Now we may of course also think in mathematical signs; yet even then thinking is tied up with what is
perceptible
to the senses.
Gottlob-Frege-Posthumous-Writings
But it is essential that they should be equiform.
However, taken strictly, we are stepping outside the confines of a spoken language designed to be heard and moving into the region of a written or printed language designed for the eye.
A sentence which an author writes down is primarily a direction for forming a spoken sentence in a language whose sequences of sounds serve as signs for expressing a sense.
So at first there is only a mediated connection set up between written signs and a sense that is expressed.
But once this connection is established, we may also regard the written or printed sentence as an immediate expression of a thought, and so as a sentence in the strict sense of the word.
In this way we obtain a language dependent on the sense of sight, which, if need be, can even be learnt by a deaf man.
In this, individual letters can be adopted as indefinitely indicating parts of a sentence.
The language we have just indicated, which I will call the object-language,1 is to serve for us as a bridge from the perceptible to the imperceptible.
It contains two different constituents: those with the form of words and the individual letters.
The former correspond to words of the spoken language, the latter have an indefinitely indicating role.
This object-language is to be distinguished from the language in which I conduct my train of thought.
That is the usual written or printed German, my meta-language.
But the sentences of the object-language are the objects to be talked about in my meta-language.
And so I must be able to designate them in my meta- language, just as in an article on astronomy the planets are designated by
1 The German editors see the distinction Frege here draws between a 'Hilfssprache' and a 'Darlegungssprache' as anticipating the distinction sub- sequently drawn by Tarski between an 'object-language' and a 'meta-language', and certainly the resemblance is close: close enough for us to have decided to avail ourselves of the expedient of using Tarski's way of speaking to translate Frege's two
notions: the reader is not to suppose that Frege anticipates this actual terminology and may judge for himself how far he anticipates Tarski's thought (trans. ).
? ? ? Logical Generality 261
their proper names 'Venus', 'Mars', etc. As such proper names of the sentences o f the object-language I use these very sentences, but enclosed in quotation marks. Moreover it follows from this that the sentences of the object-language are never given assertoric force. 'If a is a man, a is mortal' is a sentence of the object-language in which a general thought is expressed. We move from the general to the particular by substituting for the equiform indefinitely indicating letters equiform proper names. It belongs to the essence of our object-language that equiform proper names designate the same object (man). Here empty signs (names) are not proper names at all. * By substituting for the indefinitely indicating letters equiform with 'a', proper names of the form 'Napoleon', we obtain
'IfNapoleon is a man, Napoleon is mortal. '
This sentence is not however to be regarded as a conclusion, since the sentence 'If a is a man, a is mortal' is not given assertoric force, and so the thought expressed in it is not presented as one recognized to be true, for only a thought recognized as true can be made the premise o f an iriference. But an inference can be made of this, by freeing the two sentences of our object-
language from quotation marks, thus making it possible to put them forward with assertoric force.
The compound sentence ' I f Napoleon be a man, then is Napoleon mortal'1 expresses a hypothetical compound thought, composed of one condition and one consequence. The former is expressed in the sentence 'Napoleon is a man', the latter in 'Napoleon is mortal'. Strictly, however, our compound sentence contains neither a sentence equiform with 'Napoleon is a man' nor one equiform with 'Napoleon is mortal'. In this divergence between what holds at the linguistic level and what holds at the level of the thought, there emerges a defect in our object-language which is still to be remedied. I wish now to dress the thought that I expressed above in the
*I call proper names of our object-language equiform, if they are intended to be so by the writer and are meant to be of the same size, if we can recognize this to be the writer's intention even if it is imperfectly realized.
1 In German, unlike English, the order of the words in a sentence is altered when it is made into the antecedent or consequent of a hypothetical. The compound sentence with antecedent 'Napoleon is a man' and consequent 'Napoleon is mortal' transliterates 'If Napoleon a man is, is Napoleon mortal'. Throughout the preceding discussion we have translated such hypothetical sentences into natural English.
Here, where Frege is concerned with the deviation between the vernacular (German) and a language which reflects more accurately 'the level of thought', it has proved necessary to resort to a clumsy English rendering of the hypothetical sentence. The fact that English happens not to have what Frege here argues is a defective correspondence between the level of language and that of the thought, does not, of course, detract from the force or interest of the point he is here making (trans. ).
? ? 262 Logical Generality
sentence 'IfNapoleon be a man, then is Napoleon mortal', in the sentence 'If Napoleon is a man, Napoleon is mortal'; which, in what follows, I wish to call the second sentence. Similar cases are to be treated in the same way. So I want also to transform the sentence 'Ifa be a man, then is a mortal' into 'If a is a man, a is mortal', which in what follows I will call the sentence. * In the first sentence I distinguish the two individual letters equiform with 'a' from the remaining part.
*The first sentence, unlike the second, does not express a compound thought, since neither 'a is a man' nor 'a is mortal' expresses a thought. We have here really only parts of a sentence, not sentences.
? ? ? [Diary Entries on the Concept ofNumbersP [23. 3. 1924-25. 3. 1924]
23. 3. 1924 My efforts to become clear about what is meant by number have resulted in failure. We are only too easily misled by language and in this particular case the way we are misled is little short of disastrous. The sentences 'Six is an even number', 'Four is a square number', 'Five is a prime number' appear analogous to the sentences 'Sirius is a fixed star', 'Europe is a continent'-sentences whose function is to represent an object as falling under a concept. Thus the words 'six', 'four', and 'five' look like proper names of objects and 'even number', 'square number', and 'prime number', along with 'number' itself, look like concept-words. So the problem appears to be to work out more clearly the nature of the concept designated by the word 'number' and to exhibit the objects that, as it seems, are designated by number-words and numerals.
24. 3. 1924 From our earliest education onwards we are so accustomed to using the word 'number' and the number-words that we do not regard our way of using them as something that stands in need of a justification. To the mathematicians it appears beneath their dignity to concern themselves with such childish matters. But one finds amongst them the most different and contradictory statements about number and numbers. Indeed, when one has been occupied with these questions for a long time one comes to suspect that our way of using language is misleading, that number-words are not proper names of objects at all and words like 'number', 'square number' and the rest are not concept-words; and that consequently a sentence like 'Four is a square number' simply does not express that an object is subsumed under a concept and so just cannot be construed like the sentence 'Sirius is a fixed star'. But how then is it to be construed?
25. 3. 1924 At first, however, I was still captive to the error that language gives rise to. It is easy to see that a number is not a collection of things. What is a collection? A collection is a thing, a thing that is made up of things. The mathematician couples the word 'one' with the definite article
1 These remarks were taken from a diary that Frege kept from 10. 3. 24 to 9. 5. 24; the other entries are simply statements of political attitudes which cannot be counted as part of the scientific NachlajJ (ed. ). For further details concerning the nature and content of this Diary, readers are referred to Michael Dummett, Frege: Philosophy of Language, London 1973, p. xii. See also the article on Frege in the Encyclopaedia Briltanica (fifteenth edition) Trans.
? 264 [Diary Entries on the Concept ofNumber]
and speaks of 'the number one'. If now a number is a thing, we must produce the thing that is called the number one. Many people even want to call numerals numbers. If this were true, we should have to be able to specify which of these numerals was the number one. What a calamity it would then be if this One were ever to be destroyed by fire! It is already a step forward when a number is seen not as a thing but as something belonging to a thing, where the view is that different things, in spite of their differences, can possess the same One, as different leaves can, say, all possess the colour green. Now which things possess the number one? Does not the number one belong to each and every thing?
? ? ? Number 1 [September 1924]
My efforts to throw light on the questions surrounding the word 'number' and the words and signs for individual numbers seem to have ended in complete failure. Still these efforts have not been wholly in vain. Precisely because they have failed, we can learn something from them. The difficulties of these investigations are often greatly underrated. Anyone who can regard a number of a series of objects of the same kind shows that he is so far from having any real understanding that he does not yet have even an inkling of the task involved.
First of all we need to come to an understanding about the words 'number', 'numeral', 'number-word'. In everyday life one often calls the numerals numbers. In the present context we must exclude this way of speaking. The simplest thing seems to be to understand by a numeral a sign that designates a number and by a number-word a word which designates a number. Then numerals and number-words will have the same role, namely that of designating a number; only the vehicle used will be different.
What, then, are numbers themselves? What sort of thing is it that one means to designate by a number-word or numeral? Obviously it is not anything that can be perceived by the senses. A number can neither be smelt, nor tasted, nor seen, nor felt. It jars to speak of warm or yellow or bitter numbers, just as it would jar if one chose to speak of yellow or warm or bitter points in geometry. We may seek to discover something about numbers themselves from the use we make of numerals and number-words. Numerals and number-words are used, like names of objects, as proper names. The sentence 'Five is a prime number' is comparable with the sentence 'Sirius is a fixed star'. In these sentences an object (five, Sirius) is presented as falling under a concept (prime number, fixed star) (a case of an object's being subsumed under a concept). By a number, then, we are to understand an object that cannot be perceived by the senses. Even the
1 According to a remark made by the previous editors on the transcripts on which this edition is based, these comments stem 'from two drafts of which the first is dated: Sept. 1924 to 23. 111. 1924'. The date 23. 111. 1924 apparently refers to the remarks which Frege made in his diary on that day (cf. p. 263). It is also clear from remarks made by the previous editors that the first three paragraphs come from the second draft, and the last two paragraphs from the first (ed. ).
The curious dating may mean that Frege added, on Sept. 1924, the last two paragraphs to a re-working of the diary entry of 23. 111. 1924 (trans. ).
? 266 Number
objects of geometry, points, straight lines, surfaces etc. cannot really be perceived by the senses.
These investigations are especially difficult because in the very act of conducting them we are easily misled by language: by language which is, after all, an indispensable tool for carrying them out. Indeed one might think that language would first have to be freed from all logical imperfections before it was employed in such investigations. But of course the work necessary to do this can itself only be done by using this tool, for all its imperfections. Fortunately as a result of our logical work we have acquired a yardstick by which we are apprised of these defects. Such a yardstick is at work even in language, obstructed though it may be by the many illogical features that are also at work in language.
In pre-scientific discourse what are called numbers are often no more than numerals. These signs are ordered in a series beginning with the sign '1'. In this series each sign, with the exception of the last, has an immediate successor, and each sign, with the exception of the first (the sign '1') an immediate predecessor. For example 1? 2? 3? 4? 5, in which I have separated the individual numerals by asterisks.
? ? Sources of Knowledge of Mathematics and the mathematical natural Sciences1
[1924/5]
When someone comes to know something it is by his recognizing a thought to be true. For that he has first to grasp the thought. Yet I do not count the grasping of the thought as knowledge, but only the recognition of its truth, the judgement proper. What I regard as a source of knowledge is what justifies the recognition of truth, the judgement.
I distinguish the following sources of knowledge:
1. Senseperception
2. Thelogicalsourceofknowledge
3. The geometrical and temporal sources of knowledge.
Each of these is subject to its own disturbances, which detract from its value.
A. Sense Illusions
A sense impression is not in itself a judgement, but becomes important in that it is the occasion for our making a judgement. When this happens, we may be misled: in such cases we talk of 'sense illusions'. Since for the mathematical natural sciences sight is the most important sense, we will examine it in greater detail. To our consciousness, the line of vision from the eye to the object is straight. In the majority of cases, the corresponding light
1 From the correspondence between Frege and Richard Honigswald (cf. the letter from Honigswald to Frege of 24/4/1925), it emerges that Frege submitted this article to Bruno Bauch for the series edited by Honigswald and Bauch, Wissen- schaftliche Grundfragen. In the letter mentioned Honigswald gave a very favourable judgement on Frege's article, but suggested that it be essentially enlarged, 'Our concern in our venture isn't to produce a philosophical journal, in which in each issue many articles are put together, but a series of parts, each comprising between 4 to 6 sheets, and appearing on their own in bookform'. He asked Frege to resubmit the article for the Wissenschaftliche Grundfragen affer enlarging it, and listed a number of points where further reworking appeared to him to be desirable. Because of his death three months later, Frege was obviously only able to do some of the preliminary work required for this. In the NachlajJ three sheets, now lost, with notes pointing in this direction were to be found, as may be learnt from a note of the earlier editors to the Frege-Honigswald correspondence. Cf. further, Frege's letters to Honigswald of 26/4/1925 und Inter (cd. ).
? 268 Sources ofKnowledge ofMathematics and natural Sciences
ray from the object to our eye is also straight, or the deviation from a straight line is too slight to matter to us. If we notice the deviation, we speak of an optical illusion. We encounter such cases with the reflection of the light in a mirror, with diffraction or refraction of the light. Because of these illusions we might from the very outset regard visual perception as a source of knowledge that is unreliable and hence of little value, and yet it is precisely sense perception that is regarded by many as the most reliable, if not the only, source of knowledge. Of course a mirror appears to be an opening in the wall giving us a view of a neighbouring room; of course a calm stretch of water gives us the illusion of a sun which seems to be looking up at us; but people are no longer deceived by such cases, because there are at our disposal a diversity of means for correcting the judgement gained from the first impression. Of course, if there were no laws governing events, or if the laws governing events in the physical world were unknowable for us, we would lack the means for recognizing illusions for what they are and thus for rendering them harmless. The laws of nature we already know enable us to avoid being misled by sense illusions. Thus knowledge about the refraction of light tells us that many of the images we see through the microscope are wholly unreliable. In order to know the laws of nature we need perceptions that are free from illusion. And so, on its own, sense perception can be of little use to us, since to know the laws of nature we also need the other sources of knowledge: the logical and the geometrical. Thus we can only advance step by step--each extension in our knowledge of the laws of nature providing us with a further safeguard against being deceived by the senses and the purification of our perceptions helping us to a better knowledge of the laws of nature. We must be careful not to overestimate the value of sense perception, for without the other sources of knowledge, which protect us from being deceived, we could hardly get anywhere with it. We need the perceptions, but to make use of them, we also need the other sources of knowledge. Only all taken in conjunction make it possible for us to
penetrate ever deeper into mathematical physics.
For mathematics on its own, we don't need sense perception as a source
of knowledge: for it the logical and geometrical sources suffice.
At times sense perception has been an outright hindrance to the advancement of knowledge. For a long time the idea of the spherical shape of the earth was almost universally held to be completely absurd, since if it were true, on the bottom side of the earth men's heads would dangle downwards from their feet. People had been led by their senses to regard the upwards direction as everywhere the same; and even today there are difficulties in such considerations which children find difficult to surmount. It was a long time before the idea of the spherical shape of the earth gained sufficient ground for Columbus to put his trust in it and be able to embark
on his famous voyage. Its success and the subsequent circumnavigations of the earth were the victory of scientific reflection over the old view: a view
? Sources ofKnowledge ofMathematics and natural Sciences 269 that was almost overwhelmingly suggested by, and apparently unassailably
grounded in, sense perception.
B. The logical Source of Knowledge and Language
The senses present us with something external and because of this it is easier to comprehend how mistakes can occur than it is in the case of the logical source of knowledge which is wholly inside us and thus appears to be more proof against contamination. But appearances are deceptive. For our thinking is closely bound up with language and thereby with the world of the senses. Perhaps our thinking is at first a form of speaking which then becomes an imaging of speech. Silent thinking would in that case be speech which has become noiseless, taking place in the imagination.
Now we may of course also think in mathematical signs; yet even then thinking is tied up with what is perceptible to the senses. To be sure, we distinguish the sentence as the expression of a thought from the thought itself. We know we can have various expressions for the same thought. The connection of a thought with one particular sentence is not a necessary one; but that a thought of which we are conscious is connected in our mind with some sentence or other is for us men necessary. But that does not lie in the nature of the thought but in our own nature. There is no contradiction in supposing there to exist beings that can grasp the same thought as we do without needing to clad it in a form that can be perceived by the senses. But still, for us men there is this necessity. Language is a human creation; and so man had, it would appear, the capacity to shape it in conformity with the logical disposition alive in him. Certainly the logical disposition of man was at work in the formation of language but equally alongside this many other dispositions-such as the poetic disposition. And so language is not constructed from a logical blueprint.
One feature of language that threatens to undermine the reliability of thinking is its tendency to form proper names to which no objects correspond. If this happens in fiction, which everyone understands to be fiction, this has no detrimental effect. It's different if it happens in a statement which makes the claim to be strictly scientific. A particularly noteworthy example of this is the formation of a proper name after the pattern of 'the extension of the concept a', e. g. 'the extension of the concept star'. Because of the definite article, this expression appears to designate an object; but there is no object for which this phrase could be a linguistically appropriate designation. From this has arisen the paradoxes of set theory which have dealt the death blow to set theory itself. I myself was under this illusion when, in attempting to provide a logical foundation for numbers, I tried to construe numbers as sets. It is difficult to avoid an expression that has universal currency, before you learn of the mistakes it can give rise to. It
? 270 Sources ofKnowledge ofMathematics and natural Sciences
is extremely difficult, perhaps impossible, to test every expression offered us by language to see whether it is logically innocuous. So a great part of the work of a philosopher consists-or at least ought to consist-in a struggle against language. But perhaps only a few people are aware of the need for this. The same expression-'the extension of the concept star'-serves at the same time to illustrate, in yet another way, the fatal tendency of language to form apparent proper names: 'the concept star' is, of itself, one such. The definite article creates the impression that this phrase is meant to designate an object, or, what amounts to the same thing, that 'the concept star' is a proper name, whereas 'concept star' is surely a designation of a concept and thus could not be more different from a proper name. The difficulties which this idiosyncrasy of language entangles us in are incalculable.
But, isn't thinking a kind of speaking? How is it possible for thinking to be engaged in a struggle with speaking? Wouldn't that be a struggle in which thinking was at war with itself? Doesn't this spell the end to the possibility of thinking?
To be sure, if you search for the emergence of thinking in the develop- ment of an individual, you may well describe thinking as an inaudible inner speaking; but that doesn't capture the true nature of thinking. Can't a mathematician also think in formulae? The formula-language of mathematics is as much a human creation as spoken language, but is fundamentally different from it. Here those traits of spoken language which, as we have seen, lead to logical errors, can be avoided. Yet the influence of speech is so great that they aren't always avoided. Thus if we disregard how thinking occurs in the consciousness of an individual, and attend instead to the true nature of thinking, we shall not be able to equate it with speaking. In that case we shall not derive thinking from speaking; thinking will then emerge as that which has priority and we shall not be able to blame thinking for the logical defects we have noted in language.
In the formula-language of mathematics an important distinction stands out that lies concealed in verbal language. Of course mathematicians themselves are still so strongly influenced by verbal language that even in their discipline the distinction I have in mind doesn't stand out as clearly as all that. Mathematicians are compelled by the nature of their discipline to grasp a concept to which they have given the name 'function'. As early as in the upper forms of high school, pupils are introduced to trigonometric functions, and if they should then go on to study mathematics, they hear a great deal about functions, without it becoming clear to them what people mean by the word. Their teachers take great pains but in vain, and it is precisely the most gifted pupils who perhaps will understand it least, since they will notice that the definitions given do not agree with the teacher's own ~ay of speaking. One surely has some right to expect that a definition, once given, will not straightway be thrown in a corner to gather dust, but that it will be drawn upon when the expression it defines is used.
? Sources ofKnowledge ofMathematics and natural Sciences 271
But this expectation is disappointed because the alleged definition cannot begin to provide what one expects of it. For not everything can be defined; only what has been analysed into concepts can be reconstituted out of the parts yielded by the analysis. But what is simple cannot be analysed and hence not defined. If, nevertheless, someone attempts a definition, the result is nonsense. All definitions of function belong to this category. How does a child learn to understand grown-ups? Not as if it were already endowed with an understanding of a few words and grammatical constructions, so that all you would need to do would be to explain what it did not understand by means of the linguistic knowledge it already had. In reality of course children are only endowed with a capacity to learn to speak. We must be able to count on a meeting of minds with them just as in the case of animals with whom men can arrive at a mutual understanding. Neither is it possible, without a meeting o f minds, to make designations o f a logically unanalysable content intelligible to others. The word 'function' is such a designation. The way in which designations for functions are used can come to our assistance here. The sign 'sin' which we say designates the sine function, only occurs in mathematics in close combination with other signs, as in 'sin 10? ', 'sin 0? 11", 'sin a', and so is a sign in need of supplementation and therein is different from a proper name. Its content is correspondingly also in need of supplementation and therein is different from any object, for instance, even from any number; for a number, by which I don't want to understand a numerical sign, appears in mathematics as an object, e. g. the number 3.
Now it is usual in higher mathematics to permit the sign 'sin' to be
followed simply by a numerical sign or a letter standing in for one. For
instead of defining the size of an angle A in degrees, minutes and seconds, it
can simply be defined by a number as follows: Let C be a circle in the plane
of A whose centre is at the vertex of A. Let the radius of C be r. Let the sides
of A include an arc of C, whose length is b, say. Let C1 be a circle in the
plane of A, whose centre is at the vertex of A. Let the radius of C1 be r1 and
the sides of A include an arc of C1 whose length is b1? Then b1 :r1 = b :r.
Thus b/r is the same number as b/r1, and this number depends on the size
fined by the number b/r, which coincides with b1/r" and what's more a larger number corresponds to a larger angle. Thus b/r is greater than, less than or equal to b1 /r according as A is greater than, less than or equal to A 1 ? From this we may see how the number b/r (which coincides with b/r1) defines the angle A. If b/r = 1, then b = r. Thus the number 1 defines an angle for which the length b is equal to r, that is the length of the arc of C included by the sides of A is then equal to the radius of A. In the same way an angle is defined by the number 2, in which case the arc of C included by the sides of A is twice the lenath of the radius of C etc. We may also say: the number
1 of the angle A and defines that size. If instead of A we take the angle A ,
then b1/r say, takes the place of b/r, and b;/r" that of b1/r1 and in fact b1 >b if A1 >A. Andsointhatcaseb1/r>b/r. AndsothesizeoftheangleAisde-
? 272 Sources ofKnowledge ofMathematics and natural Sciences
that in this way defines the size of angle is the number yielded by measuring the arc of C included by its sides with the radius of C. In this way it is in every case fixed which number is meant when the sign 'sin' is completed by the sign for a real number. The only thing presupposed is that you know how an angle is related to its sine.
In the same way the sign 'cos' (cosine) is also in need of supplementation: it is to be completed by numerical signs, and cos 1, cos 2 and cos 3 are numbers. Thus 'cos' is neither a proper name, nor does it designate an object; but you can't deny the sign 'cos' some content. If, however, you wished to say, using the definite article, 'the content of the sign "cos"', you would convey the wrong idea, that an object was the content of the 'cos' sign. Perhaps it can be seen from this how difficult it is not to allow ourselves to be misled by language. Just because this is so difficult, it is hardly to be expected that a run-of-the-mill writer will take the trouble to avoid being misled, and linguistic usage will, to be sure, always remain as it is.
Added to this, there is also the following: mathematicians use letters to express generality, as in the sentence '(a + b) + c = (a + c) + b'. These letters here stand in for numerical signs and you arrive at the expression of a particular thought contained in the general one by substituting numerical signs for the letters. If one has in fact admitted functions, one will feel the need to express generality concerning functions too. As one uses letters instead of numerical signs so as to be able to express general thoughts concerning numbers, one will also introduce letters for the specific purpose of being able to express general thoughts concerning functions. It is customary for this purpose to use the letters J, F, g, G and also ~ and tP, which we may call function-letters. But now the function's need o f supplementation must somehow or other find expression. Now it is appropriate to introduce brackets after every function-letter, which together with that letter are to be regarded as one single sign. The space within the brackets is then the place where the sign that supplements the function-letter is to be inserted. By substituting for the function-letter in f( 1) a particular function by means of the sign 'sin', you obtain 'sin 1',just as you obtain '31' from 'a2' by substituting '3' for the letter 'a'. In each case, in so doing, yoU make the transition from an indefinitely indicating sign, that is, a letter, to one that designates determinately. If this happens in a sentence, this corresponds to the transition from a general thought to a particular one contained in it. An example ofthis is the transition from '(a- 1)? (a + 1) ? a? a- 1'to'(3- 1)? (3+1)= 3? 3- 1'. Icannotgivehereasimilar example in which a function-sign that designates definitely is substituted for an indefinitely indicating letter, since to do so I would have to presuppose certain elements of higher analysis: even so it will be clear enough what I mean, and yo1,1 will at least be able to gain some idea of the importance of the introduction of functions into mathematical investigations, and of the introduction of function-signs and function-letters into the sign-language of
? Sources ofKnowledge ofMathematics and natural Sciences 273
mathematics. It is here that the tendency of language by its use of the definite article to stamp as an object what is a function and hence a non- object, proves itself to be the source of inaccurate and misleading expressions and so also of errors of thought. Probably most of the impurities that contaminate the logical source of knowledge have their origins in this.
C. The geometrical Sources of Knowledge
From the geometrical source of knowledge flow the axioms of geometry. It is least of all liable to contamination. Yet here one has to understand the word 'axiom' in precisely its Euclidean sense. But even here people in recent works have muddied the waters by perverting-so slightly at first as to be scarcely noticeable-the old Euclidean sense, with the result that they have attached a different sense to the sentences in which the axioms have been handed down to us. For this reason I cannot emphasize strongly enough that I only mean axioms in the original Euclidean sense, when I recognize a geometrical source of knowledge in them. If we keep this firmly in mind, we need not fear that this source of knowledge will be contaminated.
From the geometrical source of knowledge flows the infinite in the genuine and strictest sense of this word. Here we are not concerned with the everyday usage according to which 'infinitely large' and 'infinitely many' imply no more than 'very large' and 'very many'. We have infinitely many points on every interval of a straight line, on every circle, and infinitely many lines through every point. That we cannot imagine the totality of these taken one at a time is neither here nor there. One man may be able to imagine more, another less. But here we are not in the domain o f psychology, of the imagination, of what is subjective, but in the domain of the objective, of what is true. It is here that geometry and philosophy come closest together. In fact they belong to one another. A philosopher who has nothing to do with geometry is only half a philosopher, and a mathematician with no element of philosophy in him is only half a mathematician. These disciplines have estranged themselves from one another to the detriment of both. This is how eventually formal arithmetic became prevalent-the view that numbers are numerals. Perhaps its time is not yet over. How do people arrive at such an idea? If someone is concerned in the science of numbers, he feels an obligation to say what is understood by numbers. Confronted by the task of explaining the concept he recognizes his inability, and without a moment's hesitation settles on the explanation that numerals are numbers. For you can of course see these things with your eyes, as you can see stones, plants and stars. You certainly have no doubt there are stones. You can have
just as little doubt there are numbers. You must only banish completely from your mind the thought that these numbers mean something or have a content. For you would then have to say what this content was, and that would lead to incredible difficulties. Just this is the advantage of formal arithmetic, that
? 274 Sources ofKnowledge ofMathematics and natural Sciences
it avoids these difficulties. That is why it cannot be emphasized strongly enough that the numbers are not the content or sense of certain signs: these very numerical signs are themselves the numbers and have no content or sense at all. People can only talk in this way if they have no glimmer of philosophical understanding. On this account, a statement of number can say nothing, and the numbers are completely useless and worthless.
It is evident that sense perception can yield nothing infinite. However many stars we may include in our inventories, there will never be infinitely many, and the same goes for us with the grains of sand on the seashore. And so, where we may legitimately claim to recognize the infinite, we have not obtained it from sense perception. For this we need a special source of knowledge, and one such is the geometrical.
Besides the spatial, the temporal must also be recognized. A source of knowledge corresponds to this too, and from this also we derive the infinite. Time stretching to infinity in both directions is like a line stretching to infinity in both directions.
? ? Numbers and Arithmetic1 [ 1924/25]
When I first set out to answer for myself the question of what is to be understood by numbers and arithmetic, I encountered-in an apparently predominant position-what was called formal arithmetic. The hallmark of formal arithmetic was the thesis 'Numbers are numerals'. How did people arrive at such a position? They felt incapable of answering the question on any rational view of what could be meant by it, they did not know how they ought to explain what is designated by the numeral '3' or the numeral '4', and then the cunning idea occurred to them of evading this question by saying 'These numerical signs do not really designate anything: they are themselves the things that we are inquiring about. ' Quite a dodge, a degree of cunning amounting, one might almost say, to genius; it's only a shame that it makes the numerals, and so the numbers themselves, completely devoid of content and quite useless. How was it possible for people not to see this? Time and again the same cunning idea occurs to people and it's very possible that there are such people to be found even today. They usually begin by assuring us that they do not intend the numerals to designate anything-no, not anything at all. And yet, it seems, in some mysterious way some content or other must manage to insinuate itself into these quite empty signs, for otherwise they would be useless. That, then, is what formal arithmetic used to be. Is it now dead? Strictly speaking, it was never alive; all the same we cannot rule out attempts to resuscitate it.
I, for my part, never had any doubt that numerals must designate something in arithmetic, if such a discipline exists at all, and that it does is surely hard to deny. We do, after all, make statements of number. In that case, what are they used to make an assertion about? For me there could be no doubt as to the answer: in a statement of number something is asserted about a concept. I was using the word 'concept' here in the sense that I still attach to it even now. To be sure, among philosophical writers this word is used in a deplorably loose way. This may be all very well for such authors, because the word is then always at hand when they need it. But, this aside, I regard the practice as pernicious.
If I say 'the number of beans in this box is six' or 'there are six beans in
1 Similarities in content to the paper 'Sources of Knowledge of Mathematics and the mathematical natural Sciences', in particular the claim of the priority for mathematics of a source of knowledge that is geometrical in nature, make it highly probable that this piece also dates from the last year of Frege's life (ed. ).
? ? 276 Numbers and Arithmetic
this box', what concept am I making an assertion about? Obviously 'bean in
this box'. *
Now numbers of different kinds have arisen in different ways and must be
distinguished accordingly. To begin with, we have what I call the kindergarten-numbers. They are, as it were, drilled into children by parents and teachers: here what people have in mind is the child's future occupation. The child is to be prepared for doing business, for buying and selling. Money has to be counted, and wares too. We have the picture of a child sitting in front of a heap of peas, picking them out one by one with his fingers, each ~me uttering a number-word. In this way something like images of numbers are formed in the child's mind. But this is an artificial process which is imposed on the child rather than one which develops naturally within him. But even if it were a natural process, there would be hardly anything to learn about the real nature of the kindergarten-numbers from the way they originate psychologically. All the same, we can go as far as to say that the series of kindergarten-numbers forms a discontinuous series, which because of this discontinuity is essentially different from the series of points on a straight line. There is always a jump from one number to the next, whereas in a series of points there is no such thing as a next point. In this respect nothing is essentially altered when the child becomes acquainted with fractions. For even after the interpolation of the rationals, the series of numbers including the rationals still has gaps in it. Anything resembling a continuum remains as impossible as ever. It is true that we can use one length to measure another with all the accuracy we need for business life, but we can do this only because the needs of business will tolerate small inaccuracies. Things are different in the strict sciences. These teach that there are infinitely many lengths that cannot be measured by a given unit of length. This is what makes the kindergarten-numbers extremely limited in their application. The labours of mathematicians have indeed led to other kinds of numbers, to the irrationals, for example; but there is no bridge which leads across from the kindergarten-numbers to the irrationals. I myself at one time held it to be possible to conquer the entire number domain, continuing along a purely logical path from the kindergarten- numbers; I have seen the mistake in this. I was right in thinking that you cannot do this if you take an empirical route. I may have arrived at this conviction as a result of the following consideration: that the series of whole numbers should eventually come to an end, that there should be a greatest whole number, is manifestly absurd. This shows that arithmetic cannot be based on sense perception; for if it could be so based, we should have to reconcile ourselves to the brute fact of the series of whole numbers comins to an end, as we may one day have to reconcile ourselves to there being no
* If something is asserted of a first level concept, what is asserted is a second level concept. And so in making a statement of number we have a second level concept.
?
1 The German editors see the distinction Frege here draws between a 'Hilfssprache' and a 'Darlegungssprache' as anticipating the distinction sub- sequently drawn by Tarski between an 'object-language' and a 'meta-language', and certainly the resemblance is close: close enough for us to have decided to avail ourselves of the expedient of using Tarski's way of speaking to translate Frege's two
notions: the reader is not to suppose that Frege anticipates this actual terminology and may judge for himself how far he anticipates Tarski's thought (trans. ).
? ? ? Logical Generality 261
their proper names 'Venus', 'Mars', etc. As such proper names of the sentences o f the object-language I use these very sentences, but enclosed in quotation marks. Moreover it follows from this that the sentences of the object-language are never given assertoric force. 'If a is a man, a is mortal' is a sentence of the object-language in which a general thought is expressed. We move from the general to the particular by substituting for the equiform indefinitely indicating letters equiform proper names. It belongs to the essence of our object-language that equiform proper names designate the same object (man). Here empty signs (names) are not proper names at all. * By substituting for the indefinitely indicating letters equiform with 'a', proper names of the form 'Napoleon', we obtain
'IfNapoleon is a man, Napoleon is mortal. '
This sentence is not however to be regarded as a conclusion, since the sentence 'If a is a man, a is mortal' is not given assertoric force, and so the thought expressed in it is not presented as one recognized to be true, for only a thought recognized as true can be made the premise o f an iriference. But an inference can be made of this, by freeing the two sentences of our object-
language from quotation marks, thus making it possible to put them forward with assertoric force.
The compound sentence ' I f Napoleon be a man, then is Napoleon mortal'1 expresses a hypothetical compound thought, composed of one condition and one consequence. The former is expressed in the sentence 'Napoleon is a man', the latter in 'Napoleon is mortal'. Strictly, however, our compound sentence contains neither a sentence equiform with 'Napoleon is a man' nor one equiform with 'Napoleon is mortal'. In this divergence between what holds at the linguistic level and what holds at the level of the thought, there emerges a defect in our object-language which is still to be remedied. I wish now to dress the thought that I expressed above in the
*I call proper names of our object-language equiform, if they are intended to be so by the writer and are meant to be of the same size, if we can recognize this to be the writer's intention even if it is imperfectly realized.
1 In German, unlike English, the order of the words in a sentence is altered when it is made into the antecedent or consequent of a hypothetical. The compound sentence with antecedent 'Napoleon is a man' and consequent 'Napoleon is mortal' transliterates 'If Napoleon a man is, is Napoleon mortal'. Throughout the preceding discussion we have translated such hypothetical sentences into natural English.
Here, where Frege is concerned with the deviation between the vernacular (German) and a language which reflects more accurately 'the level of thought', it has proved necessary to resort to a clumsy English rendering of the hypothetical sentence. The fact that English happens not to have what Frege here argues is a defective correspondence between the level of language and that of the thought, does not, of course, detract from the force or interest of the point he is here making (trans. ).
? ? 262 Logical Generality
sentence 'IfNapoleon be a man, then is Napoleon mortal', in the sentence 'If Napoleon is a man, Napoleon is mortal'; which, in what follows, I wish to call the second sentence. Similar cases are to be treated in the same way. So I want also to transform the sentence 'Ifa be a man, then is a mortal' into 'If a is a man, a is mortal', which in what follows I will call the sentence. * In the first sentence I distinguish the two individual letters equiform with 'a' from the remaining part.
*The first sentence, unlike the second, does not express a compound thought, since neither 'a is a man' nor 'a is mortal' expresses a thought. We have here really only parts of a sentence, not sentences.
? ? ? [Diary Entries on the Concept ofNumbersP [23. 3. 1924-25. 3. 1924]
23. 3. 1924 My efforts to become clear about what is meant by number have resulted in failure. We are only too easily misled by language and in this particular case the way we are misled is little short of disastrous. The sentences 'Six is an even number', 'Four is a square number', 'Five is a prime number' appear analogous to the sentences 'Sirius is a fixed star', 'Europe is a continent'-sentences whose function is to represent an object as falling under a concept. Thus the words 'six', 'four', and 'five' look like proper names of objects and 'even number', 'square number', and 'prime number', along with 'number' itself, look like concept-words. So the problem appears to be to work out more clearly the nature of the concept designated by the word 'number' and to exhibit the objects that, as it seems, are designated by number-words and numerals.
24. 3. 1924 From our earliest education onwards we are so accustomed to using the word 'number' and the number-words that we do not regard our way of using them as something that stands in need of a justification. To the mathematicians it appears beneath their dignity to concern themselves with such childish matters. But one finds amongst them the most different and contradictory statements about number and numbers. Indeed, when one has been occupied with these questions for a long time one comes to suspect that our way of using language is misleading, that number-words are not proper names of objects at all and words like 'number', 'square number' and the rest are not concept-words; and that consequently a sentence like 'Four is a square number' simply does not express that an object is subsumed under a concept and so just cannot be construed like the sentence 'Sirius is a fixed star'. But how then is it to be construed?
25. 3. 1924 At first, however, I was still captive to the error that language gives rise to. It is easy to see that a number is not a collection of things. What is a collection? A collection is a thing, a thing that is made up of things. The mathematician couples the word 'one' with the definite article
1 These remarks were taken from a diary that Frege kept from 10. 3. 24 to 9. 5. 24; the other entries are simply statements of political attitudes which cannot be counted as part of the scientific NachlajJ (ed. ). For further details concerning the nature and content of this Diary, readers are referred to Michael Dummett, Frege: Philosophy of Language, London 1973, p. xii. See also the article on Frege in the Encyclopaedia Briltanica (fifteenth edition) Trans.
? 264 [Diary Entries on the Concept ofNumber]
and speaks of 'the number one'. If now a number is a thing, we must produce the thing that is called the number one. Many people even want to call numerals numbers. If this were true, we should have to be able to specify which of these numerals was the number one. What a calamity it would then be if this One were ever to be destroyed by fire! It is already a step forward when a number is seen not as a thing but as something belonging to a thing, where the view is that different things, in spite of their differences, can possess the same One, as different leaves can, say, all possess the colour green. Now which things possess the number one? Does not the number one belong to each and every thing?
? ? ? Number 1 [September 1924]
My efforts to throw light on the questions surrounding the word 'number' and the words and signs for individual numbers seem to have ended in complete failure. Still these efforts have not been wholly in vain. Precisely because they have failed, we can learn something from them. The difficulties of these investigations are often greatly underrated. Anyone who can regard a number of a series of objects of the same kind shows that he is so far from having any real understanding that he does not yet have even an inkling of the task involved.
First of all we need to come to an understanding about the words 'number', 'numeral', 'number-word'. In everyday life one often calls the numerals numbers. In the present context we must exclude this way of speaking. The simplest thing seems to be to understand by a numeral a sign that designates a number and by a number-word a word which designates a number. Then numerals and number-words will have the same role, namely that of designating a number; only the vehicle used will be different.
What, then, are numbers themselves? What sort of thing is it that one means to designate by a number-word or numeral? Obviously it is not anything that can be perceived by the senses. A number can neither be smelt, nor tasted, nor seen, nor felt. It jars to speak of warm or yellow or bitter numbers, just as it would jar if one chose to speak of yellow or warm or bitter points in geometry. We may seek to discover something about numbers themselves from the use we make of numerals and number-words. Numerals and number-words are used, like names of objects, as proper names. The sentence 'Five is a prime number' is comparable with the sentence 'Sirius is a fixed star'. In these sentences an object (five, Sirius) is presented as falling under a concept (prime number, fixed star) (a case of an object's being subsumed under a concept). By a number, then, we are to understand an object that cannot be perceived by the senses. Even the
1 According to a remark made by the previous editors on the transcripts on which this edition is based, these comments stem 'from two drafts of which the first is dated: Sept. 1924 to 23. 111. 1924'. The date 23. 111. 1924 apparently refers to the remarks which Frege made in his diary on that day (cf. p. 263). It is also clear from remarks made by the previous editors that the first three paragraphs come from the second draft, and the last two paragraphs from the first (ed. ).
The curious dating may mean that Frege added, on Sept. 1924, the last two paragraphs to a re-working of the diary entry of 23. 111. 1924 (trans. ).
? 266 Number
objects of geometry, points, straight lines, surfaces etc. cannot really be perceived by the senses.
These investigations are especially difficult because in the very act of conducting them we are easily misled by language: by language which is, after all, an indispensable tool for carrying them out. Indeed one might think that language would first have to be freed from all logical imperfections before it was employed in such investigations. But of course the work necessary to do this can itself only be done by using this tool, for all its imperfections. Fortunately as a result of our logical work we have acquired a yardstick by which we are apprised of these defects. Such a yardstick is at work even in language, obstructed though it may be by the many illogical features that are also at work in language.
In pre-scientific discourse what are called numbers are often no more than numerals. These signs are ordered in a series beginning with the sign '1'. In this series each sign, with the exception of the last, has an immediate successor, and each sign, with the exception of the first (the sign '1') an immediate predecessor. For example 1? 2? 3? 4? 5, in which I have separated the individual numerals by asterisks.
? ? Sources of Knowledge of Mathematics and the mathematical natural Sciences1
[1924/5]
When someone comes to know something it is by his recognizing a thought to be true. For that he has first to grasp the thought. Yet I do not count the grasping of the thought as knowledge, but only the recognition of its truth, the judgement proper. What I regard as a source of knowledge is what justifies the recognition of truth, the judgement.
I distinguish the following sources of knowledge:
1. Senseperception
2. Thelogicalsourceofknowledge
3. The geometrical and temporal sources of knowledge.
Each of these is subject to its own disturbances, which detract from its value.
A. Sense Illusions
A sense impression is not in itself a judgement, but becomes important in that it is the occasion for our making a judgement. When this happens, we may be misled: in such cases we talk of 'sense illusions'. Since for the mathematical natural sciences sight is the most important sense, we will examine it in greater detail. To our consciousness, the line of vision from the eye to the object is straight. In the majority of cases, the corresponding light
1 From the correspondence between Frege and Richard Honigswald (cf. the letter from Honigswald to Frege of 24/4/1925), it emerges that Frege submitted this article to Bruno Bauch for the series edited by Honigswald and Bauch, Wissen- schaftliche Grundfragen. In the letter mentioned Honigswald gave a very favourable judgement on Frege's article, but suggested that it be essentially enlarged, 'Our concern in our venture isn't to produce a philosophical journal, in which in each issue many articles are put together, but a series of parts, each comprising between 4 to 6 sheets, and appearing on their own in bookform'. He asked Frege to resubmit the article for the Wissenschaftliche Grundfragen affer enlarging it, and listed a number of points where further reworking appeared to him to be desirable. Because of his death three months later, Frege was obviously only able to do some of the preliminary work required for this. In the NachlajJ three sheets, now lost, with notes pointing in this direction were to be found, as may be learnt from a note of the earlier editors to the Frege-Honigswald correspondence. Cf. further, Frege's letters to Honigswald of 26/4/1925 und Inter (cd. ).
? 268 Sources ofKnowledge ofMathematics and natural Sciences
ray from the object to our eye is also straight, or the deviation from a straight line is too slight to matter to us. If we notice the deviation, we speak of an optical illusion. We encounter such cases with the reflection of the light in a mirror, with diffraction or refraction of the light. Because of these illusions we might from the very outset regard visual perception as a source of knowledge that is unreliable and hence of little value, and yet it is precisely sense perception that is regarded by many as the most reliable, if not the only, source of knowledge. Of course a mirror appears to be an opening in the wall giving us a view of a neighbouring room; of course a calm stretch of water gives us the illusion of a sun which seems to be looking up at us; but people are no longer deceived by such cases, because there are at our disposal a diversity of means for correcting the judgement gained from the first impression. Of course, if there were no laws governing events, or if the laws governing events in the physical world were unknowable for us, we would lack the means for recognizing illusions for what they are and thus for rendering them harmless. The laws of nature we already know enable us to avoid being misled by sense illusions. Thus knowledge about the refraction of light tells us that many of the images we see through the microscope are wholly unreliable. In order to know the laws of nature we need perceptions that are free from illusion. And so, on its own, sense perception can be of little use to us, since to know the laws of nature we also need the other sources of knowledge: the logical and the geometrical. Thus we can only advance step by step--each extension in our knowledge of the laws of nature providing us with a further safeguard against being deceived by the senses and the purification of our perceptions helping us to a better knowledge of the laws of nature. We must be careful not to overestimate the value of sense perception, for without the other sources of knowledge, which protect us from being deceived, we could hardly get anywhere with it. We need the perceptions, but to make use of them, we also need the other sources of knowledge. Only all taken in conjunction make it possible for us to
penetrate ever deeper into mathematical physics.
For mathematics on its own, we don't need sense perception as a source
of knowledge: for it the logical and geometrical sources suffice.
At times sense perception has been an outright hindrance to the advancement of knowledge. For a long time the idea of the spherical shape of the earth was almost universally held to be completely absurd, since if it were true, on the bottom side of the earth men's heads would dangle downwards from their feet. People had been led by their senses to regard the upwards direction as everywhere the same; and even today there are difficulties in such considerations which children find difficult to surmount. It was a long time before the idea of the spherical shape of the earth gained sufficient ground for Columbus to put his trust in it and be able to embark
on his famous voyage. Its success and the subsequent circumnavigations of the earth were the victory of scientific reflection over the old view: a view
? Sources ofKnowledge ofMathematics and natural Sciences 269 that was almost overwhelmingly suggested by, and apparently unassailably
grounded in, sense perception.
B. The logical Source of Knowledge and Language
The senses present us with something external and because of this it is easier to comprehend how mistakes can occur than it is in the case of the logical source of knowledge which is wholly inside us and thus appears to be more proof against contamination. But appearances are deceptive. For our thinking is closely bound up with language and thereby with the world of the senses. Perhaps our thinking is at first a form of speaking which then becomes an imaging of speech. Silent thinking would in that case be speech which has become noiseless, taking place in the imagination.
Now we may of course also think in mathematical signs; yet even then thinking is tied up with what is perceptible to the senses. To be sure, we distinguish the sentence as the expression of a thought from the thought itself. We know we can have various expressions for the same thought. The connection of a thought with one particular sentence is not a necessary one; but that a thought of which we are conscious is connected in our mind with some sentence or other is for us men necessary. But that does not lie in the nature of the thought but in our own nature. There is no contradiction in supposing there to exist beings that can grasp the same thought as we do without needing to clad it in a form that can be perceived by the senses. But still, for us men there is this necessity. Language is a human creation; and so man had, it would appear, the capacity to shape it in conformity with the logical disposition alive in him. Certainly the logical disposition of man was at work in the formation of language but equally alongside this many other dispositions-such as the poetic disposition. And so language is not constructed from a logical blueprint.
One feature of language that threatens to undermine the reliability of thinking is its tendency to form proper names to which no objects correspond. If this happens in fiction, which everyone understands to be fiction, this has no detrimental effect. It's different if it happens in a statement which makes the claim to be strictly scientific. A particularly noteworthy example of this is the formation of a proper name after the pattern of 'the extension of the concept a', e. g. 'the extension of the concept star'. Because of the definite article, this expression appears to designate an object; but there is no object for which this phrase could be a linguistically appropriate designation. From this has arisen the paradoxes of set theory which have dealt the death blow to set theory itself. I myself was under this illusion when, in attempting to provide a logical foundation for numbers, I tried to construe numbers as sets. It is difficult to avoid an expression that has universal currency, before you learn of the mistakes it can give rise to. It
? 270 Sources ofKnowledge ofMathematics and natural Sciences
is extremely difficult, perhaps impossible, to test every expression offered us by language to see whether it is logically innocuous. So a great part of the work of a philosopher consists-or at least ought to consist-in a struggle against language. But perhaps only a few people are aware of the need for this. The same expression-'the extension of the concept star'-serves at the same time to illustrate, in yet another way, the fatal tendency of language to form apparent proper names: 'the concept star' is, of itself, one such. The definite article creates the impression that this phrase is meant to designate an object, or, what amounts to the same thing, that 'the concept star' is a proper name, whereas 'concept star' is surely a designation of a concept and thus could not be more different from a proper name. The difficulties which this idiosyncrasy of language entangles us in are incalculable.
But, isn't thinking a kind of speaking? How is it possible for thinking to be engaged in a struggle with speaking? Wouldn't that be a struggle in which thinking was at war with itself? Doesn't this spell the end to the possibility of thinking?
To be sure, if you search for the emergence of thinking in the develop- ment of an individual, you may well describe thinking as an inaudible inner speaking; but that doesn't capture the true nature of thinking. Can't a mathematician also think in formulae? The formula-language of mathematics is as much a human creation as spoken language, but is fundamentally different from it. Here those traits of spoken language which, as we have seen, lead to logical errors, can be avoided. Yet the influence of speech is so great that they aren't always avoided. Thus if we disregard how thinking occurs in the consciousness of an individual, and attend instead to the true nature of thinking, we shall not be able to equate it with speaking. In that case we shall not derive thinking from speaking; thinking will then emerge as that which has priority and we shall not be able to blame thinking for the logical defects we have noted in language.
In the formula-language of mathematics an important distinction stands out that lies concealed in verbal language. Of course mathematicians themselves are still so strongly influenced by verbal language that even in their discipline the distinction I have in mind doesn't stand out as clearly as all that. Mathematicians are compelled by the nature of their discipline to grasp a concept to which they have given the name 'function'. As early as in the upper forms of high school, pupils are introduced to trigonometric functions, and if they should then go on to study mathematics, they hear a great deal about functions, without it becoming clear to them what people mean by the word. Their teachers take great pains but in vain, and it is precisely the most gifted pupils who perhaps will understand it least, since they will notice that the definitions given do not agree with the teacher's own ~ay of speaking. One surely has some right to expect that a definition, once given, will not straightway be thrown in a corner to gather dust, but that it will be drawn upon when the expression it defines is used.
? Sources ofKnowledge ofMathematics and natural Sciences 271
But this expectation is disappointed because the alleged definition cannot begin to provide what one expects of it. For not everything can be defined; only what has been analysed into concepts can be reconstituted out of the parts yielded by the analysis. But what is simple cannot be analysed and hence not defined. If, nevertheless, someone attempts a definition, the result is nonsense. All definitions of function belong to this category. How does a child learn to understand grown-ups? Not as if it were already endowed with an understanding of a few words and grammatical constructions, so that all you would need to do would be to explain what it did not understand by means of the linguistic knowledge it already had. In reality of course children are only endowed with a capacity to learn to speak. We must be able to count on a meeting of minds with them just as in the case of animals with whom men can arrive at a mutual understanding. Neither is it possible, without a meeting o f minds, to make designations o f a logically unanalysable content intelligible to others. The word 'function' is such a designation. The way in which designations for functions are used can come to our assistance here. The sign 'sin' which we say designates the sine function, only occurs in mathematics in close combination with other signs, as in 'sin 10? ', 'sin 0? 11", 'sin a', and so is a sign in need of supplementation and therein is different from a proper name. Its content is correspondingly also in need of supplementation and therein is different from any object, for instance, even from any number; for a number, by which I don't want to understand a numerical sign, appears in mathematics as an object, e. g. the number 3.
Now it is usual in higher mathematics to permit the sign 'sin' to be
followed simply by a numerical sign or a letter standing in for one. For
instead of defining the size of an angle A in degrees, minutes and seconds, it
can simply be defined by a number as follows: Let C be a circle in the plane
of A whose centre is at the vertex of A. Let the radius of C be r. Let the sides
of A include an arc of C, whose length is b, say. Let C1 be a circle in the
plane of A, whose centre is at the vertex of A. Let the radius of C1 be r1 and
the sides of A include an arc of C1 whose length is b1? Then b1 :r1 = b :r.
Thus b/r is the same number as b/r1, and this number depends on the size
fined by the number b/r, which coincides with b1/r" and what's more a larger number corresponds to a larger angle. Thus b/r is greater than, less than or equal to b1 /r according as A is greater than, less than or equal to A 1 ? From this we may see how the number b/r (which coincides with b/r1) defines the angle A. If b/r = 1, then b = r. Thus the number 1 defines an angle for which the length b is equal to r, that is the length of the arc of C included by the sides of A is then equal to the radius of A. In the same way an angle is defined by the number 2, in which case the arc of C included by the sides of A is twice the lenath of the radius of C etc. We may also say: the number
1 of the angle A and defines that size. If instead of A we take the angle A ,
then b1/r say, takes the place of b/r, and b;/r" that of b1/r1 and in fact b1 >b if A1 >A. Andsointhatcaseb1/r>b/r. AndsothesizeoftheangleAisde-
? 272 Sources ofKnowledge ofMathematics and natural Sciences
that in this way defines the size of angle is the number yielded by measuring the arc of C included by its sides with the radius of C. In this way it is in every case fixed which number is meant when the sign 'sin' is completed by the sign for a real number. The only thing presupposed is that you know how an angle is related to its sine.
In the same way the sign 'cos' (cosine) is also in need of supplementation: it is to be completed by numerical signs, and cos 1, cos 2 and cos 3 are numbers. Thus 'cos' is neither a proper name, nor does it designate an object; but you can't deny the sign 'cos' some content. If, however, you wished to say, using the definite article, 'the content of the sign "cos"', you would convey the wrong idea, that an object was the content of the 'cos' sign. Perhaps it can be seen from this how difficult it is not to allow ourselves to be misled by language. Just because this is so difficult, it is hardly to be expected that a run-of-the-mill writer will take the trouble to avoid being misled, and linguistic usage will, to be sure, always remain as it is.
Added to this, there is also the following: mathematicians use letters to express generality, as in the sentence '(a + b) + c = (a + c) + b'. These letters here stand in for numerical signs and you arrive at the expression of a particular thought contained in the general one by substituting numerical signs for the letters. If one has in fact admitted functions, one will feel the need to express generality concerning functions too. As one uses letters instead of numerical signs so as to be able to express general thoughts concerning numbers, one will also introduce letters for the specific purpose of being able to express general thoughts concerning functions. It is customary for this purpose to use the letters J, F, g, G and also ~ and tP, which we may call function-letters. But now the function's need o f supplementation must somehow or other find expression. Now it is appropriate to introduce brackets after every function-letter, which together with that letter are to be regarded as one single sign. The space within the brackets is then the place where the sign that supplements the function-letter is to be inserted. By substituting for the function-letter in f( 1) a particular function by means of the sign 'sin', you obtain 'sin 1',just as you obtain '31' from 'a2' by substituting '3' for the letter 'a'. In each case, in so doing, yoU make the transition from an indefinitely indicating sign, that is, a letter, to one that designates determinately. If this happens in a sentence, this corresponds to the transition from a general thought to a particular one contained in it. An example ofthis is the transition from '(a- 1)? (a + 1) ? a? a- 1'to'(3- 1)? (3+1)= 3? 3- 1'. Icannotgivehereasimilar example in which a function-sign that designates definitely is substituted for an indefinitely indicating letter, since to do so I would have to presuppose certain elements of higher analysis: even so it will be clear enough what I mean, and yo1,1 will at least be able to gain some idea of the importance of the introduction of functions into mathematical investigations, and of the introduction of function-signs and function-letters into the sign-language of
? Sources ofKnowledge ofMathematics and natural Sciences 273
mathematics. It is here that the tendency of language by its use of the definite article to stamp as an object what is a function and hence a non- object, proves itself to be the source of inaccurate and misleading expressions and so also of errors of thought. Probably most of the impurities that contaminate the logical source of knowledge have their origins in this.
C. The geometrical Sources of Knowledge
From the geometrical source of knowledge flow the axioms of geometry. It is least of all liable to contamination. Yet here one has to understand the word 'axiom' in precisely its Euclidean sense. But even here people in recent works have muddied the waters by perverting-so slightly at first as to be scarcely noticeable-the old Euclidean sense, with the result that they have attached a different sense to the sentences in which the axioms have been handed down to us. For this reason I cannot emphasize strongly enough that I only mean axioms in the original Euclidean sense, when I recognize a geometrical source of knowledge in them. If we keep this firmly in mind, we need not fear that this source of knowledge will be contaminated.
From the geometrical source of knowledge flows the infinite in the genuine and strictest sense of this word. Here we are not concerned with the everyday usage according to which 'infinitely large' and 'infinitely many' imply no more than 'very large' and 'very many'. We have infinitely many points on every interval of a straight line, on every circle, and infinitely many lines through every point. That we cannot imagine the totality of these taken one at a time is neither here nor there. One man may be able to imagine more, another less. But here we are not in the domain o f psychology, of the imagination, of what is subjective, but in the domain of the objective, of what is true. It is here that geometry and philosophy come closest together. In fact they belong to one another. A philosopher who has nothing to do with geometry is only half a philosopher, and a mathematician with no element of philosophy in him is only half a mathematician. These disciplines have estranged themselves from one another to the detriment of both. This is how eventually formal arithmetic became prevalent-the view that numbers are numerals. Perhaps its time is not yet over. How do people arrive at such an idea? If someone is concerned in the science of numbers, he feels an obligation to say what is understood by numbers. Confronted by the task of explaining the concept he recognizes his inability, and without a moment's hesitation settles on the explanation that numerals are numbers. For you can of course see these things with your eyes, as you can see stones, plants and stars. You certainly have no doubt there are stones. You can have
just as little doubt there are numbers. You must only banish completely from your mind the thought that these numbers mean something or have a content. For you would then have to say what this content was, and that would lead to incredible difficulties. Just this is the advantage of formal arithmetic, that
? 274 Sources ofKnowledge ofMathematics and natural Sciences
it avoids these difficulties. That is why it cannot be emphasized strongly enough that the numbers are not the content or sense of certain signs: these very numerical signs are themselves the numbers and have no content or sense at all. People can only talk in this way if they have no glimmer of philosophical understanding. On this account, a statement of number can say nothing, and the numbers are completely useless and worthless.
It is evident that sense perception can yield nothing infinite. However many stars we may include in our inventories, there will never be infinitely many, and the same goes for us with the grains of sand on the seashore. And so, where we may legitimately claim to recognize the infinite, we have not obtained it from sense perception. For this we need a special source of knowledge, and one such is the geometrical.
Besides the spatial, the temporal must also be recognized. A source of knowledge corresponds to this too, and from this also we derive the infinite. Time stretching to infinity in both directions is like a line stretching to infinity in both directions.
? ? Numbers and Arithmetic1 [ 1924/25]
When I first set out to answer for myself the question of what is to be understood by numbers and arithmetic, I encountered-in an apparently predominant position-what was called formal arithmetic. The hallmark of formal arithmetic was the thesis 'Numbers are numerals'. How did people arrive at such a position? They felt incapable of answering the question on any rational view of what could be meant by it, they did not know how they ought to explain what is designated by the numeral '3' or the numeral '4', and then the cunning idea occurred to them of evading this question by saying 'These numerical signs do not really designate anything: they are themselves the things that we are inquiring about. ' Quite a dodge, a degree of cunning amounting, one might almost say, to genius; it's only a shame that it makes the numerals, and so the numbers themselves, completely devoid of content and quite useless. How was it possible for people not to see this? Time and again the same cunning idea occurs to people and it's very possible that there are such people to be found even today. They usually begin by assuring us that they do not intend the numerals to designate anything-no, not anything at all. And yet, it seems, in some mysterious way some content or other must manage to insinuate itself into these quite empty signs, for otherwise they would be useless. That, then, is what formal arithmetic used to be. Is it now dead? Strictly speaking, it was never alive; all the same we cannot rule out attempts to resuscitate it.
I, for my part, never had any doubt that numerals must designate something in arithmetic, if such a discipline exists at all, and that it does is surely hard to deny. We do, after all, make statements of number. In that case, what are they used to make an assertion about? For me there could be no doubt as to the answer: in a statement of number something is asserted about a concept. I was using the word 'concept' here in the sense that I still attach to it even now. To be sure, among philosophical writers this word is used in a deplorably loose way. This may be all very well for such authors, because the word is then always at hand when they need it. But, this aside, I regard the practice as pernicious.
If I say 'the number of beans in this box is six' or 'there are six beans in
1 Similarities in content to the paper 'Sources of Knowledge of Mathematics and the mathematical natural Sciences', in particular the claim of the priority for mathematics of a source of knowledge that is geometrical in nature, make it highly probable that this piece also dates from the last year of Frege's life (ed. ).
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this box', what concept am I making an assertion about? Obviously 'bean in
this box'. *
Now numbers of different kinds have arisen in different ways and must be
distinguished accordingly. To begin with, we have what I call the kindergarten-numbers. They are, as it were, drilled into children by parents and teachers: here what people have in mind is the child's future occupation. The child is to be prepared for doing business, for buying and selling. Money has to be counted, and wares too. We have the picture of a child sitting in front of a heap of peas, picking them out one by one with his fingers, each ~me uttering a number-word. In this way something like images of numbers are formed in the child's mind. But this is an artificial process which is imposed on the child rather than one which develops naturally within him. But even if it were a natural process, there would be hardly anything to learn about the real nature of the kindergarten-numbers from the way they originate psychologically. All the same, we can go as far as to say that the series of kindergarten-numbers forms a discontinuous series, which because of this discontinuity is essentially different from the series of points on a straight line. There is always a jump from one number to the next, whereas in a series of points there is no such thing as a next point. In this respect nothing is essentially altered when the child becomes acquainted with fractions. For even after the interpolation of the rationals, the series of numbers including the rationals still has gaps in it. Anything resembling a continuum remains as impossible as ever. It is true that we can use one length to measure another with all the accuracy we need for business life, but we can do this only because the needs of business will tolerate small inaccuracies. Things are different in the strict sciences. These teach that there are infinitely many lengths that cannot be measured by a given unit of length. This is what makes the kindergarten-numbers extremely limited in their application. The labours of mathematicians have indeed led to other kinds of numbers, to the irrationals, for example; but there is no bridge which leads across from the kindergarten-numbers to the irrationals. I myself at one time held it to be possible to conquer the entire number domain, continuing along a purely logical path from the kindergarten- numbers; I have seen the mistake in this. I was right in thinking that you cannot do this if you take an empirical route. I may have arrived at this conviction as a result of the following consideration: that the series of whole numbers should eventually come to an end, that there should be a greatest whole number, is manifestly absurd. This shows that arithmetic cannot be based on sense perception; for if it could be so based, we should have to reconcile ourselves to the brute fact of the series of whole numbers comins to an end, as we may one day have to reconcile ourselves to there being no
* If something is asserted of a first level concept, what is asserted is a second level concept. And so in making a statement of number we have a second level concept.
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