ordinate system on a plane, including a
specification
of the unit length for each of the axes.
Gottlob-Frege-Posthumous-Writings
g.
rH= ne(e=fl)
is a concept in the construction of which we use, among others, the function n<:,. If now for some or other meaningful proper name 'A', 'nA' were
meaningless, then
would also be meaningless and so could mean neither the True nor the False; i. e. we should have the case that it would not be determined for the object whether or not it fell under the concept
n~ = ne (e = fl)
What we have said about object-letters, also holds mutatis mutandis for function-letters; a roman function-letter, which is used as a mark of a first level function with one argument, must be replaceable wherever it occurs in sentences by the designation of a first level function with one argument without rendering the whole devoid of meaning. From this it follows that such a letter cannot occur without an argument place: and so Peano's designation 'f e vfu' must be rejected, because the letter 'f' occurs here without any argument place, so that it e. g. becomes impossible to put in its place the designation of the function <:, + 1. If the letter 'e' in 'f e vfu' was supposed to occupy the argument place of 'f(fj', you could substitute the designation of the function<:. + 1, thus getting
'(e + 1) vfu',
but as Peano construes the formula, no such substitution can occur. For the ? same reason many of Peano's designations in which a function-letter occurs , without an argument are to be rejected. They contradict the very essence of a function, its unsaturatedness or need of supplementation. Such designations, which belie the real situation, may indeed seem at first sight ? convenient, but in the end they always lead into a morass; for at some point or another their inadequacy will become painfully apparent. So here: the letter fis supposed to serve for the designation of generality. But try to move from it to particular cases, and it nearly always breaks down. And indeed a wide variety of designations have been introduced by mathematicians with an eye only on their immediate purpose; but the designations that promise best to survive are those which adapt themselves most readily to diverse requirements and can be applied over the most extensive domain, and this because they fit the subject-matter best. We shall not be able to discover such designations if we are merely concerned to cope with the case in hand and do not P~! rsue our reflections further, but only if we attain the deepest possible insight into the nature of the subject-matter.
? Logical Defects in Mathematics1 [1898/99 or later, probably not after 1903]
There is little cause for satisfaction with the state in which mathematics finds itself at present, if you have regard not to the outside, to the amount of it, but to the degree of perfection and clarity within. In this respect it leaves almost everything to be desired if you compare it with the ideal you may reasonably propose for this discipline, and when you consider that by its very nature it ought to be better fitted to approach its ideal than is any other discipline. If you ask what constitutes the value of mathematical knowledge, the answer must be: not so much what is known as how it is known, not so much its subject-matter as the degree to which it is intellectually perspicuous and affords insight into its logical interrelations. And it is just this which is lacking. Authors explain the commonest expressions such as 'function', 'variable' and 'equal' in totally different ways and these discrepancies are not just trivial but concern the very heart of the matter. It very often happens that the same word is used by one writer to name a sign, by another a content which he goes on to present as the meaning of that sign. So in the first case a word is understood to mean a material object with physical and chemical properties, such as colour and solubility in hydrochloric ncid-while in the second it is understood to mean an object with no such properties. But it isn't only in the case of different authors that we find such discrepancies--ones which affect the very foundations of the subject: not infrequently one and the same author uses a word in a way that conflicts with his own definition. It happens, for instance, that a mathematician so explains the term 'definite integral' that he wants us to understand by it a certain limit of a sum. But the same author isn't deterred in the slightest from using the expression as if he understood it to mean a combination of si)! . ns that contains the integral sign as a constituent. So, for instance, Ludwig Scheefer* says '. . . so that, according to the Riemannian definition
? IAllgemeine Untersuchungen iiber Rectification der Curven,] Acta mathematica V [1884], p. 49.
1 The editors date this essay between 1898/99 and 1903, largely by the way in which Volume I of the Grundgesetze is referred t o - t h e way in which Frege talks of the use of the letter '. ;' 'in the first volume'. They also mention that some of the material of this essay is developed in the essay Was ist eine Funktion? published in
1904. This argument for the dating seems cogent, but we think it possible to go l'urther and say that the way in which Grundgesetze Vol. I is referred to is only imclligible if Frege is producing here, as in the case of the preceding article, a draft of material intended for inclusion in Vol. 11 of the Grundgesetze (trans. ).
? ? 158
Logical Defects in Mathematics
the definite integral f~'yl + f'(x)2 dx acquires no meaning. ' Here the word 'integral' is used in both those senses in the same breath. For if there is talk of the meaning of an integral then all that can be meant is the meaning of a
sign, or combination of signs-in fact from the context we can only understand the expression that is here formed with the integral, root and plus signs combined with numerals and letters. But at the same time appeal is made to the Riemannian definition according to which an integral is the limit of a sum, and it would be difficult to understand a limit value in Riemann's sense to mean a sign whose meaning could be asked after. In Riemann we find the following:*
'If it** should have the property that however fJ and e may be chosen, it approaches a limit A indefinitely closely as fJ becomes indefinitely small, then this value is called f~ f(x) dx. If it does not have this property f~ f(x) dx has no meaning. Even in such a case, however, there have been a number of attempts to attach a meaning to this sign, and among these extensions of the concept of a definite integral one is accepted by all mathematicians. '
Here the sign 'f~flx)dx' is nowhere called an integral. Throughout this context I would put this sign in inverted commas to make it clearer that the sign alone is meant, for the definition itself asserts that otherwise a limit value is to be understood by it. Even I wouldn't have found this precaution necessary if experience had not taught me how far one must avoid anything that could in any way encourage the mathematical sickness of our time, of confusing the sign with what is signified. This sickness may well not have been so prevalent when Riemann wrote his article. And so it's easy to understand that he should regard inverted commas as dispensable, since the sense is made clear enough by the use of the phrases 'is called', 'has no meaning', and 'we understand by'. His usage would also correspond to that according to which you do not need to insert inverted commas after the phrase 'is called'. He doesn't seem to have used the expression 'the integral has a meaning'.
The following expressions are found in his writings (5): extension of the validity of the concept 'a function is integrable', 'we may speak of an integral of the functionf(x) between a and b', and also (6): 'the possibility of a definite integral'. None of these expressions gives any reason for suspecting Riemann to have been in the grip of the epidemic we have referred to. And so we must not hesitate to assume the opposite. If an astronomer said 'The planet 2. (. has a meaning: it means the planet of the solarsystemwiththegreatestmass. Whereastheplanet~hasasyet no meaning, but it could be that it will receive a meaning later', or if he said
? Ober die'Darstellbarkeit einer Funktion durch eine trigonometrisch? Reihe. Werke [ed. by H. Weber, Leipzig 18761: p. 225.
? ? That is, the sum.
? Logical Defects in Mathematics 159
'The planet '21-has a certain similarity with the number 4', one would perhaps usk oneself whether he had some kind of brain disorder. But if a mathematician adopts ways of speaking which appear not unworthy of a Dogberry, this happens apparently without any slur on his scientific reputation.
There is a great deal of talk about variables in the mathematical literature. But you may hardly conclude from this that there exists a general agreement nhout the sense of the word. To my mind it is much more likely that most who use it don't know exactly what sense they attach to it. Whether there is someone who has found a tenable explanation of the word, I don't know. Many do not explain the word at all, which is the most convenient thing to do, but would only be justifiable if there was a general agreement among people about it. At first one might think that you should supply the word 'number' and understand by 'variable number' a number with different properties at different times, so that e. g. at one time it was prime, at another n square. But that would obviously be quite beside the point, if only because 11 is doubtful whether there are any variable numbers in this sense at all. 11ere you might think of the fact that a number, e. g. 15 000, at one time was the number of inhabitants of Jena, at another not, and that we could l'onstrue this altered relation to the concept inhabitant of Jena as an nlteration in the number 15 000. However it wouldn't be the possibility of this alteration which we need in Analysis. The expression 'the number of Inhabitants of Jena varies' isn't strictly correct. One does not mean by it that 1he same number assumes different properties, but that in the course of time ever new numbers acquire the status of being the number of inhabitants of Jcna. It would be as if you wished to say of countries like England and llolland that the rulers were of variable sex, and as if someone wanted to infer from this that there were people who were now male, now female. The illusion arises from regarding a phrase like 'the King of England' as a proper nnme. But it isn't, and only becomes one if you supplement it with a time reference. In the same way, phrases like 'the number of inhabitants of Jena', 'lhe number of known moons of Jupiter' cannot be construed as proper nnmes of numbers. But once supplemented, the proper name designates a 4uite definite number, for which there can no longer be any talk of a vnriation that is of any use to Analysis. Fine, you say, but that is still only llll altered and, I will admit, improved way of talking: instead of saying 'a number varies', we must say 'in the course of time ever new numbers ll! lsume a certain status'; still that surely doesn't make any material ditl"crence. To me, however, the way of speaking doesn't appear to be so unimportant; a wrong idiom can easily lead to real confusion. And the di11tinction isn't all that slight either. V ariation always presupposes Momething permanent that varies. To say of a man that he has grown older I? to presuppose something permanent-which is designated by the proper nnme of the man--so that, despite the variation, you are acknowledging it I? the same man. Otherwise you would have to say 'a younger man has
? ? 160 Logical Defects in Mathematics
vanished and an older one appeared'. Without such a permanent we have nothing of which we can say that it has varied. If there are no variable numbers, one may not use proper names that designate them. In that case it is quite wrong to say 'the variable x'. What we then have is surely no mere difference of idiom. So it is not clear straight off what you ought to say instead in Analysis.
The scale of the changes required makes them none too easy to comprehend. Add to this the fact that it is only in kinematics that time comes into consideration, whereas variables also enter into other branches of mathematics. It is completely out of place to introduce time there. But a variation that isn't a variation in time isn't one at all in the ordinary sense of
the term. It is obviously a question here of a term of art which needs explaining, since the reference to ordinary usage is only misleading. The expression 'variable' gives an image or metaphor which, like most metaphors, at a certain point goes lame. Therefore we shall only be able to use this word without giving rise to objections after we have stipulated more precisely what it is to mean. In one of the most recent textbooks of higher Analysis* we find the following: 'By a real variable, we understand a number that is indeterminate at the outset, and which, depending on the problem in which it occurs, can assume indefinitely many real values. '
This gives rise to a host of questions. The author obviously distinguishes two classes of numbers: the determinate and the indeterminate. We may then ask, say, to which of these classes the primes belong, or whether maybe some primes are determinate numbers and others indeterminate. We may ask further whether in the case of indeterminate numbers we must distinguish between the rational and the irrational, or whether this distinction can only be applied to determinate numbers. How many indeterminate numbers are there? How are they distinguished from one another? Can you add two indeterminate numbers, and if so, how? How do you find the number that is to be regarded as their sum? The same questions arise for adding a determinate number to an indeterminate one. To which class does the sum belong? Or maybe it belongs to a third? Perhaps there is a seminal idea here which we could also find of value outside mathematics. Perhaps we could also divide men, and mathematicians in particular, into the determinate and the indeterminate. What is a value? How does an indeterminate number assume a value? Isn't a value just a number? In which case one number would assume another--or even perhaps itself? But then why does the author use both the words 'value' and 'number', if the same thing is meant? Was it perhaps to hide his misconceptions beneath the charitable cloak of darkness? Incidentally, according to our author, a determinate number can also possess a value-presumably, after it has assumed it-as we learn from the following quotation: 'In contrast with this,
? Emanuel Czuber, Vorlesungen iiber Differential und Integralrechnung, Vol. I, Leipzig, Teubner 1898.
? Logical Defects in Mathematics
161
we call a number, whether it be determinate or indeterminate, a constant, if by virtue of the problem it possesses a fixed (invariable) value. '
From this we may infer: there are both determinate and indeterminate numbers; not only the latter but also the former can assume values, which they thereupon possess. These values are either fixed and invariable, or variable. What fixed and what variable values are isn't explained, presumably because it is so simple that it needs no explanation. Much more difficult is the distinction between variable numbers and constants. That requires much more complicated definitions-these however offer such great difficulties to the understanding that the author himself has not yet been able to penetrate to the innermost depths of their conceptual content. You might imagine that the indeterminate numbers as such are not susceptible to definition, but the author continues:
'The totality of these values forms a value set, and has the particular name, the range or domain of the variable. The variable x counts as having been defined if given any real number it can be determined whether it belongs to the range or not. '
This seems to imply the following: that a variable is defined by its range, so that if you have the same range, you have the same variable. Now it is to be assumed, even if not from his definition, yet on other grounds, that where we have the equation of a curve of the third degree
y=x3
the author would speak of a variable x and a variable y, and would give the totality of real numbers as the range of each. So here we would have only one variable, which would however be designated by the two different signs 'x' and 'y'. In ? 3 we learn the following:
'If with every value of the real variable x that belongs to its range, is correlated a determinate number y, then y is thereby also in general defined as a variable, and is called a function of the real variable x. '
And so the determinate number y is defined as a number which is at the outset indeterminate, which, depending on the problem in which it occurs, can assume indefinitely many values. So the variation presumably consists in the fact that the number, which at the outset is indeterminate, gradually works its way through to becoming determinate. All the same, it is remarkable that the number y which has become determinate in this way is still a variable and can assume indefinitely many values. Which is the problem in which y occurs? And which the problem in which x occurs? Are they different problems or do we only have one problem? How prudent and well considered here is the restriction 'which belongs to its range'; otherwise it could easily befall the variable x, that it assumed values which, according to the problem in which it occurs, it cannot assume at all. It must be protected from this misfortune.
? 162 Logical Defects in Mathematics
The use of the letters 'x' and 'y' is not above question. Is 'x' a proper name, designating a variable, and 'y' the proper name of another-as '3' designates one number and '2' another? Or do these letters only indicate variables? In the first part ofthis work roman letters have been used without exception to indicate, not to designate, and this is the use that prevails in mathematics. Only a few letters such as 'e' and the Greek 'n' are used to designate-as proper names of the base of natural logarithms and of the ratio of the circumference of a circle to its diameter. In the familiar sentence
(a+ b)? c=a? c+b? c
we have the ordinary use of letters to indicate. They serve here to confer generality on the thought. They stand in the place of proper names, but are not such (pronouns). You always obtain a particular case of the general sentence when you substitute the same proper name of a number throughout for the letter a, and similarly for b and c. The sentence now says-and just this is what constitutes its generality-that a true thought is expressed in this way no matter what proper names of numbers may be substituted. So I return to the question: In the sentence 'If with every value of the real variable x, which belongs to its range, is correlated a determinate number y, then y is thereby also in general defined as a variable and is called a function of the real variable x' are the letters 'x' and 'y' used to designate or indicate? In the first case 'x' and 'y' would be proper names of different variables. However that can't be right, since these variables would then have to be made known to us. It would then have had to be said how the variable x is given, what properties it has by which you may recognize it and distinguish it from other variables. The same would have had to be said concerning y. Since nothing of that sort has been forthcoming, we can hardly assume that the letters are used here to designate. But even if we assume that the letters are used to indicate, we run into difficulties. For in that case the sentence would have to have a sense-a true sense-when we substitute numerical signs for the letters. Let us put '2', say, for 'x' and '3' for y. We then obtain
'If with every value of the real variable 2, which belongs to its range, is correlated a determinate number 3, then 3 is thereby also in general defined as a variable and is called a function of the real variable 2. '
That is completely unintelligible. What, for instance, is a value of 2? You will say the mistake consists in substituting for 'x' the sign for a determinate number; it would have to be replaced by the sign (proper name) of an indeterminate number. Unluckily, we are not yet acquainted with any indeterminate numbers at all, cannot distinguish one indeterminate number from another and so are not in a position to stipulate proper names for them. You may say 'Surely there are indeterminate numbers and they also can be distinguished and named. For instance, let there be a right-angled co?
ordinate system on a plane, including a specification of the unit length for each of the axes. Let x then be the abscissa andy the ordinate of a point. "x"
? Logical Defects in Mathematics 163
is then the proper name of one indeterminate number and "y" of another; in that case x and y will be indeterminate numbers which can be distinguished from each other, and can be spoken about by using their proper names-simply the letters 'x' and 'y'. And so it will be possible for the variables x and y in the quotation cited to be of this sort. ' I reply: give me any point on the plane whatsoever. Then this point will have a quite determinate abscissa and a quite determinate ordinate. So the question of indeterminate numbers simply doesn't arise. It may be objected that we should not take a determinate but an indeterminate point. But then we are back with the same difficulty. I doubt just as much that there are indeterminate points as that there are indeterminate numbers or indetermin- ate people.
'But the solution of the quadratic equation x2 - 4x + 3 = 0 is surely indeterminate, and so 'x' designates this indeterminate solution and hence an indeterminate number. ' I reply: this is neither true nor false, for since there is more than one solution, the definite article in the phrase 'the solution' is a logical error, and the sentence 'The problem contained in that equation permits of more than one solution' expressed correctly what was expressed erroneously above. It's a question of a property of the problem, not of a property of the solution. It is mistaken to say 'the indeterminate solution' or 'the indeterminate number which satisfied the equation': on the other hand we might perhaps call the problem an undetermining one, if this way of speaking were admitted. For this reason the formation 'x = 2 ? 1' is to be avoided like the plague. The combination of signs '2 ? 1' has neither sense nor meaning. Still the quadratic equation can be referred to as an example to show that apart from the use of letters to indicate or to designate, we have to assume yet a third. Of course this would be no argument for acknowledging indeterminate numbers. Regarded as a problem, the equation asserts nothing, and so there admittedly can be no question of using the letter x here to indicate-to confer generality on the thought. What we really have here is the designation of a concept, and the challenge to cite objects (in this case numbers) which fall under it. In that case the 'x' is used in the way I used the letter 'e' in the first volume: 1 it occupies the argument-places, so that we can recognize them as such. Once you have solved the problem, you may assert thesentence'Ifx=1orx=3,thenx2 - 4x+3=0',andheretheletter'x' is once more used to indicate as above, to make the thought general. What-
ever numerical sign may be substituted for 'x', we always obtain a true sen- tence, either because the condition is not fulfilled, in which case it is all one whether the consequent is true or false (for instance if you substitute '2'), or because the condition is fulfilled, in which case the consequent is then also true.
Now how are we to understand the use of the letters 'x' and 'y' in analytic
geometry? Of course the equation of a parabola 'y ? y - 2x = 0' isn't meant
to assert anything: rather, this equation always occurs as a constituent of -----. ----
1 Of the Grundgesetze (ed. ).
? ? 164 Logical Defects in Mathematics
sentences in which something is asserted, and in these sentences these letters will be used to indicate; for example 'If x is the abscissa of a point and y the ordinate, and if y ? y - 2x = 0, then this point lies on a parabola, which . . . etc. ' Here the co-ordinate system must be presupposed to be already given.
Thus neither do we encounter indeterminate numbers here. There are none such and that there are such is only an illusion created by a defective idiom.
The words 'function' and 'variable' are often used in conjunction with each other. We have in fact just seen how 'the function of the real variable x' has been defined as a variable. As against this Heine* says: 'A single-valued
function of a variable x means an expression which is uniquely defined for each individual rational or irrational value of x'. We are not in fact told that the letter 'x' occurs in the expression, but this must surely be assumed to be the case. xis called a variable. Whether that means the letter 'x' or what this letter means or indicates is unclear. Nor is it said what a value of xis. The most likely interpretation of Heine's meaning is surely the following: 'A single valued function of the letter "x" is a formula (a complex sign) for which it is established what every formula means which is derived from it by substituting for "x" a sign for a rational or irrational number'. Thus on Heine's view it appears that a function is not a number, but a formula, a complex sign. This explanation doesn't appear to agree with Czuber's. It is hardly worth the effort to try to reconcile the two, since, to be sure, Czuber's view is difficult to reconcile with itself. The fact that Heine shows these visible things the honour of bringing them under a concept and laying down a special word-'function'-for it, seems to imply that he regards such physical things as the objects of arithmetical investigation. But then again we have the fact that he uses the word 'define', which can here surely only mean 'stipulate a meaning for a simple or complex sign'. But if signs are taken to have meanings, it is striking that it isn't these meanings, but the signs which are supposed to be the main thing. Why then the meanings, if you don't bother with them but only with the signs? So we see obscurity and disagreement among writers of mathematics, and that where they are explaining words which are some of the most common amongst mathemati? cians. In point of fact, these questions have already been definitively resolved in our first Volume. I have only raked them up again because I suspect that many will hold my accounts to be extraordinarily difficult to understand, as compared with such accounts as you find in Czuber'a lectures or elsewhere, which they will hold to be models of clarity. I have tried to show that this illusion of clarity only lasts as long as you raise none of the questions which those accounts naturally give rise to and so fail to notice that no satisfactory answer is to be found for them.
? Die Elemente der Funktionslehre, Crelle Is Journal fiir die reine und angewandte Mathematikl, Vol. 74118721, p. 180.
? Logical Defects in Mathematics 165
But this lack of agreement isn't only to be found in the use of expressions from Analysis: it is even to be found in the case of a word like 'equal' and the sign'=', which are used throughout the highest and the lowest branches of mathematics. G. Peano* says that the opinions of writers concerning the identity sign are very diverse, and unfortunately he's quite right. The implications of what he says are enormous. For if you drop equality from nrithmetic, there's almost nothing left: and so that claim implies nothing less than that mathematicians have diverse views about the sense of the major part of their theorems. A non-mathematician on hearing this might well clap his hands to his head in amazement, and be completely unable to grasp how such a situation was possible in any science and particularly in mathematics. His amazement would grow if he learnt that this wasn't in the least felt as a major calamity, that most mathematicians believe they have rnuch more important things to do than spend their time splitting hairs in trying to remove this defect. A lot of people may think: if only concern with such questions were not so dreadfully sterile! I would like in return to ask: is mathematics then so calloused by formalism that we no longer bother at all nhout the sense of our sentences? What use are 100 000 theorems to us, if we ourselves haven't the faintest idea what we mean by them, if the man using a theorem attaches a different sense to it from the man who proved it? Ilow often is the phrase 'power series' used! But what do we understand by it? What is a power? What is a series? People don't even agree about the question whether these things are configurations which men produce with writing implements, possessing physical properties, or whether powers, series and power series are only designated by such configurations, but are 1hcmselves non-spatial and invisible. What holds the terms of a series together? That they form a whole, this being just what the series is? Or is it 1he spatial relation of juxtaposition? You will certainly receive divergent unswers to that question. And yet the difference between what is created, spatial, sensible and transitory on the one hand, and what is non-spatial, ntcmporal, non-sensible on the other, is so huge that it belongs among the areatest which can possibly exist. A science of objects of the one sort must diiTer far more profoundly from a science of objects of the other sort than, suy, astronomy and cryptography. Such questions therefore touch the very heart of arithmetic. And yet this indifference! With the divergence of opinions and modes of expression, anyone using a word such as 'number', 'function' or 'power series' should by rights state what he understands by it. But many would angrily reject the demand that they concern themselves with such tediously academic questions, in the secret fear that they couldn't
? Revue de Mathematiques (Rivista di matematica), Vol. VI, p. 61: 'Del resto le opinioni dei varii Autori, sui concetto di eguaglianza, diversificano ~tssai; ed uno studio di questa questione sarebbe assai utile, specialmente se futto coll'aiuto di simboli, anziche di parole'.
? 166 Logical Defects in Mathematics
say anything to the point. Others perhaps would boldy come straight out with an answer, but only because they were unaware of the hidden pitfalls and did not feel in the slightest the need to make their own use accord with their explanations and to test them carefully with that object in mind. What use to us are explanations when they have no intrinsic connection with a piece of work, but are only stuck on to the outside like a useless ornament?
? ? ? On Euclidean Geometry1 [1899-1906? ]
It seems worthwhile to begin by coming to an understanding about the use of certain expressions. The word 'sentence' ['Satz'F is used in diverse ways. The one that comes most readily to mind is no doubt the purely linguistic one. It is in this sense that human discourse consists of sentences. However it is not the sentence itself that really concerns us when we speak, but the sense or content which we associate with it and which we wish to communicate. Since the sense itself cannot be perceived by the senses, we have perforce, in order to communicate, to avail ourselves of something that can be perceived. So the sentence and its sense, the perceptible and the imperceptible, belong together. I call the sense of a significant sentence a thought. Thoughts are either true or false. And it is the question whether a thought is true or false that is usually the reason why, in scientific work, we nre concerned with thoughts. Now it can happen only too easily that a sign nnd its content are not clearly differentiated. ? This kind of thing happens particularly where the content itself cannot be perceived by the senses, and where, for that reason, the sign is the only thing that we can perceive. Now it is a fact that man is by nature so orientated towards what is external, towards what can be perceived, that what cannot be perceived is often ulmost completely hidden from him by what can be perceived. The distinction will come very easily to someone who knows several languages, Hince he will recall that the same sense can be expressed in different lunguages.
? Confusion of numerals and numbers.
1 These pieces form part of the controversy with Hilbert and so cannot be earlier thnn 1899, the year in which Vol. 1 of the Grundlagen der Geometrie was puhlished. Assuming the controversy came essentially to an end with Frege's last pnpcr on the foundations of geometry in 1906, it is very unlikely that the present pieces are later than 1906. However, Frege did, in subsequent unpublished writings, continue to express himself at variance with Hilbert's way of regarding axioms and definitions. See, for instance pp. 247 ff. and pp. 273 f. (ed. ).
2 The German word rendered throughout by 'sentence' is 'Satz' (plural 'Siitze') which, as Frege here remarks, is used in diverse ways. The English word does not, unfortunately, have quite the same family of uses as the German word and at Llertnin places, notably where Frege uses compound nouns formed from'Satz', the tr11nslation is (unavoidably) unintelligible. The insertions in square brackets have therefore been added to enable the reader to follow the drift of Frege's remarks (tr11ns. ).
? 168 On Euclidean Geometry
Now I have decided to use the word 'sentence' only to designate what is linguistic-that which has a thought as its sense. Thus I distinguish sentences from thoughts. Now we can understand how easy it is to ascribe to a sentence what belongs, properly speaking, to the thought and how it is that people speak of true sentences instead of true thoughts. They are then calling a sentence true when the sense of the sentence, the thought expressed in it, is true. We should note further that in grammar 'sentence' is often used somewhat differently from how I am using it here, as when one speaks of subordinate clauses [Nebensiitzen]. In many cases these do not have a complete thought as their sense, but only part of a thought.
It is particularly awkward that in mathematics it has become customary to speak of theorems [Lehrsiitzen]. This has given the word 'sentence' a chameleon-like quality: whilst we take it to refer to something linguistic, it all of a sudden changes and refers to a thought.
Many readers will have grown so accustomed to the way the word 'sentence' ['Satz'] is used in mathematical textbooks that it upsets them to find, in place of the word 'sentence', which they are used to, words like 'thought' or 'truth', to which they are not accustomed. To make it easier to follow I shall put the word 'sentence' ['Satz'] in brackets after the word 'thought'.
In the majority of cases what concerns us about a thought is whether it is true. The most appropriate name for a true thought is a truth. A science is a system of truths. A thought, once grasped, keeps pressing us for an answer to the question whether it is true. We declare our recognition of the truth of a thought, or as we may also say, our recognition of a truth, by uttering a sentence with assertoric force. Language is frequently lacking any word or part of a sentence that corresponds to the assertoric force, so that ? this often can only be felt. Subordinate clauses, which are nothing more than parts of sentences, are often uttered without assertoric force.
Whoever holds Euclidean geometry to be true will ascribe a sense to each of its theorems: he will regard each theorem as expressing a truth. He thereby implies that he recognizes that the concept-words 'point', 'line', 'plane' have a sense. In Euclidean geometry certain truths have traditionally been accorded the status of axioms. No thought that is held to be false can be accepted as an axiom, for an axiom is a truth. Furthermore, it is part of the concept of an axiom that it can be recognized as true independently of other truths.
(Truths can be inferred in accordance with the logical laws of inference. )
If a truth is given, it can be asked from what other truths its truth follows in accordance with the logical laws of inference. When this question has been answered, we can go on to ask of each of the truths that we have thus discovered from what other truths its truth follows in accordance with the logical laws of inference.
? On Euclidean Geometry 169 On Euclidean Geometry
No man can serve two masters. One cannot serve both truth and untruth. If Euclidean geometry is true, then non-Euclidean geometry is false, and if non-Euclidean geometry is true, then Euclidean geometry is false.
If given a point not lying on a line one and only one line can be drawn through that point parallel to that line then, given any line l and point P not lying on l, a line can be drawn through P parallel to l and any line that passes through P and is parallel to l will coincide with it.
Whoever acknowledges Euclidean geometry to be true must reject non- l~uclidean geometry as false, and whoever acknowledges non-Euclidean
geometry to be true must reject Euclidean geometry.
People at one time believed they practised a science, which went by the
name of alchemy; but when it was discovered that this supposed science was riddled with error, it was banished from among the sciences. Again, people at one time believed they practised a science, which went by the name of astrology. But this too was banished from among the sciences once men had seen through it and discovered that it was unscientific. The question at the present time is whether Euclidean or non-Euclidean geometry should be struck off the role of the sciences and made to line up as a museum piece alongside alchemy and astrology. If one is content to have only phantoms hovering around one, there is no need to take the matter so seriously; but in science we are subject to the necessity of seeking after truth. There it is a rase of in or out! Well, is it Euclidean or non-Euclidean geometry that should get the sack? That is the question. Do we dare to treat Euclid's elements, which have exercised unquestioned sway for 2000 years, as we have treated astrology? It is only if we do not dare to do this that we can put l~uclid's axioms forward as propositions that are neither false nor doubtful. In that case non-Euclidean geometry will have to be counted amongst the pseudo-sciences, to the study of which we still attach some slight importance, but only as historical curiosities.
? ? [Frege's Notes on Hilbert's 'Grundlagen der Geometrie'P [After 1903)
? 1
The Elements of Geometry and the five groups
of Axioms
Definition: We imagine three different systems
of things: we call the things of the first system
points and designate them by A, B, C, . . . ; we (Pim, Lim, Plim) We. call the things of the second system lines and
designate them by a, b, c . . . ; we call the things
of the third system planes and designate them by
a, /3, y, . . . ; the points are also called the ele-
ments oflinear geometry, the points and lines the
elements ofplane geometry, and the points, lines
and planes, the elements ofspatial geometry or
o f space.
We imagine the points, lines and planes as standing in certain reciprocal relations, which we designate by words such as 'lie on',
'between', 'parallel', 'congruent', 'continuous';
the exact and complete description of these relations is achieved by the axioms ofGeometry.
1 Frege's notes, which in the manuscript are written out as a continuous series of remarks, probably refer to the 2nd edition of Hilbert's Grundlagen der Geometrie, The editors have put alongside Frege's critical notes, which are set out on the right? hand side, the corresponding passages of Hilbert's text. The notes cannot refer to the 1st edition of the Grundlagen (1899), since there the theorem 4 criticized by Frege still figures as an axiom and was only proved in 1902 by E. H. Moore. Besides, Frege's last remark evidently refers to the first congruence axiom; but in the first edition ? 5 contained the parallels axiom. It is indeed possible that Freae had before him a later edition than the 2nd, but it is hardly probable, if you work on the assumption that Frege wrote down his critique during the period of his first engagement with Hilbert's work, which came to an end in 1906 (the 3rd edition first appeared in 1909) (ed. ).
rH= ne(e=fl)
is a concept in the construction of which we use, among others, the function n<:,. If now for some or other meaningful proper name 'A', 'nA' were
meaningless, then
would also be meaningless and so could mean neither the True nor the False; i. e. we should have the case that it would not be determined for the object whether or not it fell under the concept
n~ = ne (e = fl)
What we have said about object-letters, also holds mutatis mutandis for function-letters; a roman function-letter, which is used as a mark of a first level function with one argument, must be replaceable wherever it occurs in sentences by the designation of a first level function with one argument without rendering the whole devoid of meaning. From this it follows that such a letter cannot occur without an argument place: and so Peano's designation 'f e vfu' must be rejected, because the letter 'f' occurs here without any argument place, so that it e. g. becomes impossible to put in its place the designation of the function <:, + 1. If the letter 'e' in 'f e vfu' was supposed to occupy the argument place of 'f(fj', you could substitute the designation of the function<:. + 1, thus getting
'(e + 1) vfu',
but as Peano construes the formula, no such substitution can occur. For the ? same reason many of Peano's designations in which a function-letter occurs , without an argument are to be rejected. They contradict the very essence of a function, its unsaturatedness or need of supplementation. Such designations, which belie the real situation, may indeed seem at first sight ? convenient, but in the end they always lead into a morass; for at some point or another their inadequacy will become painfully apparent. So here: the letter fis supposed to serve for the designation of generality. But try to move from it to particular cases, and it nearly always breaks down. And indeed a wide variety of designations have been introduced by mathematicians with an eye only on their immediate purpose; but the designations that promise best to survive are those which adapt themselves most readily to diverse requirements and can be applied over the most extensive domain, and this because they fit the subject-matter best. We shall not be able to discover such designations if we are merely concerned to cope with the case in hand and do not P~! rsue our reflections further, but only if we attain the deepest possible insight into the nature of the subject-matter.
? Logical Defects in Mathematics1 [1898/99 or later, probably not after 1903]
There is little cause for satisfaction with the state in which mathematics finds itself at present, if you have regard not to the outside, to the amount of it, but to the degree of perfection and clarity within. In this respect it leaves almost everything to be desired if you compare it with the ideal you may reasonably propose for this discipline, and when you consider that by its very nature it ought to be better fitted to approach its ideal than is any other discipline. If you ask what constitutes the value of mathematical knowledge, the answer must be: not so much what is known as how it is known, not so much its subject-matter as the degree to which it is intellectually perspicuous and affords insight into its logical interrelations. And it is just this which is lacking. Authors explain the commonest expressions such as 'function', 'variable' and 'equal' in totally different ways and these discrepancies are not just trivial but concern the very heart of the matter. It very often happens that the same word is used by one writer to name a sign, by another a content which he goes on to present as the meaning of that sign. So in the first case a word is understood to mean a material object with physical and chemical properties, such as colour and solubility in hydrochloric ncid-while in the second it is understood to mean an object with no such properties. But it isn't only in the case of different authors that we find such discrepancies--ones which affect the very foundations of the subject: not infrequently one and the same author uses a word in a way that conflicts with his own definition. It happens, for instance, that a mathematician so explains the term 'definite integral' that he wants us to understand by it a certain limit of a sum. But the same author isn't deterred in the slightest from using the expression as if he understood it to mean a combination of si)! . ns that contains the integral sign as a constituent. So, for instance, Ludwig Scheefer* says '. . . so that, according to the Riemannian definition
? IAllgemeine Untersuchungen iiber Rectification der Curven,] Acta mathematica V [1884], p. 49.
1 The editors date this essay between 1898/99 and 1903, largely by the way in which Volume I of the Grundgesetze is referred t o - t h e way in which Frege talks of the use of the letter '. ;' 'in the first volume'. They also mention that some of the material of this essay is developed in the essay Was ist eine Funktion? published in
1904. This argument for the dating seems cogent, but we think it possible to go l'urther and say that the way in which Grundgesetze Vol. I is referred to is only imclligible if Frege is producing here, as in the case of the preceding article, a draft of material intended for inclusion in Vol. 11 of the Grundgesetze (trans. ).
? ? 158
Logical Defects in Mathematics
the definite integral f~'yl + f'(x)2 dx acquires no meaning. ' Here the word 'integral' is used in both those senses in the same breath. For if there is talk of the meaning of an integral then all that can be meant is the meaning of a
sign, or combination of signs-in fact from the context we can only understand the expression that is here formed with the integral, root and plus signs combined with numerals and letters. But at the same time appeal is made to the Riemannian definition according to which an integral is the limit of a sum, and it would be difficult to understand a limit value in Riemann's sense to mean a sign whose meaning could be asked after. In Riemann we find the following:*
'If it** should have the property that however fJ and e may be chosen, it approaches a limit A indefinitely closely as fJ becomes indefinitely small, then this value is called f~ f(x) dx. If it does not have this property f~ f(x) dx has no meaning. Even in such a case, however, there have been a number of attempts to attach a meaning to this sign, and among these extensions of the concept of a definite integral one is accepted by all mathematicians. '
Here the sign 'f~flx)dx' is nowhere called an integral. Throughout this context I would put this sign in inverted commas to make it clearer that the sign alone is meant, for the definition itself asserts that otherwise a limit value is to be understood by it. Even I wouldn't have found this precaution necessary if experience had not taught me how far one must avoid anything that could in any way encourage the mathematical sickness of our time, of confusing the sign with what is signified. This sickness may well not have been so prevalent when Riemann wrote his article. And so it's easy to understand that he should regard inverted commas as dispensable, since the sense is made clear enough by the use of the phrases 'is called', 'has no meaning', and 'we understand by'. His usage would also correspond to that according to which you do not need to insert inverted commas after the phrase 'is called'. He doesn't seem to have used the expression 'the integral has a meaning'.
The following expressions are found in his writings (5): extension of the validity of the concept 'a function is integrable', 'we may speak of an integral of the functionf(x) between a and b', and also (6): 'the possibility of a definite integral'. None of these expressions gives any reason for suspecting Riemann to have been in the grip of the epidemic we have referred to. And so we must not hesitate to assume the opposite. If an astronomer said 'The planet 2. (. has a meaning: it means the planet of the solarsystemwiththegreatestmass. Whereastheplanet~hasasyet no meaning, but it could be that it will receive a meaning later', or if he said
? Ober die'Darstellbarkeit einer Funktion durch eine trigonometrisch? Reihe. Werke [ed. by H. Weber, Leipzig 18761: p. 225.
? ? That is, the sum.
? Logical Defects in Mathematics 159
'The planet '21-has a certain similarity with the number 4', one would perhaps usk oneself whether he had some kind of brain disorder. But if a mathematician adopts ways of speaking which appear not unworthy of a Dogberry, this happens apparently without any slur on his scientific reputation.
There is a great deal of talk about variables in the mathematical literature. But you may hardly conclude from this that there exists a general agreement nhout the sense of the word. To my mind it is much more likely that most who use it don't know exactly what sense they attach to it. Whether there is someone who has found a tenable explanation of the word, I don't know. Many do not explain the word at all, which is the most convenient thing to do, but would only be justifiable if there was a general agreement among people about it. At first one might think that you should supply the word 'number' and understand by 'variable number' a number with different properties at different times, so that e. g. at one time it was prime, at another n square. But that would obviously be quite beside the point, if only because 11 is doubtful whether there are any variable numbers in this sense at all. 11ere you might think of the fact that a number, e. g. 15 000, at one time was the number of inhabitants of Jena, at another not, and that we could l'onstrue this altered relation to the concept inhabitant of Jena as an nlteration in the number 15 000. However it wouldn't be the possibility of this alteration which we need in Analysis. The expression 'the number of Inhabitants of Jena varies' isn't strictly correct. One does not mean by it that 1he same number assumes different properties, but that in the course of time ever new numbers acquire the status of being the number of inhabitants of Jcna. It would be as if you wished to say of countries like England and llolland that the rulers were of variable sex, and as if someone wanted to infer from this that there were people who were now male, now female. The illusion arises from regarding a phrase like 'the King of England' as a proper nnme. But it isn't, and only becomes one if you supplement it with a time reference. In the same way, phrases like 'the number of inhabitants of Jena', 'lhe number of known moons of Jupiter' cannot be construed as proper nnmes of numbers. But once supplemented, the proper name designates a 4uite definite number, for which there can no longer be any talk of a vnriation that is of any use to Analysis. Fine, you say, but that is still only llll altered and, I will admit, improved way of talking: instead of saying 'a number varies', we must say 'in the course of time ever new numbers ll! lsume a certain status'; still that surely doesn't make any material ditl"crence. To me, however, the way of speaking doesn't appear to be so unimportant; a wrong idiom can easily lead to real confusion. And the di11tinction isn't all that slight either. V ariation always presupposes Momething permanent that varies. To say of a man that he has grown older I? to presuppose something permanent-which is designated by the proper nnme of the man--so that, despite the variation, you are acknowledging it I? the same man. Otherwise you would have to say 'a younger man has
? ? 160 Logical Defects in Mathematics
vanished and an older one appeared'. Without such a permanent we have nothing of which we can say that it has varied. If there are no variable numbers, one may not use proper names that designate them. In that case it is quite wrong to say 'the variable x'. What we then have is surely no mere difference of idiom. So it is not clear straight off what you ought to say instead in Analysis.
The scale of the changes required makes them none too easy to comprehend. Add to this the fact that it is only in kinematics that time comes into consideration, whereas variables also enter into other branches of mathematics. It is completely out of place to introduce time there. But a variation that isn't a variation in time isn't one at all in the ordinary sense of
the term. It is obviously a question here of a term of art which needs explaining, since the reference to ordinary usage is only misleading. The expression 'variable' gives an image or metaphor which, like most metaphors, at a certain point goes lame. Therefore we shall only be able to use this word without giving rise to objections after we have stipulated more precisely what it is to mean. In one of the most recent textbooks of higher Analysis* we find the following: 'By a real variable, we understand a number that is indeterminate at the outset, and which, depending on the problem in which it occurs, can assume indefinitely many real values. '
This gives rise to a host of questions. The author obviously distinguishes two classes of numbers: the determinate and the indeterminate. We may then ask, say, to which of these classes the primes belong, or whether maybe some primes are determinate numbers and others indeterminate. We may ask further whether in the case of indeterminate numbers we must distinguish between the rational and the irrational, or whether this distinction can only be applied to determinate numbers. How many indeterminate numbers are there? How are they distinguished from one another? Can you add two indeterminate numbers, and if so, how? How do you find the number that is to be regarded as their sum? The same questions arise for adding a determinate number to an indeterminate one. To which class does the sum belong? Or maybe it belongs to a third? Perhaps there is a seminal idea here which we could also find of value outside mathematics. Perhaps we could also divide men, and mathematicians in particular, into the determinate and the indeterminate. What is a value? How does an indeterminate number assume a value? Isn't a value just a number? In which case one number would assume another--or even perhaps itself? But then why does the author use both the words 'value' and 'number', if the same thing is meant? Was it perhaps to hide his misconceptions beneath the charitable cloak of darkness? Incidentally, according to our author, a determinate number can also possess a value-presumably, after it has assumed it-as we learn from the following quotation: 'In contrast with this,
? Emanuel Czuber, Vorlesungen iiber Differential und Integralrechnung, Vol. I, Leipzig, Teubner 1898.
? Logical Defects in Mathematics
161
we call a number, whether it be determinate or indeterminate, a constant, if by virtue of the problem it possesses a fixed (invariable) value. '
From this we may infer: there are both determinate and indeterminate numbers; not only the latter but also the former can assume values, which they thereupon possess. These values are either fixed and invariable, or variable. What fixed and what variable values are isn't explained, presumably because it is so simple that it needs no explanation. Much more difficult is the distinction between variable numbers and constants. That requires much more complicated definitions-these however offer such great difficulties to the understanding that the author himself has not yet been able to penetrate to the innermost depths of their conceptual content. You might imagine that the indeterminate numbers as such are not susceptible to definition, but the author continues:
'The totality of these values forms a value set, and has the particular name, the range or domain of the variable. The variable x counts as having been defined if given any real number it can be determined whether it belongs to the range or not. '
This seems to imply the following: that a variable is defined by its range, so that if you have the same range, you have the same variable. Now it is to be assumed, even if not from his definition, yet on other grounds, that where we have the equation of a curve of the third degree
y=x3
the author would speak of a variable x and a variable y, and would give the totality of real numbers as the range of each. So here we would have only one variable, which would however be designated by the two different signs 'x' and 'y'. In ? 3 we learn the following:
'If with every value of the real variable x that belongs to its range, is correlated a determinate number y, then y is thereby also in general defined as a variable, and is called a function of the real variable x. '
And so the determinate number y is defined as a number which is at the outset indeterminate, which, depending on the problem in which it occurs, can assume indefinitely many values. So the variation presumably consists in the fact that the number, which at the outset is indeterminate, gradually works its way through to becoming determinate. All the same, it is remarkable that the number y which has become determinate in this way is still a variable and can assume indefinitely many values. Which is the problem in which y occurs? And which the problem in which x occurs? Are they different problems or do we only have one problem? How prudent and well considered here is the restriction 'which belongs to its range'; otherwise it could easily befall the variable x, that it assumed values which, according to the problem in which it occurs, it cannot assume at all. It must be protected from this misfortune.
? 162 Logical Defects in Mathematics
The use of the letters 'x' and 'y' is not above question. Is 'x' a proper name, designating a variable, and 'y' the proper name of another-as '3' designates one number and '2' another? Or do these letters only indicate variables? In the first part ofthis work roman letters have been used without exception to indicate, not to designate, and this is the use that prevails in mathematics. Only a few letters such as 'e' and the Greek 'n' are used to designate-as proper names of the base of natural logarithms and of the ratio of the circumference of a circle to its diameter. In the familiar sentence
(a+ b)? c=a? c+b? c
we have the ordinary use of letters to indicate. They serve here to confer generality on the thought. They stand in the place of proper names, but are not such (pronouns). You always obtain a particular case of the general sentence when you substitute the same proper name of a number throughout for the letter a, and similarly for b and c. The sentence now says-and just this is what constitutes its generality-that a true thought is expressed in this way no matter what proper names of numbers may be substituted. So I return to the question: In the sentence 'If with every value of the real variable x, which belongs to its range, is correlated a determinate number y, then y is thereby also in general defined as a variable and is called a function of the real variable x' are the letters 'x' and 'y' used to designate or indicate? In the first case 'x' and 'y' would be proper names of different variables. However that can't be right, since these variables would then have to be made known to us. It would then have had to be said how the variable x is given, what properties it has by which you may recognize it and distinguish it from other variables. The same would have had to be said concerning y. Since nothing of that sort has been forthcoming, we can hardly assume that the letters are used here to designate. But even if we assume that the letters are used to indicate, we run into difficulties. For in that case the sentence would have to have a sense-a true sense-when we substitute numerical signs for the letters. Let us put '2', say, for 'x' and '3' for y. We then obtain
'If with every value of the real variable 2, which belongs to its range, is correlated a determinate number 3, then 3 is thereby also in general defined as a variable and is called a function of the real variable 2. '
That is completely unintelligible. What, for instance, is a value of 2? You will say the mistake consists in substituting for 'x' the sign for a determinate number; it would have to be replaced by the sign (proper name) of an indeterminate number. Unluckily, we are not yet acquainted with any indeterminate numbers at all, cannot distinguish one indeterminate number from another and so are not in a position to stipulate proper names for them. You may say 'Surely there are indeterminate numbers and they also can be distinguished and named. For instance, let there be a right-angled co?
ordinate system on a plane, including a specification of the unit length for each of the axes. Let x then be the abscissa andy the ordinate of a point. "x"
? Logical Defects in Mathematics 163
is then the proper name of one indeterminate number and "y" of another; in that case x and y will be indeterminate numbers which can be distinguished from each other, and can be spoken about by using their proper names-simply the letters 'x' and 'y'. And so it will be possible for the variables x and y in the quotation cited to be of this sort. ' I reply: give me any point on the plane whatsoever. Then this point will have a quite determinate abscissa and a quite determinate ordinate. So the question of indeterminate numbers simply doesn't arise. It may be objected that we should not take a determinate but an indeterminate point. But then we are back with the same difficulty. I doubt just as much that there are indeterminate points as that there are indeterminate numbers or indetermin- ate people.
'But the solution of the quadratic equation x2 - 4x + 3 = 0 is surely indeterminate, and so 'x' designates this indeterminate solution and hence an indeterminate number. ' I reply: this is neither true nor false, for since there is more than one solution, the definite article in the phrase 'the solution' is a logical error, and the sentence 'The problem contained in that equation permits of more than one solution' expressed correctly what was expressed erroneously above. It's a question of a property of the problem, not of a property of the solution. It is mistaken to say 'the indeterminate solution' or 'the indeterminate number which satisfied the equation': on the other hand we might perhaps call the problem an undetermining one, if this way of speaking were admitted. For this reason the formation 'x = 2 ? 1' is to be avoided like the plague. The combination of signs '2 ? 1' has neither sense nor meaning. Still the quadratic equation can be referred to as an example to show that apart from the use of letters to indicate or to designate, we have to assume yet a third. Of course this would be no argument for acknowledging indeterminate numbers. Regarded as a problem, the equation asserts nothing, and so there admittedly can be no question of using the letter x here to indicate-to confer generality on the thought. What we really have here is the designation of a concept, and the challenge to cite objects (in this case numbers) which fall under it. In that case the 'x' is used in the way I used the letter 'e' in the first volume: 1 it occupies the argument-places, so that we can recognize them as such. Once you have solved the problem, you may assert thesentence'Ifx=1orx=3,thenx2 - 4x+3=0',andheretheletter'x' is once more used to indicate as above, to make the thought general. What-
ever numerical sign may be substituted for 'x', we always obtain a true sen- tence, either because the condition is not fulfilled, in which case it is all one whether the consequent is true or false (for instance if you substitute '2'), or because the condition is fulfilled, in which case the consequent is then also true.
Now how are we to understand the use of the letters 'x' and 'y' in analytic
geometry? Of course the equation of a parabola 'y ? y - 2x = 0' isn't meant
to assert anything: rather, this equation always occurs as a constituent of -----. ----
1 Of the Grundgesetze (ed. ).
? ? 164 Logical Defects in Mathematics
sentences in which something is asserted, and in these sentences these letters will be used to indicate; for example 'If x is the abscissa of a point and y the ordinate, and if y ? y - 2x = 0, then this point lies on a parabola, which . . . etc. ' Here the co-ordinate system must be presupposed to be already given.
Thus neither do we encounter indeterminate numbers here. There are none such and that there are such is only an illusion created by a defective idiom.
The words 'function' and 'variable' are often used in conjunction with each other. We have in fact just seen how 'the function of the real variable x' has been defined as a variable. As against this Heine* says: 'A single-valued
function of a variable x means an expression which is uniquely defined for each individual rational or irrational value of x'. We are not in fact told that the letter 'x' occurs in the expression, but this must surely be assumed to be the case. xis called a variable. Whether that means the letter 'x' or what this letter means or indicates is unclear. Nor is it said what a value of xis. The most likely interpretation of Heine's meaning is surely the following: 'A single valued function of the letter "x" is a formula (a complex sign) for which it is established what every formula means which is derived from it by substituting for "x" a sign for a rational or irrational number'. Thus on Heine's view it appears that a function is not a number, but a formula, a complex sign. This explanation doesn't appear to agree with Czuber's. It is hardly worth the effort to try to reconcile the two, since, to be sure, Czuber's view is difficult to reconcile with itself. The fact that Heine shows these visible things the honour of bringing them under a concept and laying down a special word-'function'-for it, seems to imply that he regards such physical things as the objects of arithmetical investigation. But then again we have the fact that he uses the word 'define', which can here surely only mean 'stipulate a meaning for a simple or complex sign'. But if signs are taken to have meanings, it is striking that it isn't these meanings, but the signs which are supposed to be the main thing. Why then the meanings, if you don't bother with them but only with the signs? So we see obscurity and disagreement among writers of mathematics, and that where they are explaining words which are some of the most common amongst mathemati? cians. In point of fact, these questions have already been definitively resolved in our first Volume. I have only raked them up again because I suspect that many will hold my accounts to be extraordinarily difficult to understand, as compared with such accounts as you find in Czuber'a lectures or elsewhere, which they will hold to be models of clarity. I have tried to show that this illusion of clarity only lasts as long as you raise none of the questions which those accounts naturally give rise to and so fail to notice that no satisfactory answer is to be found for them.
? Die Elemente der Funktionslehre, Crelle Is Journal fiir die reine und angewandte Mathematikl, Vol. 74118721, p. 180.
? Logical Defects in Mathematics 165
But this lack of agreement isn't only to be found in the use of expressions from Analysis: it is even to be found in the case of a word like 'equal' and the sign'=', which are used throughout the highest and the lowest branches of mathematics. G. Peano* says that the opinions of writers concerning the identity sign are very diverse, and unfortunately he's quite right. The implications of what he says are enormous. For if you drop equality from nrithmetic, there's almost nothing left: and so that claim implies nothing less than that mathematicians have diverse views about the sense of the major part of their theorems. A non-mathematician on hearing this might well clap his hands to his head in amazement, and be completely unable to grasp how such a situation was possible in any science and particularly in mathematics. His amazement would grow if he learnt that this wasn't in the least felt as a major calamity, that most mathematicians believe they have rnuch more important things to do than spend their time splitting hairs in trying to remove this defect. A lot of people may think: if only concern with such questions were not so dreadfully sterile! I would like in return to ask: is mathematics then so calloused by formalism that we no longer bother at all nhout the sense of our sentences? What use are 100 000 theorems to us, if we ourselves haven't the faintest idea what we mean by them, if the man using a theorem attaches a different sense to it from the man who proved it? Ilow often is the phrase 'power series' used! But what do we understand by it? What is a power? What is a series? People don't even agree about the question whether these things are configurations which men produce with writing implements, possessing physical properties, or whether powers, series and power series are only designated by such configurations, but are 1hcmselves non-spatial and invisible. What holds the terms of a series together? That they form a whole, this being just what the series is? Or is it 1he spatial relation of juxtaposition? You will certainly receive divergent unswers to that question. And yet the difference between what is created, spatial, sensible and transitory on the one hand, and what is non-spatial, ntcmporal, non-sensible on the other, is so huge that it belongs among the areatest which can possibly exist. A science of objects of the one sort must diiTer far more profoundly from a science of objects of the other sort than, suy, astronomy and cryptography. Such questions therefore touch the very heart of arithmetic. And yet this indifference! With the divergence of opinions and modes of expression, anyone using a word such as 'number', 'function' or 'power series' should by rights state what he understands by it. But many would angrily reject the demand that they concern themselves with such tediously academic questions, in the secret fear that they couldn't
? Revue de Mathematiques (Rivista di matematica), Vol. VI, p. 61: 'Del resto le opinioni dei varii Autori, sui concetto di eguaglianza, diversificano ~tssai; ed uno studio di questa questione sarebbe assai utile, specialmente se futto coll'aiuto di simboli, anziche di parole'.
? 166 Logical Defects in Mathematics
say anything to the point. Others perhaps would boldy come straight out with an answer, but only because they were unaware of the hidden pitfalls and did not feel in the slightest the need to make their own use accord with their explanations and to test them carefully with that object in mind. What use to us are explanations when they have no intrinsic connection with a piece of work, but are only stuck on to the outside like a useless ornament?
? ? ? On Euclidean Geometry1 [1899-1906? ]
It seems worthwhile to begin by coming to an understanding about the use of certain expressions. The word 'sentence' ['Satz'F is used in diverse ways. The one that comes most readily to mind is no doubt the purely linguistic one. It is in this sense that human discourse consists of sentences. However it is not the sentence itself that really concerns us when we speak, but the sense or content which we associate with it and which we wish to communicate. Since the sense itself cannot be perceived by the senses, we have perforce, in order to communicate, to avail ourselves of something that can be perceived. So the sentence and its sense, the perceptible and the imperceptible, belong together. I call the sense of a significant sentence a thought. Thoughts are either true or false. And it is the question whether a thought is true or false that is usually the reason why, in scientific work, we nre concerned with thoughts. Now it can happen only too easily that a sign nnd its content are not clearly differentiated. ? This kind of thing happens particularly where the content itself cannot be perceived by the senses, and where, for that reason, the sign is the only thing that we can perceive. Now it is a fact that man is by nature so orientated towards what is external, towards what can be perceived, that what cannot be perceived is often ulmost completely hidden from him by what can be perceived. The distinction will come very easily to someone who knows several languages, Hince he will recall that the same sense can be expressed in different lunguages.
? Confusion of numerals and numbers.
1 These pieces form part of the controversy with Hilbert and so cannot be earlier thnn 1899, the year in which Vol. 1 of the Grundlagen der Geometrie was puhlished. Assuming the controversy came essentially to an end with Frege's last pnpcr on the foundations of geometry in 1906, it is very unlikely that the present pieces are later than 1906. However, Frege did, in subsequent unpublished writings, continue to express himself at variance with Hilbert's way of regarding axioms and definitions. See, for instance pp. 247 ff. and pp. 273 f. (ed. ).
2 The German word rendered throughout by 'sentence' is 'Satz' (plural 'Siitze') which, as Frege here remarks, is used in diverse ways. The English word does not, unfortunately, have quite the same family of uses as the German word and at Llertnin places, notably where Frege uses compound nouns formed from'Satz', the tr11nslation is (unavoidably) unintelligible. The insertions in square brackets have therefore been added to enable the reader to follow the drift of Frege's remarks (tr11ns. ).
? 168 On Euclidean Geometry
Now I have decided to use the word 'sentence' only to designate what is linguistic-that which has a thought as its sense. Thus I distinguish sentences from thoughts. Now we can understand how easy it is to ascribe to a sentence what belongs, properly speaking, to the thought and how it is that people speak of true sentences instead of true thoughts. They are then calling a sentence true when the sense of the sentence, the thought expressed in it, is true. We should note further that in grammar 'sentence' is often used somewhat differently from how I am using it here, as when one speaks of subordinate clauses [Nebensiitzen]. In many cases these do not have a complete thought as their sense, but only part of a thought.
It is particularly awkward that in mathematics it has become customary to speak of theorems [Lehrsiitzen]. This has given the word 'sentence' a chameleon-like quality: whilst we take it to refer to something linguistic, it all of a sudden changes and refers to a thought.
Many readers will have grown so accustomed to the way the word 'sentence' ['Satz'] is used in mathematical textbooks that it upsets them to find, in place of the word 'sentence', which they are used to, words like 'thought' or 'truth', to which they are not accustomed. To make it easier to follow I shall put the word 'sentence' ['Satz'] in brackets after the word 'thought'.
In the majority of cases what concerns us about a thought is whether it is true. The most appropriate name for a true thought is a truth. A science is a system of truths. A thought, once grasped, keeps pressing us for an answer to the question whether it is true. We declare our recognition of the truth of a thought, or as we may also say, our recognition of a truth, by uttering a sentence with assertoric force. Language is frequently lacking any word or part of a sentence that corresponds to the assertoric force, so that ? this often can only be felt. Subordinate clauses, which are nothing more than parts of sentences, are often uttered without assertoric force.
Whoever holds Euclidean geometry to be true will ascribe a sense to each of its theorems: he will regard each theorem as expressing a truth. He thereby implies that he recognizes that the concept-words 'point', 'line', 'plane' have a sense. In Euclidean geometry certain truths have traditionally been accorded the status of axioms. No thought that is held to be false can be accepted as an axiom, for an axiom is a truth. Furthermore, it is part of the concept of an axiom that it can be recognized as true independently of other truths.
(Truths can be inferred in accordance with the logical laws of inference. )
If a truth is given, it can be asked from what other truths its truth follows in accordance with the logical laws of inference. When this question has been answered, we can go on to ask of each of the truths that we have thus discovered from what other truths its truth follows in accordance with the logical laws of inference.
? On Euclidean Geometry 169 On Euclidean Geometry
No man can serve two masters. One cannot serve both truth and untruth. If Euclidean geometry is true, then non-Euclidean geometry is false, and if non-Euclidean geometry is true, then Euclidean geometry is false.
If given a point not lying on a line one and only one line can be drawn through that point parallel to that line then, given any line l and point P not lying on l, a line can be drawn through P parallel to l and any line that passes through P and is parallel to l will coincide with it.
Whoever acknowledges Euclidean geometry to be true must reject non- l~uclidean geometry as false, and whoever acknowledges non-Euclidean
geometry to be true must reject Euclidean geometry.
People at one time believed they practised a science, which went by the
name of alchemy; but when it was discovered that this supposed science was riddled with error, it was banished from among the sciences. Again, people at one time believed they practised a science, which went by the name of astrology. But this too was banished from among the sciences once men had seen through it and discovered that it was unscientific. The question at the present time is whether Euclidean or non-Euclidean geometry should be struck off the role of the sciences and made to line up as a museum piece alongside alchemy and astrology. If one is content to have only phantoms hovering around one, there is no need to take the matter so seriously; but in science we are subject to the necessity of seeking after truth. There it is a rase of in or out! Well, is it Euclidean or non-Euclidean geometry that should get the sack? That is the question. Do we dare to treat Euclid's elements, which have exercised unquestioned sway for 2000 years, as we have treated astrology? It is only if we do not dare to do this that we can put l~uclid's axioms forward as propositions that are neither false nor doubtful. In that case non-Euclidean geometry will have to be counted amongst the pseudo-sciences, to the study of which we still attach some slight importance, but only as historical curiosities.
? ? [Frege's Notes on Hilbert's 'Grundlagen der Geometrie'P [After 1903)
? 1
The Elements of Geometry and the five groups
of Axioms
Definition: We imagine three different systems
of things: we call the things of the first system
points and designate them by A, B, C, . . . ; we (Pim, Lim, Plim) We. call the things of the second system lines and
designate them by a, b, c . . . ; we call the things
of the third system planes and designate them by
a, /3, y, . . . ; the points are also called the ele-
ments oflinear geometry, the points and lines the
elements ofplane geometry, and the points, lines
and planes, the elements ofspatial geometry or
o f space.
We imagine the points, lines and planes as standing in certain reciprocal relations, which we designate by words such as 'lie on',
'between', 'parallel', 'congruent', 'continuous';
the exact and complete description of these relations is achieved by the axioms ofGeometry.
1 Frege's notes, which in the manuscript are written out as a continuous series of remarks, probably refer to the 2nd edition of Hilbert's Grundlagen der Geometrie, The editors have put alongside Frege's critical notes, which are set out on the right? hand side, the corresponding passages of Hilbert's text. The notes cannot refer to the 1st edition of the Grundlagen (1899), since there the theorem 4 criticized by Frege still figures as an axiom and was only proved in 1902 by E. H. Moore. Besides, Frege's last remark evidently refers to the first congruence axiom; but in the first edition ? 5 contained the parallels axiom. It is indeed possible that Freae had before him a later edition than the 2nd, but it is hardly probable, if you work on the assumption that Frege wrote down his critique during the period of his first engagement with Hilbert's work, which came to an end in 1906 (the 3rd edition first appeared in 1909) (ed. ).
