These
disciplines
have estranged themselves from one another to the detriment of both.
Gottlob-Frege-Posthumous-Writings
).
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.
? Numbers and Arithmetic 277
stars above a certain size. But here surely the position is different: that the series of whole numbers should eventually come to an end is not just false: we find the idea absurd. So an a priori mode of cognition must be involved here. But this cognition does not have to flow from purely logical principles, as I originally assumed. There is the further possibility that it has a geometrical source. Now of course the kindergarten-numbers appear to have nothing whatever to do with geometry. But that is just a defect in the kindergarten-numbers. The more I have thought the matter over, the more convinced I have become that arithmetic and geometry have developed on the same basis-a geometrical one in fact-so that mathematics in its entirety is really geometry. Only on this view does mathematics present itself as completely homogeneous in nature. Counting, which arose psychologically out of the demands of business life, has led the learned astray.
? ? A new Attempt at a Foundation for Arithmetic1 [1924/25]
A. )2 I first repeat earlier assertions of mine that I still regard as true.
Grundgesetze I, p. 1. Arithmetic does not need to appeal to experience in its proofs. I now express this as follows: Arithmetic does not need to appeal to sense perception in its proofs.
Grundgesetze I, p. 3. A statement of number contains an assertion about a concept.
B. ) Secondly, I retract views I have expressed previously which I can no longer hold to be correct.
I have had to abandon the view that arithmetic does not need to appeal to intuition either in its proofs, understanding by intuition the geometrical source of knowledge, that is, the source from which flow the axioms of geometry. Recently, a vicious confusion has arisen over the use of the word 'axiom'. I therefore emphasize that I am using this word in its original meaning.
C. ) I distinguish the following sources of knowledge for mathematics and physics:
1. Sense perception
2. The Geometrical Source of Knowledge 3. The Logical Source of Knowledge.
1 According to early descriptions of the Frege archives the original manuscript was found in the NachlajJ together with a copy of a hand-printed version of the article Gedankengefiige which appeared in 1923-it might indeed have been written on the back of this copy. And so the earliest possible date for its composition is 1923. On the other hand there is no mention of a 'geometrical' foundation for arithmetic in Frege's brief remarks on the concept of number written in 1924. We do however find such mention in the essay 'Sources of Knowledge in Mathematics and the mathematical Natural Sciences' (pp. 267 ff. of this volume}--possibly not written before the beginning of 1925-and also, more plainly, in the short piece 'Numbers and Arithmetic (pp. 275 ff. ). This suggests that we should assign Frege's last attempt to provide a foundation for arithmetic to the same date or later (if we regard them as an attempt to fill out what he says on pp. 267 ff. , 275 ff. ) An alternative possibility would be to date Frege's new account earlier-say in Autumn 1924; that is, as a train of thought which Frege found promising, and that lies behind the later remarks about the power of 'the geometrical source of knowl- edge' for arithmetic (ed. ).
2 In the manuscript the letter 'A' together with the phrase 'preliminary remarks' appears as a heading (ed. ).
? ? A new Attempt at a Foundation for Arithmetic 279
The last of these is involved when inferences are drawn, and thus is almost always involved. Yet it seems that this on its own cannot yield us any objects. From the geometrical source of knowledge flows pure geometry. In the case of arithmetic, just as in the case of geometry, I exclude only sense perception as a source of knowledge. Everyone will grant that there is no largest whole number, i. e. that there are infinitely many whole numbers. This doesn't imply there has ever been a time at which a man has grasped infinitely many whole numbers. Rather, there are probably infinitely many whole numbers which no man has ever grasped. This knowledge cannot be derived from sense perception, since nothing infinite in the full sense of the word can flow from this source. Stars are objects of sense perception. And so it cannot be asserted with certainty that there are infinitely many of them: no more can it be asserted with certainty that there are not infinitely many stars. Since probably on its own the logical source of knowledge cannot yield numbers either, we will appeal to the geometrical source of knowledge. This is significant because it means that arithmetic and geometry, and hence the whole of mathematics flows from one and the same source of knowledge-that is the geometrical one. This is thus elevated to the status of the true source of mathematical knowledge, with, of course, the logical source of knowledge also being involved at every turn.
The Peculiarity ofGeometry
Now in geometry we speak of straight lines, just as in physics people speak of solids, say. 'Solid' is a concept and you may point at a thing, saying 'This is a solid'; by so doing you subsume the thing under the concept 'solid'. We may surely call subsumption a logical relationship. '
D. ) We may begin by outlining my plan. Departing from the usual practice, I do not want to start out from the positive whole numbers and to extend progressively the domain of what I call numbers; for there's no doubt that you are really making a logical error if you do not use 'number' with a fixed meaning, but keep understanding something different by it. That this was how the subject evolved historically is no argument to the contrary, since in mathematics we must always strive after a system that is complete in itself. If the one that has been acknowledged until now proves inadequate, it must be demolished and replaced by a new structure. Thus, right at the outset I go straight to the final goal, the general complex numbers.
If one wished to restrict oneself to the real numbers, one could take these to be ratios of intervals on a line, in which the intervals were to be regarded as oriented, and so with a distinction between a starting point and an end point. In that case one could in fact shift the interval along the line at random without altering it in a way that has any mathematical significance, but
1 It is impossible to determine whether the heading? Peculiarity of Geometry' and the paragraph that follows it arc placed here in accordance with Frcge's own instructions or at the instigation of the previous editors. We only have a note of theirs saying that this pussuge 'is to oe inserted before I>' (ed. ).
? ? 280 A new Attempt at a Foundation for Arithmetic
not switch it round. If we want to include the complex numbers in our considerations, we must adopt as our basis not a straight line, but a plane.
I call this the base plane.
I take a point in it that I call the origin, and a different point that I call the endpoint. Then there can in fact be shifts of an interval in the base plane which have no significance for our reflections, namely shifts in a parallel direction, but not rotations of the interval. If, in the way that Gauss proposed, one lets these oriented intervals in the base plane correspond to complex numbers, then the ratio of two such intervals is a complex number which is independent of the interval originally chosen as the unit length. Thus I wish to call a ratio of intervals a number; by this means I have included the complex numbers from the very outset. I say this to make it easier for the reader to see what I'm after, but in so doing I don't wish to presuppose either a knowledge of what I want to call a complex number, or a knowledge of what I want to call a ratio. These are supposed only to be explained in the exposition which follows. The reader should therefore try to forget what he hitherto thought he knew about ratios of intervals and about complex numbers; for this 'knowledge' was probably a delusion. The fundamental mistake is that people start out from the numbers they acquired as children, say through counting a heap of peas. These numbers leave us completely in the lurch even when we encounter the irrational numbers. If
you take them seriously, there are no irrational numbers. Karl Snell, a man, long since dead, who was deeply revered by me at Jena, often enunciated the principle: in mathematics, everything is to be as clear as 2 x 2 = 4. The moment there appears anything at all which is mysterious, that is a sign that not everything is in order. But he himself, when he employed Gauss's method of introducing the complex numbers, could not avoid altogether the mysterious, and he also felt this himself and was dissatisfied with the account he gave.
E. ) The ideas' I have adopted as basic are line and point. The primitive relation between points and a line which I take as basic is given by the sentence:
The point A is symmetric with the point B with respect to the line l.
F. ) Definitions.
1. If the point P is symmetric with itself with respect to the line l, then I
say
the point P is on the line l.
2. I f the point A is a symmetric with the point A 1 with respect to the line l, and if the point B is symmetric with the point B1 with respect to the line l, and if the point C is symmetric with the point C1 with respect to the line l, then I say
The triangle ABC is symmetric with the triangle A1B1C1 with respect to the line 1.
1 Against 'ideas' the previous editors have the note: in the MS as a second correction of'primitive objects'-the first being 'concepts' (ed. ).
? A new Attempt at a Foundation for Arithmetic
281
3. If there is a line, with respect to which the triangle ABC is symmetric with the triangle DEF, then I say
The triangle ABC is ~ymmetric with the triangle DEF.
4. If the triangle ABC is symmetric with the triangle DEF, and if the triangle DEF is symmetric with the triangle GHI, then I say
The triangle ABC is congruent to the triangle GH! .
5. If there is a line, which the point A is on, and which the point B is on and which the point C is on, then I say
A, B and Care collinear.
6. If the point A is symmetric with the point B with respect to the line I and if B is symmetric with the point C with respect to the line m. and if A, B and Care collinear, then I say
I is parallel to m.
7. IfthepointA isonthelineIandifthepointAisonthelinemandifIis different from m, then I say
A is the intersection ofI and m.
8. If A is the intersection of
11 and /3, C the intersection of
11 and /4, B the intersection of
12 and /3, D the intersection of
12 and /4, if /3 is parallel to /4, if
M is the intersection of /1 and /2,
if M is different from A, and B and from C, and if the triangle M MCD is congruent to the triangle PQR, then I say
The triangle MAB is similar to the triangle PQR
or, as meaning the same,
The ratio ofMA toMB is the same as the ratio ofPQ to PR.
/4
1,
If P, Q and R are points, I write the ratio of PQ to PR in the form PQ:PR.
9. I f 0 is the origin and A the endpoint (in the base plane) and the triangle OA C is similar to the triangle PQR, then I say
The point C corresponds to the ratio PQ :PR.
10.
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.
? Numbers and Arithmetic 277
stars above a certain size. But here surely the position is different: that the series of whole numbers should eventually come to an end is not just false: we find the idea absurd. So an a priori mode of cognition must be involved here. But this cognition does not have to flow from purely logical principles, as I originally assumed. There is the further possibility that it has a geometrical source. Now of course the kindergarten-numbers appear to have nothing whatever to do with geometry. But that is just a defect in the kindergarten-numbers. The more I have thought the matter over, the more convinced I have become that arithmetic and geometry have developed on the same basis-a geometrical one in fact-so that mathematics in its entirety is really geometry. Only on this view does mathematics present itself as completely homogeneous in nature. Counting, which arose psychologically out of the demands of business life, has led the learned astray.
? ? A new Attempt at a Foundation for Arithmetic1 [1924/25]
A. )2 I first repeat earlier assertions of mine that I still regard as true.
Grundgesetze I, p. 1. Arithmetic does not need to appeal to experience in its proofs. I now express this as follows: Arithmetic does not need to appeal to sense perception in its proofs.
Grundgesetze I, p. 3. A statement of number contains an assertion about a concept.
B. ) Secondly, I retract views I have expressed previously which I can no longer hold to be correct.
I have had to abandon the view that arithmetic does not need to appeal to intuition either in its proofs, understanding by intuition the geometrical source of knowledge, that is, the source from which flow the axioms of geometry. Recently, a vicious confusion has arisen over the use of the word 'axiom'. I therefore emphasize that I am using this word in its original meaning.
C. ) I distinguish the following sources of knowledge for mathematics and physics:
1. Sense perception
2. The Geometrical Source of Knowledge 3. The Logical Source of Knowledge.
1 According to early descriptions of the Frege archives the original manuscript was found in the NachlajJ together with a copy of a hand-printed version of the article Gedankengefiige which appeared in 1923-it might indeed have been written on the back of this copy. And so the earliest possible date for its composition is 1923. On the other hand there is no mention of a 'geometrical' foundation for arithmetic in Frege's brief remarks on the concept of number written in 1924. We do however find such mention in the essay 'Sources of Knowledge in Mathematics and the mathematical Natural Sciences' (pp. 267 ff. of this volume}--possibly not written before the beginning of 1925-and also, more plainly, in the short piece 'Numbers and Arithmetic (pp. 275 ff. ). This suggests that we should assign Frege's last attempt to provide a foundation for arithmetic to the same date or later (if we regard them as an attempt to fill out what he says on pp. 267 ff. , 275 ff. ) An alternative possibility would be to date Frege's new account earlier-say in Autumn 1924; that is, as a train of thought which Frege found promising, and that lies behind the later remarks about the power of 'the geometrical source of knowl- edge' for arithmetic (ed. ).
2 In the manuscript the letter 'A' together with the phrase 'preliminary remarks' appears as a heading (ed. ).
? ? A new Attempt at a Foundation for Arithmetic 279
The last of these is involved when inferences are drawn, and thus is almost always involved. Yet it seems that this on its own cannot yield us any objects. From the geometrical source of knowledge flows pure geometry. In the case of arithmetic, just as in the case of geometry, I exclude only sense perception as a source of knowledge. Everyone will grant that there is no largest whole number, i. e. that there are infinitely many whole numbers. This doesn't imply there has ever been a time at which a man has grasped infinitely many whole numbers. Rather, there are probably infinitely many whole numbers which no man has ever grasped. This knowledge cannot be derived from sense perception, since nothing infinite in the full sense of the word can flow from this source. Stars are objects of sense perception. And so it cannot be asserted with certainty that there are infinitely many of them: no more can it be asserted with certainty that there are not infinitely many stars. Since probably on its own the logical source of knowledge cannot yield numbers either, we will appeal to the geometrical source of knowledge. This is significant because it means that arithmetic and geometry, and hence the whole of mathematics flows from one and the same source of knowledge-that is the geometrical one. This is thus elevated to the status of the true source of mathematical knowledge, with, of course, the logical source of knowledge also being involved at every turn.
The Peculiarity ofGeometry
Now in geometry we speak of straight lines, just as in physics people speak of solids, say. 'Solid' is a concept and you may point at a thing, saying 'This is a solid'; by so doing you subsume the thing under the concept 'solid'. We may surely call subsumption a logical relationship. '
D. ) We may begin by outlining my plan. Departing from the usual practice, I do not want to start out from the positive whole numbers and to extend progressively the domain of what I call numbers; for there's no doubt that you are really making a logical error if you do not use 'number' with a fixed meaning, but keep understanding something different by it. That this was how the subject evolved historically is no argument to the contrary, since in mathematics we must always strive after a system that is complete in itself. If the one that has been acknowledged until now proves inadequate, it must be demolished and replaced by a new structure. Thus, right at the outset I go straight to the final goal, the general complex numbers.
If one wished to restrict oneself to the real numbers, one could take these to be ratios of intervals on a line, in which the intervals were to be regarded as oriented, and so with a distinction between a starting point and an end point. In that case one could in fact shift the interval along the line at random without altering it in a way that has any mathematical significance, but
1 It is impossible to determine whether the heading? Peculiarity of Geometry' and the paragraph that follows it arc placed here in accordance with Frcge's own instructions or at the instigation of the previous editors. We only have a note of theirs saying that this pussuge 'is to oe inserted before I>' (ed. ).
? ? 280 A new Attempt at a Foundation for Arithmetic
not switch it round. If we want to include the complex numbers in our considerations, we must adopt as our basis not a straight line, but a plane.
I call this the base plane.
I take a point in it that I call the origin, and a different point that I call the endpoint. Then there can in fact be shifts of an interval in the base plane which have no significance for our reflections, namely shifts in a parallel direction, but not rotations of the interval. If, in the way that Gauss proposed, one lets these oriented intervals in the base plane correspond to complex numbers, then the ratio of two such intervals is a complex number which is independent of the interval originally chosen as the unit length. Thus I wish to call a ratio of intervals a number; by this means I have included the complex numbers from the very outset. I say this to make it easier for the reader to see what I'm after, but in so doing I don't wish to presuppose either a knowledge of what I want to call a complex number, or a knowledge of what I want to call a ratio. These are supposed only to be explained in the exposition which follows. The reader should therefore try to forget what he hitherto thought he knew about ratios of intervals and about complex numbers; for this 'knowledge' was probably a delusion. The fundamental mistake is that people start out from the numbers they acquired as children, say through counting a heap of peas. These numbers leave us completely in the lurch even when we encounter the irrational numbers. If
you take them seriously, there are no irrational numbers. Karl Snell, a man, long since dead, who was deeply revered by me at Jena, often enunciated the principle: in mathematics, everything is to be as clear as 2 x 2 = 4. The moment there appears anything at all which is mysterious, that is a sign that not everything is in order. But he himself, when he employed Gauss's method of introducing the complex numbers, could not avoid altogether the mysterious, and he also felt this himself and was dissatisfied with the account he gave.
E. ) The ideas' I have adopted as basic are line and point. The primitive relation between points and a line which I take as basic is given by the sentence:
The point A is symmetric with the point B with respect to the line l.
F. ) Definitions.
1. If the point P is symmetric with itself with respect to the line l, then I
say
the point P is on the line l.
2. I f the point A is a symmetric with the point A 1 with respect to the line l, and if the point B is symmetric with the point B1 with respect to the line l, and if the point C is symmetric with the point C1 with respect to the line l, then I say
The triangle ABC is symmetric with the triangle A1B1C1 with respect to the line 1.
1 Against 'ideas' the previous editors have the note: in the MS as a second correction of'primitive objects'-the first being 'concepts' (ed. ).
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281
3. If there is a line, with respect to which the triangle ABC is symmetric with the triangle DEF, then I say
The triangle ABC is ~ymmetric with the triangle DEF.
4. If the triangle ABC is symmetric with the triangle DEF, and if the triangle DEF is symmetric with the triangle GHI, then I say
The triangle ABC is congruent to the triangle GH! .
5. If there is a line, which the point A is on, and which the point B is on and which the point C is on, then I say
A, B and Care collinear.
6. If the point A is symmetric with the point B with respect to the line I and if B is symmetric with the point C with respect to the line m. and if A, B and Care collinear, then I say
I is parallel to m.
7. IfthepointA isonthelineIandifthepointAisonthelinemandifIis different from m, then I say
A is the intersection ofI and m.
8. If A is the intersection of
11 and /3, C the intersection of
11 and /4, B the intersection of
12 and /3, D the intersection of
12 and /4, if /3 is parallel to /4, if
M is the intersection of /1 and /2,
if M is different from A, and B and from C, and if the triangle M MCD is congruent to the triangle PQR, then I say
The triangle MAB is similar to the triangle PQR
or, as meaning the same,
The ratio ofMA toMB is the same as the ratio ofPQ to PR.
/4
1,
If P, Q and R are points, I write the ratio of PQ to PR in the form PQ:PR.
9. I f 0 is the origin and A the endpoint (in the base plane) and the triangle OA C is similar to the triangle PQR, then I say
The point C corresponds to the ratio PQ :PR.
10.
