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.
Gottlob-Frege-Posthumous-Writings
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? ~> ~? .
? 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
? ? 244
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.
The definitions are something quite different. Their role is to bestow a meaning on a sign or word that hitherto had none. So a definition has to contain a new sign. Once a meaning has been given to this sign by the definition, the definition is transformed into an independent sentence which can be used in the development of the system as a premise for inferences. How are inferences carried out within the system?
Let us assume that we have a sentence of the form 'If A holds, so does B'. If we add to this the further sentence 'A holds', then from both premises we can infer 'B holds'. But for the conclusion to be possible, both premises have to be true. And this is why the axioms also have to be true, if they are to serve as premises. For we can draw no conclusion from something false. But it might perhaps be asked, can we not, all the same, draw consequences from a sentence which may be false, in order to see what we should get if it were true? Yes, in a certain sense this is possible. From the premises
If Fholds, so does A If A holds, so does E
1 This is how the sentence reads in the German. Since it merely repeats the first part of the preceding sentence, the editors suggest that we read 'object' in place of 'truth value', so that the sentence marks a natural inference from the one precedina (trans. ).
? we can infer
If Fholds, so does E From this and the further premise
we can infer
If E holds, so does Z
IfFholds, so does Z.
Logic in Mathematics 245
And so we can go on drawing consequences without knowing whether r is true or false. But we must notice the difference. In the earlier example the premise 'A holds' dropped out of the conclusion altogether. In this example the condition 'If Fholds' is retained throughout. We can only detach it when we have seen that it is fulfilled. In the present case 'r holds' cannot be regarded as a premise at all: what we have as a premise is
If Fholds, so does A,
and thus something of which 'r holds' is only a part. Of course this whole premise must be true; but this is possible without the condition being fulfilled, without r holding. So, strictly speaking, we simply cannot say that consequences are here being drawn from a thought that is false or doubtful; for this does not occur independently as a premise, but is only part of a premise which as such has indeed to be true, but which can be true without that part of the thought-the part which it contains as a condition-being true.
In indirect proofs it looks as if consequences are being drawn from something false. As an example, suppose we have to prove that in a triangle
the longer side subtends the greater angle. To prove:
1
A
c
If LB > LA, then AC >BC. We take as given:
I IfBC>AC,thenLA>LB. II IfBC=AC,thenLA= LB.
III If not AC >BC, and if not BC> AC, then BC=AC.
IV IfLA=LB,thennotLB>LA. V If LA> LB, then not LB >LA.
From Il and Ill there follows:
IfnotAC >BC and if not BC> AC, then LA= LB.
From this and IV we have: IfnotAC>BCandifnotBC>AC,thennot LB>L. A.
? 246 Logic in Mathematics From I and V:
IfBC> AC, then not LB >LA. From the last two sentences there follows:
If not AC >BC, then not LB > LA. And then by contraposition:
If LB >LA, then AC >BC.
To simplify matters, I shall assume that we are not speaking of triangles in general, but of a particular triangle. LA and LB may be understood as numbers, arrived at by measuring the angles by some unit-measure, as e. g. a right-angle. AC and BC may likewise be understood as numbers, arrived at by using some unit-measure for the sides, as e. g. a metre. Then the signs 'LA',' LB', 'AC', and 'BC' are to be taken as proper names of numbers.
We see that 'not AC > BC' does not occur here as a premise, but that it is contained in Ill as a part-as a condition. So strictly speaking, we cannot say that consequences are being drawn from the false thought (not AC >BC). Therefore, we ought not really to say 'suppose that not AC >BC', because this makes it look as though 'not AC >BC' was meant to serve as a premise for inference, whereas it is only a condition.
We make far too much of the peculiarity of indirect proof vis-d-vis direct proof. The truth is that the difference between them is not at all important.
The proof can also be set out in the following way. We now take as given:
I' Ifnot LA >LB then not BC> AC. II' IfnotLA= LB,thennotBC=AC.
Ill' IfnotBC>ACandifnotBC=AC,thenAC>BC. IV' If LB >LA then not LA= LB.
V' If LB >LA then not LA> LB.
From V' and I' there follows:
If LB > LA, then not BC> AC.
From this and Ill' we have:
If LB >LA and if not BC= AC, then AC >BC.
From IV' and II' there follows:
If LB >LA, then not BC= AC.
From the last two sentences we have:
If LB >LA, then AC >BC.
At no point in this proof have we entertained 'not AC >BC' even as a mere hypothesis.
? Logic in Mathematics 247
In an investigation of the foundations of geometry it may also look as if consequences are being drawn from something false or at least doubtful. Can we not put to ourselves the question: How would it be if the axiom of parallels didn't hold? Now there are two possibilities here: either no use at all is made of the axiom of parallels, but we are simply asking how far we can get with the other axioms, or we are straightforwardly supposing something which contradicts the axiom of parallels. It can only be a question of the latter case here. But it must constantly be borne in mind that what is false cannot be an axiom, at least if the word 'axiom' is being used in the traditional sense. What are we to say then? Can the axiom of parallels be acknowledged as an axiom in this sense? When a straight line intersects one of two parallel lines, does it always intersect the other? This question, strictly speaking, is one that each person can only answer for himself. I can only say: so long as I understand the words 'straight line', 'parallel' and 'intersect' as I do, I cannot but accept the parallels axiom. If someone else does not accept it, I can only assume that he understands these words differently. Their sense is indissolubly bound up with the axiom of parallels. Hence a thought which contradicts the axiom of parallels cannot be taken as a premise of an inference. But a true hypothetical thought, whose condition contradicted the axiom, could be used as a premise. This condition would then be retained in all judgements arrived at by means of our chains of inference. If at some point we arrived at a hypothetical judgement whose consequence contradicted known axioms, then we could conclude that the condition contradicting the axiom of parallels was false, and we should thereby have proved the axiom of parallels with the help of other axioms. But because it had been proved, it would lose its status as an axiom. In such a case we should really have given an indirect proof.
If, however, we went on drawing inference after inference and still did not come up against a contradiction anywhere, we should certainly become more and more inclined to regard the axiom as incapable of proof. Nevertheless we should still, strictly speaking, not have proved this to be so.
Now in his Grundlagen der Geometrie Hilbert is preoccupied with such questions as the consistency and independence of axioms. But here the sense of the word 'axiom' has shifted. For if an axiom must of necessity be true, it is impossible for axioms to be inconsistent with one another. So any discussion here would be a waste of words. But obvious though it is, it seems just not to have entered Hilbert's mind that he is not speaking of axioms in Euclid's sense at all when he discusses their consistency and independence. We could say that the word 'axiom', as he uses it, fluctuates from one sense to another without his noticing it. It is true that if we concentrate on the words of one of his axioms, the immediate impression is that we are dealing with an axiom of the Euclidean variety; but the words mislead us, because all the words have a different use from what they have in Euclid. At? 31 we
1 The quotation here is from the first edition of the Grundlagen der Geometric. Later editions give a different uxiom in plnce of thnt which Frege cites under ((. 4 (cd. ).
? 248 Logic in Mathematics
read 'Definition. The points of a straight line stand in certain relations to one another, for the description of which we appropriate the word "between". ' Now this definition is only complete once we are given the four axioms
11. 1 If A, B, C are points on a straight line, and B lies between A and C, then B lies between C and A.
11. 2 If A and C are two points on a straight line, then there is at least one point B lying between A and C, and at least one point D such that C lies between A and D.
11. 3 Given any three points on a straight line, there is one and only one which lies between the other two.
11. 4 Any four points A, B, C, D on a straight line can be so ordered that B lies between A and C and between A and D, and so that C lies between A and D and between B and D.
These axioms, then, are meant to form parts of a definition. Consequently these sentences must contain a sign which hitherto had no meaning, but which is given a meaning by all these sentences taken together. This sign is apparently the word 'between'. But a sentence that is meant to express an axiom may not contain a new sign. All the terms in it must be known to us. As long as the word 'between' remains without a sense, the sentence 'If A, B, C, are points on a straight line and B lies between A and C, then B lies between C and A' fails to express a thought.
An axiom, however, is always a true thought. Therefore, what does not express a thought, cannot express an axiom either. And yet one has the impression, on reading the first of these sentences, that it might be an axiom. But the reason for this is only that we are already accustomed to associate a sense with the word 'between'. In fact if in place of
we say
'B lies between A and C' 'B pat A nam C',
then we associate no sense with it. Instead of the so-called axiom 11. 1 we should have
'If B pat A nam C, then B pat C nam A'.
No one to whom these syllables 'pat' and 'nam' are unfamiliar will associate a sense with this apparent sentence. The same holds of the other three pseudo-axioms.
The question now arises whether, if we understand by A, B, C points on a straight line, an expression of the form
'B patA nam C'
will not at least ~ome to acquire a sense through the totality of these pseudo-sentences. I think not. We may perhaps hazard the guess that it will
come to the same thing as
? Logic in Mathematics 249 'B lies between A and C',
but it would be a guess and no more. What is to say that this matrix could not have several solutions?
But does, then, a definition have to be unambiguous? Are there not circumstances in which a certain give and take is a good thing?
Of course a2 = 4 does not determine unequivocally what a is to mean, but is there any harm in that? Well, if a is to be a proper name whose meaning is meant to be fixed, this goal will obviously not be achieved. On the contrary one can see that in this formula there is designated a concept under which
the numbers 2 and -2 fall. Once we see this, the ambiguity is harmless, but it means that we don't have a definition of an object.
If we want to compare this case with our pseudo-axiom, we have to compare the letter 'a' with 'between' or with 'pat-nam'. We must distinguish signs that designate from those which merely indicate. In fact the word 'between' or 'pat-nam' no more designates anything than does the letter 'a'. So we have here to disregard the fact that we usually associate a sense with the word 'between'. In this context it no more has a sense than does 'pat- nam'. Now to say that a sign which only indicates neither designates anything nor has a sense is not yet to say it could not contribute to the expression of a thought. It can do this by conferring generality of content on a simple sentence or on one made up of sentences.
Now there is, to be sure, a difference between our two cases; for whilst 'a' stands in for a proper name, 'pat-nam' stands in for the designation of a relation with three terms. As we call a function of one argument, whose value is always a truth-value, a concept, and a function of two arguments, whose value is always a truth-value, a relation, we can go yet a step further and call a function of three arguments whose value is always a truth-value, a relation with three terms. Then whilst the 'between-and' or the 'pat-nam' do not designate such a relation with three terms, they do indicate it, as 'a' indicates an object. Still there remains a distinction. We were able to find a concept designated in 'a2 = 4'.
What would correspond to this in the case of our pseudo-axioms? It is what I call a second level concept. In order to see more clearly what I understand by this, consider the following sentences:
'There is a positive number' 'There is a cube root of 1'.
We can see that these have something in common. A statement is being made, not about an object, however, but about a concept. In the first sentence it is the concept positive number, in the second it is the concept cube root of 1. And in each case it is asserted of the concept that it is not empty, but satisfied. It is indeed strictly a mistake to say 'The concept positive number is satisfied', for by saying this I seem to make the concept into an object, as the definite article 'the concept' shows.
'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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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.
? 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.
The definitions are something quite different. Their role is to bestow a meaning on a sign or word that hitherto had none. So a definition has to contain a new sign. Once a meaning has been given to this sign by the definition, the definition is transformed into an independent sentence which can be used in the development of the system as a premise for inferences. How are inferences carried out within the system?
Let us assume that we have a sentence of the form 'If A holds, so does B'. If we add to this the further sentence 'A holds', then from both premises we can infer 'B holds'. But for the conclusion to be possible, both premises have to be true. And this is why the axioms also have to be true, if they are to serve as premises. For we can draw no conclusion from something false. But it might perhaps be asked, can we not, all the same, draw consequences from a sentence which may be false, in order to see what we should get if it were true? Yes, in a certain sense this is possible. From the premises
If Fholds, so does A If A holds, so does E
1 This is how the sentence reads in the German. Since it merely repeats the first part of the preceding sentence, the editors suggest that we read 'object' in place of 'truth value', so that the sentence marks a natural inference from the one precedina (trans. ).
? we can infer
If Fholds, so does E From this and the further premise
we can infer
If E holds, so does Z
IfFholds, so does Z.
Logic in Mathematics 245
And so we can go on drawing consequences without knowing whether r is true or false. But we must notice the difference. In the earlier example the premise 'A holds' dropped out of the conclusion altogether. In this example the condition 'If Fholds' is retained throughout. We can only detach it when we have seen that it is fulfilled. In the present case 'r holds' cannot be regarded as a premise at all: what we have as a premise is
If Fholds, so does A,
and thus something of which 'r holds' is only a part. Of course this whole premise must be true; but this is possible without the condition being fulfilled, without r holding. So, strictly speaking, we simply cannot say that consequences are here being drawn from a thought that is false or doubtful; for this does not occur independently as a premise, but is only part of a premise which as such has indeed to be true, but which can be true without that part of the thought-the part which it contains as a condition-being true.
In indirect proofs it looks as if consequences are being drawn from something false. As an example, suppose we have to prove that in a triangle
the longer side subtends the greater angle. To prove:
1
A
c
If LB > LA, then AC >BC. We take as given:
I IfBC>AC,thenLA>LB. II IfBC=AC,thenLA= LB.
III If not AC >BC, and if not BC> AC, then BC=AC.
IV IfLA=LB,thennotLB>LA. V If LA> LB, then not LB >LA.
From Il and Ill there follows:
IfnotAC >BC and if not BC> AC, then LA= LB.
From this and IV we have: IfnotAC>BCandifnotBC>AC,thennot LB>L. A.
? 246 Logic in Mathematics From I and V:
IfBC> AC, then not LB >LA. From the last two sentences there follows:
If not AC >BC, then not LB > LA. And then by contraposition:
If LB >LA, then AC >BC.
To simplify matters, I shall assume that we are not speaking of triangles in general, but of a particular triangle. LA and LB may be understood as numbers, arrived at by measuring the angles by some unit-measure, as e. g. a right-angle. AC and BC may likewise be understood as numbers, arrived at by using some unit-measure for the sides, as e. g. a metre. Then the signs 'LA',' LB', 'AC', and 'BC' are to be taken as proper names of numbers.
We see that 'not AC > BC' does not occur here as a premise, but that it is contained in Ill as a part-as a condition. So strictly speaking, we cannot say that consequences are being drawn from the false thought (not AC >BC). Therefore, we ought not really to say 'suppose that not AC >BC', because this makes it look as though 'not AC >BC' was meant to serve as a premise for inference, whereas it is only a condition.
We make far too much of the peculiarity of indirect proof vis-d-vis direct proof. The truth is that the difference between them is not at all important.
The proof can also be set out in the following way. We now take as given:
I' Ifnot LA >LB then not BC> AC. II' IfnotLA= LB,thennotBC=AC.
Ill' IfnotBC>ACandifnotBC=AC,thenAC>BC. IV' If LB >LA then not LA= LB.
V' If LB >LA then not LA> LB.
From V' and I' there follows:
If LB > LA, then not BC> AC.
From this and Ill' we have:
If LB >LA and if not BC= AC, then AC >BC.
From IV' and II' there follows:
If LB >LA, then not BC= AC.
From the last two sentences we have:
If LB >LA, then AC >BC.
At no point in this proof have we entertained 'not AC >BC' even as a mere hypothesis.
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In an investigation of the foundations of geometry it may also look as if consequences are being drawn from something false or at least doubtful. Can we not put to ourselves the question: How would it be if the axiom of parallels didn't hold? Now there are two possibilities here: either no use at all is made of the axiom of parallels, but we are simply asking how far we can get with the other axioms, or we are straightforwardly supposing something which contradicts the axiom of parallels. It can only be a question of the latter case here. But it must constantly be borne in mind that what is false cannot be an axiom, at least if the word 'axiom' is being used in the traditional sense. What are we to say then? Can the axiom of parallels be acknowledged as an axiom in this sense? When a straight line intersects one of two parallel lines, does it always intersect the other? This question, strictly speaking, is one that each person can only answer for himself. I can only say: so long as I understand the words 'straight line', 'parallel' and 'intersect' as I do, I cannot but accept the parallels axiom. If someone else does not accept it, I can only assume that he understands these words differently. Their sense is indissolubly bound up with the axiom of parallels. Hence a thought which contradicts the axiom of parallels cannot be taken as a premise of an inference. But a true hypothetical thought, whose condition contradicted the axiom, could be used as a premise. This condition would then be retained in all judgements arrived at by means of our chains of inference. If at some point we arrived at a hypothetical judgement whose consequence contradicted known axioms, then we could conclude that the condition contradicting the axiom of parallels was false, and we should thereby have proved the axiom of parallels with the help of other axioms. But because it had been proved, it would lose its status as an axiom. In such a case we should really have given an indirect proof.
If, however, we went on drawing inference after inference and still did not come up against a contradiction anywhere, we should certainly become more and more inclined to regard the axiom as incapable of proof. Nevertheless we should still, strictly speaking, not have proved this to be so.
Now in his Grundlagen der Geometrie Hilbert is preoccupied with such questions as the consistency and independence of axioms. But here the sense of the word 'axiom' has shifted. For if an axiom must of necessity be true, it is impossible for axioms to be inconsistent with one another. So any discussion here would be a waste of words. But obvious though it is, it seems just not to have entered Hilbert's mind that he is not speaking of axioms in Euclid's sense at all when he discusses their consistency and independence. We could say that the word 'axiom', as he uses it, fluctuates from one sense to another without his noticing it. It is true that if we concentrate on the words of one of his axioms, the immediate impression is that we are dealing with an axiom of the Euclidean variety; but the words mislead us, because all the words have a different use from what they have in Euclid. At? 31 we
1 The quotation here is from the first edition of the Grundlagen der Geometric. Later editions give a different uxiom in plnce of thnt which Frege cites under ((. 4 (cd. ).
? 248 Logic in Mathematics
read 'Definition. The points of a straight line stand in certain relations to one another, for the description of which we appropriate the word "between". ' Now this definition is only complete once we are given the four axioms
11. 1 If A, B, C are points on a straight line, and B lies between A and C, then B lies between C and A.
11. 2 If A and C are two points on a straight line, then there is at least one point B lying between A and C, and at least one point D such that C lies between A and D.
11. 3 Given any three points on a straight line, there is one and only one which lies between the other two.
11. 4 Any four points A, B, C, D on a straight line can be so ordered that B lies between A and C and between A and D, and so that C lies between A and D and between B and D.
These axioms, then, are meant to form parts of a definition. Consequently these sentences must contain a sign which hitherto had no meaning, but which is given a meaning by all these sentences taken together. This sign is apparently the word 'between'. But a sentence that is meant to express an axiom may not contain a new sign. All the terms in it must be known to us. As long as the word 'between' remains without a sense, the sentence 'If A, B, C, are points on a straight line and B lies between A and C, then B lies between C and A' fails to express a thought.
An axiom, however, is always a true thought. Therefore, what does not express a thought, cannot express an axiom either. And yet one has the impression, on reading the first of these sentences, that it might be an axiom. But the reason for this is only that we are already accustomed to associate a sense with the word 'between'. In fact if in place of
we say
'B lies between A and C' 'B pat A nam C',
then we associate no sense with it. Instead of the so-called axiom 11. 1 we should have
'If B pat A nam C, then B pat C nam A'.
No one to whom these syllables 'pat' and 'nam' are unfamiliar will associate a sense with this apparent sentence. The same holds of the other three pseudo-axioms.
The question now arises whether, if we understand by A, B, C points on a straight line, an expression of the form
'B patA nam C'
will not at least ~ome to acquire a sense through the totality of these pseudo-sentences. I think not. We may perhaps hazard the guess that it will
come to the same thing as
? Logic in Mathematics 249 'B lies between A and C',
but it would be a guess and no more. What is to say that this matrix could not have several solutions?
But does, then, a definition have to be unambiguous? Are there not circumstances in which a certain give and take is a good thing?
Of course a2 = 4 does not determine unequivocally what a is to mean, but is there any harm in that? Well, if a is to be a proper name whose meaning is meant to be fixed, this goal will obviously not be achieved. On the contrary one can see that in this formula there is designated a concept under which
the numbers 2 and -2 fall. Once we see this, the ambiguity is harmless, but it means that we don't have a definition of an object.
If we want to compare this case with our pseudo-axiom, we have to compare the letter 'a' with 'between' or with 'pat-nam'. We must distinguish signs that designate from those which merely indicate. In fact the word 'between' or 'pat-nam' no more designates anything than does the letter 'a'. So we have here to disregard the fact that we usually associate a sense with the word 'between'. In this context it no more has a sense than does 'pat- nam'. Now to say that a sign which only indicates neither designates anything nor has a sense is not yet to say it could not contribute to the expression of a thought. It can do this by conferring generality of content on a simple sentence or on one made up of sentences.
Now there is, to be sure, a difference between our two cases; for whilst 'a' stands in for a proper name, 'pat-nam' stands in for the designation of a relation with three terms. As we call a function of one argument, whose value is always a truth-value, a concept, and a function of two arguments, whose value is always a truth-value, a relation, we can go yet a step further and call a function of three arguments whose value is always a truth-value, a relation with three terms. Then whilst the 'between-and' or the 'pat-nam' do not designate such a relation with three terms, they do indicate it, as 'a' indicates an object. Still there remains a distinction. We were able to find a concept designated in 'a2 = 4'.
What would correspond to this in the case of our pseudo-axioms? It is what I call a second level concept. In order to see more clearly what I understand by this, consider the following sentences:
'There is a positive number' 'There is a cube root of 1'.
We can see that these have something in common. A statement is being made, not about an object, however, but about a concept. In the first sentence it is the concept positive number, in the second it is the concept cube root of 1. And in each case it is asserted of the concept that it is not empty, but satisfied. It is indeed strictly a mistake to say 'The concept positive number is satisfied', for by saying this I seem to make the concept into an object, as the definite article 'the concept' shows.
