads', what are we making a statement about? In this case, nothing whatsoever; for the word 'Scylla' designates nothing. Nevertheless we can find a thought expressed by the sentence, and concede a sense to the word 'Scylla'. This thought, however, does not belong to the realm of truth and science but to that of myth and fiction. Such a case apart, a proper name must designate something and in a sentence containing a proper name, we are making a statement about that which it designates, about its meaning. But a proper name must have a sense as well, and this will be part of the thought of the sentence in which it occurs. From this we can see that it is possible for two signs to designate the same thing and yet, because they have different senses, not to be interchangeable as far as the thought-content of sentences in which they occur is concerned. But the fact that they are not interchangeable may be the reason why some people have refused to
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acknowledge that they designate the same number. But we have now seen this reason won't hold water; and we maintain our position that the equals sign in arithmetic is to be construed as a sign of identity.
Moreover we can find confirmation for this in the lecture notes. One of the concerns there is to investigate how the number domain must be extended for subtraction to be always possible. In this connection we read 'In that case (a -a) must also have a meaning: it has the meaning that it leaves unchanged the value of whatever number it is added to. '
Here the value of a numerical magnitude is distinguished from the magnitude itself: this value is meant to be the same after the addition of (a- a) as before. We must now take it that what Weierstrass is calling the value of a numerical magnitude is really a number. The number has there- fore remained the same. So we arrive at the view that according to Weier- strass equal numerical magnitudes have the same value. Therefore in making the transition from Weierstrass's numerical magnitudes to their values, we are at the same time making the transition from Weierstrass's equality to identity. If now, as is probable, Weierstrass means by 'value of a numerical magnitude' what is ordinarily called number, then with these numbers we also arrive at identity.
So the situation is as follows. First for Weierstrass the distinction between what he calls a numerical magnitude and a number in arithmetic is blurred. But he still cannot avoid introducing numbers themselves under the guise of values of numerical magnitudes and thus distinguishing between a numerical magnitude and its value; at the same time it incidentally emerges that numerical magnitudes have the same value according as they are, in Weierstrass's terms, equal to one another. But, on close inspection, the equals sign does not occur in arithmetic between names of Weierstrass's numerical magnitudes, but between names of numbers proper, which Weierstrass introduces, albeit covertly, under the name of values of numerical magnitudes.
In this way therefore the conception of number as a series of things of the same kind, as a mass, heap or whole consisting of parts of the same kind, is very closely bound up with the view that the equals sign is not used only as a sign for identity. But then as soon as, by some logical sleight of hand, we get numbers in the sense of arithmetic, as it is inevitable we should, the equals sign is at the same time transmuted into an identity sign. So it is not to be wondered at if there is a constant fluctuation from one conception to another.
We have something analogous in the case of the plus sign. This first made its appearance when addition was defined. According to this definition a + b is to designate the numerical magnitude which results by adding one after another all the units of b to the units of a. So the plus sign here occurs between signs of numerical magnitudes. But in the case of multiplication we read 'If one designates the units of b by a and forms
b tipes
I ~--~, ? ? ? ,
a+ a+ a+ . . . + a
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Here the plus sign occurs between signs of units, and by a unit must be understood a member of a series of things of the same kind. So we must assume that it is Weierstrass's view that even a single thing is to be regarded as a series of things of the same kind, as a numerical magnitude, that is, as a series consisting of a single member. So in Weierstrass's sense even a bean is to be regarded as a numerical magnitude. Let us now take a bean and designate it by 'cf. Let us take another bean and designate it by 'P'? If we now put bean Pnext to bean a; we obtain a new series of things of the same kind, and this Weierstrass will no doubt designate by 'a+ p? . If we now take a further bean, designate it by 'y' and put it next to the numerical magnitude a + p, consisting of beans a and p we obtain by addition a new numerical magnitude which, following Weierstrass, we designate by '(a+ {J) + y'. In this way we shall, according to Weierstrass, be able to form the name of a numerical magnitude from the names of its elements or units by means of the plus sign. But what can 'a+ et designate on this account? We can of course put bean Pnext to bean a to form a series of things of the same kind, but how do we manage to put bean a alongside itself? Presumably a will have to be so good as to occur more than once.
Let us now take the planets Jupiter, Saturn, Uranus, and Neptune as an example of a series of things of the same kind. Let us suppose this series to be designated by
? ~ + Q + 0 + ~ '.
Let us call this numerical magnitude b. ~ is therefore a unit of b and so is Q and so also is 0, and so finally is~ .
Fortunately none of these units is repeated. Can we now say 'If we designate the units of b by " 2j. " and form
"~ + ~ + ~ + ~"'? Does this really designate the same as
'~ + Q + 0 + ~ ' ?
It is not permissible to designate different things by the same sign, for the first thing that we must require of our signs is that they should be unambiguous. Obviously the plus sign cannot be employed here, if one wishes to use it as it is normally used in arithmetic. We do write' 1 + 1 + 1 + 1'; but here the first unit-sign means the same as the second, the same as the third, and the same as the fourth. We do not have here different things which form a series, a group or a heap: we just have the number one. It is clear from this that the plus sign cannot correspond to the 'and' of speech. If we say 'Schiller and Goethe are poets', we are not really connecting the proper names by 'and', but the sentences 'Schiller is a poet' and 'Goethe is a poet', which have been telescoped into one. It is different with the sentence 'Siemens and Halske have built the first major telegraph network'. ' Here we
1 In Frege's time 'Siemens and Halske' formed the name of a company, now known as 'Siemens A. G. ' It seems clear, however, that Frege does not here intend 'Siemens and Halske' to be undentood as the name of the company. The example is
? ? 228 Logic in Mathematics
don't have a telescoped form of two sentences, but 'Siemens and Halske' designates a compound object about which a statement is being made, and the word 'and' is used to help form the sign for this object. It is only this 'and' which is to be compared with the plus sign. But this comparison also shows at once that the cases 'Siemens and Halske' or 'Earth and Moon' and '1 + 1' are quite different.
Here we see throughout a crude and mathematically unworkable conception in conflict with the only viable one.
The conception of number as a series of things of the same kind, as a group, heap, etc. is very closely connected with the view which takes the sign of equality to designate partial agreement only, and with the view of the plus sign as synonymous with 'and'. But these views collapse at the first serious attempt to use them as a foundation for arithmetic. If we make such an attempt, we are always obliged to smuggle in something which is in conflict with these views. They provide us with examples of what a definition should not be. It is only if we do not go so far as to try and construct a system with the help of these definitions that we can deceive ourselves as to how utterly unworkable they are. And from this we may derive the principle that a definition must prove itself in the construction of a scientific system.
We shall now go into the definition ofconcepts more closely. The simplest case of the occurrence of a concept is that of a sentence whose grammatical subject is a proper name. We may say that in such a sentence an object is subsumed under a concept, namely that object of which the grammatical subject is the proper name. The remaining, predicative, part of the sentence means a concept. I say therefore: the concept is predicative in character, it is in need of supplementation, just as the predicative part of the sentence always demands a grammatical subject, being manifestly incomplete without it. Because it is thus incomplete, or in need of supplementation, we cannot have the predicative part occurring by itself on one side of a definition, but we have to supplement the predicative part with something which takes the place of the grammatical subject. For this purpose we may take the letter 'a'. On the right and left sides of the definition we now have an expression which contains the letter 'a', and we wish to stipulate by its means that both sides of the definition are to have the same sense, no matter what meaningful proper name be put in the place of a. We make use of this letter in order to give generality to our stipulation.
Now the simplest case of the definition of a concept is where one concept is combined with another to form a new concept in the following way. We begin by connecting two sentences with 'and'; the compound sentence, like each of its clauses, can be regarded as the expression of a thought-the one we have compounded-just because we can affirm or deny it as a whole. If
rather to be compared with 'Bunsen and KirchhotT laid the foundations of spectrum analysis', which occurs in one of Frege's letters to Russell. See p. 222 of Gottlob Frege Wissenschqftlicher Briefwechsel (Felix Meiner 1976) (trans. ).
? ? ? a 1s pos1tive
If a is a multiple of an integer greater than 1, then a is that integer
and
a is an integer
and
a is greater that 1
=a is a prime number.
Logic in Mathematics 229
we affirm it, we thereby affirm each of the clauses; if we deny it, we leave it open whether the first or second clause is false or whether both are.
Let us take the case '8 is a perfect cube and 8 is positive'.
In this compound sentence something is asserted of the number 8. We can construe what is present in the sentence over and above the numeral '8' as a sign for a concept, and introduce by definition a new sign for it, as in
a is a cube}
. and. . = a is a positive cube.
{
Here the letter 'a' occurs twice on the left. There is no objection to this; but on the right-hand side, where the dejiniendum occurs, the letter 'a' should occur only once. For if 'a' occurred in different places, it would be possible to fill them with different proper names, and thus obtain an undefined expression; but we must make it impossible for an expression to occur which has no sense.
Here we have a concept (that of a positive cube) that is formed by putting together the component concepts (perfect cube and positive). These we call the characteristic marks of this compound concept.
The case where a proper name occurs as the grammatical subject of a sentence, the predicate of which designates a concept, is the simplest language affords, but is not the only possible one. Whenever a proper name occurs in an assertoric sentence, we can regard the remaining part as a concept-sign. A concept-sign always stands in need of completion by a proper name or a sign standing in for a proper name, such as a letter.
The definition ofprime number will serve as an example. We stipulate
Here on the left we have the dejiniens, on the right the definiendum. We are here saying: No matter what meaningful proper name we put in place of the letter 'a', the expression on the right is always to have the same sense as the expression on the left.
It is an imperative requirement that a concept have sharp boundaries. What this means is that it must hold of every object either that it falls under the concept or that it does not. We cannot allow a third case in which it is somehow undecided or indeterminate whether an object falls under the concept.
This yields a requirement for concept-signs. Such a sign, when completed by a proper name, must always result in a sentence. And this sentence must always express a thouaht which is either true or false. If therefore a new
? 230 Logic in Mathematics
concept-sign is introduced by definition, this requirement has to be satisfied. This happens automatically if the defining side of the definition consists of a properly constructed concept-sign that is completed by a letter which stands in for a proper name and serves to give the necessary generality to the definition. Of course a definition may not be conditional. The stipulation that the dejiniendum is to have in every case the same sense as the dejiniens may not be made to depend on a condition; for then nothing is determined in the case where the condition is not satisfied. The dejiniendum is then without sense. Thus if we replace the indicating letter by the name of an object which fails to satisfy the condition, we shall not have, on the right of the definition, a sentence which expresses a thought that is either true or false, but a senseless combination of signs.
In general, therefore, we must have no truck with conditional definitions. In any case such a definition, if it should be employed, would require a special kind ofjustification.
We move on to the definitions o f relations. If two proper names occur in a sentence, the remaining part of the sentence can be regarded as a sign for a relation, as for example in
'3 is greater than 2',
where the numerals '3' and '2' are to be regarded as proper names. When we define a relation we need two indicating signs to stand in for the names of the objects which stand in the relation. For the same reason that we have seen to hold for the definition of concepts each of these letters may occur only once on the side that is defined. Let us take as an example the definition of congruence between numbers.
(a-b)isamultipleof7 ) and
1aisaninteger and
b is an integer
= aiscongruenttobmodulo7.
The purpose of the letters a and b is to make the stipulation general. We say by this that no matter what meaningful proper names are substituted for 'a' and 'b', the expression on the right-hand side is always to have the same sense as the expression on the left. So if e. g. we put' 16' for 'a' and 2 for 'b' we get
(16- 2) is a multiple of 7 ) and
16 is ~n integer and
2 is an integer
= 16 is congruent to 2 modulo 7.
? Logic in Mathematics 231
So long as the indefinitely indicating letters are not replaced by proper names, the left-hand side has no sense on its own and neither does the right- hand side. But it is necessary that everything on the left-hand side, apart from the letters, should be understood, so that the left-hand side always has a sense when meaningful proper names are substituted for 'a' and 'b'. From this it follows that the minus sign must not only be defined for the case where it stands between numerals, because otherwise the sentence ' ( a - b) is a multiple of 7' would not always have a sense when we substitute meaningful proper names for 'a' and 'b'. In that case it would follow that the expression
'a is congruent to b modulo 7'
would not always have a sense either.
The combination of signs '(16 - 2)' is a proper name of a number.
Accordingly the sign '(a - b)' stands in for a proper name. We obtain a proper name from it by replacing 'a' by a proper name and 'b' likewise. Therefore in the sentence
'(16- 2) is a multiple of 7'
the sign '(16 - 2)' has a meaning: it means the number 14. Therefore the sign '(16 - 2)' has a sense as well, which is a part of the sense of the above sentence, and this sense is the contribution which the sign '(16 - 2)' makes to the expression of the thought. So we may say that the sign 'a - b' acquires a sense by our replacing each of the two letters by a meaningful proper name. In this respect the sign ' ( a - b)' is on all fours with
'(a- b) is a multiple of 7'
By replacing 'a' here by '16' and 'b' by '2' we obtain a sense, namely the
sense of the sentence
'(16- 2)isamultipleof7'
and this sense is a thought. '(16 - 2)' has a sense too; but this is not a
thought: it is only part of the thought. Still the compound sign '(a- b)' and '(a- b) is a multiple of 7'
agree in both acquiring a sense if the letters are replaced by meaningful proper names. The sign '(a - b)' at the same time acquires a meaning by such a replacement.
