Only a sign which as yet has no sense can have a sense arbitrarily
assigned
to it.
Gottlob-Frege-Posthumous-Writings
In no other discipline does inference play so large a part, although inferences do occur here and there in other disciplines.
Part of the mathematician's activity, besides drawing inferences, is to give definitions.
Most disciplines are not concerned with the latter at all; only in jurisprudence is it of some importance, for although its subject-matter is quite different, it is in several respects close to mathematics.
Jurisprudence takes its materials from history and psychology and for this reason these must claim to have some share in it.
And there is nothing resembling this with mathematics.
Inferring and defining are subject to logical laws. From this it follows that logic is of greater importance to mathematics than to any other science. If one counts logic as part of philosophy, there will be a specially close
bond between mathematics and philosophy, and this is confirmed by the history of these sciences (Plato, Descartes, Leibniz, Newton, Kant).
But are there perhaps modes ofinference peculiar to mathematics which, for that very reason, do not belong to logic? Here one may point to the inference by mathematical induction from n to n + 1. Well, even a mode of inference peculiar to mathematics must be subject to a law and this law, if it is not logical in nature, will belong to mathematics, and can be ranked with the theorems or axioms of this science. For instance, mathematical
induction rests on the law that can be expressed as follows:
If the number 1 has the property c[J and if it holds generally for every positive whole number n that if it has the property c[J then n + 1 has the
property C/J, then every positive whole number has the property C/J.
If this law can be proved, it will be included amongst the theorems of mathematics; if it cannot, it will be included amongst the axioms. If one
draws inferences by mathematical induction, then one is actually making an application of this theorem or axiom; that is, this truth is taken as a premise of an inference. For example: the proof of the proposition (a + b) + n = a+ (b + n).
So likewise in other cases one can reduce a mode of inference that is peculiar to mathematics to a general law, if not a law of logic, then one of
1 The dating of the previous editors has been adopted (ed. ).
? Tracing the chains of inference backwards
mathematics. And from this law one can then draw consequences m accordance with general logical laws.
Now let us examine somewhat more closely what takes place m mathematics, beginning with inference.
We may distinguish two kinds of inferences: inferences from two premises and inferences from one premise.
Now we make advances in mathematics by choosing as the premises of an inference one or two propositions that have already been recognised as true. The conclusion obtained from these is a new truth of mathematics. And this can in turn be used, alone or together with another truth, in drawing further conclusions. It would be possible to call each truth thus obtained a theorem. But usually a truth is only called a theorem when it has not merely been obtained as the result of an inference, but is itself in turn used as a premise in the development of the science, and that not just for one but for a number of inferences. In this way chains of inference are formed connecting truths; and the further the science develops the longer and more numerous become the chains of inference and the greater the diversity of the theorems.
But one can also trace the chains of inference backwards by asking from what truths each theorem has been inferred. As the diversity of theorems becomes greater as we go forward along the chains of inference, so, as we step backwards, the circle of theorems closes in more and more. Whereas it appears that there is no limit to the number of steps forward we can take, when we go backwards we must eventually come to an end by arriving at truths which cannot themselves be inferred in turn from other truths. Going backwards we come up against the axioms and postulates. We may come up against definitions as well, but we shall take a closer look at that later. If we start from a theorem and trace the chains of inference backwards until we arrive at other theorems or at axioms, postulates or definitions, we discover chains of inference starting from known theorems, axioms, postulates or definitions and terminating with the theorem in question.
The totality of these inference-chains constitutes the proofof the theorem. We may say that a proof starts from propositions that are accepted as true and leads via chains of inferences to the theorem. But it can also happen that a proof consists only of a single chain of inference. In most cases a proof will proceed via truths which are not called theorems for the simple reason that they occur only in this proof, and are not used elsewhere. A proof does not only serve to convince us of the truth of what is proved: it also serves to reveal logical relations between truths. This is why we already find in Euclid proofs of truths that appear to stand in no need of proof because they are obvious without one.
Science demands that we prove whatever is suceptible of proof and that we do n,ot rest until we come up against something unprovable. It must endeavour to make the circle of unprovable primitive truths as small as possible, for the whole of mathematics is contained in these primitive truths
Proof
Primitive truths
204 Logic in Mathematics
? Logic in Mathematics 205
as in a kernel. Our only concern is to generate the whole of mathematics from this kernel. The essence of mathematics has to be defined by this kernel of truths, and until we have learnt what these primitive truths are, we cannot be clear about the nature of mathematics. If we assume that we have succeeded in discovering these primitive truths, and that mathematics has been developed from them, then it will appear as a system of truths that are connected with one another by logical inference.
Euclid had an inkling of this idea of a system; but he failed to realize it and it almost seems as if at the present time we were further from this goal than ever. We see mathematicians each pursuing his own work on some fragment of the subject, but these fragments do not fit together into a system; indeed the idea of a system seems almost to have been lost. And yet the striving after a system is a justified one. We cannot long remain content with the fragmentation that prevails at present. Order can only be created by a system. But in order to construct a system it is necessary that in any step forward we take we should be aware of the logical inferences involved.
When an inference is being drawn, we must know what its premises are. We must not allow the premises to be confused with the laws of inference, which are purely logical; otherwise the logical purity of the inferences will be lost and it would not be possible, in the confusion of premises with laws of inference, clearly to distinguish the former. But if we have no clear recognition of what the premises are, we can have no certainty of arriving at the primitive truths, and failing that we cannot construct a system. For this reason we must avoid such expressions as 'a moment's reflection shows that' or 'as we can easily see'. We must put the moment's reflection into
words so that we can see what inferences it consists of and what premises it makes use of. In mathematics we must never rest content with the fact that something is obvious or that we are convinced of something, but we must strive to obtain a clear insight into the network of inferences that support our conviction. Only in this way can we discover what the primitive truths are, and only in this way can a system be constructed.
Let us now take a closer look at the axioms, postulates and definitions:
The axioms are truths as are the theorems, but they are truths for which no proof can be given in our system, and for which no proof is needed. It follows from this that there are no false axioms, and that we cannot accept a thought as an axiom if we are in doubt about its truth; for it is either false and hence not an axiom, or it is true but stands in need of proof and hence is not an axiom. Not every truth for which no proof is required is an axiom, for such a truth might still be proved in our system. Whether a truth is an axiom depends therefore on the system, and it is possible for a truth to be an nxiom in one system and not in another. That is to say, it is conceivable that there should be a truth A and a truth B, each of which can be proved from the other in conjunction with truths C, D, E, F, whilst the truths C, D, E, F nre not sufficient on their own to prove either A or B. If now C, D, E, F may serve as axioms, then we have the choice of regarding A, C, D, E, F as
The system
of mathematic! i
The axiom?
? ? Parenthetical remark on sentence and thought
axioms and B as a theorem, orB, C, D, E, F as axioms, and A as a theorem. We can see from this that the possibility of one system does not necessarily rule out the possibility of an alternative system, and that we may have a choice between different systems. So it is really only relative to a particular system that one can speak of something as an axiom.
Here, in passing, I may say something about the expressions 'thought' and 'sentence'. I use the word 'sentence' to refer to a sign that is normally complex, whether it is made up of sounds or written signs. Of course this sign must have a sense. Here I am only considering sentences in which we state or assert something. We can translate sentences into another language. The sentence in the other language is different from the original one, for its constituents (component sounds) are different and are put together differently; but if the translation is correct, it will express the same sense and of course it is really the sense that concerns us. The sentence is of value to us because of the sense that we grasp in it, which is recognizably the same in the translation too. I call this sense a thought. What we prove is not a sen- tence, but a thought. And it is neither here nor there which language is used in giving the proof. It is true that in mathematics we often speak of proofs of a theorem [Lehrsatz], understanding by the word 'sentence' ['Satz'] what I am calling a thought, or perhaps not distinguishing clearly between the expression in words or signs and the thought expressed. 1 A thought cannot be perceived by the senses, but in the sentence it is represented by what can be heard or seen. For this reason I do not use 'Lehrsatz' but 'Theorem', and not 'Grundsatz' but 'Axiom', understanding by theorems and axioms true thoughts. This, however, is to imply that a thought is not something subjective, is not the product of any form of mental activity; for the thought that we have in Pythagoras's theorem is the same for everybody, and its truth is quite independent of its being thought by so-and-so or indeed by anyone at all. We are not to regard thinking as the act of producing a thought, but as that of grasping a thought.
Postulates seem at first sight to be essentially different from axioms. In Euclid we have the postulate 'Let it be postulated that a straight line may be drawn from any point to any other'.
This is obviously introduced with a view to making constructions. The postulates, so it seems, present the simplest procedures for making every construction, and postulate their possibility. At first we might perhaps think that none of this is of any help in providing proofs, but only for solving problems. But this would be a mistake, for sometimes an auxiliary line is needed for a proof, and sometimes an auxiliary point, an auxiliary number-an auxiliary object of some kind. In the proof of a theorem an auxiliary object is one of which nothing is said in the theorem, but which is
1 The play that Frege is here making on the words 'Lehrsatz' and 'Satz' is lost in translation, so we have enclosed them in square brackets after their English equivalents (trans. ).
Postulates
206 Logic in Mathematics
? Logic in Mathematics 207
required for the proof, so that this would collapse if there were no such object. And if there is no such object, it seems that we must be able to create one and we need a postulate to ensure that this is possible. But what in actual fact is this drawing a line? It is not, at any rate, a line in the geometrical sense that we are creating when we make a stroke with a pencil. And how in this way are we to connect a point in the interior of Sirius with a point in Rigel? Our postulate cannot refer to any such external procedure. It refers rather to something conceptual. But what is here in question is not a subjective, psychological possibility, but an objective one. Surely the truth of a theorem cannot really depend on something we do, when it holds quite independently of us. So the only way of regarding the matter is that by drawing a straight line we merely become ourselves aware of what obtains independently of us. So the content of our postulate is essentially this, that given any two points there is a straight line connecting them. So a postulate is a truth as is an axiom, its only peculiarity being that it asserts the existence of something with certain properties. From this it follows that there is no real need to distinguish axioms and postulates. A postulate can be regarded as a special case of an axiom.
We come to definitions. Definitions proper must be distinguished from illustrative examples. In the first stages of any discipline we cannot avoid the use of ordinary words. But these words are, for the most part, not really appropriate for scientific purposes, because they are not precise enough and fluctuate in their use. Science needs technical terms that have precise and fixed meanings, and in order to come to an understanding about these meanings and exclude possible misunderstandings, we give examples illustrating their use. Of course in so doing we have again to use ordinary words, and these may display defects similar to those which the examples are intended to remove. So it seems that we shall then have to do the same thing over again, providing new examples. Theoretically one will never really achieve one's goal in this way. In practice, however, we do manage to come to an understanding about the meanings of words. Of course we have to be able to count on a meeting of minds, on others guessing what we have in mind. But all this precedes the construction of a system and does not helong within a system. In constructing a system it must be assumed that the words have precise meanings and that we know what they are. Hence we can at this point leave illustrative examples out of account and turn our nttention to the construction of a system.
In constructing a system the same group of signs, whether they are Hounds or combinations of sounds (spoken signs) or written signs, may occur over and over again. This gives us a reason for introducing a simple Nign to replace such a group of signs with the stipulation that this simple sign is always to take the place of that group of signs. As a sentence is generally 11 complex sign, so the thought expressed by it is complex too: in fact it is put together in such a way that parts of the thought correspond to parts of the sentence. So as a general rule when a group of signs occurs in a sentence
Definitions
Illustrative examples
Definition
proper
? 208 Logic in Mathematics
it will have a sense which is part of the thought expressed. Now when a simple sign is thus introduced to replace a group of signs, such a stipulation is a definition. The simple sign thereby acquires a sense which is the same as that of the group of signs. Definitions are not absolutely essential to a system. We could make do with the original group o f signs. The introduction of a simple sign adds nothing to the content; it only makes for ease and simplicity of expression. So definition is really only concerned with signs. We shall call the simple sign the definiendum, and the complex group of signs which it replaces the definiens. The definiendum acquires its sense only from the definiens. This sense is built up out of the senses of the parts of the definiens. When we illustrate the use of a sign, we do not build its sense up out of simpler constituents in this way, but treat it as simple. All we do is to guard against misunderstanding where an expression is ambiguous.
A sign has a meaning once one has been bestowed upon it by definition, and the definition goes over into a sentence asserting an identity. Of course the sentence is really only a tautology and does not add to our knowledge. It contains a truth which is so self-evident that it appears devoid of content, and yet in setting up a system it is apparently used as a premise. I say apparently, for what is thus presented in the form of a conclusion makes no addition to our knowledge; all it does in fact is to effect an alteration of expression, and we might dispense with this if the resultant simplification of expression did not strike us as desirable. In fact it is not possible to prove something new from a definition alone that would be unprovable without it. When something that looks like a definition really makes it possible to prove something which could not be proved before, then it is no mere definition but must conceal something which would have either to be proved as a theorem or accepted as an axiom. Of course it may look as if a definition makes it possible to give a new proof. But here we have to distinguish between a sentence and the thought it expresses. If the definiens occurs in a sentence and we replace it by. the definiendum, this does not affect the thought at all. It is true we get a different sentence if we do this, but we do
not get a different thought. Of course we need the definition if, in the proof of this thought, we want it to assume the form of the second sentence. But if the thought can be proved at all, it can also be proved in such a way that it assumes the form of the first sentence, and in that case we have no need of the definition. So if we take the sentence as that which is proved, a definition may be essential, but not if we regard the thought as that which is to be proved.
It appears from this that definition is, after all, quite inessential. In fact considered from a logical point of view it stands out as something wholly inessential and dispensable. Now of course I can see that strong exception will be taken to this. We can imagine someone saying: Surely we are undertaking ~ logical analysis when we give a definition. You might as well say that it doesn't matter whether I carry out a chemical analysis of a body in order to see what elements it is composed of, as say that it is immaterial
? Logic in Mathematics 209
whether I carry out a logical analysis of a logical structure in order to find out what its constituents are or leave it unanalysed as if it were simple, when it is in fact complex. It is surely impossible to make out that the activity of defining something is without any significance when we think of the considerable intellectual effort required to furnish a good definition. -There is certainly something right about this, but before I go into it more closely, I want to stress the following point. To be without logical significance is still by no means to be without psychological significance. When we examine what actually goes on in our mind when we are doing intellectual work, we find that it is by no means always the case that a thought is present to our consciousness which is clear in all its parts. For example, when we use the word 'integral', are we always conscious of everything appertaining to its sense? I believe that this is only very seldom the case. Usually just the word is present to our consciousness, allied no doubt with a more or less dim awareness that this word is a sign which has a sense, and that we can, if we wish, call this sense to mind. But we are usually content with the knowledge that we can do this. If we tried to call to mind everything appertaining to the sense of this word, we should make no headway. Our minds are simply not comprehensive enough. We often need to use a sign with which we associate a very complex sense. Such a sign seems, so to speak, a receptacle for the sense, so that we can carry it with us, while being always aware that we can open this receptacle should we have need of what it contains. It follows from this that a thought, as I understand the word, is in no way to be identified with a content of my consciousness. If therefore we need such signs-signs in which, as it were, we conceal a very complex sense as in a receptacle-we also need definitions so that we can cram this sense into the receptacle and also take it out again. So if from a logical point of view definitions are at hottom quite inessential, they are nevertheless of great importance for thinking as this actually takes place in human beings.
An objection was mentioned above which arose from the consideration that it is by means of definitions that we perform logical analyses. In the development of science it can indeed happen that one has used a word, a sign, an expression, over a long period under the impression that its sense is simple until one succeeds in analysing it into simpler logical constituents. By means of such an analysis, we may hope to reduce the number of axioms; for it may not be possible to prove a truth containing a complex constituent so long as that constituent remains unanalysed; but it may be possible, given nn analysis, to prove it from truths in which the elements of the analysis occur. This is why it seems that a proof may be possible by means of a definition, if it provides an analysis, which would not be possible without this analysis, and this seems to contradict what we said earlier. Thus what seemed to be an axiom before the analysis can appear as a theorem after the analysis.
But how does one judge whether a logical analysis is correct? We cannot prove it to be so. The most one can be certain of is that as far as the form of
? 210 Logic in Mathematics
words goes we have the same sentence after the analysis as before. But that the thought itself also remains the same is problematic. When we think that we have given a logical analysis of a word or sign that has been in use over a long period, what we have is a complex expression the sense of whose parts is known to us. The sense of the complex expression must be yielded by that of its parts. But does it coincide with the sense of the word with the long
established use? I believe that we shall only be able to assert that it does when this is self-evident. And then what we have is an axiom. But that the simple sign that has been in use over a long period coincides in sense with that of the complex expression that we have formed, is just what the definition was meant to stipulate.
We have therefore to distinguish two quite different cases:
(1) We construct a sense out of its constituents and introduce an entirely new sign to express this sense. This may be called a 'constructive definition', but we prefer to call it a 'definition' tout court.
(2) We have a simple sign with a long established use. We believe that we can give a logical analysis of its sense, obtaining a complex expression which in our opinion has the same sense. We can only allow something as a constituent of a complex expression if it has a sense we recognize. The sense of the complex expression must be yielded by the way in which it is put together. That it agrees with the sense of the long established simple sign is not a matter for arbitrary stipulation, but can only be recognized by an immediate insight. No doubt we speak of a definition in this case too. It might be called an 'analytic definition' to distinguish it from the first case. But it is better to eschew the word 'definition' altogether in this case, because what we should here like to call a definition is really to be regarded as an axiom. In this second case there remains no room for an arbitrary stipulation, because the simple sign already has a sense.
Only a sign which as yet has no sense can have a sense arbitrarily assigned to it. So we shall stick to our original way of speaking and call only a constructive definition a definition. According to that a definition is an arbitrary stipulation which confers a sense on a simple sign which previously had none. This sense has, of course, to be expressed by a complex sign whose sense results from the way it is put together.
Now we still have to consider the difficulty we come up against in giving a logical analysis when it is problematic whether this analysis is correct.
Let us assume that A is the long-established sign (expression) whose sense we have attempted to analyse logically by constructing a complex expression that gives the analysis. Since we are not certain whether the analysis is successful, we are not prepared to present the complex expression as one which can be replaced by the simple sign A. If it is our intention to put forward a definition proper, we are not entitled to choose the sign . 4, which already has a sense, but we must choose a fresh sign B, say, which has the sense of the complex expression only in virtue of the definition. The question now is whether A and B have the same sense. But we can bypa11
? Logic in Mathematics 211
this question altogether if we are constructing a new system from the bottom up; in that case we shall make no further use of the sign A-we shall only use B. We have introduced the sign B to take the place of the complex expression in question by arbitrary fiat and in this way we have conferred a sense on it. This is a definition in the proper sense, namely a constructive definition.
If we have managed in this way to construct a system for mathematics without any need for the sign A, we can leave the matter there; there is no need at all to answer the question concerning the sense in which-whatever it may be-this sign had been used earlier. In this way we court no objections. However it may be felt expedient to use sign A instead of sign B. But if we do this, we must treat it as an entirely new sign which had no sense prior to the definition. We must therefore explain that the sense in which this sign was used before the new system was constructed is no longer of any concern to us, that its sense is to be understood purely from the constructive definition that we have given. In constructing the new system we can take no account, logically speaking, of anything in mathematics that existed prior to the new system. Everything has to be made anew from the ground up. Even anything that we may have accomplished by our analytical activities is to be regarded only as preparatory work which does not itself make any appearance in the new system itself.
Perhaps there still remains a certain unclarity. How is it possible, one may ask, that it should be doubtful whether a simple sign has the same sense as a complex expression if we know not only the sense of the simple sign, but can recognize the sense of the complex one from the way it is put together? The fact is that if we really do have a clear grasp of the sense of the simple sign, then it cannot be doubtful whether it agrees with the sense of the complex expression. If this is open to question although we can clearly recognize the sense of the complex expression from the way it is put together, then the reason must lie in the fact that we do not have a clear grasp of the sense of the simple sign, but that its outlines are confused as if we saw it through a mist. The effect of the logical analysis of which we spoke will then be precisely this-to articulate the sense clearly. Work of this kind is very useful; it does not, however, form part of the construction of the system, but must take place beforehand. Before the work of construction is begun, the building stones have to be carefully prepared so ns to be usable; i. e. the words, signs, expressions, which are to be used, must have a clear sense, so far as a sense is not to be conferred on them in the Nystem itself by means of a constructive definition.
We stick then to our original conception: a definition is an arbitrary . vtlpulation by which a new sign is introduced to take the place of a complex expression whose sense we know from the way it is put together. A sign which hitherto had no sense acquires the sense of a complex expression by definition.
? ? 212 Logic in Mathematics
When we look around us at the writings of mathematicians, we come acros! many things which look like definitions, and are even called such, without really being definitions. Such definitions are to be compared with thosf stucco-embellishments on buildings which look as though they supported something whereas in reality they could be removed without the slightest detriment to the building. We can recognize such definitions by the fact that no use is made of them, that no proof ever draws upon them. But if a wore or sign which has been introduced by definition is used in a theorem, thf only way in which it can make its appearance there is by applying thf definition or the identity which follows immediately from it. If such an application is never made, then there must be a mistake somewhere. Of course the application may be tacit. That is why it is so important, if we are to have a clear insight into what is going on, for us to be able to recognize the premises of every inference which occurs in a proof and the law of inference in accordance with which it takes place. So long as proofs are drawn up in conformity with the practice which is everywhere current at the present time, we cannot be certain what is really used in the proof, what it rests on. And so we cannot tell either whether a definition is a mere stucco- , definition which serves only as an ornament, and is only included because it is in fact usual to do so, or whether it has a deeper justification. That is why it is so important that proofs should be drawn up in accordance with the requirements we have laid down.
We can characterize another kind of inadmissible definition by a metaphor from algebra. Let us assume that three unknowns x, y, z occur in three equations. Then they can be determined by means of these equations. , Strictly speaking, however, they are determined only for the case where there is only one solution. In a similar way the words 'point', 'straight line',, 'surface' may occur in several sentences. Let us assume that these words have as yet no sense. It may be required to find a sense for each of these words such that the sentences in question express true thoughts. But have, we here provided a means for determining the sense uniquely? At any rate not in general; and in most cases it must remain undecided how many solutions are possible. But if it can be proved that only one solution is' possible, then this is given by assigning, via a constructive definition, a sense in turn to each of the words that needs defining. But we cannot regard as a definition the system of sentences in each of which there occur several of the expressions that need defining.
A special case of this is where only one sign, which has as yet no sense, occurs in one or more sentences. Let us assume that the other constituents of the sentences are known. The question is now what sense has to be given to this sign for the sentences to have a sense such that the thoughts expressed in them are true. This case is to be compared to that in which the letter x occ. urs in one or more equations whose other constituents are known, where the problem is: what meaning do we have to give the letter Jt for the equations to express true thoughts? If there are several equations,j
? Logic in Mathematics 213
this problem will usually be insoluble. It is obvious that in general no number whatsoever is determined in this way. And it is like this with the case in hand. No sense accrues to a sign by the mere fact that it is used in one or more sentences, the other constituents of which are known. In algebra we have the advantage that we can say something about the possible solutions and how many there are-an advantage one does not have in the general case. But a sign must not be ambiguous. Freedom from ambiguity is the most important requirement for a system of signs which is to be used for scientific purposes. One surely needs to know what one is talking about and the statements one is making, what thoughts one is expressing.
Now it is true that there have even been people, who have fancied themselves logicians, who have held that concept-words (nomina appel- /ativa) are distinguished from proper names by the fact that they are
ambiguous. The word 'man', for example, means Plato as well as Socrates and Charlemagne. The word 'number' designates the number 1 as well as the number 2, and so on. Nothing is more wrong-headed. Of course I can use the words 'this man' to designate now this man, now that man. But still on each single occasion I mean them to designate just one man. The sentences of our everyday language leave a good deal to guesswork. It is the surrounding circumstances that enable us to make the right guess. The sentence I utter does not always contain everything that is necessary; a great deal has to be supplied by the context, by the gestures I make and the direction of my eyes. But a language that is intended for scientific employment must not leave anything to guesswork. A concept-word combined with the demonstrative pronoun or definite article often has in this way the logical status of a proper name in that it serves to designate a single determinate object. But then it is not the concept-word alone, but the whole consisting of the concept-word together with the demonstrative pronoun and accompanying circumstances which has to be understood as a proper name. We have an actual concept-word when it is not accompanied by the definite article or demonstrative pronoun and is accompanied either by no article or by the indefinite article, or when it is combined with 'all', 'no' and 'some'. We must not think that I mean to assert something about an African chieftain from darkest Africa who is wholly unknown to me, when I say 'All men are mortal'. I am not saying anything about either this man or that man, but I am subordinating the concept man to the concept of what is mortal. In the sentence 'Plato is mortal' we have an instance of subsumption, in the sentence 'All men are mortal' one of subordination. What is being spoken about here is a concept, not an individual thing. We must not think either that the sense of the sentence 'Cato is mortal' is contained in that of the sentence 'All men are mortal', so that by uttering the latter sentence I should at the same time have expressed the thought contained in the former sentence. The matter is rather as follows. By the sentence 'All men are mortal' I say 'If anything is a man, it is mortal'. By an inference from the general to the particular, I obtain from this the sentence 'If Cato is a man,
? 214 Logic in Mathematics
then Cato is mortal'. Now I still need a second premise, namely 'Cato is a man'. From these two premises I infer 'Cato is mortal'.
Since therefore we need inferences and a second premise, the thought that Cato is mortal is not included in what is expressed by the sentence 'All men are mortal', and so 'man' is not an ambiguous word which amongst its many meanings has that which we designate by the proper name 'Plato'. On the contrary, a concept-word simply serves to designate a concept. And a concept is quite different from an individual. If I say 'Plato is a man', I am not as it were giving Plato a new name-the name 'man'-but I am saying that Plato falls under the concept man. Likewise we have two quite different cases when I give the definition '2 + 1 = 3' and when I say '2 + 1 is a prime number'. In the first case I confer on the sign '3', which is so far empty, a sense and a meaning by saying that it is to mean the same as the combination of signs '2 + 1'. In the second case I am subsuming the meaning of '2 + 1' under the concept prime number. I do not give it a new name by doing that. The fact therefore that I subsume different objects under the same concept does not make the concept-word ambiguous. So in the sentences
'2 is a prime number' '3 is a prime number' '5 is a prime number'
the word 'prime number' is not somehow ambiguous because 2, 3, 5 are different numbers; for 'prime number' is not a name which is given to these numbers.
It is of the essence of a concept to be predicative. If an empty proper name occurs in a sentence, the other parts of which are known, so that the sentence has a sense once a sense is given to that proper name, then, so long as the proper name remains empty, the sentence contains the possibility of a statement, but we do not have an object about which anything is being said. So the sentence 'x is a prime number', does indeed contain the possibility of a statement, but so long as no meaning is given to the letter 'x', we do not have an object about which anything is being said. Another way of putting this would be to say: we have a concept but we have no object subsumed under it. If we take as a further instance the sentence 'x increased by 2 is divisible by 4' then we have a concept again. We can take these two concepts as characteristic marks of a new concept by putting together the sentences 'x is a prime number' and 'x increased by 2 is divisible by 4'. Under this concept there falls only one object-the number 2. But a concept under which only one object falls is still a concept; this does not make the expression for it into a proper name.
Our position is this: we cannot recognize sentences containing an empty sign, the otrn:r constituents of which are known, as definitions. But such sentences can have an explanatory role by providing a clue to what is to be understood by the sign or word in question.
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I have read that verbal definitions are considered faulty, and 1t ts argued that we should really have no further truck with such definitions. By way of example reference was made to a definition given by me, but it was not said what a verbal definition was. 1 Of course every definition makes use of words or signs. Perhaps what is meant by a verbal definition is one in which the definiens contains a word which is a mere word as such, having no sense. Certainly this should not be allowed, but from the fact that the reader attaches no sense to a word it does not follow that the author of the definition has attached no sense to it. The insistence on sense is absolutely justified, and all the more so since many mathematicians seem to prove what are merely sentences without bothering whether they have a sense and what sense it is they have.
How little value is commonly placed on sense and definitions can be seen from the sharply conflicting accounts that mathematicians give of what number is. (We are speaking here of the natural numbers. ) Weierstrass says 'Number is a series of things of the same kind'. Another says that certain conventional shapes produced by writing, such as 2 and 3, are numbers. 2 A third is of the opinion: if I hear the clock strike three I see nothing in this of what three is. Therefore it cannot be anything visible. If I see three lines, then I hear nothing in this of what three is. Therefore it can be nothing audible either. An axiom is not a visible thing and so if we speak of three axioms, the three here is nothing visible either. Number cannot be anything whatever which can be perceived by the senses. 3
Obviously each of these attaches a different sense to the word 'number'. So the arithmetics of these three mathematicians must be quite different. A sentence from the first mathematician must have a quite different sense from the equivalent-sounding sentence of the second mathematician. This resembles what it would be like if botanists were not agreed about what they wished to understand by a plant, so that for one botanist a plant was, say, an organically developing structure, for another a human artefact, and for a third something that was not perceptible by the senses at all. Such a situation would certainly not give rise to a common science of botany.
But why should it not be possible to lay it down that by a number is to be meant a series of things of the same kind? Admittedly we can raise objections to such a course. For one thing, it may be thought that the sense of the word 'series' is not firmly enough established. Are we to think in this connection of a spatial ordering, or of a temporal ordering or of a spatia- temporal one perhaps? Further it is not clear what we are to understand by 'of the same kind'. For example, are the notes of a scale of the same kind qua
1 Frege is here referring to the article Ober die Stellung der Definition in der Axiomatik by A. Schoenjliess in Jahresbericht der deutschen Mathematiker- Vereinlgung XX (1911), pp. 222-255 (ed. ).
2 The reference is to the so-called 'formal' theory of arithmetic (arithmetic as a game with signs) held by Frege's contemporaries E. He/ne and J. Thomae (ed. ).
3 The reference is probably to G. Cantor (ed. ).
? 216 Logic in Mathematics
notes, or are they of the same kind only if they have the same pitch? But let us assume that explanations were given which cleared these matters up. A train is a series of objects of the same kind "'hich moves along rails on wheels. It may be thought that the engine is nevertheless something of a different kind. Still that makes no essential difference. And so such a number comes steaming here from Berlin. Let us assume that the science of these numbers has been set up. There is no doubt that it must be entirely different from the science in which certain shapes that one makes on a writing surface with a writing instrument are called numbers. Even if the form of words is the same, the thought expressed must be quite different. Now it is striking that the sentences of these fundamentally different sciences, each of which is called arithmetic, are constituted by precisely the same words. And it is even more striking that the practitioners of these sciences have no inkling that their sciences are fundamentally different. They all believe that they are doing arithmetic, and the same arithmetic at that, the same number theory, although what one of them is calling a number has no resemblance at all to what another is calling a number.
How is this possible? One would almost think that mathematicians regard the words used, the form of an expression, as the essential thing, and the thought expressed as quite inessential. Perhaps they think 'The thoughts contained in sentences are really no concern of mathematicians-they are a matter for philosophers; and everything to do with philosophy is of course extremely imprecise, uncertain and essentially unscientific. A mathematician who remains true to his scientific calling will have nothing to do with it. True, it can happen to even the best of them, in a moment of weakness, to let a definition slip, or something which looks like one, but we should not accord any significance to that. It is all one with a man sneezing. Really the only thing that matters is that they should all agree on the words and formulae they use. That is enough for a mathematician who has not been infected by philosophy. '
But is that then a science which proves sentences without knowing what it proves? But is it the case in actual fact that scientists do agree in the words they use? Are not mathematical works written in different languages, and are they not translated into other languages? In which case, of course, we no longer have the same form of words. But there must be something else which is preserved. And what can this be but the sense? So the thought, the sense of a sentence, cannot after all be wholly irrelevant. And does one not feel in the depth of one's being that the thoughts are the essential thing-that it is in fact these alone that we are concerned about?
But how do they come to be treated as irrelevant? How can one possibly imagine that two quite different sciences should really have the same content? Is it only because they are both called arithmetic and both treat of numbers, althpugh what is called a number in the one is quite different from what is called a number in the other? Or is not the explanation rather that we have really to do with the same science; that this man does attach the
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same sense to the word number as that man, only he doesn't manage to get hold of it properly? Perhaps the sense appears to both through such a haze that when they make to get hold of it, they miss it. One of them makes a grasp to the right perhaps and the other to the left, and so although they mean to get hold of the same thing, they fail to do so. How thick the fog must be for this to be possible! But it must surely show up in the proofs that they have not got hold of the same sense. Yes, it could not fail to, if the proofs were drawn up in a logically rigorous way, with no gaps in the chains of inference. But this is just what we shall fail to find. If no use is ever made of a definition, there might as well not be one. However wide it may be of the actual target, no-one will notice. Another mathematician's shot may miscarry on the other side; but since he makes no use of his definition, this too might as well not be there. We can see in this way how definitions, which seem to be utterly irreconcilable, lie peacefully alongside one another like the animals in paradise. I f only things were really like that!
Inferring and defining are subject to logical laws. From this it follows that logic is of greater importance to mathematics than to any other science. If one counts logic as part of philosophy, there will be a specially close
bond between mathematics and philosophy, and this is confirmed by the history of these sciences (Plato, Descartes, Leibniz, Newton, Kant).
But are there perhaps modes ofinference peculiar to mathematics which, for that very reason, do not belong to logic? Here one may point to the inference by mathematical induction from n to n + 1. Well, even a mode of inference peculiar to mathematics must be subject to a law and this law, if it is not logical in nature, will belong to mathematics, and can be ranked with the theorems or axioms of this science. For instance, mathematical
induction rests on the law that can be expressed as follows:
If the number 1 has the property c[J and if it holds generally for every positive whole number n that if it has the property c[J then n + 1 has the
property C/J, then every positive whole number has the property C/J.
If this law can be proved, it will be included amongst the theorems of mathematics; if it cannot, it will be included amongst the axioms. If one
draws inferences by mathematical induction, then one is actually making an application of this theorem or axiom; that is, this truth is taken as a premise of an inference. For example: the proof of the proposition (a + b) + n = a+ (b + n).
So likewise in other cases one can reduce a mode of inference that is peculiar to mathematics to a general law, if not a law of logic, then one of
1 The dating of the previous editors has been adopted (ed. ).
? Tracing the chains of inference backwards
mathematics. And from this law one can then draw consequences m accordance with general logical laws.
Now let us examine somewhat more closely what takes place m mathematics, beginning with inference.
We may distinguish two kinds of inferences: inferences from two premises and inferences from one premise.
Now we make advances in mathematics by choosing as the premises of an inference one or two propositions that have already been recognised as true. The conclusion obtained from these is a new truth of mathematics. And this can in turn be used, alone or together with another truth, in drawing further conclusions. It would be possible to call each truth thus obtained a theorem. But usually a truth is only called a theorem when it has not merely been obtained as the result of an inference, but is itself in turn used as a premise in the development of the science, and that not just for one but for a number of inferences. In this way chains of inference are formed connecting truths; and the further the science develops the longer and more numerous become the chains of inference and the greater the diversity of the theorems.
But one can also trace the chains of inference backwards by asking from what truths each theorem has been inferred. As the diversity of theorems becomes greater as we go forward along the chains of inference, so, as we step backwards, the circle of theorems closes in more and more. Whereas it appears that there is no limit to the number of steps forward we can take, when we go backwards we must eventually come to an end by arriving at truths which cannot themselves be inferred in turn from other truths. Going backwards we come up against the axioms and postulates. We may come up against definitions as well, but we shall take a closer look at that later. If we start from a theorem and trace the chains of inference backwards until we arrive at other theorems or at axioms, postulates or definitions, we discover chains of inference starting from known theorems, axioms, postulates or definitions and terminating with the theorem in question.
The totality of these inference-chains constitutes the proofof the theorem. We may say that a proof starts from propositions that are accepted as true and leads via chains of inferences to the theorem. But it can also happen that a proof consists only of a single chain of inference. In most cases a proof will proceed via truths which are not called theorems for the simple reason that they occur only in this proof, and are not used elsewhere. A proof does not only serve to convince us of the truth of what is proved: it also serves to reveal logical relations between truths. This is why we already find in Euclid proofs of truths that appear to stand in no need of proof because they are obvious without one.
Science demands that we prove whatever is suceptible of proof and that we do n,ot rest until we come up against something unprovable. It must endeavour to make the circle of unprovable primitive truths as small as possible, for the whole of mathematics is contained in these primitive truths
Proof
Primitive truths
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as in a kernel. Our only concern is to generate the whole of mathematics from this kernel. The essence of mathematics has to be defined by this kernel of truths, and until we have learnt what these primitive truths are, we cannot be clear about the nature of mathematics. If we assume that we have succeeded in discovering these primitive truths, and that mathematics has been developed from them, then it will appear as a system of truths that are connected with one another by logical inference.
Euclid had an inkling of this idea of a system; but he failed to realize it and it almost seems as if at the present time we were further from this goal than ever. We see mathematicians each pursuing his own work on some fragment of the subject, but these fragments do not fit together into a system; indeed the idea of a system seems almost to have been lost. And yet the striving after a system is a justified one. We cannot long remain content with the fragmentation that prevails at present. Order can only be created by a system. But in order to construct a system it is necessary that in any step forward we take we should be aware of the logical inferences involved.
When an inference is being drawn, we must know what its premises are. We must not allow the premises to be confused with the laws of inference, which are purely logical; otherwise the logical purity of the inferences will be lost and it would not be possible, in the confusion of premises with laws of inference, clearly to distinguish the former. But if we have no clear recognition of what the premises are, we can have no certainty of arriving at the primitive truths, and failing that we cannot construct a system. For this reason we must avoid such expressions as 'a moment's reflection shows that' or 'as we can easily see'. We must put the moment's reflection into
words so that we can see what inferences it consists of and what premises it makes use of. In mathematics we must never rest content with the fact that something is obvious or that we are convinced of something, but we must strive to obtain a clear insight into the network of inferences that support our conviction. Only in this way can we discover what the primitive truths are, and only in this way can a system be constructed.
Let us now take a closer look at the axioms, postulates and definitions:
The axioms are truths as are the theorems, but they are truths for which no proof can be given in our system, and for which no proof is needed. It follows from this that there are no false axioms, and that we cannot accept a thought as an axiom if we are in doubt about its truth; for it is either false and hence not an axiom, or it is true but stands in need of proof and hence is not an axiom. Not every truth for which no proof is required is an axiom, for such a truth might still be proved in our system. Whether a truth is an axiom depends therefore on the system, and it is possible for a truth to be an nxiom in one system and not in another. That is to say, it is conceivable that there should be a truth A and a truth B, each of which can be proved from the other in conjunction with truths C, D, E, F, whilst the truths C, D, E, F nre not sufficient on their own to prove either A or B. If now C, D, E, F may serve as axioms, then we have the choice of regarding A, C, D, E, F as
The system
of mathematic! i
The axiom?
? ? Parenthetical remark on sentence and thought
axioms and B as a theorem, orB, C, D, E, F as axioms, and A as a theorem. We can see from this that the possibility of one system does not necessarily rule out the possibility of an alternative system, and that we may have a choice between different systems. So it is really only relative to a particular system that one can speak of something as an axiom.
Here, in passing, I may say something about the expressions 'thought' and 'sentence'. I use the word 'sentence' to refer to a sign that is normally complex, whether it is made up of sounds or written signs. Of course this sign must have a sense. Here I am only considering sentences in which we state or assert something. We can translate sentences into another language. The sentence in the other language is different from the original one, for its constituents (component sounds) are different and are put together differently; but if the translation is correct, it will express the same sense and of course it is really the sense that concerns us. The sentence is of value to us because of the sense that we grasp in it, which is recognizably the same in the translation too. I call this sense a thought. What we prove is not a sen- tence, but a thought. And it is neither here nor there which language is used in giving the proof. It is true that in mathematics we often speak of proofs of a theorem [Lehrsatz], understanding by the word 'sentence' ['Satz'] what I am calling a thought, or perhaps not distinguishing clearly between the expression in words or signs and the thought expressed. 1 A thought cannot be perceived by the senses, but in the sentence it is represented by what can be heard or seen. For this reason I do not use 'Lehrsatz' but 'Theorem', and not 'Grundsatz' but 'Axiom', understanding by theorems and axioms true thoughts. This, however, is to imply that a thought is not something subjective, is not the product of any form of mental activity; for the thought that we have in Pythagoras's theorem is the same for everybody, and its truth is quite independent of its being thought by so-and-so or indeed by anyone at all. We are not to regard thinking as the act of producing a thought, but as that of grasping a thought.
Postulates seem at first sight to be essentially different from axioms. In Euclid we have the postulate 'Let it be postulated that a straight line may be drawn from any point to any other'.
This is obviously introduced with a view to making constructions. The postulates, so it seems, present the simplest procedures for making every construction, and postulate their possibility. At first we might perhaps think that none of this is of any help in providing proofs, but only for solving problems. But this would be a mistake, for sometimes an auxiliary line is needed for a proof, and sometimes an auxiliary point, an auxiliary number-an auxiliary object of some kind. In the proof of a theorem an auxiliary object is one of which nothing is said in the theorem, but which is
1 The play that Frege is here making on the words 'Lehrsatz' and 'Satz' is lost in translation, so we have enclosed them in square brackets after their English equivalents (trans. ).
Postulates
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? Logic in Mathematics 207
required for the proof, so that this would collapse if there were no such object. And if there is no such object, it seems that we must be able to create one and we need a postulate to ensure that this is possible. But what in actual fact is this drawing a line? It is not, at any rate, a line in the geometrical sense that we are creating when we make a stroke with a pencil. And how in this way are we to connect a point in the interior of Sirius with a point in Rigel? Our postulate cannot refer to any such external procedure. It refers rather to something conceptual. But what is here in question is not a subjective, psychological possibility, but an objective one. Surely the truth of a theorem cannot really depend on something we do, when it holds quite independently of us. So the only way of regarding the matter is that by drawing a straight line we merely become ourselves aware of what obtains independently of us. So the content of our postulate is essentially this, that given any two points there is a straight line connecting them. So a postulate is a truth as is an axiom, its only peculiarity being that it asserts the existence of something with certain properties. From this it follows that there is no real need to distinguish axioms and postulates. A postulate can be regarded as a special case of an axiom.
We come to definitions. Definitions proper must be distinguished from illustrative examples. In the first stages of any discipline we cannot avoid the use of ordinary words. But these words are, for the most part, not really appropriate for scientific purposes, because they are not precise enough and fluctuate in their use. Science needs technical terms that have precise and fixed meanings, and in order to come to an understanding about these meanings and exclude possible misunderstandings, we give examples illustrating their use. Of course in so doing we have again to use ordinary words, and these may display defects similar to those which the examples are intended to remove. So it seems that we shall then have to do the same thing over again, providing new examples. Theoretically one will never really achieve one's goal in this way. In practice, however, we do manage to come to an understanding about the meanings of words. Of course we have to be able to count on a meeting of minds, on others guessing what we have in mind. But all this precedes the construction of a system and does not helong within a system. In constructing a system it must be assumed that the words have precise meanings and that we know what they are. Hence we can at this point leave illustrative examples out of account and turn our nttention to the construction of a system.
In constructing a system the same group of signs, whether they are Hounds or combinations of sounds (spoken signs) or written signs, may occur over and over again. This gives us a reason for introducing a simple Nign to replace such a group of signs with the stipulation that this simple sign is always to take the place of that group of signs. As a sentence is generally 11 complex sign, so the thought expressed by it is complex too: in fact it is put together in such a way that parts of the thought correspond to parts of the sentence. So as a general rule when a group of signs occurs in a sentence
Definitions
Illustrative examples
Definition
proper
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it will have a sense which is part of the thought expressed. Now when a simple sign is thus introduced to replace a group of signs, such a stipulation is a definition. The simple sign thereby acquires a sense which is the same as that of the group of signs. Definitions are not absolutely essential to a system. We could make do with the original group o f signs. The introduction of a simple sign adds nothing to the content; it only makes for ease and simplicity of expression. So definition is really only concerned with signs. We shall call the simple sign the definiendum, and the complex group of signs which it replaces the definiens. The definiendum acquires its sense only from the definiens. This sense is built up out of the senses of the parts of the definiens. When we illustrate the use of a sign, we do not build its sense up out of simpler constituents in this way, but treat it as simple. All we do is to guard against misunderstanding where an expression is ambiguous.
A sign has a meaning once one has been bestowed upon it by definition, and the definition goes over into a sentence asserting an identity. Of course the sentence is really only a tautology and does not add to our knowledge. It contains a truth which is so self-evident that it appears devoid of content, and yet in setting up a system it is apparently used as a premise. I say apparently, for what is thus presented in the form of a conclusion makes no addition to our knowledge; all it does in fact is to effect an alteration of expression, and we might dispense with this if the resultant simplification of expression did not strike us as desirable. In fact it is not possible to prove something new from a definition alone that would be unprovable without it. When something that looks like a definition really makes it possible to prove something which could not be proved before, then it is no mere definition but must conceal something which would have either to be proved as a theorem or accepted as an axiom. Of course it may look as if a definition makes it possible to give a new proof. But here we have to distinguish between a sentence and the thought it expresses. If the definiens occurs in a sentence and we replace it by. the definiendum, this does not affect the thought at all. It is true we get a different sentence if we do this, but we do
not get a different thought. Of course we need the definition if, in the proof of this thought, we want it to assume the form of the second sentence. But if the thought can be proved at all, it can also be proved in such a way that it assumes the form of the first sentence, and in that case we have no need of the definition. So if we take the sentence as that which is proved, a definition may be essential, but not if we regard the thought as that which is to be proved.
It appears from this that definition is, after all, quite inessential. In fact considered from a logical point of view it stands out as something wholly inessential and dispensable. Now of course I can see that strong exception will be taken to this. We can imagine someone saying: Surely we are undertaking ~ logical analysis when we give a definition. You might as well say that it doesn't matter whether I carry out a chemical analysis of a body in order to see what elements it is composed of, as say that it is immaterial
? Logic in Mathematics 209
whether I carry out a logical analysis of a logical structure in order to find out what its constituents are or leave it unanalysed as if it were simple, when it is in fact complex. It is surely impossible to make out that the activity of defining something is without any significance when we think of the considerable intellectual effort required to furnish a good definition. -There is certainly something right about this, but before I go into it more closely, I want to stress the following point. To be without logical significance is still by no means to be without psychological significance. When we examine what actually goes on in our mind when we are doing intellectual work, we find that it is by no means always the case that a thought is present to our consciousness which is clear in all its parts. For example, when we use the word 'integral', are we always conscious of everything appertaining to its sense? I believe that this is only very seldom the case. Usually just the word is present to our consciousness, allied no doubt with a more or less dim awareness that this word is a sign which has a sense, and that we can, if we wish, call this sense to mind. But we are usually content with the knowledge that we can do this. If we tried to call to mind everything appertaining to the sense of this word, we should make no headway. Our minds are simply not comprehensive enough. We often need to use a sign with which we associate a very complex sense. Such a sign seems, so to speak, a receptacle for the sense, so that we can carry it with us, while being always aware that we can open this receptacle should we have need of what it contains. It follows from this that a thought, as I understand the word, is in no way to be identified with a content of my consciousness. If therefore we need such signs-signs in which, as it were, we conceal a very complex sense as in a receptacle-we also need definitions so that we can cram this sense into the receptacle and also take it out again. So if from a logical point of view definitions are at hottom quite inessential, they are nevertheless of great importance for thinking as this actually takes place in human beings.
An objection was mentioned above which arose from the consideration that it is by means of definitions that we perform logical analyses. In the development of science it can indeed happen that one has used a word, a sign, an expression, over a long period under the impression that its sense is simple until one succeeds in analysing it into simpler logical constituents. By means of such an analysis, we may hope to reduce the number of axioms; for it may not be possible to prove a truth containing a complex constituent so long as that constituent remains unanalysed; but it may be possible, given nn analysis, to prove it from truths in which the elements of the analysis occur. This is why it seems that a proof may be possible by means of a definition, if it provides an analysis, which would not be possible without this analysis, and this seems to contradict what we said earlier. Thus what seemed to be an axiom before the analysis can appear as a theorem after the analysis.
But how does one judge whether a logical analysis is correct? We cannot prove it to be so. The most one can be certain of is that as far as the form of
? 210 Logic in Mathematics
words goes we have the same sentence after the analysis as before. But that the thought itself also remains the same is problematic. When we think that we have given a logical analysis of a word or sign that has been in use over a long period, what we have is a complex expression the sense of whose parts is known to us. The sense of the complex expression must be yielded by that of its parts. But does it coincide with the sense of the word with the long
established use? I believe that we shall only be able to assert that it does when this is self-evident. And then what we have is an axiom. But that the simple sign that has been in use over a long period coincides in sense with that of the complex expression that we have formed, is just what the definition was meant to stipulate.
We have therefore to distinguish two quite different cases:
(1) We construct a sense out of its constituents and introduce an entirely new sign to express this sense. This may be called a 'constructive definition', but we prefer to call it a 'definition' tout court.
(2) We have a simple sign with a long established use. We believe that we can give a logical analysis of its sense, obtaining a complex expression which in our opinion has the same sense. We can only allow something as a constituent of a complex expression if it has a sense we recognize. The sense of the complex expression must be yielded by the way in which it is put together. That it agrees with the sense of the long established simple sign is not a matter for arbitrary stipulation, but can only be recognized by an immediate insight. No doubt we speak of a definition in this case too. It might be called an 'analytic definition' to distinguish it from the first case. But it is better to eschew the word 'definition' altogether in this case, because what we should here like to call a definition is really to be regarded as an axiom. In this second case there remains no room for an arbitrary stipulation, because the simple sign already has a sense.
Only a sign which as yet has no sense can have a sense arbitrarily assigned to it. So we shall stick to our original way of speaking and call only a constructive definition a definition. According to that a definition is an arbitrary stipulation which confers a sense on a simple sign which previously had none. This sense has, of course, to be expressed by a complex sign whose sense results from the way it is put together.
Now we still have to consider the difficulty we come up against in giving a logical analysis when it is problematic whether this analysis is correct.
Let us assume that A is the long-established sign (expression) whose sense we have attempted to analyse logically by constructing a complex expression that gives the analysis. Since we are not certain whether the analysis is successful, we are not prepared to present the complex expression as one which can be replaced by the simple sign A. If it is our intention to put forward a definition proper, we are not entitled to choose the sign . 4, which already has a sense, but we must choose a fresh sign B, say, which has the sense of the complex expression only in virtue of the definition. The question now is whether A and B have the same sense. But we can bypa11
? Logic in Mathematics 211
this question altogether if we are constructing a new system from the bottom up; in that case we shall make no further use of the sign A-we shall only use B. We have introduced the sign B to take the place of the complex expression in question by arbitrary fiat and in this way we have conferred a sense on it. This is a definition in the proper sense, namely a constructive definition.
If we have managed in this way to construct a system for mathematics without any need for the sign A, we can leave the matter there; there is no need at all to answer the question concerning the sense in which-whatever it may be-this sign had been used earlier. In this way we court no objections. However it may be felt expedient to use sign A instead of sign B. But if we do this, we must treat it as an entirely new sign which had no sense prior to the definition. We must therefore explain that the sense in which this sign was used before the new system was constructed is no longer of any concern to us, that its sense is to be understood purely from the constructive definition that we have given. In constructing the new system we can take no account, logically speaking, of anything in mathematics that existed prior to the new system. Everything has to be made anew from the ground up. Even anything that we may have accomplished by our analytical activities is to be regarded only as preparatory work which does not itself make any appearance in the new system itself.
Perhaps there still remains a certain unclarity. How is it possible, one may ask, that it should be doubtful whether a simple sign has the same sense as a complex expression if we know not only the sense of the simple sign, but can recognize the sense of the complex one from the way it is put together? The fact is that if we really do have a clear grasp of the sense of the simple sign, then it cannot be doubtful whether it agrees with the sense of the complex expression. If this is open to question although we can clearly recognize the sense of the complex expression from the way it is put together, then the reason must lie in the fact that we do not have a clear grasp of the sense of the simple sign, but that its outlines are confused as if we saw it through a mist. The effect of the logical analysis of which we spoke will then be precisely this-to articulate the sense clearly. Work of this kind is very useful; it does not, however, form part of the construction of the system, but must take place beforehand. Before the work of construction is begun, the building stones have to be carefully prepared so ns to be usable; i. e. the words, signs, expressions, which are to be used, must have a clear sense, so far as a sense is not to be conferred on them in the Nystem itself by means of a constructive definition.
We stick then to our original conception: a definition is an arbitrary . vtlpulation by which a new sign is introduced to take the place of a complex expression whose sense we know from the way it is put together. A sign which hitherto had no sense acquires the sense of a complex expression by definition.
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When we look around us at the writings of mathematicians, we come acros! many things which look like definitions, and are even called such, without really being definitions. Such definitions are to be compared with thosf stucco-embellishments on buildings which look as though they supported something whereas in reality they could be removed without the slightest detriment to the building. We can recognize such definitions by the fact that no use is made of them, that no proof ever draws upon them. But if a wore or sign which has been introduced by definition is used in a theorem, thf only way in which it can make its appearance there is by applying thf definition or the identity which follows immediately from it. If such an application is never made, then there must be a mistake somewhere. Of course the application may be tacit. That is why it is so important, if we are to have a clear insight into what is going on, for us to be able to recognize the premises of every inference which occurs in a proof and the law of inference in accordance with which it takes place. So long as proofs are drawn up in conformity with the practice which is everywhere current at the present time, we cannot be certain what is really used in the proof, what it rests on. And so we cannot tell either whether a definition is a mere stucco- , definition which serves only as an ornament, and is only included because it is in fact usual to do so, or whether it has a deeper justification. That is why it is so important that proofs should be drawn up in accordance with the requirements we have laid down.
We can characterize another kind of inadmissible definition by a metaphor from algebra. Let us assume that three unknowns x, y, z occur in three equations. Then they can be determined by means of these equations. , Strictly speaking, however, they are determined only for the case where there is only one solution. In a similar way the words 'point', 'straight line',, 'surface' may occur in several sentences. Let us assume that these words have as yet no sense. It may be required to find a sense for each of these words such that the sentences in question express true thoughts. But have, we here provided a means for determining the sense uniquely? At any rate not in general; and in most cases it must remain undecided how many solutions are possible. But if it can be proved that only one solution is' possible, then this is given by assigning, via a constructive definition, a sense in turn to each of the words that needs defining. But we cannot regard as a definition the system of sentences in each of which there occur several of the expressions that need defining.
A special case of this is where only one sign, which has as yet no sense, occurs in one or more sentences. Let us assume that the other constituents of the sentences are known. The question is now what sense has to be given to this sign for the sentences to have a sense such that the thoughts expressed in them are true. This case is to be compared to that in which the letter x occ. urs in one or more equations whose other constituents are known, where the problem is: what meaning do we have to give the letter Jt for the equations to express true thoughts? If there are several equations,j
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this problem will usually be insoluble. It is obvious that in general no number whatsoever is determined in this way. And it is like this with the case in hand. No sense accrues to a sign by the mere fact that it is used in one or more sentences, the other constituents of which are known. In algebra we have the advantage that we can say something about the possible solutions and how many there are-an advantage one does not have in the general case. But a sign must not be ambiguous. Freedom from ambiguity is the most important requirement for a system of signs which is to be used for scientific purposes. One surely needs to know what one is talking about and the statements one is making, what thoughts one is expressing.
Now it is true that there have even been people, who have fancied themselves logicians, who have held that concept-words (nomina appel- /ativa) are distinguished from proper names by the fact that they are
ambiguous. The word 'man', for example, means Plato as well as Socrates and Charlemagne. The word 'number' designates the number 1 as well as the number 2, and so on. Nothing is more wrong-headed. Of course I can use the words 'this man' to designate now this man, now that man. But still on each single occasion I mean them to designate just one man. The sentences of our everyday language leave a good deal to guesswork. It is the surrounding circumstances that enable us to make the right guess. The sentence I utter does not always contain everything that is necessary; a great deal has to be supplied by the context, by the gestures I make and the direction of my eyes. But a language that is intended for scientific employment must not leave anything to guesswork. A concept-word combined with the demonstrative pronoun or definite article often has in this way the logical status of a proper name in that it serves to designate a single determinate object. But then it is not the concept-word alone, but the whole consisting of the concept-word together with the demonstrative pronoun and accompanying circumstances which has to be understood as a proper name. We have an actual concept-word when it is not accompanied by the definite article or demonstrative pronoun and is accompanied either by no article or by the indefinite article, or when it is combined with 'all', 'no' and 'some'. We must not think that I mean to assert something about an African chieftain from darkest Africa who is wholly unknown to me, when I say 'All men are mortal'. I am not saying anything about either this man or that man, but I am subordinating the concept man to the concept of what is mortal. In the sentence 'Plato is mortal' we have an instance of subsumption, in the sentence 'All men are mortal' one of subordination. What is being spoken about here is a concept, not an individual thing. We must not think either that the sense of the sentence 'Cato is mortal' is contained in that of the sentence 'All men are mortal', so that by uttering the latter sentence I should at the same time have expressed the thought contained in the former sentence. The matter is rather as follows. By the sentence 'All men are mortal' I say 'If anything is a man, it is mortal'. By an inference from the general to the particular, I obtain from this the sentence 'If Cato is a man,
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then Cato is mortal'. Now I still need a second premise, namely 'Cato is a man'. From these two premises I infer 'Cato is mortal'.
Since therefore we need inferences and a second premise, the thought that Cato is mortal is not included in what is expressed by the sentence 'All men are mortal', and so 'man' is not an ambiguous word which amongst its many meanings has that which we designate by the proper name 'Plato'. On the contrary, a concept-word simply serves to designate a concept. And a concept is quite different from an individual. If I say 'Plato is a man', I am not as it were giving Plato a new name-the name 'man'-but I am saying that Plato falls under the concept man. Likewise we have two quite different cases when I give the definition '2 + 1 = 3' and when I say '2 + 1 is a prime number'. In the first case I confer on the sign '3', which is so far empty, a sense and a meaning by saying that it is to mean the same as the combination of signs '2 + 1'. In the second case I am subsuming the meaning of '2 + 1' under the concept prime number. I do not give it a new name by doing that. The fact therefore that I subsume different objects under the same concept does not make the concept-word ambiguous. So in the sentences
'2 is a prime number' '3 is a prime number' '5 is a prime number'
the word 'prime number' is not somehow ambiguous because 2, 3, 5 are different numbers; for 'prime number' is not a name which is given to these numbers.
It is of the essence of a concept to be predicative. If an empty proper name occurs in a sentence, the other parts of which are known, so that the sentence has a sense once a sense is given to that proper name, then, so long as the proper name remains empty, the sentence contains the possibility of a statement, but we do not have an object about which anything is being said. So the sentence 'x is a prime number', does indeed contain the possibility of a statement, but so long as no meaning is given to the letter 'x', we do not have an object about which anything is being said. Another way of putting this would be to say: we have a concept but we have no object subsumed under it. If we take as a further instance the sentence 'x increased by 2 is divisible by 4' then we have a concept again. We can take these two concepts as characteristic marks of a new concept by putting together the sentences 'x is a prime number' and 'x increased by 2 is divisible by 4'. Under this concept there falls only one object-the number 2. But a concept under which only one object falls is still a concept; this does not make the expression for it into a proper name.
Our position is this: we cannot recognize sentences containing an empty sign, the otrn:r constituents of which are known, as definitions. But such sentences can have an explanatory role by providing a clue to what is to be understood by the sign or word in question.
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I have read that verbal definitions are considered faulty, and 1t ts argued that we should really have no further truck with such definitions. By way of example reference was made to a definition given by me, but it was not said what a verbal definition was. 1 Of course every definition makes use of words or signs. Perhaps what is meant by a verbal definition is one in which the definiens contains a word which is a mere word as such, having no sense. Certainly this should not be allowed, but from the fact that the reader attaches no sense to a word it does not follow that the author of the definition has attached no sense to it. The insistence on sense is absolutely justified, and all the more so since many mathematicians seem to prove what are merely sentences without bothering whether they have a sense and what sense it is they have.
How little value is commonly placed on sense and definitions can be seen from the sharply conflicting accounts that mathematicians give of what number is. (We are speaking here of the natural numbers. ) Weierstrass says 'Number is a series of things of the same kind'. Another says that certain conventional shapes produced by writing, such as 2 and 3, are numbers. 2 A third is of the opinion: if I hear the clock strike three I see nothing in this of what three is. Therefore it cannot be anything visible. If I see three lines, then I hear nothing in this of what three is. Therefore it can be nothing audible either. An axiom is not a visible thing and so if we speak of three axioms, the three here is nothing visible either. Number cannot be anything whatever which can be perceived by the senses. 3
Obviously each of these attaches a different sense to the word 'number'. So the arithmetics of these three mathematicians must be quite different. A sentence from the first mathematician must have a quite different sense from the equivalent-sounding sentence of the second mathematician. This resembles what it would be like if botanists were not agreed about what they wished to understand by a plant, so that for one botanist a plant was, say, an organically developing structure, for another a human artefact, and for a third something that was not perceptible by the senses at all. Such a situation would certainly not give rise to a common science of botany.
But why should it not be possible to lay it down that by a number is to be meant a series of things of the same kind? Admittedly we can raise objections to such a course. For one thing, it may be thought that the sense of the word 'series' is not firmly enough established. Are we to think in this connection of a spatial ordering, or of a temporal ordering or of a spatia- temporal one perhaps? Further it is not clear what we are to understand by 'of the same kind'. For example, are the notes of a scale of the same kind qua
1 Frege is here referring to the article Ober die Stellung der Definition in der Axiomatik by A. Schoenjliess in Jahresbericht der deutschen Mathematiker- Vereinlgung XX (1911), pp. 222-255 (ed. ).
2 The reference is to the so-called 'formal' theory of arithmetic (arithmetic as a game with signs) held by Frege's contemporaries E. He/ne and J. Thomae (ed. ).
3 The reference is probably to G. Cantor (ed. ).
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notes, or are they of the same kind only if they have the same pitch? But let us assume that explanations were given which cleared these matters up. A train is a series of objects of the same kind "'hich moves along rails on wheels. It may be thought that the engine is nevertheless something of a different kind. Still that makes no essential difference. And so such a number comes steaming here from Berlin. Let us assume that the science of these numbers has been set up. There is no doubt that it must be entirely different from the science in which certain shapes that one makes on a writing surface with a writing instrument are called numbers. Even if the form of words is the same, the thought expressed must be quite different. Now it is striking that the sentences of these fundamentally different sciences, each of which is called arithmetic, are constituted by precisely the same words. And it is even more striking that the practitioners of these sciences have no inkling that their sciences are fundamentally different. They all believe that they are doing arithmetic, and the same arithmetic at that, the same number theory, although what one of them is calling a number has no resemblance at all to what another is calling a number.
How is this possible? One would almost think that mathematicians regard the words used, the form of an expression, as the essential thing, and the thought expressed as quite inessential. Perhaps they think 'The thoughts contained in sentences are really no concern of mathematicians-they are a matter for philosophers; and everything to do with philosophy is of course extremely imprecise, uncertain and essentially unscientific. A mathematician who remains true to his scientific calling will have nothing to do with it. True, it can happen to even the best of them, in a moment of weakness, to let a definition slip, or something which looks like one, but we should not accord any significance to that. It is all one with a man sneezing. Really the only thing that matters is that they should all agree on the words and formulae they use. That is enough for a mathematician who has not been infected by philosophy. '
But is that then a science which proves sentences without knowing what it proves? But is it the case in actual fact that scientists do agree in the words they use? Are not mathematical works written in different languages, and are they not translated into other languages? In which case, of course, we no longer have the same form of words. But there must be something else which is preserved. And what can this be but the sense? So the thought, the sense of a sentence, cannot after all be wholly irrelevant. And does one not feel in the depth of one's being that the thoughts are the essential thing-that it is in fact these alone that we are concerned about?
But how do they come to be treated as irrelevant? How can one possibly imagine that two quite different sciences should really have the same content? Is it only because they are both called arithmetic and both treat of numbers, althpugh what is called a number in the one is quite different from what is called a number in the other? Or is not the explanation rather that we have really to do with the same science; that this man does attach the
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same sense to the word number as that man, only he doesn't manage to get hold of it properly? Perhaps the sense appears to both through such a haze that when they make to get hold of it, they miss it. One of them makes a grasp to the right perhaps and the other to the left, and so although they mean to get hold of the same thing, they fail to do so. How thick the fog must be for this to be possible! But it must surely show up in the proofs that they have not got hold of the same sense. Yes, it could not fail to, if the proofs were drawn up in a logically rigorous way, with no gaps in the chains of inference. But this is just what we shall fail to find. If no use is ever made of a definition, there might as well not be one. However wide it may be of the actual target, no-one will notice. Another mathematician's shot may miscarry on the other side; but since he makes no use of his definition, this too might as well not be there. We can see in this way how definitions, which seem to be utterly irreconcilable, lie peacefully alongside one another like the animals in paradise. I f only things were really like that!
