We construct a number containing all the
elements
oftwo numbers a and bformedfrom the same basic element.
Gottlob-Frege-Posthumous-Writings
[Draft towards a Review o f Cantor's Lehre vom Transfiniten] 71
this difficult task, we ask the first 'What general concept (p. 56) have you arrived at? ' Non-mathematician that he is, he answers 'Pure Being'. The second thinks rather 'Pure nothingness', the third-I suspect a pupil of Cantor's-'The cardinal number one'. A fourth is perhaps left with the woeful feeling that everything has evaporated, a fifth-surely a pupil of Cantor's-hears an inner voice whispering that graphite and wood, the wnstituents of the pencil, are 'constitutive elements', and so he arrives at the general concept called the cardinal number two. Now why shouldn't one man come out with the answer and another with another? Whether in fact Cantor's definitions have the sharpness and precision their author boasts of is accordingly doubtful to me. But perhaps we got such varying replies because it was a pencil we carried out our experiment with. It may be said 'But a pencil isn't a set'. Why not? Well then, let us look at the moon. 'The moon is not a set either! ' What a pity! The cardinal number one would be only too happy to come into existence at any place and at any time, and the moon seemed the very thing to assist at the birth. Well then, let us take a heap of sand. Oh dear, there's someone already trying to separate the grains. 'You are surely not going to try and count them all! That is strictly forbidden! You have to arrive at the number by a single act of abstraction' (footnote top. 15). 'But in order to be able to abstract from the nature of a grain of sand, I must surely first have looked at it, grasped it, come to know it! ' 'That's quite unnecessary. What would happen to the infinite cardinals in that case? By the time you had looked at the last grain, you would be bound to have forgotten the first ones. I must emphasize once more that you are meant to arrive at the number by a single act of abstraction. Of course for that you need the help of supernatural powers. Surely you don't imagine you can bring it off by ordinary abstraction. When you look at books, some in 4uarto, some in octavo, some thick, some thin, some in Gothic type and some in Roman and you abstract from these properties which distinguish them, and thus arrive at, say, the concept "book", this, when you come down to it, is no great feat. Allow me to clarify for you the difference between ordinary abstraction and the higher, supernatural, kind.
With ordinary abstraction we start out by comparing objects a, b, c, and lind that they agree in many properties but differ in others. We abstract from the latter and arrive at a concept tP under which a and b and c all fall. Now this concept has neither the properties abstracted from nor those common to a, band c. The concept "book", for instance, no more consists of printed sheets-although the individual books we started by comparing do consist of such-than the concept "female mammal" bears young or suckles them with milk secreted from its glands; for it has no glands. Things ure quite different with supernatural abstraction. Here we have, for instance, u heap of sand . . . 1
1 At this point the manuscript breaks ofT (cd. ).
? ? On the Concept of Number1 [1891/92]
[A Criticism ofBiermann]
In my Grundlagen (? 68) I called the concept F equal in number to the concept G if it is possible to correlate one-to-one the objects falling under F with those falling under G and then gave the following definition:
The number2 belonging to the concept F is the extension of the concept 'equal in number to the concept F'.
The following discussion will show that this definition gives the right results when applied, by deriving the basic properties of the numbers from it. But first we need to clarify a few points and meet some objections. The way the word number is used outside mathematics does not reveal the sense of the word with the clarity that is indispensable for scientific purposes. Things are, so to speak, dragged into the number, lock, stock and barrel; people imagine a number of trees as something rather like a group or row of trees, so that the trees themselves belong to the number. On this view, the number
of peas on the table would not only be changed, say, if I filed something off one of them or if I took away one and put another in its place; it would also be changed if I simply shuffled the peas about; for a change in spatial relations means that the whole is changed, as a heap of sand is changed if someone spreads it out without adding or taking away a grain of sand. If we ask what the number 2 is, likely enough the answer will be 'two things'. Now put an apple and a pear in front of someone who has given this answer and say 'Here you have your number 2'. Perhaps he will begin to hesitate at this point, but he will be even more unsure if we ask him to multiply this number 2 by the number 1 which we could give him, say, in the shape of a
1 According to notes on the transcripts on which this edition is based, both the following papers were found, unseparated, in a folder under the heading: 'On the Concept of Number'. They were probably written in the years 1891/92 since the section beginning on p. 87 is a preliminary draft of Frege's article Ober Begriff und Gegenstand, published in 1892. The section dealing with Biermann may have been written earlier. This is indicated by the use of the word 'Inhalt' (p. 85) where Frege's later practice would require 'Bedeutung' (ed. ).
2
Here and in the first five sentences of the paragraph following the word translated 'number' is 'Anzahf, which strictly means 'natural number'. Throughout most of the rest of the paper Frege uses 'Zahf, which has the same broad application as our 'number' (trans. ).
? ? On the Concept ofNumber 73
strawberry. It would, indeed, be very pleasant if we could conjure up a five- pound note by multiplying a cork by a nail. All this is pure childishness, of course, and that we have to bother with such things is a sufficient indictment of the times we live in. Yet even mathematicians will stoop to giving such definitions as:*
'The concept of number may be defined as the idea of a plurality composed of things of the same kind. When we use the term "one" for each of the elements of the same kind, counting the elements or units of the set consists in assigning the new terms two, three, etc. , to one and one-and one, etc. Number is the idea of the groups of elements designated by these terms. '
Expressions such as 'things of the same kind', 'elements', 'one', 'unit' are all jostled together here, as though the third section of my Grundlagen had never been written. Do these expressions have the same meaning, or not? If they do, why this plethora of terms? If they don't, what is the difference between them? ** At any rate, we may at least assume that 'elements' is intended to mean the same as 'things of the same kind' and 'group' the same as 'plurality'. Strange what importance is so often attached to a mere change in expression when presenting these rudiments. It seems almost as though the expressions 'one' and 'element of our group' are intended to have the same meaning, and yet the replacement of the latter by the former is stressed as an important step. Why does counting the elements or units of a set not consist in assigning new terms to element and element-and element? Or is that what it actually does? Let us assume, to clarify matters, that there is a lion standing beside a lioness lying on the ground and together they form a picturesque group of things of the same kind; both animals do, of course, belong to the species felis Ieo. At first we have nothing more than just this group. But now comes the highly significant act of designating both the lion und the lioness by the term one; then follows the no less significant act of assigning the new term two to one and one (and thus, indubitably, to our group); and we are very happy to have acquired the number two at this juncture. For surely what we need most of all is to have the terms and words; once we have them we can say: 'Number is the idea of groups of elements designated by these terms. ' Accordingly, in our case the number two would presumably be the idea of the group oflions. We might still ask what the word 'one' actually means in that case. Does it designate a number? According to Biermann's notion, one would then be a group of elements. Now we have just used the term one to designate the lioness. Since Ihe lioness is not an idea we must presumably accept the idea of the lioness
? Otto Biermann, Theorie der analytischen Funktionen [Leipzig 1887] ? I.
? ? I'll wager Biermann did not know this himself at the time and still does not know.
? ? 74
On the Concept ofNumber
as the meaning of the word 'one'. The lioness is made up of molecules, after all, and so may well be seen as a group.
Perhaps Biermann also takes the word 'group' in such a way that a single thing is also a group, no matter whether it is composite or not, although the contrary is indicated by the fact that Biermann speaks of elements, things of the same kind, and stresses the way a number is composed.
Admittedly, we also used one to designate the lion, and this could give us pause should we find the idea [image] 1 of the lion different from that of the lioness; on the other hand, the idea [image] we have can be so blurred that the idea [image] of the lion merges with that of the lioness. Hence, just as two is the idea [image] of our group of lions, one would be an idea [image] though a pretty blurred one, of a lion. What degree of vagueness we would have to assume admittedly remains uncertain. As a second example, let us take the Laocoon group, the well-known Greek sculpture found in 1506. We might well doubt whether in this case the components may be assumed to be of the same kind.
Biermann does not state to what extent the similarity he requires allows scope for differences. If the coincidence were exact, should we ever get beyond the number one? Presumably, then, a certain degree of latitude must be allowed. We also speak of two cats even if they are not the same colour, or two coins even if one is made of gold and the other of nickel. No doubt we shall always be able to discover some sort of similarity, even if it consists in nothing more than each of the elements having the property of being like itself in every respect. What then is the point of this condition of being of the same kind, when either it is always fulfilled, or we never know when it is fulfilled? Biermann does not know himself. He is onto something, but doesn't know what. He could have found out the answer from my Grundlagen, just as he could have learnt a great deal more that he does not know, but he must have thought in his heart of hearts: metaphysica sunt, non leguntur.
Let us return to the Laocoon group. Since the whole thing is made of marble and since, moreover, the parts of the group represent living beings, presumably there is nothing to prevent us from regarding the elements as being sufficiently alike. So what we do first-and this is very significant indeed-is to designate Laocoon, or more precisely his marble image, and each of his sons, and the serpent as well, by the term one. Then we again assign terms to one and one-and one . . . All right, what actually is one in this case? The idea [image] of a living being sculpted in marble which is so blurred we are not able to distinguish whether what is represented is the figure of a snake or a human being, an older or a younger man? That seems
1 The difficult word 'Vorstellung' we have rendered throughout as 'idea', sometimes enclosing the word 'image' in brackets afterwards where this sense seems more appropriate. Owing to the influence of Kant, the term was of course frequently used in German philosophical writings of Frege's time. The standard English translation of ? Vorstellung' as it occurs in Kant is 'representation' (trans. ).
? On the Concept ofNumber 75
a trifle implausible. In that case, 'one' would presumably have to have a different meaning from what it had in the previous example. Or must the idea [image] be taken to be so blurred that it is quite impossible to say what it is an idea of? Is there only one one? Or are there many ones? Could we say in the latter case 'Laocoon is a one' or 'The idea of Laocoon is a one'? Or can Biermann give us an example of something that is a one? What does it mean when someone says: this or that is a one? And if there are several ones, we must still be able to express the fact that something belongs to the species one. The phrase 'by designating each of the elements of the same kind by one' may lead someone to suppose that one is a title like 'Sir', for instance. We have the privilege of conferring this title, and everything and everyone receiving it is simply a one as a result. Then we could never go wrong in calling something one, just as a reigning monarch can never go wrong when he bestows a title on someone. Of course, this title 'One' would he pretty worthless; you cannot imagine it amounting to anything much. At 1he same time, conferring this title would be a source of revenue for us, in so far as it is through this that we come into possession of the numbers. If there were only one one, if, that is, 'one' were a proper name, what would 'one and one' mean? Clearly, one again; for what can 'Charlemagne and t 'harlemagne' mean, if not Charlemagne? Then there must be several ones after all, and our earlier conjecture that one is a vague idea is possibly false.
At this stage the most obvious thing seems to be to regard 'one' as a title. We have conferred this title on Laocoon, his sons and the serpent, and we uow assign a new term to the Laocoon group. Very well, but which? We must not imagine for one moment that there is no need for a new term, since we could simply call our group the Laocoon group. That would clearly not he the right term to assign to it. Biermann knows best what term we need to use here. Perhaps he would fix on the term 'four'. Others would perhaps prefer 'one'; but why should the same thing not have different titles?
Now Biermann says 'Number is the idea of the groups of elements designated by those terms'. Fine! We designate the group itself-so here the Luocoon group-by the term 'four' or 'one' or by both; but the number is not the group itself which we designate by the number word, it is the idea of 11. Is the idea of the Laocoon group the number one or the number four or what number? Why not one and four at the same time, as the whim takes us'! But we must have made a mistake! The whole group consists of molecules of calcium carbonate. Here we have elements of our group that nrc of the same kind. Possibly it would have been better if we had designated 0111. :h of these molecules by the term one. Well, we go ahead and do this and ? o ure able to designate the Laocoon group by a new term. The number is now the idea of the group designated by this term (our old Laocoon group uvcr again). It is only at this point that we begin to see how subtle the term 'idea' is. To be sure, the Laocoon group is the same; but now we have a dill"crcnt idea of it. That is why we now have a different number. If our idea uf a plurality composed of things of the same kind, e. g. a heap of sand,
? ? 76 On the Concept ofNumber
changes, the number changes as well; for the number is simply the idea of the heap of sand. If, for instance, we spread the heap of sand out, the idea of the heap of sand when spread out is clearly a different idea from that of the original heap. These ideas are numbers and they are different; this means that they are different numbers. To arrive at this result there is no need to add or take away a grain of sand.
Let us assume a child and a painter are looking at a group, consisting of an apple, a pear and a nut. Following Biermann's instructions they may both have reached the point of designating this group by the new term three. Suppose now a cloth be spread over this plurality composed of things of the same kind; what ideas of them are left behind in the minds of the observers? Evidently quite different ones. In the case of the child ideas of taste will predominate, in the case of the painter ideas of the colours and how they shade off, of the shadows and outlines, etc. , so that the two of them, if Biermann's definition were correct, would have quite different numbers, that is, different ideas of the group of elements designated by the word three. But I suppose Biermann would say that all this is psychology, that these considerations have nothing to do with arithmetic. Just so! But how did we get here? Via the term 'idea' which Biermann uses and which simply does belong to psychology. Any attempt to exclude psychological considerations from mathematics has my full approval. But let us do the job properly. Away with the word idea! To be sure, we must then dispense with such interesting psychological propositions as 'Through number we possess the idea of a plurality or set of similar things'. 1 The full implications of this proposition can only be realized when it is applied to examples; thus we possess the idea of a heap of sand through the idea of a heap of sand; we possess the idea of the constellation of Orion through the idea of the constellation of Orion; we possess the idea of the Laocoon group through
the idea of the Laocoon group. But let us not make an issue of whether arithmetic is the proper place for this psychological truth. Let us rather rejoice over truths wherever we find them. And let us look to see whether Biermann's book does not yield yet more meat for psychology. Sure enough! We learn something about the composition of ideas:
'Two numbers formed from an indeterminate basic element e or the abstract unit 1 . . . ', etc. 2
If only we knew what an 'indeterminate basic element' is! Perhaps we shall glean it by comparing this with a later passage:
?
We construct a number containing all the elements oftwo numbers a and bformedfrom the same basic element. '3
1 Cf. Biermaqn, p. I.
2 Ibid. 3 Ibid.
? ? On the Concept ofNumber 77
Thus we have, say, two heaps of peas,* and we make a single heap with them. What is the basic element the heap is made up of? A pea? A single pea? Which pea? An indeterminate pea. I confess that I have not yet seen an indeterminate pea, but I can well imagine that Biermann has the concept pea in mind; or would it be an idea of a pea? I trust I do Biermann no injustice if I assume that he has never broached this question. Though we cannot, to be sure, say that a heap of peas is composed of an indeterminate pea, or of the concept pea, or of an idea of a pea, we can say that it is composed of peas, i. e. of objects falling under the concept pea. Therefore, in view of the fact that Biermann's terminology is somewhat imprecise, we may assume that his phrase 'number formed from an indeterminate basic clement' is intended to mean: 'number whose components fall under one concept' and that is only another term for something that has already given us food for thought-elements of the same kind; evidently, Biermann is fond of presenting us with the same thing in different guises so that we do not find it too easy to achieve clarity. But we are forgetting, and perhaps Biermann is, too, that it is not the heap of peas that is a number, but the idea of the heap. The number, i. e. the idea, is not, however, made up of peas. What is an element of a number? After what has just been said everyone must think: an element of the group whose idea is the number. Let us consider Biermann's formulation once more: 'We construct a number containing all the elements of two numbers a and b formed from the same basic element. ' That is to say: we form the idea of a group, which idea contains all the elements of two numbers. In our example we had to take the peas as the elements of the numbers. Thus we formed a leguminous idea which may possibly be of some use as a nitrogenous food. But I think it would be better if we left aside the question of forming this idea: the feat is beyond us. By element of a number should we perhaps understand: idea of a component of the group whose idea is the number? The elements of the number in our example would then not be the peas, but the ideas of the peas. True, Biermann had previously spoken only of elements of the group whose idea is the number; but that may well have slipped his mind, which would be only human after all; or possibly he wants to catch a superficial critic off his
? No doubt Biermann is thinking 'How vulgar to talk about peas, apples und the like when we are supposedly dealing with scientific issues. It sounds for all the world like someone talking to little children! ' Well, I am indeed trying to show that we are only left with puerilities once we go to the core of his argument and strip off the semblance of learning he is able to create by using terms like 'element', 'group', 'idea', 'assign', and so on. There is no hetter place to hide the most childish confusions than in the most learned- NilUnding terms. That is why the cruder the examples we use to throw light on these terms, the more pitifully apparent it becomes how utterly obscure they are. Biermann's account is able to retain its aura of learning only hccause he forbears applying it to particular cases. It would be out of place to he too serious here.
? ? 78 On the Concept ofNumber
guard, for such a reader might well be beguiled by the use of the same word 'element' into thinking that we are still dealing here with elements of the group. We may surely allow that the idea of a heap of peas contains ideas of peas. At all events it will contain, in addition, the idea of a certain proximity which we indicate, of course, by the word 'heap'. True, Biermann nowhere says that the idea we are to form should contain only the elements of the numbers a and b; if we took him at his word, we could go on adding as many ideas as we liked not containing elements of the numbers a and b; but that would hardly measure up to his view. But is there any need at all for a new heap or an idea of it? * We simply form an idea containing the ideas of the two original heaps, that is, the idea of a group of the two heaps whose ideas are the numbers a and b. Unfortunately, this idea would also contain an element that does not occur in a and b-an idea of the spatial relationship of the two heaps-and here, of course, we may choose any one of many such relationships. Or are the two numbers a and b, which are ideas after all, supposed to merge into one idea, in keeping with the psychological principle of the fusing of likes, much as, say, the ideas [images] of two similar faces merge together? Unfortunately, the result would turn out somewhat blurred. ** Yet I hear Biermann crying out in despair 'This eternal psychologizing! If only I had not used the word "idea"! I did not mean it as seriously as all that! ' But Biermann also says things like 'we become aware', 'we arrive at the concept of a set', 'we abstract'. And this is already quite enough to lead us off into psychological irrelevancies. 'We' is not an object of mathematics at all, just as little as our ideas are. Truths in mathematics are eternal and not dependent on whether we are alive or dead or become aware of them.
I can well imagine that Biermann has used the word 'idea' in much the same way as we use 'Esteemed Sir' and 'Honoured Sir', to make what he says sound weightier and more impressive without actually changing its sense. There is more than one circle of society in which people do not feel fully clothed if they lack a title. And likewise it is possible that a certain modesty-which, incidentally, redounds to his credit-has prevented Biermann from taking the peas with all their adventitious little wrinkles and introducing them into mathematics without dressing them up. For my part, I am more for things in their raw and natural state and prefer the following to Biermann's formulation: Number is a plurality composed of things of the same kind, or numbers are groups of elements. I concede that this sounds somewhat less impressive; the first formulation, in particular, has a touch of
*That [i. e. that we need an idea of a new heap] would even be wrong, for the ideas of proximity which are elements of the ideas ofthe original heaps of peas would be quite missing from the idea of a new heap of peas. Yet the idea we are to form is supposed to contain all the elements of the numbers a and b.
** As we know, this process may be emulated by photographic means.
? On the Concept ofNumber 79
tautology about it and might make you suspect the hand of a girl at a finishing school; but that is the very reason I prefer it. And if the word 'idea' is really intended to serve as nothing more than an ornament, I shall be able to adopt my formulation without departing in essentials from what Biermann thinks. Granted, the statement that the Laocoon group may bear the title one as well as the title four now sadly reduces to nothing: But this is offset by the fact that from this point on mathematics really has something to get its teeth into.
As is well known, this discipline is concerned with numbers. Now heaps of peas, of sand, and other heaps are numbers; herds of sheep, of cows and of other animals are numbers, too. Consequently, all these heaps and herds are objects of mathematics. Indeed, we may perhaps say that mathematics is concerned with all possible things; a window is one, a house with many windows is one, the country in which there are many houses is one. * Now if every such one is a number, then the window is a number, etc. No doubt Biermann will say 'Just so! Mathematics is concerned with all possible things in respect of what is number about them. ' The striking thing, however, is that herds of sheep are seldom mentioned in this discipline. I helieve they do not even appear in Biermann's book at all. Does my memory deceive me, or have I really only read about herds of sheep-if I have read about them at all in mathematical books-in the sets of examples given to illustrate the application of mathematical propositions? But I am probably putting words into Biermann's mouth that he has never thought of uttering. Number is not something attaching to the herds; the herds themselves just as they are, skin and bone and dirt, are numbers. It looks as though I have got confused here with J. S. Mill's view according to which a number is a property of an aggregate-that is, the way an aggregate is put together. I must confess that there were times, as I was struggling through Biermann's obscurities, when this view seemed to me full of insight. But it appears that light is now beginning to penetrate these regions of darkness. Let us take
Biermann's formula: 'Two numbers formed from an indeterminate basic clement or the abstract unit 1 are equal, when to each element of the one there belongs an element of the other'1 and apply it e. g. to herds of sheep. llow clear everything now becomes! Two herds of sheep are equal when to each sheep of one herd there belongs a sheep of the other. Admittedly, when a sheep A belongs to a sheep B is something we are not told. Let us turn to the difficult question of whether it is conceivable that a herd of sheep is equal
*'Omnia una sunt', a Latinist would say, if not deterred by his feeling for the language, which would here be confirmed by the nature of things as well. Apparently, Biermann has not yet got round to asking himself what underlies this phenomenon of language; for he can say 'units' as though it were the same thing.
1 Hiermann, p. I.
? ? 80 On the Concept ofNumber
to a constellation of stars. The one thing we do at least know is that both are numbers. The only question we still have to settle is whether they are formed from the same basic element. * I believe we have already worked out what Biermann means: when he says that a number is formed from an indeterminate basic element, he means that a number is formed from objects falling under one concept, and the 'indeterminate basic element' then corresponds to the concept. In this case we can point to such a concept: heavy, inert body. Both the sheep and the stars fall under this concept. There can, therefore, presumably be no doubt that the herd of sheep and the constellation are formed from 'the same indeterminate basic element'. Now it is surely conceivable that every star in the constellation belongs to a sheep in the herd, and so it is also conceivable that a herd of sheep should be equal to a constellation. We must not say here that they may of course be equal in respect of the number of solid inert bodies out of which they are made up; for the herd of sheep is itself one of the numbers and the constellation itself is the other. We have already established that according to Biermann number is not a property in respect of which the herd is interchangeable with the constellation. True, we may say, this beetle and the bark of this tree
are equal' so far as their colour is concerned; but here neither the beetle nor the bark are a colour; moreover we do not have two colours, but one and the same. So according to Biermann this case is quite different from that of the numbers; for even if the phrase 'idea of a group' were to mean something quite different from the group itself, it still would not mean a property of the group. And even if, quite contrary to normal usage, Biermann were to use the term 'idea' in such a way that the idea of a group was a property of it, the proposition 'a number is the idea of a group' would amount to the same as 'a number is the number of a group': that is to say, a number is that property of a group which we call idea or number.
We still need to emphasize that according to Biermann's definition the word 'equal' does not mean complete coincidence: a number may be equal to another without being the same; a herd of sheep may be equal to a constellation without being the constellation itself. The question now arises what the number words mean: the most obvious answer would be that the number word 'two', for example, designates one (and only one) number, so that we may say: two is a number, three is a number, and so on. Two and three would be related to the concept of number in the same way as, say, Archimedes, Euclid and Diophantus are related to the concept of mathematician. If we say this, however, we should certainly get into difficulties. Let us again imagine a group consisting of a lion standing and a
*It is not clear from Biermann's wording whether or not this condition must be fulfilled; what we have is only: 'from a', and not 'from the same'. To be on the safe side, we will assume that it must.
1 Here English idiom requires 'alike' rather than 'equal', but in German the same word-'g/eich'-does duty for both (trans. ).
? On the Concept ofNumber 81
lioness lying on the ground. This group is a number. Let us assign to it the number word 'two' as its proper name. Then in future we shall mean our group of lions when we say 'two'. Let us now think of the Goethe-Schiller memorial in Weimar. We surely cannot give the Goethe-Schiller group the same name as the group of lions. To do this could lead to some singularly unfortunate mistakes in identity! We do not seem able to manage with the number words alone. But Biermann has a simple way of getting round this difficulty; as we call Homer, Virgil and Goethe poets, so we might with equal justification call both the Goethe-Schiller group and the group oflions lwo. If we call something two, that shows we want to allocate it to a certain species: we want to say that it has a certain property or properties. In just 1he same way by calling someone a poet I acknowledge he has certain properties characteristic of being a poet,* or by calling a thing blue I attribute a certain property to it or assign it to some species or other. I-ikewise by calling a group three I would be saying that it has a certain property. As we call the properties green, blue, yellow colours, so we could rail two and three numbers. But wait! Here we are again on the same false ! rack as before. We cannot repeat often enough: number is a group or plurality composed of things of the same kind; therefore numbers are the subjects of the properties expressed by the number words. We are no more JUstified in asserting that two is a number than we are in counting green as a u1loured object instead of a colour. Thus we are now able to say: two, three, four, etc. , are properties of groups with constituents of the same kind, which t-:roups or sets are called numbers. A somewhat more elevated way of putting the same thing is to say: two, three, four, etc. , are properties of ideas or groups with constituents of the same kind, and these ideas are called numbers. The most remarkable thing for the layman, for me myself and perhaps even for Biermann in all this, and the most astonishing, is that the number words do not designate numbers but properties of numbers. The equation 2 = 1 + 1 is generally considered to be true. Biermann's definition of a = b is not applicable here, since it relates to numbers, i. e. groups, whereas for us the sign '2' on the left does not mean a number here but a property of a number. Let us see whether we fare any better with the plus Nign. Biermann says: 'We construct a number containing all the elements of 1wo numbers a and b formed from the same basic element' . . . 'The resultant number we designate by a+ b . . . '1
We do not learn from this what 2 + 3 and 2 + 1 mean, for 1, 2 and 3 are not numbers at all. But perhaps we learn something else which merits our nttention. As we have seen, the constellation Orion is a number, the belt in this constellation is likewise a number and, moreover, a different one. If we understand Biermann correctly, both numbers are formed from 'the same
* We need not ask here whether one or more than one property is involved in the word 'two'.
1 Hiermann, p. I.
? ? ? 82 On the Concept ofNumber
basic element', for the elements of the belt are also the elements of Orion. Let us then form a number containing all the elements of Orion and of its belt. What could be simpler! We do not even have to bother constructing this number: it is already there. The constellation of Orion is itself this number. So let us designate Orion by a and its belt by b; then we have a+ b = a, whilst at the same time we have remained happily in accord with Biermann's definitions of the plus sign and the equals sign. But that's enough of pretending to be more stupid than we are! After all, Biermann is writing for people already familiar with these matters who will read the right thing into his words even when they are false or devoid of sense! * After all, the words are there so that the reader may understand what is meant, despite them. We have to bear in mind the condition that no element may be common to both numbers. It follows that no number may ever be added to itself: presumably we should have to say that a + a is devoid of sense; yet a little further on Biermann's text reads: a + a + a . . . (b-times). Let us see how that comes about. Biermann says: 'If we substitute the number a for each of the elements of a number b we obtain the sum a + a + a . . . (b-times), which we designate in short by a x b or ab. '1 The sense of this definition is not easy to grasp; let us take an example to clarify what is at stake: let b be the Goethe-Schiller group in the square in front of the theatre in Weimar, which according to Biermann is indubitably a number. Let a be the constel- lation of Orion. Now let us first of all substitute Orion for the figure of Goethe. It is no easy thing that Biermann is asking of us; his arithmetic would seem to be designed for gods. For him personally all this is child's play; we shall witness still greater feats. F o r the figure o f Schiller let us now substitute-well, what? Why, Orion, too! ** And what do we get then?
Orion + Orion + Orion (Goethe-Schiller-group times).
Who would have thought it! A little later Biermann says: 'In a+ a+ . . . +a (b-times) we may pick one out of each group of a elements and their combination yields b'. If Biermann had not said it, I would not have believed
*It would have been more to the purpose, in my view, if ? 1 had been left unwritten.
this difficult task, we ask the first 'What general concept (p. 56) have you arrived at? ' Non-mathematician that he is, he answers 'Pure Being'. The second thinks rather 'Pure nothingness', the third-I suspect a pupil of Cantor's-'The cardinal number one'. A fourth is perhaps left with the woeful feeling that everything has evaporated, a fifth-surely a pupil of Cantor's-hears an inner voice whispering that graphite and wood, the wnstituents of the pencil, are 'constitutive elements', and so he arrives at the general concept called the cardinal number two. Now why shouldn't one man come out with the answer and another with another? Whether in fact Cantor's definitions have the sharpness and precision their author boasts of is accordingly doubtful to me. But perhaps we got such varying replies because it was a pencil we carried out our experiment with. It may be said 'But a pencil isn't a set'. Why not? Well then, let us look at the moon. 'The moon is not a set either! ' What a pity! The cardinal number one would be only too happy to come into existence at any place and at any time, and the moon seemed the very thing to assist at the birth. Well then, let us take a heap of sand. Oh dear, there's someone already trying to separate the grains. 'You are surely not going to try and count them all! That is strictly forbidden! You have to arrive at the number by a single act of abstraction' (footnote top. 15). 'But in order to be able to abstract from the nature of a grain of sand, I must surely first have looked at it, grasped it, come to know it! ' 'That's quite unnecessary. What would happen to the infinite cardinals in that case? By the time you had looked at the last grain, you would be bound to have forgotten the first ones. I must emphasize once more that you are meant to arrive at the number by a single act of abstraction. Of course for that you need the help of supernatural powers. Surely you don't imagine you can bring it off by ordinary abstraction. When you look at books, some in 4uarto, some in octavo, some thick, some thin, some in Gothic type and some in Roman and you abstract from these properties which distinguish them, and thus arrive at, say, the concept "book", this, when you come down to it, is no great feat. Allow me to clarify for you the difference between ordinary abstraction and the higher, supernatural, kind.
With ordinary abstraction we start out by comparing objects a, b, c, and lind that they agree in many properties but differ in others. We abstract from the latter and arrive at a concept tP under which a and b and c all fall. Now this concept has neither the properties abstracted from nor those common to a, band c. The concept "book", for instance, no more consists of printed sheets-although the individual books we started by comparing do consist of such-than the concept "female mammal" bears young or suckles them with milk secreted from its glands; for it has no glands. Things ure quite different with supernatural abstraction. Here we have, for instance, u heap of sand . . . 1
1 At this point the manuscript breaks ofT (cd. ).
? ? On the Concept of Number1 [1891/92]
[A Criticism ofBiermann]
In my Grundlagen (? 68) I called the concept F equal in number to the concept G if it is possible to correlate one-to-one the objects falling under F with those falling under G and then gave the following definition:
The number2 belonging to the concept F is the extension of the concept 'equal in number to the concept F'.
The following discussion will show that this definition gives the right results when applied, by deriving the basic properties of the numbers from it. But first we need to clarify a few points and meet some objections. The way the word number is used outside mathematics does not reveal the sense of the word with the clarity that is indispensable for scientific purposes. Things are, so to speak, dragged into the number, lock, stock and barrel; people imagine a number of trees as something rather like a group or row of trees, so that the trees themselves belong to the number. On this view, the number
of peas on the table would not only be changed, say, if I filed something off one of them or if I took away one and put another in its place; it would also be changed if I simply shuffled the peas about; for a change in spatial relations means that the whole is changed, as a heap of sand is changed if someone spreads it out without adding or taking away a grain of sand. If we ask what the number 2 is, likely enough the answer will be 'two things'. Now put an apple and a pear in front of someone who has given this answer and say 'Here you have your number 2'. Perhaps he will begin to hesitate at this point, but he will be even more unsure if we ask him to multiply this number 2 by the number 1 which we could give him, say, in the shape of a
1 According to notes on the transcripts on which this edition is based, both the following papers were found, unseparated, in a folder under the heading: 'On the Concept of Number'. They were probably written in the years 1891/92 since the section beginning on p. 87 is a preliminary draft of Frege's article Ober Begriff und Gegenstand, published in 1892. The section dealing with Biermann may have been written earlier. This is indicated by the use of the word 'Inhalt' (p. 85) where Frege's later practice would require 'Bedeutung' (ed. ).
2
Here and in the first five sentences of the paragraph following the word translated 'number' is 'Anzahf, which strictly means 'natural number'. Throughout most of the rest of the paper Frege uses 'Zahf, which has the same broad application as our 'number' (trans. ).
? ? On the Concept ofNumber 73
strawberry. It would, indeed, be very pleasant if we could conjure up a five- pound note by multiplying a cork by a nail. All this is pure childishness, of course, and that we have to bother with such things is a sufficient indictment of the times we live in. Yet even mathematicians will stoop to giving such definitions as:*
'The concept of number may be defined as the idea of a plurality composed of things of the same kind. When we use the term "one" for each of the elements of the same kind, counting the elements or units of the set consists in assigning the new terms two, three, etc. , to one and one-and one, etc. Number is the idea of the groups of elements designated by these terms. '
Expressions such as 'things of the same kind', 'elements', 'one', 'unit' are all jostled together here, as though the third section of my Grundlagen had never been written. Do these expressions have the same meaning, or not? If they do, why this plethora of terms? If they don't, what is the difference between them? ** At any rate, we may at least assume that 'elements' is intended to mean the same as 'things of the same kind' and 'group' the same as 'plurality'. Strange what importance is so often attached to a mere change in expression when presenting these rudiments. It seems almost as though the expressions 'one' and 'element of our group' are intended to have the same meaning, and yet the replacement of the latter by the former is stressed as an important step. Why does counting the elements or units of a set not consist in assigning new terms to element and element-and element? Or is that what it actually does? Let us assume, to clarify matters, that there is a lion standing beside a lioness lying on the ground and together they form a picturesque group of things of the same kind; both animals do, of course, belong to the species felis Ieo. At first we have nothing more than just this group. But now comes the highly significant act of designating both the lion und the lioness by the term one; then follows the no less significant act of assigning the new term two to one and one (and thus, indubitably, to our group); and we are very happy to have acquired the number two at this juncture. For surely what we need most of all is to have the terms and words; once we have them we can say: 'Number is the idea of groups of elements designated by these terms. ' Accordingly, in our case the number two would presumably be the idea of the group oflions. We might still ask what the word 'one' actually means in that case. Does it designate a number? According to Biermann's notion, one would then be a group of elements. Now we have just used the term one to designate the lioness. Since Ihe lioness is not an idea we must presumably accept the idea of the lioness
? Otto Biermann, Theorie der analytischen Funktionen [Leipzig 1887] ? I.
? ? I'll wager Biermann did not know this himself at the time and still does not know.
? ? 74
On the Concept ofNumber
as the meaning of the word 'one'. The lioness is made up of molecules, after all, and so may well be seen as a group.
Perhaps Biermann also takes the word 'group' in such a way that a single thing is also a group, no matter whether it is composite or not, although the contrary is indicated by the fact that Biermann speaks of elements, things of the same kind, and stresses the way a number is composed.
Admittedly, we also used one to designate the lion, and this could give us pause should we find the idea [image] 1 of the lion different from that of the lioness; on the other hand, the idea [image] we have can be so blurred that the idea [image] of the lion merges with that of the lioness. Hence, just as two is the idea [image] of our group of lions, one would be an idea [image] though a pretty blurred one, of a lion. What degree of vagueness we would have to assume admittedly remains uncertain. As a second example, let us take the Laocoon group, the well-known Greek sculpture found in 1506. We might well doubt whether in this case the components may be assumed to be of the same kind.
Biermann does not state to what extent the similarity he requires allows scope for differences. If the coincidence were exact, should we ever get beyond the number one? Presumably, then, a certain degree of latitude must be allowed. We also speak of two cats even if they are not the same colour, or two coins even if one is made of gold and the other of nickel. No doubt we shall always be able to discover some sort of similarity, even if it consists in nothing more than each of the elements having the property of being like itself in every respect. What then is the point of this condition of being of the same kind, when either it is always fulfilled, or we never know when it is fulfilled? Biermann does not know himself. He is onto something, but doesn't know what. He could have found out the answer from my Grundlagen, just as he could have learnt a great deal more that he does not know, but he must have thought in his heart of hearts: metaphysica sunt, non leguntur.
Let us return to the Laocoon group. Since the whole thing is made of marble and since, moreover, the parts of the group represent living beings, presumably there is nothing to prevent us from regarding the elements as being sufficiently alike. So what we do first-and this is very significant indeed-is to designate Laocoon, or more precisely his marble image, and each of his sons, and the serpent as well, by the term one. Then we again assign terms to one and one-and one . . . All right, what actually is one in this case? The idea [image] of a living being sculpted in marble which is so blurred we are not able to distinguish whether what is represented is the figure of a snake or a human being, an older or a younger man? That seems
1 The difficult word 'Vorstellung' we have rendered throughout as 'idea', sometimes enclosing the word 'image' in brackets afterwards where this sense seems more appropriate. Owing to the influence of Kant, the term was of course frequently used in German philosophical writings of Frege's time. The standard English translation of ? Vorstellung' as it occurs in Kant is 'representation' (trans. ).
? On the Concept ofNumber 75
a trifle implausible. In that case, 'one' would presumably have to have a different meaning from what it had in the previous example. Or must the idea [image] be taken to be so blurred that it is quite impossible to say what it is an idea of? Is there only one one? Or are there many ones? Could we say in the latter case 'Laocoon is a one' or 'The idea of Laocoon is a one'? Or can Biermann give us an example of something that is a one? What does it mean when someone says: this or that is a one? And if there are several ones, we must still be able to express the fact that something belongs to the species one. The phrase 'by designating each of the elements of the same kind by one' may lead someone to suppose that one is a title like 'Sir', for instance. We have the privilege of conferring this title, and everything and everyone receiving it is simply a one as a result. Then we could never go wrong in calling something one, just as a reigning monarch can never go wrong when he bestows a title on someone. Of course, this title 'One' would he pretty worthless; you cannot imagine it amounting to anything much. At 1he same time, conferring this title would be a source of revenue for us, in so far as it is through this that we come into possession of the numbers. If there were only one one, if, that is, 'one' were a proper name, what would 'one and one' mean? Clearly, one again; for what can 'Charlemagne and t 'harlemagne' mean, if not Charlemagne? Then there must be several ones after all, and our earlier conjecture that one is a vague idea is possibly false.
At this stage the most obvious thing seems to be to regard 'one' as a title. We have conferred this title on Laocoon, his sons and the serpent, and we uow assign a new term to the Laocoon group. Very well, but which? We must not imagine for one moment that there is no need for a new term, since we could simply call our group the Laocoon group. That would clearly not he the right term to assign to it. Biermann knows best what term we need to use here. Perhaps he would fix on the term 'four'. Others would perhaps prefer 'one'; but why should the same thing not have different titles?
Now Biermann says 'Number is the idea of the groups of elements designated by those terms'. Fine! We designate the group itself-so here the Luocoon group-by the term 'four' or 'one' or by both; but the number is not the group itself which we designate by the number word, it is the idea of 11. Is the idea of the Laocoon group the number one or the number four or what number? Why not one and four at the same time, as the whim takes us'! But we must have made a mistake! The whole group consists of molecules of calcium carbonate. Here we have elements of our group that nrc of the same kind. Possibly it would have been better if we had designated 0111. :h of these molecules by the term one. Well, we go ahead and do this and ? o ure able to designate the Laocoon group by a new term. The number is now the idea of the group designated by this term (our old Laocoon group uvcr again). It is only at this point that we begin to see how subtle the term 'idea' is. To be sure, the Laocoon group is the same; but now we have a dill"crcnt idea of it. That is why we now have a different number. If our idea uf a plurality composed of things of the same kind, e. g. a heap of sand,
? ? 76 On the Concept ofNumber
changes, the number changes as well; for the number is simply the idea of the heap of sand. If, for instance, we spread the heap of sand out, the idea of the heap of sand when spread out is clearly a different idea from that of the original heap. These ideas are numbers and they are different; this means that they are different numbers. To arrive at this result there is no need to add or take away a grain of sand.
Let us assume a child and a painter are looking at a group, consisting of an apple, a pear and a nut. Following Biermann's instructions they may both have reached the point of designating this group by the new term three. Suppose now a cloth be spread over this plurality composed of things of the same kind; what ideas of them are left behind in the minds of the observers? Evidently quite different ones. In the case of the child ideas of taste will predominate, in the case of the painter ideas of the colours and how they shade off, of the shadows and outlines, etc. , so that the two of them, if Biermann's definition were correct, would have quite different numbers, that is, different ideas of the group of elements designated by the word three. But I suppose Biermann would say that all this is psychology, that these considerations have nothing to do with arithmetic. Just so! But how did we get here? Via the term 'idea' which Biermann uses and which simply does belong to psychology. Any attempt to exclude psychological considerations from mathematics has my full approval. But let us do the job properly. Away with the word idea! To be sure, we must then dispense with such interesting psychological propositions as 'Through number we possess the idea of a plurality or set of similar things'. 1 The full implications of this proposition can only be realized when it is applied to examples; thus we possess the idea of a heap of sand through the idea of a heap of sand; we possess the idea of the constellation of Orion through the idea of the constellation of Orion; we possess the idea of the Laocoon group through
the idea of the Laocoon group. But let us not make an issue of whether arithmetic is the proper place for this psychological truth. Let us rather rejoice over truths wherever we find them. And let us look to see whether Biermann's book does not yield yet more meat for psychology. Sure enough! We learn something about the composition of ideas:
'Two numbers formed from an indeterminate basic element e or the abstract unit 1 . . . ', etc. 2
If only we knew what an 'indeterminate basic element' is! Perhaps we shall glean it by comparing this with a later passage:
?
We construct a number containing all the elements oftwo numbers a and bformedfrom the same basic element. '3
1 Cf. Biermaqn, p. I.
2 Ibid. 3 Ibid.
? ? On the Concept ofNumber 77
Thus we have, say, two heaps of peas,* and we make a single heap with them. What is the basic element the heap is made up of? A pea? A single pea? Which pea? An indeterminate pea. I confess that I have not yet seen an indeterminate pea, but I can well imagine that Biermann has the concept pea in mind; or would it be an idea of a pea? I trust I do Biermann no injustice if I assume that he has never broached this question. Though we cannot, to be sure, say that a heap of peas is composed of an indeterminate pea, or of the concept pea, or of an idea of a pea, we can say that it is composed of peas, i. e. of objects falling under the concept pea. Therefore, in view of the fact that Biermann's terminology is somewhat imprecise, we may assume that his phrase 'number formed from an indeterminate basic clement' is intended to mean: 'number whose components fall under one concept' and that is only another term for something that has already given us food for thought-elements of the same kind; evidently, Biermann is fond of presenting us with the same thing in different guises so that we do not find it too easy to achieve clarity. But we are forgetting, and perhaps Biermann is, too, that it is not the heap of peas that is a number, but the idea of the heap. The number, i. e. the idea, is not, however, made up of peas. What is an element of a number? After what has just been said everyone must think: an element of the group whose idea is the number. Let us consider Biermann's formulation once more: 'We construct a number containing all the elements of two numbers a and b formed from the same basic element. ' That is to say: we form the idea of a group, which idea contains all the elements of two numbers. In our example we had to take the peas as the elements of the numbers. Thus we formed a leguminous idea which may possibly be of some use as a nitrogenous food. But I think it would be better if we left aside the question of forming this idea: the feat is beyond us. By element of a number should we perhaps understand: idea of a component of the group whose idea is the number? The elements of the number in our example would then not be the peas, but the ideas of the peas. True, Biermann had previously spoken only of elements of the group whose idea is the number; but that may well have slipped his mind, which would be only human after all; or possibly he wants to catch a superficial critic off his
? No doubt Biermann is thinking 'How vulgar to talk about peas, apples und the like when we are supposedly dealing with scientific issues. It sounds for all the world like someone talking to little children! ' Well, I am indeed trying to show that we are only left with puerilities once we go to the core of his argument and strip off the semblance of learning he is able to create by using terms like 'element', 'group', 'idea', 'assign', and so on. There is no hetter place to hide the most childish confusions than in the most learned- NilUnding terms. That is why the cruder the examples we use to throw light on these terms, the more pitifully apparent it becomes how utterly obscure they are. Biermann's account is able to retain its aura of learning only hccause he forbears applying it to particular cases. It would be out of place to he too serious here.
? ? 78 On the Concept ofNumber
guard, for such a reader might well be beguiled by the use of the same word 'element' into thinking that we are still dealing here with elements of the group. We may surely allow that the idea of a heap of peas contains ideas of peas. At all events it will contain, in addition, the idea of a certain proximity which we indicate, of course, by the word 'heap'. True, Biermann nowhere says that the idea we are to form should contain only the elements of the numbers a and b; if we took him at his word, we could go on adding as many ideas as we liked not containing elements of the numbers a and b; but that would hardly measure up to his view. But is there any need at all for a new heap or an idea of it? * We simply form an idea containing the ideas of the two original heaps, that is, the idea of a group of the two heaps whose ideas are the numbers a and b. Unfortunately, this idea would also contain an element that does not occur in a and b-an idea of the spatial relationship of the two heaps-and here, of course, we may choose any one of many such relationships. Or are the two numbers a and b, which are ideas after all, supposed to merge into one idea, in keeping with the psychological principle of the fusing of likes, much as, say, the ideas [images] of two similar faces merge together? Unfortunately, the result would turn out somewhat blurred. ** Yet I hear Biermann crying out in despair 'This eternal psychologizing! If only I had not used the word "idea"! I did not mean it as seriously as all that! ' But Biermann also says things like 'we become aware', 'we arrive at the concept of a set', 'we abstract'. And this is already quite enough to lead us off into psychological irrelevancies. 'We' is not an object of mathematics at all, just as little as our ideas are. Truths in mathematics are eternal and not dependent on whether we are alive or dead or become aware of them.
I can well imagine that Biermann has used the word 'idea' in much the same way as we use 'Esteemed Sir' and 'Honoured Sir', to make what he says sound weightier and more impressive without actually changing its sense. There is more than one circle of society in which people do not feel fully clothed if they lack a title. And likewise it is possible that a certain modesty-which, incidentally, redounds to his credit-has prevented Biermann from taking the peas with all their adventitious little wrinkles and introducing them into mathematics without dressing them up. For my part, I am more for things in their raw and natural state and prefer the following to Biermann's formulation: Number is a plurality composed of things of the same kind, or numbers are groups of elements. I concede that this sounds somewhat less impressive; the first formulation, in particular, has a touch of
*That [i. e. that we need an idea of a new heap] would even be wrong, for the ideas of proximity which are elements of the ideas ofthe original heaps of peas would be quite missing from the idea of a new heap of peas. Yet the idea we are to form is supposed to contain all the elements of the numbers a and b.
** As we know, this process may be emulated by photographic means.
? On the Concept ofNumber 79
tautology about it and might make you suspect the hand of a girl at a finishing school; but that is the very reason I prefer it. And if the word 'idea' is really intended to serve as nothing more than an ornament, I shall be able to adopt my formulation without departing in essentials from what Biermann thinks. Granted, the statement that the Laocoon group may bear the title one as well as the title four now sadly reduces to nothing: But this is offset by the fact that from this point on mathematics really has something to get its teeth into.
As is well known, this discipline is concerned with numbers. Now heaps of peas, of sand, and other heaps are numbers; herds of sheep, of cows and of other animals are numbers, too. Consequently, all these heaps and herds are objects of mathematics. Indeed, we may perhaps say that mathematics is concerned with all possible things; a window is one, a house with many windows is one, the country in which there are many houses is one. * Now if every such one is a number, then the window is a number, etc. No doubt Biermann will say 'Just so! Mathematics is concerned with all possible things in respect of what is number about them. ' The striking thing, however, is that herds of sheep are seldom mentioned in this discipline. I helieve they do not even appear in Biermann's book at all. Does my memory deceive me, or have I really only read about herds of sheep-if I have read about them at all in mathematical books-in the sets of examples given to illustrate the application of mathematical propositions? But I am probably putting words into Biermann's mouth that he has never thought of uttering. Number is not something attaching to the herds; the herds themselves just as they are, skin and bone and dirt, are numbers. It looks as though I have got confused here with J. S. Mill's view according to which a number is a property of an aggregate-that is, the way an aggregate is put together. I must confess that there were times, as I was struggling through Biermann's obscurities, when this view seemed to me full of insight. But it appears that light is now beginning to penetrate these regions of darkness. Let us take
Biermann's formula: 'Two numbers formed from an indeterminate basic clement or the abstract unit 1 are equal, when to each element of the one there belongs an element of the other'1 and apply it e. g. to herds of sheep. llow clear everything now becomes! Two herds of sheep are equal when to each sheep of one herd there belongs a sheep of the other. Admittedly, when a sheep A belongs to a sheep B is something we are not told. Let us turn to the difficult question of whether it is conceivable that a herd of sheep is equal
*'Omnia una sunt', a Latinist would say, if not deterred by his feeling for the language, which would here be confirmed by the nature of things as well. Apparently, Biermann has not yet got round to asking himself what underlies this phenomenon of language; for he can say 'units' as though it were the same thing.
1 Hiermann, p. I.
? ? 80 On the Concept ofNumber
to a constellation of stars. The one thing we do at least know is that both are numbers. The only question we still have to settle is whether they are formed from the same basic element. * I believe we have already worked out what Biermann means: when he says that a number is formed from an indeterminate basic element, he means that a number is formed from objects falling under one concept, and the 'indeterminate basic element' then corresponds to the concept. In this case we can point to such a concept: heavy, inert body. Both the sheep and the stars fall under this concept. There can, therefore, presumably be no doubt that the herd of sheep and the constellation are formed from 'the same indeterminate basic element'. Now it is surely conceivable that every star in the constellation belongs to a sheep in the herd, and so it is also conceivable that a herd of sheep should be equal to a constellation. We must not say here that they may of course be equal in respect of the number of solid inert bodies out of which they are made up; for the herd of sheep is itself one of the numbers and the constellation itself is the other. We have already established that according to Biermann number is not a property in respect of which the herd is interchangeable with the constellation. True, we may say, this beetle and the bark of this tree
are equal' so far as their colour is concerned; but here neither the beetle nor the bark are a colour; moreover we do not have two colours, but one and the same. So according to Biermann this case is quite different from that of the numbers; for even if the phrase 'idea of a group' were to mean something quite different from the group itself, it still would not mean a property of the group. And even if, quite contrary to normal usage, Biermann were to use the term 'idea' in such a way that the idea of a group was a property of it, the proposition 'a number is the idea of a group' would amount to the same as 'a number is the number of a group': that is to say, a number is that property of a group which we call idea or number.
We still need to emphasize that according to Biermann's definition the word 'equal' does not mean complete coincidence: a number may be equal to another without being the same; a herd of sheep may be equal to a constellation without being the constellation itself. The question now arises what the number words mean: the most obvious answer would be that the number word 'two', for example, designates one (and only one) number, so that we may say: two is a number, three is a number, and so on. Two and three would be related to the concept of number in the same way as, say, Archimedes, Euclid and Diophantus are related to the concept of mathematician. If we say this, however, we should certainly get into difficulties. Let us again imagine a group consisting of a lion standing and a
*It is not clear from Biermann's wording whether or not this condition must be fulfilled; what we have is only: 'from a', and not 'from the same'. To be on the safe side, we will assume that it must.
1 Here English idiom requires 'alike' rather than 'equal', but in German the same word-'g/eich'-does duty for both (trans. ).
? On the Concept ofNumber 81
lioness lying on the ground. This group is a number. Let us assign to it the number word 'two' as its proper name. Then in future we shall mean our group of lions when we say 'two'. Let us now think of the Goethe-Schiller memorial in Weimar. We surely cannot give the Goethe-Schiller group the same name as the group of lions. To do this could lead to some singularly unfortunate mistakes in identity! We do not seem able to manage with the number words alone. But Biermann has a simple way of getting round this difficulty; as we call Homer, Virgil and Goethe poets, so we might with equal justification call both the Goethe-Schiller group and the group oflions lwo. If we call something two, that shows we want to allocate it to a certain species: we want to say that it has a certain property or properties. In just 1he same way by calling someone a poet I acknowledge he has certain properties characteristic of being a poet,* or by calling a thing blue I attribute a certain property to it or assign it to some species or other. I-ikewise by calling a group three I would be saying that it has a certain property. As we call the properties green, blue, yellow colours, so we could rail two and three numbers. But wait! Here we are again on the same false ! rack as before. We cannot repeat often enough: number is a group or plurality composed of things of the same kind; therefore numbers are the subjects of the properties expressed by the number words. We are no more JUstified in asserting that two is a number than we are in counting green as a u1loured object instead of a colour. Thus we are now able to say: two, three, four, etc. , are properties of groups with constituents of the same kind, which t-:roups or sets are called numbers. A somewhat more elevated way of putting the same thing is to say: two, three, four, etc. , are properties of ideas or groups with constituents of the same kind, and these ideas are called numbers. The most remarkable thing for the layman, for me myself and perhaps even for Biermann in all this, and the most astonishing, is that the number words do not designate numbers but properties of numbers. The equation 2 = 1 + 1 is generally considered to be true. Biermann's definition of a = b is not applicable here, since it relates to numbers, i. e. groups, whereas for us the sign '2' on the left does not mean a number here but a property of a number. Let us see whether we fare any better with the plus Nign. Biermann says: 'We construct a number containing all the elements of 1wo numbers a and b formed from the same basic element' . . . 'The resultant number we designate by a+ b . . . '1
We do not learn from this what 2 + 3 and 2 + 1 mean, for 1, 2 and 3 are not numbers at all. But perhaps we learn something else which merits our nttention. As we have seen, the constellation Orion is a number, the belt in this constellation is likewise a number and, moreover, a different one. If we understand Biermann correctly, both numbers are formed from 'the same
* We need not ask here whether one or more than one property is involved in the word 'two'.
1 Hiermann, p. I.
? ? ? 82 On the Concept ofNumber
basic element', for the elements of the belt are also the elements of Orion. Let us then form a number containing all the elements of Orion and of its belt. What could be simpler! We do not even have to bother constructing this number: it is already there. The constellation of Orion is itself this number. So let us designate Orion by a and its belt by b; then we have a+ b = a, whilst at the same time we have remained happily in accord with Biermann's definitions of the plus sign and the equals sign. But that's enough of pretending to be more stupid than we are! After all, Biermann is writing for people already familiar with these matters who will read the right thing into his words even when they are false or devoid of sense! * After all, the words are there so that the reader may understand what is meant, despite them. We have to bear in mind the condition that no element may be common to both numbers. It follows that no number may ever be added to itself: presumably we should have to say that a + a is devoid of sense; yet a little further on Biermann's text reads: a + a + a . . . (b-times). Let us see how that comes about. Biermann says: 'If we substitute the number a for each of the elements of a number b we obtain the sum a + a + a . . . (b-times), which we designate in short by a x b or ab. '1 The sense of this definition is not easy to grasp; let us take an example to clarify what is at stake: let b be the Goethe-Schiller group in the square in front of the theatre in Weimar, which according to Biermann is indubitably a number. Let a be the constel- lation of Orion. Now let us first of all substitute Orion for the figure of Goethe. It is no easy thing that Biermann is asking of us; his arithmetic would seem to be designed for gods. For him personally all this is child's play; we shall witness still greater feats. F o r the figure o f Schiller let us now substitute-well, what? Why, Orion, too! ** And what do we get then?
Orion + Orion + Orion (Goethe-Schiller-group times).
Who would have thought it! A little later Biermann says: 'In a+ a+ . . . +a (b-times) we may pick one out of each group of a elements and their combination yields b'. If Biermann had not said it, I would not have believed
*It would have been more to the purpose, in my view, if ? 1 had been left unwritten.
