This
difficulty
lies in the expressions we use.
Gottlob-Frege-Posthumous-Writings
The sign '(a - b)' at the same time acquires a meaning by such a replacement.
If we replace 'a' by '16' and 'b' by '2', this meaning is the number 14.
One can now go on to ask whether the sentence
'(16- 2) is a multiple of 7'
has not only a sense, but also a meaning.
Let us take for comparison the sentence 'Etna is higher than Vesuvius'.
With this sentence we associate a sense, a thought; we understand it, we can translate it into other languages. In this sentence we have the proper name
? 232 Logic in Mathematics
'Etna', which makes a contribution to the sense of the whole sentence, to the thought. This contribution is a part of the thought, it is the sense of the word 'Etna'. But we are not making a statement about this sense, but about a mountain, which is not part of the thought. One who holds an idealist theory of knowledge will no doubt say 'That is wrong. Etna is only an idea in your mind. ' Anyone who utters the sentence 'Etna is higher than Vesuvius' understands it in the sense that it is meant to assert something about an object that is quite independent of the speaker. Now the idealist may say that it is wrong to hold that the name 'Etna' designates something. If that were so, the speaker, whilst believing himself to be operating in the realm of truth, would be lost in the realm of myth and fiction. But the idealist is not justified in turning the thought round like this, as if the speaker meant to use the name 'Etna' to designate one of his ideas, and communicate something about this. Either the speaker designates what he means to designate by the word 'Etna' or he designates nothing at all by this name, and it is meaningless.
It is therefore essential first, that the name 'Etna' should have a sense, for otherwise the sentence would not have a sense, would not express a thought, and secondly, that the name 'Etna' should have a meaning, for otherwise we should be lost in fiction. The latter of course is essential only if we wish to operate in the realm of science. In the case of fiction it does not matter whether the people who occur in it are, as we should say, historical personages. Or speaking more precisely, 'whether the personal proper names occurring in fiction are meaningful'.
Now if we were concerned only with the sense of 'Etna is higher than Vesuvius', we should have no reason for requiring that the name 'Etna' should have a meaning as well; for in order that the sentence have a sense, it is only necessary for the name 'Etna' to have a sense; the meaning contributes nothing to the thought expressed. If therefore we are concerned that the name 'Etna' should designate something, we shall also be concerned with the meaning of the sentence as a whole. That the name should designate something matters to us if and only if we are concerned with truth in the scientific sense. So our sentence will have a meaning when and only when the thought expressed in it is true or false. The meaning of a sentence must be something which remains the same, if one of the parts is replaced by something having the same meaning. We return now to the sentence '(16- 2)isamultipleof7'.
The sign '(16 - 2)' is a proper name of a number. '(17 - 3)' designates the same number, but '(17- 3)' does not have the same sense as '(16- 2)', Thus the sense of the sentence '(17 - 3) is a multiple of 7' is also different from the sense of the sentence '(16 - 2) is a multiple of 7'; and likewise the sense of the syntence '16 is congruent to 2 modulo 7' is different from the sense of the sentence '17 is congruent to 3 modulo 7'. But the sentence
'( 17 - 3) is a multiple of 7' must have the same meaning as that of the sen?
? Logic in Mathematics 233
tence '(16- 2) is a multiple of 7'. Now what is not altered by replacing the sign '(16 - 2)' by the sign '(17 - 3)', whose meaning is the same, is what I call the truth-value. These sentences are either both true or both false. In our example they are both true, but it is easy to construct a different example in which they are both false. We only need to take the number 3 in place of the number 7.
We say accordingly that sentences have the same meaning if they are both true, or if they are both false. On the other hand, they have a different meaning, if one is true and the other false. If a sentence is true, I say its meaning is the True. If a sentence is false, I say its meaning is the False. If a sentence is neither true or false, it has no meaning. Nevertheless it may still have a sense, and in such a case I say: it belongs to the realm of fiction.
For brevity I have called a sentence true or false though it would certainly be more correct to say that the thought expressed in the sentence is true or false.
But this, surely, strikes a discordant note. If I say 'the thought that (16 - 2) is a multiple of 7 is true', I am treating true as a property of the thought, whereas it has emerged that the thought is the sense and the True the meaning of the sentence. Of course treating truth as a property of sen- tences or of thoughts is in accordance with linguistic usage. If we say 'The sentence "3 > 2" is true', then the form of words is such that we are saying something about a sentence: we are saying that it has a certain property, a property we designate by the word 'true'. And if we say 'The thought that 3 > 2 is true' the corresponding thing holds of the thought. Still the predicate true is quite different from other predicates such as green, salt, rational, for what we mean by the sentence 'The thought that 3 > 2 is true' can be more simply said by the sentence '3 is greater than 2'. Thus we do not need the word 'true' at all to say this. And we can see that really nothing at all is added to the sense by this predicate. In order to put something forward as
true, we do not need a special predicate: we only need the assertoric force with which the sentence is uttered.
When we utter an assertoric sentence, we do not always utter it with assertoric force. An actor on the stage and poet reading from his works will both give frequent utterance to assertoric sentences, but the circumstances show that their utterances do not have assertoric force. They only act as if they were making assertions. In our definition, too,
(a- b) is a multiple of 7 ) and
la is an integer = a is congruent to b modulo 7 and
b is an integer
we do not utter the separate parts '(a- b) is a multiple of 7', 'a is an integer',
? 234 Logic in Mathematics
'b is an integer', with assertoric force nor do we do so in a case where the
letters 'a' and 'b' are replaced by proper names. We may even say (16- 3)isamultipleof7) )
and
l16 ~s an ;~ger = 16 is congruent to 3 modulo 7
3 ts an mteger
although some of the clauses are false; for we mean only to put forward the right-hand side of the equation as having the same sense as the left-hand side, without making a judgement about the truth of the clauses.
If a man says something with assertoric force which he knows to be false, then he is lying. This is not so with an actor on the stage, when he says something false. He is not lying, because assertoric force is lacking. And if an actor on the stage says 'it is true that 3 is greater than 2' he is no more making an assertion than if he says '3 is greater then 2'. Whether an assertion is being made, therefore, has nothing at all to do with the word 'true': it is solely a matter of the assertoric force with which the sentence is uttered. So to say of a sentence, or thought, that it is true is really quite different from saying of sea water, for example, that it is salt. In the latter case we add something essential by the predicate, in the former we do not.
Showing, as it does, that truth is not a property of sentences or thoughts, as language might lead one to suppose, this consideration confirms that a thought is related to its truth value as the sense of a sign is to its meaning.
Wehaveseenthat'(a- b)'and'(a- b)isamultipleof7'areakinto one another in that both acquire a sense and a meaning as a result of our putting meaningful proper names for 'a' and 'b'. What makes a difference between them is that the sense which '(a - b) acquires in this way is only part of a thought, whereas the sense which '(a - b) is a multiple of 7' ac? quires in this way is a thought. If we begin by just replacing 'b' by the proper name '2', we obtain '(a - 2)', '(a - 2) is a multiple of 7'. What is present in: the second combination of signs over and above the letter 'a' is the sign of a concept. And we may construe the sentence '(16- 2) is a multiple of 7' aa' consisting of the proper name '16' together with this concept-sign, so that in j this sentence we are asserting the concept in question of the number 16,. What we have is the subsumption of an object under a concept.
We can, in an analogous way, regard what is present in 'a - 2', apart from the letter 'a', as a sign. On this view, then,' 1 6 - 2' will be composed of: the proper name' 16' and this sign, which like the concept-sign above, is in need of supplementation. What it designates must be in need ot: supplementation, just as the concept is. We call it a function. The concept? ' sign, when supplemented by a proper name, yields a proper name. In our, case the function-sign, when supplemented by the proper names '2', '3', '4'1 , yields respectively the proper names ' 2 - 2'. '3 - 2'. '4 - 2'.
? The objects
is the False,
3- 2isamultipleof7*
16- 2 is a multiple of 7*
2 - 3 - 4 -
2 is the value of our function for the argument 2, 2 is the value of our function for the argument 3, 2 is the value of our function for the argument 4.
Logic in Mathematics 235
2- 2,3- 2,4- 2.
of which these proper names are the signs, we call the values of our function. Thus
But what we obtain from the sentence
'(a - 2) is a multiple of 7'
by replacing 'a' by a proper name, is also to be understood as a proper name; for it designates a truth value and such an entity is to be regarded as an object. Thus
the True. So there is a far-reaching agreement between the cases in which we speak of a function and the cases in which we speak of a concept; and it seems appropriate to understand a concept as a function-namely, a function whose value is always a truth value. So if the concept above is understood as a function, then the False is the value of this function for the argument 3, and the True is the value of the function for the argument 16. What we should otherwise say occurred as a logical subject is here presented as an argument.
It is not possible to give a definition of what a function is, because we have here to do with something simple and unanalysable. It is only possible to hint at what is meant and to make it clearer by relating it to what is known. Instead of a definition we must provide illustrations; here of course we must count on a meeting of minds.
There often seems to be unclarity about what a function is. In this connection the word 'variable' is often used. This makes it look at first as if there were two kinds of number, constant or ordinary numbers and variable numbers. The former, it seems, are designated by the familiar signs for numbers, the latter by the letters 'x', 'y', 'z'. But this cannot be reconciled with the way we proceed in Analysis. When we have the letter 'x' combined with other signs as in
'x- 2'
*It is understood that these sentences are here uttered without assertoric force.
? 236 Logic in Mathematics
Analysis requires that it be possible to substitute different number-signs for
this 'x' as in
'3- 2','4- 2','5- 2',etc.
Buthere we cannot properly speak of anything altering; for if we say that something alters, the thing which alters must be recognizable as the same throughout the alteration. If a monarch grows older, he alters. But we can only say this because he can be recognized as the same in spite of the alteration. When, on the other hand, a monarch dies and his successor mounts the throne, we cannot say that the former has been transformed into the latter; for the new monarch is just not the same as the old one. Putting '3', '4', '5' in turn for 'x' in ' x - 2' is comparable with this. We do not have here the same thing assuming different properties in the course of time: we have quite different numbers. Now if the letter 'x' designated a variable number, we should have to be able to recognize it again as the same number even though its properties were different. But 4 is not the same number as 3. So there is nothing at all that we could designate by the name 'x'. If it means 3, it does not mean 4, and if it means 4, it does not mean 3. In arithmetic and Analysis letters serve to confer generality of content on sentences. This is no less true when it is concealed by the fact that the greater part of the proof is set out in words. In such a case we must take everything into consideration, and not just what goes on in the arithmetical formulae. We say, for instance, 'Let a designate such-and-such and b such-and-such' and take this to be the point at which we begin our inquiry. But what in fact we have here are antecedents
'if a is such-and-such', 'if b is such-and-such',
and they have to be introduced as such or attached in thought to each of the sentences which follow, and these letters, whose role is merely an indicating one, make the whole general. It is only when, as we say, an unknown is designated by 'x' that we have a somewhat different case. E. g. let the question be to solve the equation
'x2 - 4 = 0'
We obtain the solutions 2 or -2. But even here we may present the equation together with its solution in the form of a general sentence: 'If x2 - 4 = 0, then x = 2 or x = -2'. We may take this opportunity to point out that the sign '? yl4? is to be rejected out of hand. Here people have not taken sufficient care in using language as a guide. The proper place for the word 'or' is between sentences: 'x is equal to 2 or x is equal to -2'. But we contract the two sentences into 'x is equal to plus 2 or minus 2' and accordingly write 'x = ? /4'; however'? /4' doesn't designate anything at all; it isn't a meaningful sign. What one can say is
'2isequalto+y'4or2isequalto- yf4? ,
? Logic in Mathematics 237 where the assertoric force extends over the whole sentence, the two clauses
being uttered without assertoric force. Equally one can say '-2 is equal to +y'4or -2 is equal to -/4'
but '2 is equal to ? /4' has no sense.
At this point we may go into the concept of the square root of 4. If we
think of '2? 2 = 4' as resulting from ? ~? ~ = 4' by replacing the letter? ~? by the numeral '2', then we are seeing '2 ? 2 = 4' as composed of the name '2' and a concept-sign, which as such is in need of supplementation, and so we can read '2? 2 = 4' as '2 is a square root of 4'. We can likewise read '(-2) ? (-2) = 4' as '(-2) is a square root of 4'. But we must not read the equation '2 = /4' as '2 is a square root of 4'. For we cannot allow the sign '/4' to be equivocal. It is absolutely ruled out that a sign be equivocal or ambiguous. if the sign '/4' were equivocal, we should not be able to say whether the sentence '2 = /4' were true, and just on this account this combination of signs could not properly be called a sentence at all, because it would be indeterminate which thought it expresses. Signs must be so defined that it is determinate what '/4' means, whether it is the number 2 or some other number. We have come to see that the equals sign is a sign for identity. And this is how it has to be understood in '2 = J4' too. 'y4' means an object and '2' means an object. We may adopt the reading '2 is the positive square root of 4'. And so the 'is' is to be understood here as a sign for identity, not as a mere copula.
'2 = /4'may not be read as '2 is a square root of 4'; for the 'is' here would be the copula. If I judge '2 is a square root of 4', I am subsuming the object 2 under a concept. This is the case we have whenever the grammatical subject is a proper name, with the predicate consisting of 'is' together with a substantive accompanied by the indefinite article. In such a case the 'is' is always the copula and the substantive a nomen appellativum. And then an object is being subsumed under a concept. Identity is something quite different. And yet people sometimes write down an equals sign when what we have is a case of subsumption. The sign '/4' is not incomplete in any way, but has the stamp of a proper name. So it is absolutely impossible for it to designate a concept and it cannot be rendered verbally by a nomen appellativum with or without an indefinite article. When what stands to the left of an equality sign is a proper name, then what stands to the right must be a proper name as well, or become such once the indicating letters in it are replaced by meaningful signs.
However, let us return from this digression to the matter in hand. Where they do not stand for an unknown, letters in arithmetic have the role of conferring generality of content on sentences, not of designating a variable number; for there are no variable numbers. Every alteration takes place in time. The laws of number, however, are timeless and eternal. Time does not enter into arithmetic or Analysis. Time can come in only when it is a matter
of applying arithmetic. The number 3 has always been u prime number und
? 238
Logic in Mathematics
will always remain such. How could a change be possible here? It feels incongruous to speak of a variable number and so people prefer to say 'variable magnitude', as if that was a great improvement. Of course an iron rod grows longer when it is heated, and shorter when it cools down-it changes in time. If we measure its length in millimeters, we get now this number, now that. If we now say 'the number which gives the length of this rod at time t', we have an expression containing an indefinitely indicating letter 't'. If instead of't' we put the proper name of a time instant, it becomes the proper name of a number. This is on all fours with what we have in the
case of the expression ' x - 2'. This likewise becomes a proper name if we put the name of a number for 'x'. In both cases we have a function which may yield different values when saturated by different arguments. Iron rods and time, when you come down to it, are of no account to arithmetic; for this is concerned neither with pebbles, nor with peppermints, nor with railway trains, nor with rows of books, nor with iron rods, nor with time instants. These are things which may come into the applications of mathematics, but they do not have any role in constructing mathematics as a system.
In the light of all this, we can see that there is no place in arithmetic either for variable numbers or for variable magnitudes. 'Magnitude' is either a subterfuge for number, in which case variable magnitudes no more exist than variable numbers, or 'magnitude' is understood in such a way that we can properly speak of variable magnitudes; but in that case they do not belong in arithmetic.
If the letters 'x' and 'y' designated different variables, we should have to be able to say how these are distinguished; but this no one can do. We have only to keep before our mind the fact that we are concerned with pure arithmetic, not with its applications. We may perhaps seek a way out by taking the view that the letters 'x' and 'y' are not signs for what is variable, but are themselves the things that vary. But if we do this, we run foul of the established use of our signs. In the case of the equals sign, for example, it is always presupposed that the simple or complex sign which occurs on the left is either a meaningful proper name, or will become such once the indicating letters in it are replaced by designating signs.
Hence it is impossible to explain what a functi()n is by referring to what is called a variable. It is rather the case that when we seek to make clear to ourselves what a variable is, we come back again and again to what we have called a function, thus recognizing that variables are not a proper part ofthe subject-matter ofarithmetic.
(We have seen that a concept can be construed as a special case of a function. We have made it a requirement that concepts have sharp boundaries. There is an analogous requirement for the more general case of a function. )
A great deal of unclarity still prevails over what a function is. In particular, it is easy to confuse a function with the value of a function, as if one wrote
fx=f,
? Logic in Mathematics 239
using the letter on one side to indicate a function, and on the other side to indicate the value of the function. There is admittedly a difficulty which accounts for its being so extremely hard to grasp the true nature of a function.
This difficulty lies in the expressions we use. We say 'the function' and 'the concept', expressions which we can hardly avoid but which are inappropriate. The definite article gives these expressions the form of proper names in the logical sense, as if they were meant to designate objects, when this is precisely what they are not meant to do. The very nature of concepts and functions-their unsaturatedness-is thus concealed. Language forces an inappropriate expression on us. This is a situation which, unfortunately, can hardly be avoided, but we can render it harmless by always bearing the inappropriateness of language in mind. At the same time we shall also avoid confusing the value of a function with a function.
Now we sometimes speak of a function when what we have in mind are cases like (1 + x )2. Here' 1 + x' occurs in the argument-place of the squared function. But '1 + x' does not designate a function at all, but only indicates thevalueofafunctionindefinitely. Ifin'(1 +x)2', weput,say,'3'inplaceof 'x', then we get (1 + 3)2, and here the value of the function 1 + ~for the argument 3 is the argument of the squared function. But this argument is an object, a number. Here a function is compounded out of two functions by taking the value of the first function for a certain argument as an argument of the second function. In this connection we must persist in emphasizing the fundamental difference of object and function. No function-name can stand in a place where an object-name, a proper name, stands and conversely a proper name cannot stand where a function-name stands.
Even where we have a function whose value is the same for every argument, this value must be distinguished from the function. So the function
1 + ~- ~
is different from the number 1 itself. We must not say 'We have 1 + ~- ~ = 1 and the equals sign is the identity sign, and so the function 1 +~-~just is the number 1', for when we say 'the function 1 +~-~. the letter? ~? is not part of the function-sign; for the proper name '1 + 3 - 3' is composed of the function-name and the proper name '3', and the letter? ~? does not occur in it at all. In the sentence '1 + a - a = 1', however, the letter a has the role of conferring generality of content on the sentence, whereas when I say 'the function 1 + ~ - ~? the role of the letter '~? is to enable us to recognize the places where the supplementing proper name is to be put. In order to form the differential quotient of a function for the argument 3 we subtract the value of the function for the argument 3 from the value of the function for the argument (3 + k) and divide the difference by k, and so on. For the
function I + ~- ~this is represented in the formula
11 + (3 + k ) - (3 + k ) l - 1 1 + 3 - 31 k
? 240 Logic in Mathematics
But in the case of the proper name' 1' there are simply no places in which we can first put '3 + k' and then '3'. The prescription can only be carried out where we have a function.
It is a mistake to define the number one by saying 'One is a single thing', because on account of the indefinite article 'a single thing' has to be understood as a concept-word. But in that case the word 'is' is the copula and belongs with the predicate. We then have the object the number one being subsumed under a concept. But that is no definition. A definition always stipulates that a new sign or word is to mean the same as a complex sign we already understand. When the word 'is' is used in a definition, it is to be understood as a sign for identity, not as a copula. If now a proper name occurs to the left of the identity sign, such a sign must occur on the right too; 'a single thing' is, however, a nomen appellativum.
As functions of one argument are fundamentally different from objects, so functions of two arguments are fundamentally different from functions of
one argument.
By supplementing the sign for a function of one argument with a proper
name, we obtain a proper name. So e. g. from the function-sign? ~- 2' and thepropername'3'weobtainthenewpropername'3- 2'. Theletter? ~? in ? ~- 2' only serves to keep open the place for the supplementing argu- ment-sign. In the same way, by supplementing the concept-sign? ~> 0' with '1' we obtain '1 > 0' and this is a name of the True.
A function of two arguments is doubly in need of supplementation. In ? ~- r; we have a sign for a function of two arguments. The letters? ~? and 'C' are meant to keep open the places for the argument-signs. The difference between the letters? ~? and 'C' is to show that a different argument-sign may
be put in the two places. By putting the proper name '2' in the C-argument place we obtain ? ~ - 2', which is a function-sign for a function of one argument. In the same way the relation-sign? ~> C' yields the concept-sign '~ > 0'. So as a result of being partly saturated, functions of two arguments yield functions of one argument and relations yield concepts. A further way in which this can happen is through abolishing the difference between the argument-places. If I write? ~- ~? ,I indicate, by using the letter? ~? in both places, that the same proper name is to be put in both places, and so what I have is the name of a function of only one argument. When I call this the 'name of a function', this is to be taken cum grano salis. The proper name which we obtain by supplementing this function with a proper name, e. g. '3 - 3', does not contain the letter? ~, although it contains the function-name in question. This ? ~? is therefore not a constituent of the function-name but only enables us to recognize how the function-sign is combined with the proper name supplementing it. This? ~ gives us a pointer for how to use the function-name. We can similarly form a concept-sign from a relation-sign by abolishing the difference between the argument-places. Thus from the relation-sign? ~> C' we obtain the concept-sign? ~> ~? .
? Logic in Mathematics
241
We have seen that the value of one function can occur as the argument of a second function. We may call the former the inner function, the latter the outer function. So from the name of a function of two arguments and a concept-sign we can obtain a relation-sign in which the concept is the outer function. E. g. let? ~- C' be the sign for the function of two arguments and? ~ is a multiple of 7' the concept-sign. Then '(~- ()is a multiple of 7' will be the new relation-sign.
A concept must have sharp boundaries; i. e. it must hold of every object either that it falls under a concept or does not. We may not have a case in which this is indeterminate. From this there follows something correspond- ing for the case of a relation; for of course by partially saturating a relation we obtain a concept. This must have sharp boundaries. And indeed any concept obtained by partly saturating a relation must have sharp boundaries. This means in other words: Every object must either stand or not stand in the relation to every object. We must exclude a third possibility. If we apply this to the relation (~ - 0 is a multiple of 7, it follows that a meaningful proper name must always result from the complex sign ? ~- C' by replacing the letters '~, and 'C' by meaningful proper names, and so not only when signs for numbers are inserted. Therefore the minus sign has to be defined in such a way that whatever meaningful proper names are put to the right and left of it, the whole combination of signs always has a meaning. So we arrive at the general requirements:
Every sign for a function of one argument has to be defined in such a way that the result always has a meaning whatever meaningful argument-sign is taken to supplement it.
Every sign for a function of two arguments has to be defined in such a way that the result has a meaning whatever meaningful argument-signs are used to supplement it.
E. g. we could stipulate that the value of the functions ~- Cis always to be the False, if one of the two arguments is not a number, whatever the other argument may be. Of course, we should then also have to know what a number is.
(We can stipulate likewise that the value of the function ~ > Cis to be the False, if one of the two arguments is not a real number, whatever the other argument may be. )
But it is precisely on this issue that views have changed. Originally the numbers recognized were the positive integers, then fractions were added, then negative numbers, irrational numbers, and complex numbers. So in the course of time wider and wider concepts came to be associated with the word 'number'. Bound up with this was the fact that the addition sign changed its meaning too. And the same happened with other arithmetical signs. Needless to say, this is a process which logic must condemn and which is all the more dangerous, the less one is aware of the shift taking place. The progress of the history of the sciences runs counter to the demands of logic. We must always distinguish between history and system.
? 242 Logic in Mathematics
In history we have development; a system is static. Systems can be constructed. But what is once standing must remain, or else the whole system must be dismantled in order that a new one may be constructed. Science only comes to fruition in a system. We shall never be able to do without systems. Only through a system can we achieve complete clarity
and order. No science is in such command of its subject-matter as mathematics and can work it up into such a perspicuous form; but perhaps also no science can be so enveloped in obscurity as mathematics, if it fails to construct a system.
As a science develops a certain system may prove no longer to be adequate, not because parts of it are recognized to be false but because we wish, quite rightly, to assemble a large mass of detail under a more comprehensive point of view in order to obtain greater command of the material and a simpler way of formulating things. In such a case we shall be led to introduce more comprehensive, i. e. superordinate, concepts and relations. What now suggests itself is that we should, as people say, extend our concepts. Of course this is an ine11. act way of speaking, for when you come down to it, we do not alter a concept; what we do rather is to associate a different concept with a concept-word or concept-sign-a concept to which the original concept is subordinate. The sense does not alter, nor does the sign, but the correlation between sign and sense is different. In this way it can happen that sentences which meant the True before the shift, mean the False afterwards. Former proofs lose their cogency. Everything begins to totter. We shaH avoid aii these disasters if, instead of providing old expressions or signs with new meanings, we introduce whoiiy new signs for the new concepts we have introduced. But this is not usually what happens; we continue instead to use the same signs. If we have a system with definitions that are of some use and aren't merely there as ornaments, but are taken seriously, this puts a stop to such shifts taking place. We have then an alternative: either to introduce completely new designations for the new concepts, relations, functions which occur, or to abandon the system so as to erect a new one. In fact we have at present no system in arithmetic. All we have are movements in that direction. Definitions are set up, but it doesn't so much as enter the author's head to take them seriously and to hold himself bound by them. So there is nothing to place any check on our associating, quite unwittingly, a different meaning with a sign or word.
We begin by using the addition sign only where it stands between signs for positive integers, and we define how it is to be used for this case, holding ourselves free to complete the definition for other cases later; but this piecemeal mode of definition is inadmissible; for as long as a sign is incompletely defined, it is possible to form signs with it that are to be taken as concept-sjgns, although they cannot be admitted as such because the concept designated would not have sharp boundaries and so could not be recognized as a concept. An example of such a concept-sign would be '3 + <! = 5'. Now one can show that 2 falls under this concept, since 3 + 2 = 5.
? Logic in Mathematics 243
But whether there are other objects besides this one, and if so which, that fall under the concept would have to be left quite undecided whilst the addition sign remained incompletely defined. Now it will probably not be possible to construct a system without ascending by stages from the simpler to the more difficult cases-much as things have developed historically. But in doing this we do not have to commit the error of retaining the same sign '+' throughout these changes. E. g. we may use the sign 'I' when what is in question is just the addition of positive integers, but define it completely so that the value of the function ~ I Cis determined whatever is taken as the ~- and the C-argument. E. g. we may stipulate that the value of this function is to be the False when one of the two arguments is something other than a positive integer.
So piecemeal definition and what is referred to as the extension of con- cepts by stages must be rejected. Definitions must be given once and for all; for whilst the definition of a concept remains incomplete, the concept itself does not have sharp boundaries and cannot be acknowledged as such.
Let us take one more look at the ground we have just covered
A sentence has a sense and we call the sense of an assertoric sentence a thought. A sentence is uttered either with assertoric force or without. It is not enough for science that a sentence should only have a sense; it must have a truth-value too and this we call the meaning of the sentence. If a sentence only has a sense, but no meaning, it belongs to fiction, and not to science.
Language has the power to express, with comparatively few means such a profusion of thoughts that no one could possibly command a view of them all. What makes this possible is that a thought has parts out of which it is constructed and that these parts correspond to parts of sentences, by which they are expressed. The simplest case is that of a thought which consists of a complete part and an unsaturated one. The latter we may also call the predicative part. Each of these parts must equally have a meaning, if the whole sentence is to have a meaning, a truth-value. We call the meaning of the complete part an object, that of the part which is in need of supplementation, which is unsaturated or predicative, we call a concept. We may call the way in which object and concept are combined in a sentence the subsumption of the object under the concept. Objects and concepts are fundamentally different. We call the complete part of a sentence the proper name of the object it designates. The part of a sentence that is in need of supplementation we call a concept-word or concept-sign. It is a necessary requirement for concepts that they have sharp boundaries. Both parts of a sentence, the proper name and the concept-word, may in turn be complex. The proper name may itself consist of a complete part and a part in need of supplementation. The former is again a proper name and designates an object; the latter we call a function-sign. As a result of completing a
concept-sign with a proper name we obtain a sentence, whose meaning is a
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Logic in Mathematics
truth-value. As a result of supplementing a function-sign with a proper name we obtain a proper name, whose meaning is an object. We obtain the same perspective on both if we count a concept as a function, namely a function whose value is always a truth-value, and if we count a truth-value as an object. Then a concept is a function whose value is always a truth-value. '
But a function-sign may be complex too: it may be composed of a complete part which is again a proper name, and a part that is doubly in need of supplementation-what is a name or sign of a function of two arguments. A function of two arguments whose value is always a truth- value, we call a relation. The requirement that a concept have sharp boundaries corresponds to the more general requirement that the name of a function of one argument, when supplemented with a meaningful proper name, must in turn yield a meaningful proper name. And the same holds mutatis mutandis for functions of two arguments.
Let us take a look at something that came still earlier. We realized the necessity of constructing mathematics as a system, which is not to rule out the possibility of there being different systems. It turned out that the foundations of a system are
1. the axioms and
2. thedefinitions.
The axioms of a system serve as premises for the inferences by means of
which the system is built up, but they do not figure as inferred truths. Since they are intended as premises, they have to be true. An axiom that is not true is a contradiction in terms. An axiom must not contain any term with which we are unfamiliar.
'(16- 2) is a multiple of 7'
has not only a sense, but also a meaning.
Let us take for comparison the sentence 'Etna is higher than Vesuvius'.
With this sentence we associate a sense, a thought; we understand it, we can translate it into other languages. In this sentence we have the proper name
? 232 Logic in Mathematics
'Etna', which makes a contribution to the sense of the whole sentence, to the thought. This contribution is a part of the thought, it is the sense of the word 'Etna'. But we are not making a statement about this sense, but about a mountain, which is not part of the thought. One who holds an idealist theory of knowledge will no doubt say 'That is wrong. Etna is only an idea in your mind. ' Anyone who utters the sentence 'Etna is higher than Vesuvius' understands it in the sense that it is meant to assert something about an object that is quite independent of the speaker. Now the idealist may say that it is wrong to hold that the name 'Etna' designates something. If that were so, the speaker, whilst believing himself to be operating in the realm of truth, would be lost in the realm of myth and fiction. But the idealist is not justified in turning the thought round like this, as if the speaker meant to use the name 'Etna' to designate one of his ideas, and communicate something about this. Either the speaker designates what he means to designate by the word 'Etna' or he designates nothing at all by this name, and it is meaningless.
It is therefore essential first, that the name 'Etna' should have a sense, for otherwise the sentence would not have a sense, would not express a thought, and secondly, that the name 'Etna' should have a meaning, for otherwise we should be lost in fiction. The latter of course is essential only if we wish to operate in the realm of science. In the case of fiction it does not matter whether the people who occur in it are, as we should say, historical personages. Or speaking more precisely, 'whether the personal proper names occurring in fiction are meaningful'.
Now if we were concerned only with the sense of 'Etna is higher than Vesuvius', we should have no reason for requiring that the name 'Etna' should have a meaning as well; for in order that the sentence have a sense, it is only necessary for the name 'Etna' to have a sense; the meaning contributes nothing to the thought expressed. If therefore we are concerned that the name 'Etna' should designate something, we shall also be concerned with the meaning of the sentence as a whole. That the name should designate something matters to us if and only if we are concerned with truth in the scientific sense. So our sentence will have a meaning when and only when the thought expressed in it is true or false. The meaning of a sentence must be something which remains the same, if one of the parts is replaced by something having the same meaning. We return now to the sentence '(16- 2)isamultipleof7'.
The sign '(16 - 2)' is a proper name of a number. '(17 - 3)' designates the same number, but '(17- 3)' does not have the same sense as '(16- 2)', Thus the sense of the sentence '(17 - 3) is a multiple of 7' is also different from the sense of the sentence '(16 - 2) is a multiple of 7'; and likewise the sense of the syntence '16 is congruent to 2 modulo 7' is different from the sense of the sentence '17 is congruent to 3 modulo 7'. But the sentence
'( 17 - 3) is a multiple of 7' must have the same meaning as that of the sen?
? Logic in Mathematics 233
tence '(16- 2) is a multiple of 7'. Now what is not altered by replacing the sign '(16 - 2)' by the sign '(17 - 3)', whose meaning is the same, is what I call the truth-value. These sentences are either both true or both false. In our example they are both true, but it is easy to construct a different example in which they are both false. We only need to take the number 3 in place of the number 7.
We say accordingly that sentences have the same meaning if they are both true, or if they are both false. On the other hand, they have a different meaning, if one is true and the other false. If a sentence is true, I say its meaning is the True. If a sentence is false, I say its meaning is the False. If a sentence is neither true or false, it has no meaning. Nevertheless it may still have a sense, and in such a case I say: it belongs to the realm of fiction.
For brevity I have called a sentence true or false though it would certainly be more correct to say that the thought expressed in the sentence is true or false.
But this, surely, strikes a discordant note. If I say 'the thought that (16 - 2) is a multiple of 7 is true', I am treating true as a property of the thought, whereas it has emerged that the thought is the sense and the True the meaning of the sentence. Of course treating truth as a property of sen- tences or of thoughts is in accordance with linguistic usage. If we say 'The sentence "3 > 2" is true', then the form of words is such that we are saying something about a sentence: we are saying that it has a certain property, a property we designate by the word 'true'. And if we say 'The thought that 3 > 2 is true' the corresponding thing holds of the thought. Still the predicate true is quite different from other predicates such as green, salt, rational, for what we mean by the sentence 'The thought that 3 > 2 is true' can be more simply said by the sentence '3 is greater than 2'. Thus we do not need the word 'true' at all to say this. And we can see that really nothing at all is added to the sense by this predicate. In order to put something forward as
true, we do not need a special predicate: we only need the assertoric force with which the sentence is uttered.
When we utter an assertoric sentence, we do not always utter it with assertoric force. An actor on the stage and poet reading from his works will both give frequent utterance to assertoric sentences, but the circumstances show that their utterances do not have assertoric force. They only act as if they were making assertions. In our definition, too,
(a- b) is a multiple of 7 ) and
la is an integer = a is congruent to b modulo 7 and
b is an integer
we do not utter the separate parts '(a- b) is a multiple of 7', 'a is an integer',
? 234 Logic in Mathematics
'b is an integer', with assertoric force nor do we do so in a case where the
letters 'a' and 'b' are replaced by proper names. We may even say (16- 3)isamultipleof7) )
and
l16 ~s an ;~ger = 16 is congruent to 3 modulo 7
3 ts an mteger
although some of the clauses are false; for we mean only to put forward the right-hand side of the equation as having the same sense as the left-hand side, without making a judgement about the truth of the clauses.
If a man says something with assertoric force which he knows to be false, then he is lying. This is not so with an actor on the stage, when he says something false. He is not lying, because assertoric force is lacking. And if an actor on the stage says 'it is true that 3 is greater than 2' he is no more making an assertion than if he says '3 is greater then 2'. Whether an assertion is being made, therefore, has nothing at all to do with the word 'true': it is solely a matter of the assertoric force with which the sentence is uttered. So to say of a sentence, or thought, that it is true is really quite different from saying of sea water, for example, that it is salt. In the latter case we add something essential by the predicate, in the former we do not.
Showing, as it does, that truth is not a property of sentences or thoughts, as language might lead one to suppose, this consideration confirms that a thought is related to its truth value as the sense of a sign is to its meaning.
Wehaveseenthat'(a- b)'and'(a- b)isamultipleof7'areakinto one another in that both acquire a sense and a meaning as a result of our putting meaningful proper names for 'a' and 'b'. What makes a difference between them is that the sense which '(a - b) acquires in this way is only part of a thought, whereas the sense which '(a - b) is a multiple of 7' ac? quires in this way is a thought. If we begin by just replacing 'b' by the proper name '2', we obtain '(a - 2)', '(a - 2) is a multiple of 7'. What is present in: the second combination of signs over and above the letter 'a' is the sign of a concept. And we may construe the sentence '(16- 2) is a multiple of 7' aa' consisting of the proper name '16' together with this concept-sign, so that in j this sentence we are asserting the concept in question of the number 16,. What we have is the subsumption of an object under a concept.
We can, in an analogous way, regard what is present in 'a - 2', apart from the letter 'a', as a sign. On this view, then,' 1 6 - 2' will be composed of: the proper name' 16' and this sign, which like the concept-sign above, is in need of supplementation. What it designates must be in need ot: supplementation, just as the concept is. We call it a function. The concept? ' sign, when supplemented by a proper name, yields a proper name. In our, case the function-sign, when supplemented by the proper names '2', '3', '4'1 , yields respectively the proper names ' 2 - 2'. '3 - 2'. '4 - 2'.
? The objects
is the False,
3- 2isamultipleof7*
16- 2 is a multiple of 7*
2 - 3 - 4 -
2 is the value of our function for the argument 2, 2 is the value of our function for the argument 3, 2 is the value of our function for the argument 4.
Logic in Mathematics 235
2- 2,3- 2,4- 2.
of which these proper names are the signs, we call the values of our function. Thus
But what we obtain from the sentence
'(a - 2) is a multiple of 7'
by replacing 'a' by a proper name, is also to be understood as a proper name; for it designates a truth value and such an entity is to be regarded as an object. Thus
the True. So there is a far-reaching agreement between the cases in which we speak of a function and the cases in which we speak of a concept; and it seems appropriate to understand a concept as a function-namely, a function whose value is always a truth value. So if the concept above is understood as a function, then the False is the value of this function for the argument 3, and the True is the value of the function for the argument 16. What we should otherwise say occurred as a logical subject is here presented as an argument.
It is not possible to give a definition of what a function is, because we have here to do with something simple and unanalysable. It is only possible to hint at what is meant and to make it clearer by relating it to what is known. Instead of a definition we must provide illustrations; here of course we must count on a meeting of minds.
There often seems to be unclarity about what a function is. In this connection the word 'variable' is often used. This makes it look at first as if there were two kinds of number, constant or ordinary numbers and variable numbers. The former, it seems, are designated by the familiar signs for numbers, the latter by the letters 'x', 'y', 'z'. But this cannot be reconciled with the way we proceed in Analysis. When we have the letter 'x' combined with other signs as in
'x- 2'
*It is understood that these sentences are here uttered without assertoric force.
? 236 Logic in Mathematics
Analysis requires that it be possible to substitute different number-signs for
this 'x' as in
'3- 2','4- 2','5- 2',etc.
Buthere we cannot properly speak of anything altering; for if we say that something alters, the thing which alters must be recognizable as the same throughout the alteration. If a monarch grows older, he alters. But we can only say this because he can be recognized as the same in spite of the alteration. When, on the other hand, a monarch dies and his successor mounts the throne, we cannot say that the former has been transformed into the latter; for the new monarch is just not the same as the old one. Putting '3', '4', '5' in turn for 'x' in ' x - 2' is comparable with this. We do not have here the same thing assuming different properties in the course of time: we have quite different numbers. Now if the letter 'x' designated a variable number, we should have to be able to recognize it again as the same number even though its properties were different. But 4 is not the same number as 3. So there is nothing at all that we could designate by the name 'x'. If it means 3, it does not mean 4, and if it means 4, it does not mean 3. In arithmetic and Analysis letters serve to confer generality of content on sentences. This is no less true when it is concealed by the fact that the greater part of the proof is set out in words. In such a case we must take everything into consideration, and not just what goes on in the arithmetical formulae. We say, for instance, 'Let a designate such-and-such and b such-and-such' and take this to be the point at which we begin our inquiry. But what in fact we have here are antecedents
'if a is such-and-such', 'if b is such-and-such',
and they have to be introduced as such or attached in thought to each of the sentences which follow, and these letters, whose role is merely an indicating one, make the whole general. It is only when, as we say, an unknown is designated by 'x' that we have a somewhat different case. E. g. let the question be to solve the equation
'x2 - 4 = 0'
We obtain the solutions 2 or -2. But even here we may present the equation together with its solution in the form of a general sentence: 'If x2 - 4 = 0, then x = 2 or x = -2'. We may take this opportunity to point out that the sign '? yl4? is to be rejected out of hand. Here people have not taken sufficient care in using language as a guide. The proper place for the word 'or' is between sentences: 'x is equal to 2 or x is equal to -2'. But we contract the two sentences into 'x is equal to plus 2 or minus 2' and accordingly write 'x = ? /4'; however'? /4' doesn't designate anything at all; it isn't a meaningful sign. What one can say is
'2isequalto+y'4or2isequalto- yf4? ,
? Logic in Mathematics 237 where the assertoric force extends over the whole sentence, the two clauses
being uttered without assertoric force. Equally one can say '-2 is equal to +y'4or -2 is equal to -/4'
but '2 is equal to ? /4' has no sense.
At this point we may go into the concept of the square root of 4. If we
think of '2? 2 = 4' as resulting from ? ~? ~ = 4' by replacing the letter? ~? by the numeral '2', then we are seeing '2 ? 2 = 4' as composed of the name '2' and a concept-sign, which as such is in need of supplementation, and so we can read '2? 2 = 4' as '2 is a square root of 4'. We can likewise read '(-2) ? (-2) = 4' as '(-2) is a square root of 4'. But we must not read the equation '2 = /4' as '2 is a square root of 4'. For we cannot allow the sign '/4' to be equivocal. It is absolutely ruled out that a sign be equivocal or ambiguous. if the sign '/4' were equivocal, we should not be able to say whether the sentence '2 = /4' were true, and just on this account this combination of signs could not properly be called a sentence at all, because it would be indeterminate which thought it expresses. Signs must be so defined that it is determinate what '/4' means, whether it is the number 2 or some other number. We have come to see that the equals sign is a sign for identity. And this is how it has to be understood in '2 = J4' too. 'y4' means an object and '2' means an object. We may adopt the reading '2 is the positive square root of 4'. And so the 'is' is to be understood here as a sign for identity, not as a mere copula.
'2 = /4'may not be read as '2 is a square root of 4'; for the 'is' here would be the copula. If I judge '2 is a square root of 4', I am subsuming the object 2 under a concept. This is the case we have whenever the grammatical subject is a proper name, with the predicate consisting of 'is' together with a substantive accompanied by the indefinite article. In such a case the 'is' is always the copula and the substantive a nomen appellativum. And then an object is being subsumed under a concept. Identity is something quite different. And yet people sometimes write down an equals sign when what we have is a case of subsumption. The sign '/4' is not incomplete in any way, but has the stamp of a proper name. So it is absolutely impossible for it to designate a concept and it cannot be rendered verbally by a nomen appellativum with or without an indefinite article. When what stands to the left of an equality sign is a proper name, then what stands to the right must be a proper name as well, or become such once the indicating letters in it are replaced by meaningful signs.
However, let us return from this digression to the matter in hand. Where they do not stand for an unknown, letters in arithmetic have the role of conferring generality of content on sentences, not of designating a variable number; for there are no variable numbers. Every alteration takes place in time. The laws of number, however, are timeless and eternal. Time does not enter into arithmetic or Analysis. Time can come in only when it is a matter
of applying arithmetic. The number 3 has always been u prime number und
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Logic in Mathematics
will always remain such. How could a change be possible here? It feels incongruous to speak of a variable number and so people prefer to say 'variable magnitude', as if that was a great improvement. Of course an iron rod grows longer when it is heated, and shorter when it cools down-it changes in time. If we measure its length in millimeters, we get now this number, now that. If we now say 'the number which gives the length of this rod at time t', we have an expression containing an indefinitely indicating letter 't'. If instead of't' we put the proper name of a time instant, it becomes the proper name of a number. This is on all fours with what we have in the
case of the expression ' x - 2'. This likewise becomes a proper name if we put the name of a number for 'x'. In both cases we have a function which may yield different values when saturated by different arguments. Iron rods and time, when you come down to it, are of no account to arithmetic; for this is concerned neither with pebbles, nor with peppermints, nor with railway trains, nor with rows of books, nor with iron rods, nor with time instants. These are things which may come into the applications of mathematics, but they do not have any role in constructing mathematics as a system.
In the light of all this, we can see that there is no place in arithmetic either for variable numbers or for variable magnitudes. 'Magnitude' is either a subterfuge for number, in which case variable magnitudes no more exist than variable numbers, or 'magnitude' is understood in such a way that we can properly speak of variable magnitudes; but in that case they do not belong in arithmetic.
If the letters 'x' and 'y' designated different variables, we should have to be able to say how these are distinguished; but this no one can do. We have only to keep before our mind the fact that we are concerned with pure arithmetic, not with its applications. We may perhaps seek a way out by taking the view that the letters 'x' and 'y' are not signs for what is variable, but are themselves the things that vary. But if we do this, we run foul of the established use of our signs. In the case of the equals sign, for example, it is always presupposed that the simple or complex sign which occurs on the left is either a meaningful proper name, or will become such once the indicating letters in it are replaced by designating signs.
Hence it is impossible to explain what a functi()n is by referring to what is called a variable. It is rather the case that when we seek to make clear to ourselves what a variable is, we come back again and again to what we have called a function, thus recognizing that variables are not a proper part ofthe subject-matter ofarithmetic.
(We have seen that a concept can be construed as a special case of a function. We have made it a requirement that concepts have sharp boundaries. There is an analogous requirement for the more general case of a function. )
A great deal of unclarity still prevails over what a function is. In particular, it is easy to confuse a function with the value of a function, as if one wrote
fx=f,
? Logic in Mathematics 239
using the letter on one side to indicate a function, and on the other side to indicate the value of the function. There is admittedly a difficulty which accounts for its being so extremely hard to grasp the true nature of a function.
This difficulty lies in the expressions we use. We say 'the function' and 'the concept', expressions which we can hardly avoid but which are inappropriate. The definite article gives these expressions the form of proper names in the logical sense, as if they were meant to designate objects, when this is precisely what they are not meant to do. The very nature of concepts and functions-their unsaturatedness-is thus concealed. Language forces an inappropriate expression on us. This is a situation which, unfortunately, can hardly be avoided, but we can render it harmless by always bearing the inappropriateness of language in mind. At the same time we shall also avoid confusing the value of a function with a function.
Now we sometimes speak of a function when what we have in mind are cases like (1 + x )2. Here' 1 + x' occurs in the argument-place of the squared function. But '1 + x' does not designate a function at all, but only indicates thevalueofafunctionindefinitely. Ifin'(1 +x)2', weput,say,'3'inplaceof 'x', then we get (1 + 3)2, and here the value of the function 1 + ~for the argument 3 is the argument of the squared function. But this argument is an object, a number. Here a function is compounded out of two functions by taking the value of the first function for a certain argument as an argument of the second function. In this connection we must persist in emphasizing the fundamental difference of object and function. No function-name can stand in a place where an object-name, a proper name, stands and conversely a proper name cannot stand where a function-name stands.
Even where we have a function whose value is the same for every argument, this value must be distinguished from the function. So the function
1 + ~- ~
is different from the number 1 itself. We must not say 'We have 1 + ~- ~ = 1 and the equals sign is the identity sign, and so the function 1 +~-~just is the number 1', for when we say 'the function 1 +~-~. the letter? ~? is not part of the function-sign; for the proper name '1 + 3 - 3' is composed of the function-name and the proper name '3', and the letter? ~? does not occur in it at all. In the sentence '1 + a - a = 1', however, the letter a has the role of conferring generality of content on the sentence, whereas when I say 'the function 1 + ~ - ~? the role of the letter '~? is to enable us to recognize the places where the supplementing proper name is to be put. In order to form the differential quotient of a function for the argument 3 we subtract the value of the function for the argument 3 from the value of the function for the argument (3 + k) and divide the difference by k, and so on. For the
function I + ~- ~this is represented in the formula
11 + (3 + k ) - (3 + k ) l - 1 1 + 3 - 31 k
? 240 Logic in Mathematics
But in the case of the proper name' 1' there are simply no places in which we can first put '3 + k' and then '3'. The prescription can only be carried out where we have a function.
It is a mistake to define the number one by saying 'One is a single thing', because on account of the indefinite article 'a single thing' has to be understood as a concept-word. But in that case the word 'is' is the copula and belongs with the predicate. We then have the object the number one being subsumed under a concept. But that is no definition. A definition always stipulates that a new sign or word is to mean the same as a complex sign we already understand. When the word 'is' is used in a definition, it is to be understood as a sign for identity, not as a copula. If now a proper name occurs to the left of the identity sign, such a sign must occur on the right too; 'a single thing' is, however, a nomen appellativum.
As functions of one argument are fundamentally different from objects, so functions of two arguments are fundamentally different from functions of
one argument.
By supplementing the sign for a function of one argument with a proper
name, we obtain a proper name. So e. g. from the function-sign? ~- 2' and thepropername'3'weobtainthenewpropername'3- 2'. Theletter? ~? in ? ~- 2' only serves to keep open the place for the supplementing argu- ment-sign. In the same way, by supplementing the concept-sign? ~> 0' with '1' we obtain '1 > 0' and this is a name of the True.
A function of two arguments is doubly in need of supplementation. In ? ~- r; we have a sign for a function of two arguments. The letters? ~? and 'C' are meant to keep open the places for the argument-signs. The difference between the letters? ~? and 'C' is to show that a different argument-sign may
be put in the two places. By putting the proper name '2' in the C-argument place we obtain ? ~ - 2', which is a function-sign for a function of one argument. In the same way the relation-sign? ~> C' yields the concept-sign '~ > 0'. So as a result of being partly saturated, functions of two arguments yield functions of one argument and relations yield concepts. A further way in which this can happen is through abolishing the difference between the argument-places. If I write? ~- ~? ,I indicate, by using the letter? ~? in both places, that the same proper name is to be put in both places, and so what I have is the name of a function of only one argument. When I call this the 'name of a function', this is to be taken cum grano salis. The proper name which we obtain by supplementing this function with a proper name, e. g. '3 - 3', does not contain the letter? ~, although it contains the function-name in question. This ? ~? is therefore not a constituent of the function-name but only enables us to recognize how the function-sign is combined with the proper name supplementing it. This? ~ gives us a pointer for how to use the function-name. We can similarly form a concept-sign from a relation-sign by abolishing the difference between the argument-places. Thus from the relation-sign? ~> C' we obtain the concept-sign? ~> ~? .
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241
We have seen that the value of one function can occur as the argument of a second function. We may call the former the inner function, the latter the outer function. So from the name of a function of two arguments and a concept-sign we can obtain a relation-sign in which the concept is the outer function. E. g. let? ~- C' be the sign for the function of two arguments and? ~ is a multiple of 7' the concept-sign. Then '(~- ()is a multiple of 7' will be the new relation-sign.
A concept must have sharp boundaries; i. e. it must hold of every object either that it falls under a concept or does not. We may not have a case in which this is indeterminate. From this there follows something correspond- ing for the case of a relation; for of course by partially saturating a relation we obtain a concept. This must have sharp boundaries. And indeed any concept obtained by partly saturating a relation must have sharp boundaries. This means in other words: Every object must either stand or not stand in the relation to every object. We must exclude a third possibility. If we apply this to the relation (~ - 0 is a multiple of 7, it follows that a meaningful proper name must always result from the complex sign ? ~- C' by replacing the letters '~, and 'C' by meaningful proper names, and so not only when signs for numbers are inserted. Therefore the minus sign has to be defined in such a way that whatever meaningful proper names are put to the right and left of it, the whole combination of signs always has a meaning. So we arrive at the general requirements:
Every sign for a function of one argument has to be defined in such a way that the result always has a meaning whatever meaningful argument-sign is taken to supplement it.
Every sign for a function of two arguments has to be defined in such a way that the result has a meaning whatever meaningful argument-signs are used to supplement it.
E. g. we could stipulate that the value of the functions ~- Cis always to be the False, if one of the two arguments is not a number, whatever the other argument may be. Of course, we should then also have to know what a number is.
(We can stipulate likewise that the value of the function ~ > Cis to be the False, if one of the two arguments is not a real number, whatever the other argument may be. )
But it is precisely on this issue that views have changed. Originally the numbers recognized were the positive integers, then fractions were added, then negative numbers, irrational numbers, and complex numbers. So in the course of time wider and wider concepts came to be associated with the word 'number'. Bound up with this was the fact that the addition sign changed its meaning too. And the same happened with other arithmetical signs. Needless to say, this is a process which logic must condemn and which is all the more dangerous, the less one is aware of the shift taking place. The progress of the history of the sciences runs counter to the demands of logic. We must always distinguish between history and system.
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In history we have development; a system is static. Systems can be constructed. But what is once standing must remain, or else the whole system must be dismantled in order that a new one may be constructed. Science only comes to fruition in a system. We shall never be able to do without systems. Only through a system can we achieve complete clarity
and order. No science is in such command of its subject-matter as mathematics and can work it up into such a perspicuous form; but perhaps also no science can be so enveloped in obscurity as mathematics, if it fails to construct a system.
As a science develops a certain system may prove no longer to be adequate, not because parts of it are recognized to be false but because we wish, quite rightly, to assemble a large mass of detail under a more comprehensive point of view in order to obtain greater command of the material and a simpler way of formulating things. In such a case we shall be led to introduce more comprehensive, i. e. superordinate, concepts and relations. What now suggests itself is that we should, as people say, extend our concepts. Of course this is an ine11. act way of speaking, for when you come down to it, we do not alter a concept; what we do rather is to associate a different concept with a concept-word or concept-sign-a concept to which the original concept is subordinate. The sense does not alter, nor does the sign, but the correlation between sign and sense is different. In this way it can happen that sentences which meant the True before the shift, mean the False afterwards. Former proofs lose their cogency. Everything begins to totter. We shaH avoid aii these disasters if, instead of providing old expressions or signs with new meanings, we introduce whoiiy new signs for the new concepts we have introduced. But this is not usually what happens; we continue instead to use the same signs. If we have a system with definitions that are of some use and aren't merely there as ornaments, but are taken seriously, this puts a stop to such shifts taking place. We have then an alternative: either to introduce completely new designations for the new concepts, relations, functions which occur, or to abandon the system so as to erect a new one. In fact we have at present no system in arithmetic. All we have are movements in that direction. Definitions are set up, but it doesn't so much as enter the author's head to take them seriously and to hold himself bound by them. So there is nothing to place any check on our associating, quite unwittingly, a different meaning with a sign or word.
We begin by using the addition sign only where it stands between signs for positive integers, and we define how it is to be used for this case, holding ourselves free to complete the definition for other cases later; but this piecemeal mode of definition is inadmissible; for as long as a sign is incompletely defined, it is possible to form signs with it that are to be taken as concept-sjgns, although they cannot be admitted as such because the concept designated would not have sharp boundaries and so could not be recognized as a concept. An example of such a concept-sign would be '3 + <! = 5'. Now one can show that 2 falls under this concept, since 3 + 2 = 5.
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But whether there are other objects besides this one, and if so which, that fall under the concept would have to be left quite undecided whilst the addition sign remained incompletely defined. Now it will probably not be possible to construct a system without ascending by stages from the simpler to the more difficult cases-much as things have developed historically. But in doing this we do not have to commit the error of retaining the same sign '+' throughout these changes. E. g. we may use the sign 'I' when what is in question is just the addition of positive integers, but define it completely so that the value of the function ~ I Cis determined whatever is taken as the ~- and the C-argument. E. g. we may stipulate that the value of this function is to be the False when one of the two arguments is something other than a positive integer.
So piecemeal definition and what is referred to as the extension of con- cepts by stages must be rejected. Definitions must be given once and for all; for whilst the definition of a concept remains incomplete, the concept itself does not have sharp boundaries and cannot be acknowledged as such.
Let us take one more look at the ground we have just covered
A sentence has a sense and we call the sense of an assertoric sentence a thought. A sentence is uttered either with assertoric force or without. It is not enough for science that a sentence should only have a sense; it must have a truth-value too and this we call the meaning of the sentence. If a sentence only has a sense, but no meaning, it belongs to fiction, and not to science.
Language has the power to express, with comparatively few means such a profusion of thoughts that no one could possibly command a view of them all. What makes this possible is that a thought has parts out of which it is constructed and that these parts correspond to parts of sentences, by which they are expressed. The simplest case is that of a thought which consists of a complete part and an unsaturated one. The latter we may also call the predicative part. Each of these parts must equally have a meaning, if the whole sentence is to have a meaning, a truth-value. We call the meaning of the complete part an object, that of the part which is in need of supplementation, which is unsaturated or predicative, we call a concept. We may call the way in which object and concept are combined in a sentence the subsumption of the object under the concept. Objects and concepts are fundamentally different. We call the complete part of a sentence the proper name of the object it designates. The part of a sentence that is in need of supplementation we call a concept-word or concept-sign. It is a necessary requirement for concepts that they have sharp boundaries. Both parts of a sentence, the proper name and the concept-word, may in turn be complex. The proper name may itself consist of a complete part and a part in need of supplementation. The former is again a proper name and designates an object; the latter we call a function-sign. As a result of completing a
concept-sign with a proper name we obtain a sentence, whose meaning is a
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truth-value. As a result of supplementing a function-sign with a proper name we obtain a proper name, whose meaning is an object. We obtain the same perspective on both if we count a concept as a function, namely a function whose value is always a truth-value, and if we count a truth-value as an object. Then a concept is a function whose value is always a truth-value. '
But a function-sign may be complex too: it may be composed of a complete part which is again a proper name, and a part that is doubly in need of supplementation-what is a name or sign of a function of two arguments. A function of two arguments whose value is always a truth- value, we call a relation. The requirement that a concept have sharp boundaries corresponds to the more general requirement that the name of a function of one argument, when supplemented with a meaningful proper name, must in turn yield a meaningful proper name. And the same holds mutatis mutandis for functions of two arguments.
Let us take a look at something that came still earlier. We realized the necessity of constructing mathematics as a system, which is not to rule out the possibility of there being different systems. It turned out that the foundations of a system are
1. the axioms and
2. thedefinitions.
The axioms of a system serve as premises for the inferences by means of
which the system is built up, but they do not figure as inferred truths. Since they are intended as premises, they have to be true. An axiom that is not true is a contradiction in terms. An axiom must not contain any term with which we are unfamiliar.
