A new Attempt at a
Foundation
for Arithmetic
281
3.
281
3.
Gottlob-Frege-Posthumous-Writings
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. Theorem: If the triangle OAC is similar to the triangle PQR, and if the triangle OAD is similar to the triangle PQR, D is the same point as C; or
If the point C corresponds to the ratio PQ:PR and the point D corresponds to the ratio PQ :PR, then D is the same point as C.
? INDEX
This index is largely based on the index prepared for the German edition by Gottfried Gabriel, the main difference being that it contains no references to editorial matter. Thus the present index-with the exception of one or two references to translators' footnotes-treats only of Frege's text. Since Frege's thought revolves around relatively few basic concepts, so that terms such as concept, object, meaning, sense recur again and again throughout these writings, there is a danger of making the index too comprehensive and of course some attempt has to be made to confine references under a given term to the more significant of its occurences. In this respect, although we have made a few additions, we have in general adopted a slightly more restrictive policy than that of the German edition, believing that this would produce a more useful index. Possibly the index as it stands now is still too long, and we are, of course, aware that there will be no general agreement about which occurrences of a term are the more significant and about the point at which references become too numerous to be of much use.
We have sometimes included, under a given word, references to parts of the text where the relevant notion is plainly in question though the word itself does not occur there. For instance, unsaturatedness is clearly being discussed when Frege uses phrases like 'predicative nature' or 'in need of supplementation'.
For the most part, cross references in this index are to main catchwords, and signify either that the word at stake is used in one of the subsidiary catchwords under that main one, or that one will discover the most significant references to the notion involved by looking up the references under the second catchword.
We are extremely grateful to Roger Matthews for the generous assistance he has given us in the preparation of this index.
Abstract, 69fT.
Achelis, T. , 146 Acknowledge, v. true
Active, 107, 141, ! 43 Addition (logical), 33f. , 38, 46 Addition sign,
arithmetical-, v. plus sign
logical-, 10, 35f. , 48fT.
Aesthetics, 128, 252
Affection of the ego, 54fT. , 58f. , 61, 64fT. Affirmation, 15
Aggregate, 181IT.
Algorithm, 12
All (v. everything, generalization), 63, 105f. ,
120, 213f. , 259 Alteration v. variation
Ambiguous, 123f. , 213
Analyse (analysis), 208fT.
Analysis (classical), 119, 159, 235fT. , 255 And (v. plus sign), 12, 48, 50f. , 86, 188, 200,
227ff.
Antecedent (v. condition), 19, 152fT. , 186fT. ,
199, 253f.
Antinomy (v. paradox), 176, 182
Aristotle, 15 ,
Arithmetic, 13,222, 237f. , 242, 256f. , 275fT.
equality sign in -, v. equals sign
formula-language of -, v. language
use of letters in-, v. letter Art, work of, 126, 130fT. , 139 Article
formula-
definite -, 94f. , lOOfT. , 114f. , 122, 163, 178fT. , 182, 193, 213, 239, 249, 269f. , 272f.
indefinite-, 94, 104, 120,237
Assert, assertion (v. statement), 2, 20, 52,
129, 139, 149, 251 Assertion
-(s) in fiction (v. fiction), 130
Assertoric force, 168, 177, 185, 198f. , 233f. ,
251, 261 Associative law, 38
Associationofideas, 126,131, 144f. , 174 Auxiliary (v. copula), 62f. , 91
- object, 206f.
Axiom, 203, 205fT. , 209f. , 244, 248
-(s) of geometry, ! 68f. , 170fT. , 247f. , 273, 278
'Beautiful', sense of (v. true), 126, 128, 13lf. , 252
Beauty, judgements of, 126, 13lf. Begri! Jsschrlft, 9fT. , 47fT. , 198
? Being (v. copula), 59, 61f. , 64, 65f. Berkeley, 105
Biermann, 0. , 70, 72fT.
Boole, 9fT. , 47fT.
Cantor, 68fT.
Calculus ratiocinator, 9fT.
Cause, psychological, v. ground Characteristic mark, 101fT. , 111fT. , 229 Chemistry, analysis in, 36f.
Class, 10, lSf. , 34,184
Clause (subordinate), 168, 198 Colouring, v. thought
Common name, 123f.
Complete (v. saturated/unsaturated)
meaning that is - in itself, 119 Commutative law, 38
Concavity (v. generality), 20f.
Concept, 17f. , 32fT. , 87fT. , 118fT. , 154, 184,
193, 214, 234, 238f. , 243
empty -(s), 124, 179f.
-(s) equal in extension, 1Sf. , 118, 121f. ,
182
-(s) equal in number, 72
--expression, v. concept-word
extension of -(s), 106, 118fT. , 181fT. , 184 falling of a concept under a higher -, 93,
108, 110fT.
falling of an object under a -, 18, 87fT. ,
118, 123f. , 179, 182f. , 193, 213f. , 228,
237,243,254,263,265,279 first/second level-, 108, 110fT. , 250, 254 mark of a - , v. characteristic mark
-(s) must have sharp boundaries, 152fT. ,
19Sf. , 229f. , 241, 243f.
--name, v. concept-word
negative-, 17
--sign, v. concept-word
subordination of -(s), 15, 18, 63, 91, 93,
97, 112, 181f. , 186, 193, 213,254 unsaturated nature of -(s), 87fT. , 119fT. ,
- of axioms, 247
-of a concept, 179fT. Constant, 161 Content, 12, 85
possible - of judgement, 11, 46, 47f. , 51, 99
-of a sentence, 197f.
--stroke, 11, 39, 52
Continuum, 276
Contraposition, 153f.
Copula, 62fT. , 90fT. , 113, 174, 177,237,240 Czuber, E. , 160fT.
Darmstaedter, L. , 253fT.
Dedekind, 127, 136
Deduction-sign (Peano's), 152fT.
Definition, 69f. , 88f. , 96, 102, 152fT. , 203,
207fT. , 215, 217, 222f. , 240,244, 248f. ,
256f. , 270f. analytic-, 210f.
- o f a concept, 228f. conditional -, 230 constructive-, 21Of. piecemeal-, 243
- of a relation, 230f.
Demonstrative pronoun, 178fT. , 213 Denial, v. negation
Descartes, 203
Designate, v. sentence-part, sign -deutig, 123f.
Discipline (v. science), 203 Distinction, 142
Division, logical, 48
Each (v. all), 63
Empiricism, 105
Empty places (argument-places) (v. func-
tion), 119, 121 Endpoint, 280f.
Epistemology, 3
Equality in number (of concepts), 72
Equals sign, 86, 91, 165, 222fT. , 226, 228,
236f.
Boole's -, 15, 35fT. , 48f. , 52
Equation (v. identity), 46, 121, 182 logical-, 113, 115
Equipollence, 197
Equivocal (v. ambiguous), 123f.
Error, 132
Ethics, 4, 128, 252
Euclid, 169, 205,206,247
Everything, 188
Evolution, theory of, 4, 174
Existence (v. 'there is', being), 53fT. , 101,
107
Existential judgement, 14, 20, 55, 63, 66,
254 Experience, 53fT.
object of-, 65 Explanation, 36
177f. , 228
widening of -(s), 242
- - w o r d , 118, 176fT. , 201,
213f. , 229f. , meaningless/meaningful-, 122fT. , 180f.
234, 240f. , 243
Concept-script (v. Begriffsschrift), 6, 9fT. , 47fT. , 67, 142, 184, 188, 195,253 primitive laws of-, 37, 39fT. , 46, 50
primitive signs of-, 36, 46, 48f. , 50 Condition (v. antecedent), 38, 41fT. , 152fT. ,
163,200
Conditional stroke, 11f. , 35fT. , 52, 152fT. ,
186f. , 188 Congruent, v. triangles
Congruent to a modulus, 22 Conjunction (v. and), 188f. , 191 Consequence, v. condition Consequent, v.
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. Theorem: If the triangle OAC is similar to the triangle PQR, and if the triangle OAD is similar to the triangle PQR, D is the same point as C; or
If the point C corresponds to the ratio PQ:PR and the point D corresponds to the ratio PQ :PR, then D is the same point as C.
? INDEX
This index is largely based on the index prepared for the German edition by Gottfried Gabriel, the main difference being that it contains no references to editorial matter. Thus the present index-with the exception of one or two references to translators' footnotes-treats only of Frege's text. Since Frege's thought revolves around relatively few basic concepts, so that terms such as concept, object, meaning, sense recur again and again throughout these writings, there is a danger of making the index too comprehensive and of course some attempt has to be made to confine references under a given term to the more significant of its occurences. In this respect, although we have made a few additions, we have in general adopted a slightly more restrictive policy than that of the German edition, believing that this would produce a more useful index. Possibly the index as it stands now is still too long, and we are, of course, aware that there will be no general agreement about which occurrences of a term are the more significant and about the point at which references become too numerous to be of much use.
We have sometimes included, under a given word, references to parts of the text where the relevant notion is plainly in question though the word itself does not occur there. For instance, unsaturatedness is clearly being discussed when Frege uses phrases like 'predicative nature' or 'in need of supplementation'.
For the most part, cross references in this index are to main catchwords, and signify either that the word at stake is used in one of the subsidiary catchwords under that main one, or that one will discover the most significant references to the notion involved by looking up the references under the second catchword.
We are extremely grateful to Roger Matthews for the generous assistance he has given us in the preparation of this index.
Abstract, 69fT.
Achelis, T. , 146 Acknowledge, v. true
Active, 107, 141, ! 43 Addition (logical), 33f. , 38, 46 Addition sign,
arithmetical-, v. plus sign
logical-, 10, 35f. , 48fT.
Aesthetics, 128, 252
Affection of the ego, 54fT. , 58f. , 61, 64fT. Affirmation, 15
Aggregate, 181IT.
Algorithm, 12
All (v. everything, generalization), 63, 105f. ,
120, 213f. , 259 Alteration v. variation
Ambiguous, 123f. , 213
Analyse (analysis), 208fT.
Analysis (classical), 119, 159, 235fT. , 255 And (v. plus sign), 12, 48, 50f. , 86, 188, 200,
227ff.
Antecedent (v. condition), 19, 152fT. , 186fT. ,
199, 253f.
Antinomy (v. paradox), 176, 182
Aristotle, 15 ,
Arithmetic, 13,222, 237f. , 242, 256f. , 275fT.
equality sign in -, v. equals sign
formula-language of -, v. language
use of letters in-, v. letter Art, work of, 126, 130fT. , 139 Article
formula-
definite -, 94f. , lOOfT. , 114f. , 122, 163, 178fT. , 182, 193, 213, 239, 249, 269f. , 272f.
indefinite-, 94, 104, 120,237
Assert, assertion (v. statement), 2, 20, 52,
129, 139, 149, 251 Assertion
-(s) in fiction (v. fiction), 130
Assertoric force, 168, 177, 185, 198f. , 233f. ,
251, 261 Associative law, 38
Associationofideas, 126,131, 144f. , 174 Auxiliary (v. copula), 62f. , 91
- object, 206f.
Axiom, 203, 205fT. , 209f. , 244, 248
-(s) of geometry, ! 68f. , 170fT. , 247f. , 273, 278
'Beautiful', sense of (v. true), 126, 128, 13lf. , 252
Beauty, judgements of, 126, 13lf. Begri! Jsschrlft, 9fT. , 47fT. , 198
? Being (v. copula), 59, 61f. , 64, 65f. Berkeley, 105
Biermann, 0. , 70, 72fT.
Boole, 9fT. , 47fT.
Cantor, 68fT.
Calculus ratiocinator, 9fT.
Cause, psychological, v. ground Characteristic mark, 101fT. , 111fT. , 229 Chemistry, analysis in, 36f.
Class, 10, lSf. , 34,184
Clause (subordinate), 168, 198 Colouring, v. thought
Common name, 123f.
Complete (v. saturated/unsaturated)
meaning that is - in itself, 119 Commutative law, 38
Concavity (v. generality), 20f.
Concept, 17f. , 32fT. , 87fT. , 118fT. , 154, 184,
193, 214, 234, 238f. , 243
empty -(s), 124, 179f.
-(s) equal in extension, 1Sf. , 118, 121f. ,
182
-(s) equal in number, 72
--expression, v. concept-word
extension of -(s), 106, 118fT. , 181fT. , 184 falling of a concept under a higher -, 93,
108, 110fT.
falling of an object under a -, 18, 87fT. ,
118, 123f. , 179, 182f. , 193, 213f. , 228,
237,243,254,263,265,279 first/second level-, 108, 110fT. , 250, 254 mark of a - , v. characteristic mark
-(s) must have sharp boundaries, 152fT. ,
19Sf. , 229f. , 241, 243f.
--name, v. concept-word
negative-, 17
--sign, v. concept-word
subordination of -(s), 15, 18, 63, 91, 93,
97, 112, 181f. , 186, 193, 213,254 unsaturated nature of -(s), 87fT. , 119fT. ,
- of axioms, 247
-of a concept, 179fT. Constant, 161 Content, 12, 85
possible - of judgement, 11, 46, 47f. , 51, 99
-of a sentence, 197f.
--stroke, 11, 39, 52
Continuum, 276
Contraposition, 153f.
Copula, 62fT. , 90fT. , 113, 174, 177,237,240 Czuber, E. , 160fT.
Darmstaedter, L. , 253fT.
Dedekind, 127, 136
Deduction-sign (Peano's), 152fT.
Definition, 69f. , 88f. , 96, 102, 152fT. , 203,
207fT. , 215, 217, 222f. , 240,244, 248f. ,
256f. , 270f. analytic-, 210f.
- o f a concept, 228f. conditional -, 230 constructive-, 21Of. piecemeal-, 243
- of a relation, 230f.
Demonstrative pronoun, 178fT. , 213 Denial, v. negation
Descartes, 203
Designate, v. sentence-part, sign -deutig, 123f.
Discipline (v. science), 203 Distinction, 142
Division, logical, 48
Each (v. all), 63
Empiricism, 105
Empty places (argument-places) (v. func-
tion), 119, 121 Endpoint, 280f.
Epistemology, 3
Equality in number (of concepts), 72
Equals sign, 86, 91, 165, 222fT. , 226, 228,
236f.
Boole's -, 15, 35fT. , 48f. , 52
Equation (v. identity), 46, 121, 182 logical-, 113, 115
Equipollence, 197
Equivocal (v. ambiguous), 123f.
Error, 132
Ethics, 4, 128, 252
Euclid, 169, 205,206,247
Everything, 188
Evolution, theory of, 4, 174
Existence (v. 'there is', being), 53fT. , 101,
107
Existential judgement, 14, 20, 55, 63, 66,
254 Experience, 53fT.
object of-, 65 Explanation, 36
177f. , 228
widening of -(s), 242
- - w o r d , 118, 176fT. , 201,
213f. , 229f. , meaningless/meaningful-, 122fT. , 180f.
234, 240f. , 243
Concept-script (v. Begriffsschrift), 6, 9fT. , 47fT. , 67, 142, 184, 188, 195,253 primitive laws of-, 37, 39fT. , 46, 50
primitive signs of-, 36, 46, 48f. , 50 Condition (v. antecedent), 38, 41fT. , 152fT. ,
163,200
Conditional stroke, 11f. , 35fT. , 52, 152fT. ,
186f. , 188 Congruent, v. triangles
Congruent to a modulus, 22 Conjunction (v. and), 188f. , 191 Consequence, v. condition Consequent, v.
