The task of logic is to set up laws according to which a judgement is
justified
by others, irrespective of whether these are themselves true.
Gottlob-Frege-Posthumous-Writings
).
? 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. ).
Hilbert
? 1 (Definition? )
System.
Relations are described.
? [Frege's Notes on Hilbert's 'Grundlagen der Geometrie'] 171
? 2
The 1st Axiom Group: Axioms of connection
The axioms of this group establish a connection between the concepts point, line and plane, defined above and are as follows:
I 1. Two distinct points A and B always define a line.
? 2 Concepts defined (Points)
I 1. Define. Two. Iden- tity denied.
Better:
Instead of 'define' we will also use other turns
of phrase, such as a 'passes through' A 'and A lies on a (A point, a
through' B, a 'connects' A 'and' or 'with' B. IfA is a point which together with another point defines the line a, we also use the phrases: A 'lies on' a, A 'is a point of' a, 'There is the point' A 'on' a etc. IfA lies on the line a and also on the line b, we also use the phrase: 'the lines' a 'and' b 'have the point A in common' etc.
? 3
The 2nd Axiom Group: Axioms of ordering
The Axioms of this group define the concept 'between' and make it possible, by using this concept, to order the points of a line, in a plane and in space.
line) IfAisapoint,andBisa point, then there is a line on which A lies and on which B lies etc.
more than one theorem in I 1.
? 3 Axioms define con- cept between
Definition. The points of a line stand in The word 'between' used certain relations to one another, for whose todescribe
description we single out the word 'between'.
11 1. /fA, B, and Care points ofa line and B lies between A and C then B lies between C and A.
11 2. IfA and C are two points ofa line then there is always at least one point B which lies between A and C, and at least one point D such that C lies between A and D.
11 3. Given any three points ofa line there is always one and only one which lies between the other two.
Definition. We consider two points A and B on a line a; we call the system of the two points A and Ban interval and designate it AB or BA. The points between A and B are called points of the interval AB or are said to lie within the interval AB; the points A and B are called
Def. System of the two points
Interval
? 172 [Frege's Notes on Hilbert's 'Grundlagen der Geometrie'] endpoints of the interval AB. All other points of
the line a are said to lie outside the interval AB.
? 4
Consequences of the axioms of connection and ordering
From the 1st and 2nd sets of axioms the following theorems follow:
Theorem 3. Between any two points of a line there are always infinitely many points.
Theorem 4. Given any four points on a line, they can always be designated by A, B, C, and D in such a way that the point designated by B lies between A and C, and also between A and D, and that further the point C lies between A and D and also between B and D.
Theorem 5. (Generalization of Theorem 4). Given any finite number of points on a line, they can always be designated by A,B, C,D,E, . . . K, in such a way that the point designated by B lies between A on the one side and C, D, E, . . . K on the other, that C lies between A and B on the one side and D, E, . . . K on the other, and that D lies between A, B, C on the one side and E . . . K on the other etc. In addition to this mode of designation there is only the converse mode K . . . E, D, C, B, A with this feature.
? 4 Theorem 3. Between any two points of a line there are always in- finitely many points. 'Infinitely many' is not explained.
There is no axiom in whose phrasing 'infi- nitely many' occurs. Theorem 4. The letters and mode of designation are not part of the con- tent of the theorem.
Theorem 5. Dots and etc. do not belong to the content of the theorem. 'Finite number'. We should have to borrow from arithmetic some sentence or other con- taining the expression 'finite number'.
Theorem 6. Any line a lying in a plane a, Theorem 6. 'Region',
divides the points lying on this plane a and not on a into two regions with the following properties: any point A of the one region defines with any point B of the other an interval AB within which lies a point of a, whereas any two points A and 4' of the same region define an interval AA' containing no point of a.
'divide' have not oc- curred.
? lFrege's Notes on Hilbert's 'Grundlagen der Geometrie'] 173
? 5
The 3rd Axiom Group: Axioms of congruence
rn1. IfA,BaretwopointsofalineaandA'is ? 5'Onecanfind' a point o f the same or another line a', one can
always find on a given side of the line a' ofA'
one and only one point B', such that the interval
AB is congruent or equal to the interval A' B', in signs:
AB=A'B'.
Every interval is congruent to itself, that is we always have
AB=AB and AB=BA.
? ? [17 Key Sentences on LogicP [1906 or earlier]
1. The connections which constitute the essence of thinking are of a different order from associations of ideas.
2. The difference is not a mere matter of the presence of some ancillary thought from which the connections in the former case derive their status.
3. In the case of thinking it is not really ideas that are connected, but things, properties, concepts, relations.
4. A thought always contains something reaching out beyond the particular case so that this is presented to us as falling under something general.
5. In language the distinctive character of a thought finds expression in the copula or personal ending of the verb.
6. A criterion for whether a mode of connection constitutes a thought is that it makes sense to . ask whether it is true or untrue. Associations of ideas are neither true nor untrue.
7. What true is, I hold to be indefinable.
8. The expression in language for a thought is a sentence. We also speak in
an extended sense of the truth of a sentence.
9. A sentence can be true or untrue only if it is an expression f~. r~
thought. The sentence 'Leo Sachse is a man' is the expression of a thought only if 'Leo Sachse' designates something. And so too the sentence 'this table is round' is the expression of a thought only if the words 'this table' are not empty sounds but designate something specific for me.
11. '2 times 2 is 4' is true and will continue to be so even if, as a result of Darwinian evolution, human beings were to come to assert that 2 times 2 is 5. Every truth is eternal and independent of being thought by any? one and of the psychological make-up of anyone thinking it.
1 According to a note of Heinrich Scholz's, the manuscript should be dated around 1906. But it could have formed part of Frege's plans for a text book on logic (cf. pp. 1 ff. , 126 ff. ) and in that case its date would be much earlier. A further argument for an earlier dating is that, according to notes made by the editors pre? ceding Scholz, the manuscript was found together with the preparatory material for
the dialogue with Piinjer (pp. 53 ff. of this volume), where the name 'Leo Sachse' occurs again (ed. ).
? [17 Key Sentences on Logic] 175
12. Logic only becomes possible with the conviction that there is a difference between truth and untruth.
13. We justify a judgement either by going back to truths that have been recognized already or without having recourse to other judgements. Only the first case, inference, is the concern of Logic.
14. The theory of concepts and of judgement is only preparatory to the theory of inference.
15.
The task of logic is to set up laws according to which a judgement is justified by others, irrespective of whether these are themselves true.
16. Following the laws of logic can guarantee the truth of a judgement only
insofar as our original grounds for making it, reside in judgements that
are true.
17. Nopsychologicalinvestigationcanjustifythelawsoflogic.
? ? On Schoenflies: Die logischen Paradoxien der Mengenlehre1
Concept and object,
[1906)
Plan of critique of Schoenflies etc.
nomen appelativum, nomen proprium.
Analysis of a sentence, predicative nature of a concept. Function, sharp boundaries, independent of objects, consistency not to be insisted on. Subsumption, subordination. Mutual subordination. Relation. Identity. First and second level relations.
Aggregate, extension of a concept. Inbegriff2 (belong to, include). System, series, set, class.
How applied in criticizing Schoenflies' statements.
Can the extension of a concept fall under a concept, whose extension it is?
It does not need to be all-encompassing.
Russell's contradiction cannot be eliminated in Schoenflies' way. Concepts which coincide in extension, although this extension falls under the one, but not the other.
Remedy from extensions of second level concepts impossible.
Set theory in ruins.
My concept-script in the main not dependent on it. (Contrast with other
similar projects. )
1 Frege obviously intended this essay for publication in the Jahresbericht der deutschen Mathematiker-Vereinigung. Whether it was rejected by the editor, or, whether because it remained a fragment, it was not submitted by Frege, is not known. -It is dated by Frege's opening remarks (ed. ).
2 As far as we can see, this word does not have a sense in German which fits the context. Inbegri. ff usually means 'essence' or 'embodiment' (as in 'He is the very embodiment of health'). It is for us impossible to determine from these fragmentary notes to what use Frege was putting the term, and we thought it better to leave it untranslated than to put in a probably false conjecture. Frege obviously has in mind the different relation of an object to the extension of a concept under which it falls, and to an aggregate of which it is a part; but further than that we leave for the reader to decide (trans. ).
? ? On Schoenflies: Die Logischen Paradoxien der Mengenlehre 177 [Discussion]
The article by S, Ober die logischen Paradoxien der Mengenlehre* induces me to make the following remarks, in which I repeat much that I have already discussed previously, since it does not seem to be widely known. I fail to find in S and also in Korselt** the sharp distinction between concept and object. ? ? ? In the signs, a proper name (nomen proprium) corresponds to an object, a concept-word or concept-sign (nomen appellativum) to a concept. A sentence such as 'Two is a prime' can be analysed into two essentially different component parts: into 'two' and 'is a prime'. The former appears complete, the latter in need of supplementation, unsaturated. 'Two'-at least in this sentence-is a proper name, its meaning is an object, which can also be designated with greater prolixity by 'the number two'. The object, too, appears as a complete whole, whereas the predicative part has something unsaturated in its meaning as well. We count the copula 'is' as belonging to this part of the sentence. But there is usually something combined with it which here must be disregarded: assertoric force. We can of course express a thought, without stating it to be true. The thought is strictly the same, whether we merely express it or whether we also put it forward as true. Thus assertoric force, which is often connected with the copula or else with the grammatical predicate, does not belong to the expression of the thought, and so may be disregarded here.
This predicative component part of our sentence which we have described in this way, is also meaningful. We call it a concept-word or nomen appellativum, even though it is not customary to include the copula in this. Just as it itself appears unsaturated, there is also something unsaturated in the realm of meanings corresponding to it: we call this a concept. This unsaturatedness of one of the components is necessary, since otherwise the parts do not hold together. Of course two complete wholes can stand in a relation to one another; but then this relation is a third element-and one that is doubly unsaturated! In the case of a concept we can also call the unsaturatedness its predicative nature. But in this connection it is necessary to point out an imprecision forced on us by language, which, if we are not conscious of it, will prevent us from recognizing the heart of the matter: i. e. we can scarcely avoid using such expressions as 'the concept prime'. Here there is no trace left of unsaturatedness, of the predicative nature. Rather, the expression is constructed in a way which precisely parallels 'the poet Schiller'. So language brands a concept as an object, since the only way it can fit the designation for a concept into its grammatical structure is as a proper name. But in so doing, strictly speaking it falsifies matters. In the same way, the word 'concept' itself is, taken strictly, already defective, since
? The current Jahresbericht, Vol. XV, p. 19 (Jan. 1906).
? ? The current Jahresbericht, Vol. XV, p. 215 (March-April 1906). ? ? ? Cf. My essay 'Concept and Object'.
? ? 178 On Schoenjltes: Die Logischen Paradoxien der Mengenlehre
the phrase 'is a concept' requires a proper name as grammatical subject; and so, strictly speaking, it requires something contradictory, since no proper name can designate a concept; or perhaps better still, something nonsensical. It is no objection to say that surely the grammatical predicate 'is rectangular' can be combined with the grammatical subject 'every square', which isn't a proper name; for even the sentence 'every square is rectangular' can only make sense in virtue of the fact that you can assert of an object that it is rectangular, either rightly or wrongly, but in either case significantly. By a proper name I understand the sign of an object, independently of the question whether it be a simple word or sign, or a complex one, provided only that it designates the object determinately.
In the sentence 'Two is a prime' we find a relation designated: that of subsumption. We may also say the object falls under the concept prime, but if we do so, we must not forget the imprecision of linguistic expression we have just mentioned. This also creates the impression that the relation of subsumption is a third element supervenient upon the object and the concept. This isn't the case: the unsaturatedness of the concept brings it about that the object, in effecting the saturation, engages immediately with the concept, without need of any special cement. Object and concept are fundamentally made for each other, and in subsumption we have their fundamental union.
We call a concept empty if no object falls under it. The concept-word for an empty concept never yields a true sentence,? no matter what proper name may saturate it, or in other words: no matter what proper name may be attached as a grammatical subject to the concept-word as predicate. A concept under which one and only one object falls must still be distinguished from the latter; its sign is a nomen appellativum, not a nomen proprium.
With the help of the definite article or demonstrative, language forms proper names out of concept-words. So, for instance, the phrase 'this A' on p. 20 of the Schoenfties article is a proper name. If forming a proper name in this way is to be legitimate, the concept whose designation is used in its formation must satisfy two conditions:
1. It may not be empty.
2. Only one object may fall under it.
If the first condition is unsatisfied, there is no object at all to which the proper name would be ascribed. If the second is unsatisfied, there are indeed several such objects, but none of them is determined as the one meant to be designated by the proper name. In science the purpose of a proper name is to designate an object determinately; if this purpose is unfulfilled, the proper name has no justification in science. How things may be in ordinary
? For brevity, I here call a sentence expressing a true thought, a true sentence.
? On Schoenjlies: Die Logischen Paradoxien der Mengenlehre 179
language is no concern of ours here. Our first requirement includes the requirement that the concept be consistent.
But this requirement doesn't mean that every concept ought to be consistent, only that a concept must be consistent if you wish to form a proper name from the concept-word together with the definite article or demonstrative. But for this purpose, this requirement doesn't go far enough: the concept is not to be empty, no matter what the reason for its being empty may be.
But if we ask, under what conditions a concept is admissible in science. the first thing to stress is that consistency is no~ such a condition. The only requirement to be made of a concept is that it should have sharp boundaries; that is, that for every object it holds that it either falls under the concept or does not do so. Essentially this is nothing but the requirement that the principle of non-contradiction should hold. But the admissibility of a concept is entirely independent of the question whether objects fall under it, and if so which, or in other words; whether there be objects, and if so which, of which it can be truly asserted. For, before we can raise such questions, we already need the concept. The requirement that a concept be consistent encounters great difficulties. The only apparent way to show a concept has this property, is to cite an object falling under the concept. But to do that, you already need the concept.
Schoenfties obviously only lays down this requirement as a result of his confounding concepts and objects. It is of course self-evident that an object t:annot have inconsistent properties. That the concept of a right-angled equilateral pentagon contains a contradiction does not make it inadmissible. For we can see no reason why a man should not be able to say of an object that it is not a right-angled equilateral pentagon, or why he should not be permitted to say there are no right-angled equilateral pentagons. And before he arrives at such judgements, he must consider the matter, and to do that he requires this concept. It is completely wrongheaded to imagine that every contradiction is immediately recognizable; frequently the contradiction lies deeply buried and is only discovered by a lengthy chain of inference: throughout which you need the concept. The only thing is, given a concept, you may not presuppose without further question that an object falls under it. But someone does that when, by means of a definite article or demonstrative, he forms out of a concept-word a proper name destined for use in science.
Let us now examine Schoenflies' sentences in the light of what we have said so far. Earlier on p. 20 we read,
'I. The A has the property B, and
2. The A does not have the property B. '
The definite article before the 'A' can only be understood in such a way that 'the A' is being thought of as a proper name. In logic it must be presupposed that every proper name is meaningful; that is, that it serves its purpose of
? 180 On Schoenjlies: Die Logischen Paradoxien der Mengenlehre
designating an object. For a sentence containing a meaningless proper name either expresses no thought at all, or it expresses a thought that belongs to myth or fiction. In either case it falls outside the domain governed by the laws of logic. For instance 'the A' would be a name that was in this way inadmissible for scientific use if it were formed by means of the definite article from a name of a concept (nomen appellativum), under which either no object or more than one fell. The mistake would then not lie in the use of
the concept-word in itself, but in forming a proper name from it. Let us now assume 'the A' is a meaningful proper name, and look at the concept or logical predicate 'has the property B'. We must presuppose for this concept that it has sharp boundaries in the sense given above; for only in that case can it be logically recognized as a concept. Then indeed the principle of non- contradiction holds, i. e. that of the two sentences cited always one and only one is right, that is, expresses a true thought; and indeed this holds even if the concept of the predicate is contradictory; for then the second sentence is right, but the first not.
And so the conditions for the validity of the principle of non-contradiction in S's formulation consist in the fact that proper names as well as concept- words satisfy logical requirements. But these do not include the require- ment that a concept-word should designate a consistent concept. Since only concepts with sharp boundaries are to be recognized as meaningful by logic, we may also say: proper names and concept-words must be meaningful.
Now if A is the concept of a right-angled equilateral pentagon, 'this A' is an inadmissible-meaningless-proper name, and nothing whatever can be legitimately inferred either from the sentence 'This A has the property B' or from 'This A does not have the property B'; for, strictly, we do not infer from sentences at all, but from thoughts. And only true thoughts are admissible premises of inferences. Also it isn't strictly sentences, it is thoughts which have contradictory counterparts. If you always bear this in mind, that we cannot legitimately infer from sentences, but only from true thoughts, and that proper names and concept-words must be meaningful, then the case of the failure of the principle of non-contradiction or of indirect proof cannot obtain. The failure of the principle of non- contradiction cannot serve as a criterion for the admissibility of a concept; for then you could never avail yourself of indirect proof. We should always have to reckon with its failing. But before we go on to proofs at all, we must have assured ourselves that the proper names and concept-words we employ are admissible. From the failure of indirect proof we can only conclude that we have made a mistake in this respect, which ought not to have occurred at all. Now whether this mistake consists in the use of a meaningless proper name or a meaningless concept-word, cannot be generally answered. At any rate, an inconsistent concept is not to be rejected of itself. Only if some- one makes a proper name from the corresponding concept-word by means of the definite article or demonstrative, does he fall into error. But whether it is a mistake of this kind or of another kind which is to hlame for the failure of the indirect prm~f. needs special investigation in each case.
? On Schoenjlies: Die Logischen Paradoxien der Mengenlehre 181
Now Schoenfties states that the concept of the set of all sets which do not contain themselves as elements is not consistent. It must first be remarked that 'the set of all sets which do not contain themselves as elements' is not a concept-word, it is a proper name, and it can only be a question of whether this proper name is meaningful. It is evidently not formed from the nomen appellativum is a set ofall sets which do not contain themselves as elements with the definite article; for the use of such a predicate has not preceded it: the expression makes its debut together with the definite article. And so we have here a different case from the previous one with the phrase 'this A'. Hence our proper name requires a special scrutiny. We may distinguish two different ways of using the word 'set', going with two different conceptions, which are probably most plainly identified by the words 'aggregate' and 'extension of a concept'. But frequently these conceptions do not occur in their pure form, but mixed together and this makes for unclarity. The aggregative conception is the first to offer itself, but the requirements of mathematics pull towards the opposite side, and so confusions easily arise.
The word 'aggregate' is connected with 'grex'. An aggregate is something like a herd, a whole whose parts are like one another. But since the agreement is never perfect, and since on the other hand it is surely always possible to find some respect in which the parts do agree, the similarity of the parts is unfitted to serve as a characteristic mark. We can say after all that every object in which we may distinguish parts is an aggregate, for instance the solar system, a heap of sand, a grain of sand, a piece of music, and a corporation. For objects to be recognised as parts of a whole, they have to be held together by relations or interactions of some kind or another. Such relations may be spatial, temporal, physical, psychical, legal, even
intervals of pitch. And strictly we ought really to have a different word for 'part' in each of the cases; for obviously a head is a part of a man in a differ- ent sense from that in which the man himself is part of a corporation. It not infrequently occurs that parts of an aggregate are themselves aggregates. A grain of sand is part of a heap of sand, a silicic acid molecule part of the 11,rain of sand and also part of the heap of sand. What is part of a part is part of the whole, at least if the word 'part' is taken in essentially the same sense. lu this description of an aggregate we perhaps fail to find the precision that is to be expected elsewhere in mathematics. At the same time we have to say that aggregate is not really a mathematical concept. But frequently the word ? ! 11enge' [set) is patently used in the sense which agrees with that given here l'or aggregate. I need only mention 'Menschenmenge' [crowd). But so understood, the word 'Menge' can surely claim no place in mathematics.
I move on to the extension of a concept. The word itself indicates that we ure not here dealing with something spatial and physical, but something lo~~,ical. By means of our logical faculties we lay hold upon the extension of a concept, by starting out from the concept.
Let the letters' C/J' and ? 'I" stand in for concept-words (nomina appellativa). Then we designate subordination in sentences of the form 'If something is a 1/l. then it is 11 'P'. In Sl? ntenccs of the form 'If sorncthin11, is 11 1/1, then it is n 'P
? 182 On Schoenjlies: Die Logischen Paradoxien der Mengenlehre
and if something is a 'P then it is a 4J' we designate mutual subordination, a second level relation, which has strong affinities with the first level relation of equality (identity). The properties of equality, that is, which we express in the sentences 'a =a', 'if a~~ b, then b =a', 'if a= band b = c, then a= c', have their analogues for the case of that second level relation.
? 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. ).
Hilbert
? 1 (Definition? )
System.
Relations are described.
? [Frege's Notes on Hilbert's 'Grundlagen der Geometrie'] 171
? 2
The 1st Axiom Group: Axioms of connection
The axioms of this group establish a connection between the concepts point, line and plane, defined above and are as follows:
I 1. Two distinct points A and B always define a line.
? 2 Concepts defined (Points)
I 1. Define. Two. Iden- tity denied.
Better:
Instead of 'define' we will also use other turns
of phrase, such as a 'passes through' A 'and A lies on a (A point, a
through' B, a 'connects' A 'and' or 'with' B. IfA is a point which together with another point defines the line a, we also use the phrases: A 'lies on' a, A 'is a point of' a, 'There is the point' A 'on' a etc. IfA lies on the line a and also on the line b, we also use the phrase: 'the lines' a 'and' b 'have the point A in common' etc.
? 3
The 2nd Axiom Group: Axioms of ordering
The Axioms of this group define the concept 'between' and make it possible, by using this concept, to order the points of a line, in a plane and in space.
line) IfAisapoint,andBisa point, then there is a line on which A lies and on which B lies etc.
more than one theorem in I 1.
? 3 Axioms define con- cept between
Definition. The points of a line stand in The word 'between' used certain relations to one another, for whose todescribe
description we single out the word 'between'.
11 1. /fA, B, and Care points ofa line and B lies between A and C then B lies between C and A.
11 2. IfA and C are two points ofa line then there is always at least one point B which lies between A and C, and at least one point D such that C lies between A and D.
11 3. Given any three points ofa line there is always one and only one which lies between the other two.
Definition. We consider two points A and B on a line a; we call the system of the two points A and Ban interval and designate it AB or BA. The points between A and B are called points of the interval AB or are said to lie within the interval AB; the points A and B are called
Def. System of the two points
Interval
? 172 [Frege's Notes on Hilbert's 'Grundlagen der Geometrie'] endpoints of the interval AB. All other points of
the line a are said to lie outside the interval AB.
? 4
Consequences of the axioms of connection and ordering
From the 1st and 2nd sets of axioms the following theorems follow:
Theorem 3. Between any two points of a line there are always infinitely many points.
Theorem 4. Given any four points on a line, they can always be designated by A, B, C, and D in such a way that the point designated by B lies between A and C, and also between A and D, and that further the point C lies between A and D and also between B and D.
Theorem 5. (Generalization of Theorem 4). Given any finite number of points on a line, they can always be designated by A,B, C,D,E, . . . K, in such a way that the point designated by B lies between A on the one side and C, D, E, . . . K on the other, that C lies between A and B on the one side and D, E, . . . K on the other, and that D lies between A, B, C on the one side and E . . . K on the other etc. In addition to this mode of designation there is only the converse mode K . . . E, D, C, B, A with this feature.
? 4 Theorem 3. Between any two points of a line there are always in- finitely many points. 'Infinitely many' is not explained.
There is no axiom in whose phrasing 'infi- nitely many' occurs. Theorem 4. The letters and mode of designation are not part of the con- tent of the theorem.
Theorem 5. Dots and etc. do not belong to the content of the theorem. 'Finite number'. We should have to borrow from arithmetic some sentence or other con- taining the expression 'finite number'.
Theorem 6. Any line a lying in a plane a, Theorem 6. 'Region',
divides the points lying on this plane a and not on a into two regions with the following properties: any point A of the one region defines with any point B of the other an interval AB within which lies a point of a, whereas any two points A and 4' of the same region define an interval AA' containing no point of a.
'divide' have not oc- curred.
? lFrege's Notes on Hilbert's 'Grundlagen der Geometrie'] 173
? 5
The 3rd Axiom Group: Axioms of congruence
rn1. IfA,BaretwopointsofalineaandA'is ? 5'Onecanfind' a point o f the same or another line a', one can
always find on a given side of the line a' ofA'
one and only one point B', such that the interval
AB is congruent or equal to the interval A' B', in signs:
AB=A'B'.
Every interval is congruent to itself, that is we always have
AB=AB and AB=BA.
? ? [17 Key Sentences on LogicP [1906 or earlier]
1. The connections which constitute the essence of thinking are of a different order from associations of ideas.
2. The difference is not a mere matter of the presence of some ancillary thought from which the connections in the former case derive their status.
3. In the case of thinking it is not really ideas that are connected, but things, properties, concepts, relations.
4. A thought always contains something reaching out beyond the particular case so that this is presented to us as falling under something general.
5. In language the distinctive character of a thought finds expression in the copula or personal ending of the verb.
6. A criterion for whether a mode of connection constitutes a thought is that it makes sense to . ask whether it is true or untrue. Associations of ideas are neither true nor untrue.
7. What true is, I hold to be indefinable.
8. The expression in language for a thought is a sentence. We also speak in
an extended sense of the truth of a sentence.
9. A sentence can be true or untrue only if it is an expression f~. r~
thought. The sentence 'Leo Sachse is a man' is the expression of a thought only if 'Leo Sachse' designates something. And so too the sentence 'this table is round' is the expression of a thought only if the words 'this table' are not empty sounds but designate something specific for me.
11. '2 times 2 is 4' is true and will continue to be so even if, as a result of Darwinian evolution, human beings were to come to assert that 2 times 2 is 5. Every truth is eternal and independent of being thought by any? one and of the psychological make-up of anyone thinking it.
1 According to a note of Heinrich Scholz's, the manuscript should be dated around 1906. But it could have formed part of Frege's plans for a text book on logic (cf. pp. 1 ff. , 126 ff. ) and in that case its date would be much earlier. A further argument for an earlier dating is that, according to notes made by the editors pre? ceding Scholz, the manuscript was found together with the preparatory material for
the dialogue with Piinjer (pp. 53 ff. of this volume), where the name 'Leo Sachse' occurs again (ed. ).
? [17 Key Sentences on Logic] 175
12. Logic only becomes possible with the conviction that there is a difference between truth and untruth.
13. We justify a judgement either by going back to truths that have been recognized already or without having recourse to other judgements. Only the first case, inference, is the concern of Logic.
14. The theory of concepts and of judgement is only preparatory to the theory of inference.
15.
The task of logic is to set up laws according to which a judgement is justified by others, irrespective of whether these are themselves true.
16. Following the laws of logic can guarantee the truth of a judgement only
insofar as our original grounds for making it, reside in judgements that
are true.
17. Nopsychologicalinvestigationcanjustifythelawsoflogic.
? ? On Schoenflies: Die logischen Paradoxien der Mengenlehre1
Concept and object,
[1906)
Plan of critique of Schoenflies etc.
nomen appelativum, nomen proprium.
Analysis of a sentence, predicative nature of a concept. Function, sharp boundaries, independent of objects, consistency not to be insisted on. Subsumption, subordination. Mutual subordination. Relation. Identity. First and second level relations.
Aggregate, extension of a concept. Inbegriff2 (belong to, include). System, series, set, class.
How applied in criticizing Schoenflies' statements.
Can the extension of a concept fall under a concept, whose extension it is?
It does not need to be all-encompassing.
Russell's contradiction cannot be eliminated in Schoenflies' way. Concepts which coincide in extension, although this extension falls under the one, but not the other.
Remedy from extensions of second level concepts impossible.
Set theory in ruins.
My concept-script in the main not dependent on it. (Contrast with other
similar projects. )
1 Frege obviously intended this essay for publication in the Jahresbericht der deutschen Mathematiker-Vereinigung. Whether it was rejected by the editor, or, whether because it remained a fragment, it was not submitted by Frege, is not known. -It is dated by Frege's opening remarks (ed. ).
2 As far as we can see, this word does not have a sense in German which fits the context. Inbegri. ff usually means 'essence' or 'embodiment' (as in 'He is the very embodiment of health'). It is for us impossible to determine from these fragmentary notes to what use Frege was putting the term, and we thought it better to leave it untranslated than to put in a probably false conjecture. Frege obviously has in mind the different relation of an object to the extension of a concept under which it falls, and to an aggregate of which it is a part; but further than that we leave for the reader to decide (trans. ).
? ? On Schoenflies: Die Logischen Paradoxien der Mengenlehre 177 [Discussion]
The article by S, Ober die logischen Paradoxien der Mengenlehre* induces me to make the following remarks, in which I repeat much that I have already discussed previously, since it does not seem to be widely known. I fail to find in S and also in Korselt** the sharp distinction between concept and object. ? ? ? In the signs, a proper name (nomen proprium) corresponds to an object, a concept-word or concept-sign (nomen appellativum) to a concept. A sentence such as 'Two is a prime' can be analysed into two essentially different component parts: into 'two' and 'is a prime'. The former appears complete, the latter in need of supplementation, unsaturated. 'Two'-at least in this sentence-is a proper name, its meaning is an object, which can also be designated with greater prolixity by 'the number two'. The object, too, appears as a complete whole, whereas the predicative part has something unsaturated in its meaning as well. We count the copula 'is' as belonging to this part of the sentence. But there is usually something combined with it which here must be disregarded: assertoric force. We can of course express a thought, without stating it to be true. The thought is strictly the same, whether we merely express it or whether we also put it forward as true. Thus assertoric force, which is often connected with the copula or else with the grammatical predicate, does not belong to the expression of the thought, and so may be disregarded here.
This predicative component part of our sentence which we have described in this way, is also meaningful. We call it a concept-word or nomen appellativum, even though it is not customary to include the copula in this. Just as it itself appears unsaturated, there is also something unsaturated in the realm of meanings corresponding to it: we call this a concept. This unsaturatedness of one of the components is necessary, since otherwise the parts do not hold together. Of course two complete wholes can stand in a relation to one another; but then this relation is a third element-and one that is doubly unsaturated! In the case of a concept we can also call the unsaturatedness its predicative nature. But in this connection it is necessary to point out an imprecision forced on us by language, which, if we are not conscious of it, will prevent us from recognizing the heart of the matter: i. e. we can scarcely avoid using such expressions as 'the concept prime'. Here there is no trace left of unsaturatedness, of the predicative nature. Rather, the expression is constructed in a way which precisely parallels 'the poet Schiller'. So language brands a concept as an object, since the only way it can fit the designation for a concept into its grammatical structure is as a proper name. But in so doing, strictly speaking it falsifies matters. In the same way, the word 'concept' itself is, taken strictly, already defective, since
? The current Jahresbericht, Vol. XV, p. 19 (Jan. 1906).
? ? The current Jahresbericht, Vol. XV, p. 215 (March-April 1906). ? ? ? Cf. My essay 'Concept and Object'.
? ? 178 On Schoenjltes: Die Logischen Paradoxien der Mengenlehre
the phrase 'is a concept' requires a proper name as grammatical subject; and so, strictly speaking, it requires something contradictory, since no proper name can designate a concept; or perhaps better still, something nonsensical. It is no objection to say that surely the grammatical predicate 'is rectangular' can be combined with the grammatical subject 'every square', which isn't a proper name; for even the sentence 'every square is rectangular' can only make sense in virtue of the fact that you can assert of an object that it is rectangular, either rightly or wrongly, but in either case significantly. By a proper name I understand the sign of an object, independently of the question whether it be a simple word or sign, or a complex one, provided only that it designates the object determinately.
In the sentence 'Two is a prime' we find a relation designated: that of subsumption. We may also say the object falls under the concept prime, but if we do so, we must not forget the imprecision of linguistic expression we have just mentioned. This also creates the impression that the relation of subsumption is a third element supervenient upon the object and the concept. This isn't the case: the unsaturatedness of the concept brings it about that the object, in effecting the saturation, engages immediately with the concept, without need of any special cement. Object and concept are fundamentally made for each other, and in subsumption we have their fundamental union.
We call a concept empty if no object falls under it. The concept-word for an empty concept never yields a true sentence,? no matter what proper name may saturate it, or in other words: no matter what proper name may be attached as a grammatical subject to the concept-word as predicate. A concept under which one and only one object falls must still be distinguished from the latter; its sign is a nomen appellativum, not a nomen proprium.
With the help of the definite article or demonstrative, language forms proper names out of concept-words. So, for instance, the phrase 'this A' on p. 20 of the Schoenfties article is a proper name. If forming a proper name in this way is to be legitimate, the concept whose designation is used in its formation must satisfy two conditions:
1. It may not be empty.
2. Only one object may fall under it.
If the first condition is unsatisfied, there is no object at all to which the proper name would be ascribed. If the second is unsatisfied, there are indeed several such objects, but none of them is determined as the one meant to be designated by the proper name. In science the purpose of a proper name is to designate an object determinately; if this purpose is unfulfilled, the proper name has no justification in science. How things may be in ordinary
? For brevity, I here call a sentence expressing a true thought, a true sentence.
? On Schoenjlies: Die Logischen Paradoxien der Mengenlehre 179
language is no concern of ours here. Our first requirement includes the requirement that the concept be consistent.
But this requirement doesn't mean that every concept ought to be consistent, only that a concept must be consistent if you wish to form a proper name from the concept-word together with the definite article or demonstrative. But for this purpose, this requirement doesn't go far enough: the concept is not to be empty, no matter what the reason for its being empty may be.
But if we ask, under what conditions a concept is admissible in science. the first thing to stress is that consistency is no~ such a condition. The only requirement to be made of a concept is that it should have sharp boundaries; that is, that for every object it holds that it either falls under the concept or does not do so. Essentially this is nothing but the requirement that the principle of non-contradiction should hold. But the admissibility of a concept is entirely independent of the question whether objects fall under it, and if so which, or in other words; whether there be objects, and if so which, of which it can be truly asserted. For, before we can raise such questions, we already need the concept. The requirement that a concept be consistent encounters great difficulties. The only apparent way to show a concept has this property, is to cite an object falling under the concept. But to do that, you already need the concept.
Schoenfties obviously only lays down this requirement as a result of his confounding concepts and objects. It is of course self-evident that an object t:annot have inconsistent properties. That the concept of a right-angled equilateral pentagon contains a contradiction does not make it inadmissible. For we can see no reason why a man should not be able to say of an object that it is not a right-angled equilateral pentagon, or why he should not be permitted to say there are no right-angled equilateral pentagons. And before he arrives at such judgements, he must consider the matter, and to do that he requires this concept. It is completely wrongheaded to imagine that every contradiction is immediately recognizable; frequently the contradiction lies deeply buried and is only discovered by a lengthy chain of inference: throughout which you need the concept. The only thing is, given a concept, you may not presuppose without further question that an object falls under it. But someone does that when, by means of a definite article or demonstrative, he forms out of a concept-word a proper name destined for use in science.
Let us now examine Schoenflies' sentences in the light of what we have said so far. Earlier on p. 20 we read,
'I. The A has the property B, and
2. The A does not have the property B. '
The definite article before the 'A' can only be understood in such a way that 'the A' is being thought of as a proper name. In logic it must be presupposed that every proper name is meaningful; that is, that it serves its purpose of
? 180 On Schoenjlies: Die Logischen Paradoxien der Mengenlehre
designating an object. For a sentence containing a meaningless proper name either expresses no thought at all, or it expresses a thought that belongs to myth or fiction. In either case it falls outside the domain governed by the laws of logic. For instance 'the A' would be a name that was in this way inadmissible for scientific use if it were formed by means of the definite article from a name of a concept (nomen appellativum), under which either no object or more than one fell. The mistake would then not lie in the use of
the concept-word in itself, but in forming a proper name from it. Let us now assume 'the A' is a meaningful proper name, and look at the concept or logical predicate 'has the property B'. We must presuppose for this concept that it has sharp boundaries in the sense given above; for only in that case can it be logically recognized as a concept. Then indeed the principle of non- contradiction holds, i. e. that of the two sentences cited always one and only one is right, that is, expresses a true thought; and indeed this holds even if the concept of the predicate is contradictory; for then the second sentence is right, but the first not.
And so the conditions for the validity of the principle of non-contradiction in S's formulation consist in the fact that proper names as well as concept- words satisfy logical requirements. But these do not include the require- ment that a concept-word should designate a consistent concept. Since only concepts with sharp boundaries are to be recognized as meaningful by logic, we may also say: proper names and concept-words must be meaningful.
Now if A is the concept of a right-angled equilateral pentagon, 'this A' is an inadmissible-meaningless-proper name, and nothing whatever can be legitimately inferred either from the sentence 'This A has the property B' or from 'This A does not have the property B'; for, strictly, we do not infer from sentences at all, but from thoughts. And only true thoughts are admissible premises of inferences. Also it isn't strictly sentences, it is thoughts which have contradictory counterparts. If you always bear this in mind, that we cannot legitimately infer from sentences, but only from true thoughts, and that proper names and concept-words must be meaningful, then the case of the failure of the principle of non-contradiction or of indirect proof cannot obtain. The failure of the principle of non- contradiction cannot serve as a criterion for the admissibility of a concept; for then you could never avail yourself of indirect proof. We should always have to reckon with its failing. But before we go on to proofs at all, we must have assured ourselves that the proper names and concept-words we employ are admissible. From the failure of indirect proof we can only conclude that we have made a mistake in this respect, which ought not to have occurred at all. Now whether this mistake consists in the use of a meaningless proper name or a meaningless concept-word, cannot be generally answered. At any rate, an inconsistent concept is not to be rejected of itself. Only if some- one makes a proper name from the corresponding concept-word by means of the definite article or demonstrative, does he fall into error. But whether it is a mistake of this kind or of another kind which is to hlame for the failure of the indirect prm~f. needs special investigation in each case.
? On Schoenjlies: Die Logischen Paradoxien der Mengenlehre 181
Now Schoenfties states that the concept of the set of all sets which do not contain themselves as elements is not consistent. It must first be remarked that 'the set of all sets which do not contain themselves as elements' is not a concept-word, it is a proper name, and it can only be a question of whether this proper name is meaningful. It is evidently not formed from the nomen appellativum is a set ofall sets which do not contain themselves as elements with the definite article; for the use of such a predicate has not preceded it: the expression makes its debut together with the definite article. And so we have here a different case from the previous one with the phrase 'this A'. Hence our proper name requires a special scrutiny. We may distinguish two different ways of using the word 'set', going with two different conceptions, which are probably most plainly identified by the words 'aggregate' and 'extension of a concept'. But frequently these conceptions do not occur in their pure form, but mixed together and this makes for unclarity. The aggregative conception is the first to offer itself, but the requirements of mathematics pull towards the opposite side, and so confusions easily arise.
The word 'aggregate' is connected with 'grex'. An aggregate is something like a herd, a whole whose parts are like one another. But since the agreement is never perfect, and since on the other hand it is surely always possible to find some respect in which the parts do agree, the similarity of the parts is unfitted to serve as a characteristic mark. We can say after all that every object in which we may distinguish parts is an aggregate, for instance the solar system, a heap of sand, a grain of sand, a piece of music, and a corporation. For objects to be recognised as parts of a whole, they have to be held together by relations or interactions of some kind or another. Such relations may be spatial, temporal, physical, psychical, legal, even
intervals of pitch. And strictly we ought really to have a different word for 'part' in each of the cases; for obviously a head is a part of a man in a differ- ent sense from that in which the man himself is part of a corporation. It not infrequently occurs that parts of an aggregate are themselves aggregates. A grain of sand is part of a heap of sand, a silicic acid molecule part of the 11,rain of sand and also part of the heap of sand. What is part of a part is part of the whole, at least if the word 'part' is taken in essentially the same sense. lu this description of an aggregate we perhaps fail to find the precision that is to be expected elsewhere in mathematics. At the same time we have to say that aggregate is not really a mathematical concept. But frequently the word ? ! 11enge' [set) is patently used in the sense which agrees with that given here l'or aggregate. I need only mention 'Menschenmenge' [crowd). But so understood, the word 'Menge' can surely claim no place in mathematics.
I move on to the extension of a concept. The word itself indicates that we ure not here dealing with something spatial and physical, but something lo~~,ical. By means of our logical faculties we lay hold upon the extension of a concept, by starting out from the concept.
Let the letters' C/J' and ? 'I" stand in for concept-words (nomina appellativa). Then we designate subordination in sentences of the form 'If something is a 1/l. then it is 11 'P'. In Sl? ntenccs of the form 'If sorncthin11, is 11 1/1, then it is n 'P
? 182 On Schoenjlies: Die Logischen Paradoxien der Mengenlehre
and if something is a 'P then it is a 4J' we designate mutual subordination, a second level relation, which has strong affinities with the first level relation of equality (identity). The properties of equality, that is, which we express in the sentences 'a =a', 'if a~~ b, then b =a', 'if a= band b = c, then a= c', have their analogues for the case of that second level relation.
