, 186, 193, 213,254
unsaturated
nature of -(s), 87fT.
Gottlob-Frege-Posthumous-Writings
, 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. condition Consistency
Index 283
? 284 Index
Extension? , v. concept
Extensionalist logicians v. intensionalist/
extensionalist logicians
Factor (common), 22 Fallacy, 34f. , 143, ! 55
False (v. true), 127, 138, 149
the-, 195, 233, 255
reject as-, 8, 149, 185, 197f.
Feeling, 129, 198
Fick, A. , 17
Fiction, 118, 122, 129f. , 191fT. , 225, 232,
269 Fischer, K. , 84
Following (in a series), 22, 38
Formal arithmetic, 165, 215, 273f. , 275 Formula-language
- of algebra, 2, 6
- of arithmetic, 13f. , 142
- Boole's logical-, 13fT. , 47fT. - ofgeometry, 13
- o f mathematics, 13f. , 47, 270
Function, 24fT. , 119, 155f. , 157fT. , 184, 195, 234f. , 238fT. , 270fT.
argument, argument-place, of a -, 119, 121, 154fT.
- of one, of two, of three arguments, 240fT. , 244, 250
continuous -, 24f.
--letter (v. letter), 121, 154, 156,272 --sign (--name) 119, 234, 239fT. , 243f. ,
271f.
unsaturatedness of a -, 97fT. , 119, 156,
254,272
value of a-, 119, 235, 238f.
Gauss, 280
Generality, 11, ! BIT. , 52, 153fT. , IG2, 187fT. ,
194f. , 199fT. , 230f. , 236fT. , 258fT. , 272 - of an equation between functions, 121 notation for -, 18fT.
Generalization, 191
Geometry (v. sources of knowledge), 170fT. ,
273f. , 277, 278fT. Euclidean/non-Euclidea'l-, 167fT.
God, v. ontological proof of the existence of-
Good, 252
Grammar, 6f. , 141fT. , 146f. Grassmann, R. , 34 Ground, liT. , 147
Heine, E. , 164
Here, 135
Hereditary property, 28 Hilbert, 170fT. , 247fT. History, 258
historical/ahistorical -, 3 Husserl, 123f. ' Hypothetical
-judgement, 15, 52, 185, 198
- sentence [conditional sentence], 153f. ,
185, 188, 198fT. , 258fT. -thought, 188f. , 200,258, 261
I (v. affectionoftheego), 127,135
Idea (v. image, representation, association of - s ) , 54fT. , 73fT. , 104f. , 126fT. , 139f. ,
143fT. , 148, 174, 198 act of forming an -, 57 object of an-, 56fT. , 64f. -ofan-, 57
Idealism, 105, 130, 143f. , 232 Identity (v. equals sign, equation)
If
law of-, 62
relation of-, l2lf. , 182 sign of-, 10,226, 237, 240
. . . then . . . (v. hypothetical sentence), 153, 189fT.
Illustrative examples, 207, 214, 235 Image, 58, 74
Indefinitely indicating (v. letter)
- sentence-part, v. sentence-part
-sign, v. sign
Individual (thing), 213
Induction (mathematical), 31, 203
Inference (v. fallacy), 15, 27, 34f. , 118, 175,
180, 203fT. , 212, 244fT. , 261
chain of-(s), 204
- from the general to the particular,
258fT. , 272
laws of actual- /laws of correct-, 3fT.
Infinite, 273f. , 279 absolute-, 68f. actual-, 68f. potential-, 68f.
Inner perception, 58
Integral, definite, 15 7f. lntensionalist/Extensionalist logicians, 118,
122f. Interjection, 135, 140
Intuition, 32, 58, 2 78 Is, v. copula
[as expression of existence], 53, 62, 63f.
[as expression of identity], 90fT. , 113fT. It,62, 190,260
Jevons, W. S. , 10, 35f. , 48
Judgement, ! IT. , 7, 11, 16f. , 45f. , 47, 51f. ,
59f. , 119, 139, 149, 151, 185, 194,
197fT. , 251, 253f. , 267 aesthetic-, 126f. , 131f.
content of possible-, v. content hypothetical-, v. hypothetical particular-, v. particular singular-, v. singular --stroke,! ! , 18fT. , 51f. , 195,198
1urisprudence, 203
Justification (v. judgement, ground), 2ff. ,
147, 175
? Kant, 203
Kerry, B. , 87fT.
Knowledge (v. recognition)
-a priori, 277
geometrical source of-, 277, 278fT. sources of- (v. sense perception), 266fT. temporal source of-, v. time
Korselt, A. , I77
Language (v.
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. condition Consistency
Index 283
? 284 Index
Extension? , v. concept
Extensionalist logicians v. intensionalist/
extensionalist logicians
Factor (common), 22 Fallacy, 34f. , 143, ! 55
False (v. true), 127, 138, 149
the-, 195, 233, 255
reject as-, 8, 149, 185, 197f.
Feeling, 129, 198
Fick, A. , 17
Fiction, 118, 122, 129f. , 191fT. , 225, 232,
269 Fischer, K. , 84
Following (in a series), 22, 38
Formal arithmetic, 165, 215, 273f. , 275 Formula-language
- of algebra, 2, 6
- of arithmetic, 13f. , 142
- Boole's logical-, 13fT. , 47fT. - ofgeometry, 13
- o f mathematics, 13f. , 47, 270
Function, 24fT. , 119, 155f. , 157fT. , 184, 195, 234f. , 238fT. , 270fT.
argument, argument-place, of a -, 119, 121, 154fT.
- of one, of two, of three arguments, 240fT. , 244, 250
continuous -, 24f.
--letter (v. letter), 121, 154, 156,272 --sign (--name) 119, 234, 239fT. , 243f. ,
271f.
unsaturatedness of a -, 97fT. , 119, 156,
254,272
value of a-, 119, 235, 238f.
Gauss, 280
Generality, 11, ! BIT. , 52, 153fT. , IG2, 187fT. ,
194f. , 199fT. , 230f. , 236fT. , 258fT. , 272 - of an equation between functions, 121 notation for -, 18fT.
Generalization, 191
Geometry (v. sources of knowledge), 170fT. ,
273f. , 277, 278fT. Euclidean/non-Euclidea'l-, 167fT.
God, v. ontological proof of the existence of-
Good, 252
Grammar, 6f. , 141fT. , 146f. Grassmann, R. , 34 Ground, liT. , 147
Heine, E. , 164
Here, 135
Hereditary property, 28 Hilbert, 170fT. , 247fT. History, 258
historical/ahistorical -, 3 Husserl, 123f. ' Hypothetical
-judgement, 15, 52, 185, 198
- sentence [conditional sentence], 153f. ,
185, 188, 198fT. , 258fT. -thought, 188f. , 200,258, 261
I (v. affectionoftheego), 127,135
Idea (v. image, representation, association of - s ) , 54fT. , 73fT. , 104f. , 126fT. , 139f. ,
143fT. , 148, 174, 198 act of forming an -, 57 object of an-, 56fT. , 64f. -ofan-, 57
Idealism, 105, 130, 143f. , 232 Identity (v. equals sign, equation)
If
law of-, 62
relation of-, l2lf. , 182 sign of-, 10,226, 237, 240
. . . then . . . (v. hypothetical sentence), 153, 189fT.
Illustrative examples, 207, 214, 235 Image, 58, 74
Indefinitely indicating (v. letter)
- sentence-part, v. sentence-part
-sign, v. sign
Individual (thing), 213
Induction (mathematical), 31, 203
Inference (v. fallacy), 15, 27, 34f. , 118, 175,
180, 203fT. , 212, 244fT. , 261
chain of-(s), 204
- from the general to the particular,
258fT. , 272
laws of actual- /laws of correct-, 3fT.
Infinite, 273f. , 279 absolute-, 68f. actual-, 68f. potential-, 68f.
Inner perception, 58
Integral, definite, 15 7f. lntensionalist/Extensionalist logicians, 118,
122f. Interjection, 135, 140
Intuition, 32, 58, 2 78 Is, v. copula
[as expression of existence], 53, 62, 63f.
[as expression of identity], 90fT. , 113fT. It,62, 190,260
Jevons, W. S. , 10, 35f. , 48
Judgement, ! IT. , 7, 11, 16f. , 45f. , 47, 51f. ,
59f. , 119, 139, 149, 151, 185, 194,
197fT. , 251, 253f. , 267 aesthetic-, 126f. , 131f.
content of possible-, v. content hypothetical-, v. hypothetical particular-, v. particular singular-, v. singular --stroke,! ! , 18fT. , 51f. , 195,198
1urisprudence, 203
Justification (v. judgement, ground), 2ff. ,
147, 175
? Kant, 203
Kerry, B. , 87fT.
Knowledge (v. recognition)
-a priori, 277
geometrical source of-, 277, 278fT. sources of- (v. sense perception), 266fT. temporal source of-, v. time
Korselt, A. , I77
Language (v.
