No More Learning

1 In what follows Frege is referring to the review of Schroder's Vorlesungen iiber die Algebra der Logik (Exakte Logik) I (Leipzig 1890), which Husserl had written for the Gottingischen Gelehrten Anzeigen (pp. 243-278, April 1891) (ed. ).
2 In the place referred to by Frege Schroder fixes on the adjectives ending in 'deutig' as terms for the sizes of extensions of concepts. Schroder speaks generally of names and calls proper names 'eindeutig', common names like 'my hand' 'zweideutig' [having two meanings], common names in general 'mehrdeutig' or 'vieldeutig' [having many meanings]. The corresponding formations with 'sinnig' are employed by Schroder to distinguish terms whose use is precisely fixed {'einsinnig' or 'univocal'), from terms with multiple meanings ('doppelsinnig' ! having a double sense], 'mehrsinnig' or 'equivocal') and from formations without sense ('unsinnig'; 'round square' in Schroder's example). With Husserl Frege chiefly criticizes Schroder for calling a name like 'round square' 'undeutig' when for this label to apply the? name is surely already presupposed as being significant as Much, so that it cannot at the same time be designated as 'unsinnlg' (ed. ).
? ? ? 124 [Comments on Sense and Meaning]
object falls under the corresponding concept. * On this view it would be possible for a common name to be undeutig too, like 'the round square', without its being defective. Schroder, however, calls it unsinnig as well and is thus untrue to his own way of speaking; for according to this the 'round square' would have to be called einsinnig, and Husserl was right when he called it a univocal common name; for 'univocal' and 'equivocal' correspond to Schroder's 'einsinnig' and 'mehrsinnii. Husserl says (p. 250) 'Obviously he confuses two quite different questions here, namely (1) whether a name has a Bedeutung (a 'Sinn'); and (2) whether there does or does not exist an object corresponding to the name'. This distinction is inadequate. The word 'common name' leads to the mistaken assumption that a common name is related to objects in essentially the same way as is a proper name, the difference being only that the latter names just one thing whilst the former is usually applicable to more than one. But this is false, and that is why I prefer 'concept-word' to 'common name'. A proper name must at least have a sense (as I use the word); otherwise it would be an empty sequence of sounds and it would be wrong to call it a name. But if it is to have a use in science we must require that it have a meaning too, that it designates or names an object. Thus it is via a sense, and only via a sense that a proper name is related to an object. 1 _
A concept-word must have a sense too and if it is to have a use in science, a meaning; but this consists neither of one object nor of a plurality of objects: it is a concept. Now in the case of a concept it can of course again be asked whether one object falls under it, or more than one or none. But
this relates directly to the concept and nothing else. So a concept-word can be absolutely impeccable, logically speaking, without there being an object to which it is related through its sense and meaning (the concept itself). As we see, this relation to an object is more indirect and inessential, so that there seems little point in dividing concept-words up according as no object falls under the corresponding concepts or one object or more than one.
* If, as Husserl says in the first footnote to p. 252, a distributive name is one 'whose Bedeutung is such that it designates any one of a plurality of things', then a concept-word (common name) is at any rate not a distributive name.
1 Since Schroder and Husserl did not distinguish, in the way Frege did, between the Sinn and Bedeutung of an expression, we have thought it best in this paragraph to preserve the actual German where these terms or (more commonly) their cognates with 'sinnig' and 'deutig' occur in quotation from these authors, or where Frege himself uses the latter in alluding to their views. We have given what help we could to the reader by providing renderings in square brackets; he should only remember not to attribute to the words 'sense' and 'meaning', as they occur in these renderings, the significance they have in the main body of the text, where they are of course used to render Frege's 'Sinn' and 'Bedeutung' (trans. ).
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Logic must demand not only of proper names but of concept-words as well that the step from the word to the sense and from the sense to the meaning be determinate beyond any doubt. Otherwise we should not be entitled to speak of a meaning at all. Of course this holds for all signs and combinations of signs with the same function as proper names or concept- words.
? ? [128]
[129]
[129] [130] [131]
[131f. ]
Logic 1 [1897]
The word 'true' specifies the goal. Logic is concerned with the predicate 'true' in a special way. The word 'true' characterizes logic. True cannot be defined; we cannot say: an idea is true if it agrees with reality.
True primitive and simple. This feature of our predicate is to be brought out by comparing it with others. Predicating it is always included in predicating anything whatever. To locate the domain of application of the predicate 'true'. Not applicable to what is material. It is most frequently ascribed to sentences-but only to assertoric sentences. Not, however, to sentences as series of sounds.
Translation.
We do not need to consider mock assertions in logic.
The sense of a sentence is called a thought. The predicate 'true' applies to thoughts. Does it also apply to ideas? Even where an idea is called true, it is really a thought to which this predicate is ascribed. A thought is not an idea and is not composed of ideas. Thoughts and ideas are fundamentally different. By associating ideas we never arrive at anything that could be true.
The proper means of expression for a thought is a sentence. On the other hand, a sentence is hardly an appropriate vehicle for conveying ideas. By contrast, pictures and musical compositions are unsuited
for expressing thoughts. The predicate 'true' compared with 'beautiful'. The latter admits of degree, not the former. What is beautiful is beautiful only for him who experiences it as such. There is no disputing tastes. What is true is true in itself; nothing is beautiful in itself. The assumption of a normal human being always underlies objective judgements of beauty. However, what is normal?
The objective sense of 'beautiful' can only be based on the subjective sense. Nor is it of any use to replace the assumption of a normal human being by that of an ideal one. A work of art is a structure of ideas within us. Each of us has his own. Aesthetic judgements don't contradict one another. Anyone who asserts that it -is only our recognizing a thing as true that makes it so, would, by so doing, contradict the content of his own assertion. In reality he could assert
[132f. ]
1 The date mentioned on p. 147 and the quotation on p. J58 make it probable that this essay was composed in 1897 (ed. ). '
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nothing. Every opinion would then be unjustified; there would be no science. Properly speaking, there would be nothing that is true. The independence of being recognized by us is integral to the sense of the word 'true'. Nor do thoughts have to be thought by us in order to be true. Laws of nature are discovered (not invented).
l133f. ) Thoughts are independent of our thinking. A thought does not belong specially to the person who thinks it, as an idea does to the person who has it: whoever thinks it encounters it in the same way, as the same thought. Otherwise two people would never attach the same thought to the same sentence. A contradiction between the assertions of different people would be impossible. A dispute about the truth of something would be futile. There would be no common ground to fight on. As each man makes a judgement about his poem, if the
judgement is an aesthetic one, so each man would make a judgement about his thought, if the thought were related to a sentence as the auditory ideas of the spoken sounds are related to the sound waves. If a thought were something mental, then its truth could only consist in a relation to something external, and that this relation obtained would be a thought into the truth of which we could inquire.
l134f. ) Treadmill. A thought is something impersonal. Writing on a wall. Objection: what of a sentence like 'I am cold'? The spoken word often needs to be supplemented. The word 'I' does not always designate the same person. A sentence containing 'I' can be cast into a more appropriate form. Interjections are different. The words 'now' and 'here' analogous to 'I'. Identity of the speaker essential where a subjective judgement of taste is concerned.
[135) Objection:myuseoftheword'thought'isoutoftheordinary. 1136] Footnote. Dedekind's way of using it agrees with mine.
[137) Thoughts are not generated by, but grasped by, thinking.
l137f. ) A thought not something spatial, material. Only in a special sense is it something actual.
[138) False thoughts are also independent of the speaker. The predicate 'true' is predicated in predicating anything. In an assertoric sentence the expression of a thought and the recognition of its truth usually go hand in hand. This does not have to be so. An assertoric sentence does not always contain an assertion. Grasping a thought usually precedes the recognition of truth. Judging, asserting.
ll39f. ] A sentence is also meant to have an effect on the imagination and feelings. It is able to do this because it consists of heard sounds. Onomatopoeia. Words also have an effect on the imagination through the sense they have. But ideas and sense must not be confused. A word by itself does not determine an idea. Different ideas answer to the same word. Words furnish hints to the imagination.
t140] The means available to the poet. 'Dog' and 'cur' can be substituted
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Logic
[141 j
for one another without altering the thought. What distinguishes them is of the nature of an interjection. Criterion. To distinguish thoughts that are expressed from those that are merely evoked in us. A sad tone of voice, 'ab', 'unfortunately'.
Changes in language give rise to borderline cases. 1
Introduction
The predicate true, thoughts, consequences for the treatment of logic
When entering upon the study of a science, we need to have some idea, if only a provisional one, of its nature. We want to have in sight a goal to strive towards; we want some point to aim at that will guide our steps in the right direction. The word 'true' can be used to indicate such a goal for logic, just as can 'good' for ethics and 'beautiful' for aesthetics. Of course all the sciences have truth as their goal, but logic is concerned with the predicate 'true' in a quite special way, namely in a way analogous to that in which physics has to do with the predicates 'heavy' and 'warm' or chemistry with the predicates 'acid' and 'alkaline'. There is, however, the difference that these sciences have to take into account other properties besides these we have mentioned, and that there is no one property by which their nature is so completely characterized as logic is by the word 'true'.
Like ethics, logic can also be called a normative science. How must I think in order to reach the goal, truth? We expect logic to give us the answer to this question, but we do not demand of it that it should go into what is peculiar to each branch of knowledge and its subject-matter. On the. contrary, the task we assign logic is only that of saying what holds with the utmost generality for all thinking, whatever its subject-matter. We must assume that the rules for our thinking and for our holding something to be true are prescribed by the laws of truth. The former are given along with the latter. Consequently we can also say: logic is the science of the most general laws of truth. The reader may find that he can form no very precise impression from this description of what is meant. The author's inadequacy and the awkwardness of language are probably to blame for this. But it il only a question of giving a rough indication of the goal of logic. What is still lacking in the account will have to be made good as we go on.
Now it would be futile to employ a definition in order to make it clearer what is to be understood by 'true'. If, for example, we wished to say 'an idea is true if it agrees with reality' nothing would have been achieved, since in order to apply this definition we should have to decide whether some idea or other did agree with reality. Thus we should have to presuppose the very thing that is being defined. The same would hold of any definition of the form 'A is true if and only if it has such-and-such properties or stands in
1 Frege only took the table of contents as far as p. 141 (ed. ).
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such-and-such a relation to such-and-such a thing'. In each case in hand it would always come back to the question whether it is true that A has such- and-such properties, or stands in such-and-such a relation to such-and-such a thing. Truth is obviously something so primitive and simple that it is not possible to reduce it to anything still simpler. Consequently we have no alternative but to bring out the peculiarity of our predicate by comparing it with others. What, in the first place, distinguishes it from all other predicates is that predicating it is always included in predicating anything whatever.
If I assert that the sum of 2 and 3 is 5, then I thereby assert that it is true that 2 and 3 make 5. So I assert that it is true that my idea of Cologne Cathedral agrees with reality, ifl assert that it agrees with reality. Therefore it is really by using the form of an assertoric sentence that we assert truth, and to do this we do not need the word 'true'. Indeed we can say that even where we use the form of expression 'it is true that . . . 'the essential thing is really the assertoric form of the sentence.
We now ask: what can the predicate 'true' be applied to? The issue here is to delimit the range of application of the word. Whatever else may be the case, the word cannot be applied to anything that is material. If there is any doubt about this, it could arise only for works of art. But if we speak of truth in connection with these, then we are surely using the word with a different meaning from the one that is meant here. In any case it is only as a work of art that a thing is called true. If a thing had come into existence through the
blind play of natural forces, our predicate would be clearly inappropriate. Por the same reason we are excluding from consideration the use that is made by, say, an art critic when he calls feelings and experiences true.
No one would deny that our predicate is, for the most part, ascribed to sentences. We are not, however, concerned with sentences expressing wishes, questions, requests and commands, but only with assertoric sentences, sentences that is to say, in which we communicate facts and propound mathematical laws or laws of nature.
Further, it is clear that we do not, properly speaking, ascribe truth to the series of sounds which constitute a sentence, but to its sense; for, on the one hand, the truth of a sentence is preserved when it is correctly translated into another language, and, on the other hand, it is at least conceivable that the same series of sounds should have a true sense in one language and a false sense in another.
We are here including under the word 'sentence' the main clause of a sentence and clauses that are subordinate to it.
In the cases which alone concern logic the sense of an assertoric sentence is either true or false, and then we have what we call a thought proper. But there remains a third case of which at least some mention must be made here.
The sentence 'Scylla has six heads' is not true, but the sentence 'Scylla does not have six heads' is not true either; for it to be true the proper name 'Scylla' would have to designate something. Perhaps we think that the name
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'Scylla' does designate something, namely an idea. In that case the first question to ask is 'Whose idea? ' We often speak as if one and the same idea occurred to different men, but that is false, at least if the word 'idea' is used in the psychological sense: each man has his own idea. But then an idea does not have heads, and so one cannot cut heads off an idea either. The name 'Scylla' does not therefore designate an idea. Names that fail to fulfil the usual role of a proper name, which is to name something, may be called mock proper names.
Although the tale of Willian Tell is a legend and not history and the name 'William Tell' is a mock proper name, we cannot deny it a sense. But the sense of the sentence 'William Tell shot an apple off his son's head' is no more true than is that of the sentence 'William Tell did not shoot an apple off his son's head'. I do not say, however, that this sense is false either, but I characterize it as fictitious. This may elucidate the sense in which I am using the word 'false', which is as little susceptible of a definition proper as is the word 'true'.
If the idealist theory of knowledge is correct then all the sciences would belong to the realm of fiction. Indeed one might try to reinterpret all sentences in such a way that they were about ideas. By doing this, however, their sense would be completely changed and we should obtain quite a different science; this new science would be a branch of psychology.
Instead of speaking of 'fiction', we could speak of 'mock thoughts'. Thus if the sense of an assertoric sentence is not true, it is either false or fictitious, and it will generally be the latter if it contains a mock proper name. * The writer, in common with, for example, the painter, has his eye on appearances. Assertions in fiction are not to be taken seriously: they are only mock assertions. Even the thoughts are not to be taken seriously as in the sciences: they are only mock thoughts. If Schiller's Don Car/os were to be regarded as a piece of history, then to a large extent the drama would be false. But a work of fiction is not meant to be taken seriously in this way ati all: it's all play. Even the proper names in the drama, though they: correspond to names of historical personages, are mock proper names; they are not meant to be taken seriously in the work. We have a similar thing in the case of an historical painting. As a work of art it simply does not claim to give a visual representation of things that actually happened. A picture that was intended to portray some significant moment in history with photographic accuracy would not be a work of art in the higher sense of the word, but would be comparable rather to an anatomical drawing in a scientific work.
The logician does not have to bother with mock thoughts, just as 1 physicist, who sets out to investigate thunder, will not pay any attention to stage-thunder. When we speak of thoughts in what follows we mean thoughts proper, thoughts that are either true or false.
*We have an exception where a mock proper name occurs within a clause in indirect speech.
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The sense of an assertoric sentence I call a thought. Examples of thoughts are laws of nature, mathematical laws, historical facts: all these find expression in assertoric sentences. I can now be more precise and say: The predicate 'true' applies to thoughts.
Of course we speak of true ideas1 as well. By an idea we understand a picture that is called up by the imagination: unlike a perception it does not consist of present impressions, but of the reactivated traces of past impressions or actions. Like any other picture, an idea is not true in itself, hut only in relation to something to which it is meant to correspond. If it is said that a picture is meant to represent Cologne Cathedral, fair enough; it can then be asked whether this intention is realized; if there is no reference to an intention to depict something, there can be no question of the truth of a picture. It can be seen from this that the predicate true is not really conferred on the idea itself, but on the thought that the idea depicts a certain object. And this thought is not an idea, nor is it made up of ideas in any way. Thoughts are fundamentally different from ideas (in the psychological sense). The idea of a red rose is something different from the thought that this rose is red. Associate ideas or run them together as we may, we shall still finish up with an idea and never with something that could be true. This difference also comes out in the modes we have of communicating. The proper means of expression for a thought is a sentence. But a sentence is hardly an appropriate vehicle for conveying an idea. I have only to remind you how inadequate any description is by comparison with a pictorial representation. Things are not so bad where it is a matter of representing sounds, since we have the resources of onomatopoeia; but onomatopoeia has nothing whatever to do with the expression of thoughts, and whilst in translation the play of sounds is easily lost, the thought must be preserved if we are to speak of a translation at all. Conversely, pictures and musical
compositions without accompanying words are hardly suited for expressing thoughts. It is true that we may associate all kinds of thoughts with some work of art or other but there is no necessary connection between the two, and we are not surprised when someone else associates different thoughts with it.
In order to shed a clearer light on the peculiarity of the predicate true, let us compare it with the predicate beautiful. We can see, to begin with, that what is beautiful admits of degree, but what is true does not. We can think two objects beautiful, and yet think one more beautiful than the other. On the other hand, if two thoughts are true, one is not more true than the other. And here there emerges the essential difference that what is true is true
1 In the German, 'Vorstellungen'. Throughout this essay the difficult word ? Vorstellung' has been generally rendered by 'idea'. Admittedly this makes certain passages read unnaturally, but the gist of what Frege is saying should be clear if the reader bears in mind the explanation he gives here of how the term 'VorstellunJ. :' is hcing used (trans. ).
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independently of our recognizing it as such, but what is beautiful is beautiful only for him who experiences it as such. What is beautiful for one person is not necessarily beautiful for another. There is no disputing tastes. Where truth is concerned, there is the possibility of error, but not where beauty is concerned. By the very fact that I consider something beautiful it is beautiful for me. But something does not have to be true because I consider it to be true, and if it is not true in itself, it is not true for me either. Nothing is beautiful in itself: it is only beautiful for some being experiencing it and this is necessarily implicit in any aesthetic judgement. Now it is true that we also make judgements of this kind which seem to lay claim to being objective. Whether we are aware of it or not the assumption of a normal human being always underlies such judgements, and each one of us cannot help but think that he himself is so close to the normal human being that he believes he can speak in his name. What, then, we mean by 'This rose is beautiful' is 'This rose is beautiful for a normal human being'. But what is normal? That depends on the circle of human beings one has in mind. If there is some remote mountain valley where nearly all the people have goitres, then having a goitre will be looked on as normal there, and those who lack such an adornment will be considered ugly. How is a negro from the heart of Africa to be weaned from the view that the narrow nose of the European is ugly, whereas the broad nose of the negro is beautiful? And cannot a negro qua negro be just as normal as a white man qua white man? Cannot a child be
just as normal as a grown-up? The ideas that are awakened in us by the power of association have a great influence on the judgements a man forms of what is beautiful, and these ideas depend upon what he has absorbed in earlier life. But this varies from person to person. And even if we managed to define a normal human being and so 'beautiful' in an objective sense, it would still be only possible to do this on the basis of the subjective sense. Far from having rid ourselves of this, we would have recognized it as the root sense. We could not alter the situation by trying to substitute an ideal human being for a normal one. In the absence of experiences and ideas there would be no instance of anything subjectively beautiful and therefore no instance of anything objectively beautiful either. There is therefore much to be said for the view that the real work of art is a structure of ideas within us and that the external thing-the painting, the statue-is only a means for producing the real work of art in us. On this view, anyone who enjoys a work of art has his own work of art, with the consequence that there is no contradiction whatever between varying aesthetic judgements. Hence: dl gustibus non disputandum.
If anyone tried to contradict the statement that what is true is true independently of our recognizing it as such, he would by his very assertion contradict what he had asserted; he would be in a similar position to the Cretan who said that all Cretans are liars.
To elaborate: if something were true only for him who held it to be true, there would be no contradiction between the opinions of different people. So
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to be consistent, any person holding this view would have no right whatever to contradict the opposite view; he would have to espouse the principle: non disputandum est. He would not be able to assert anything at all in the normal sense, and even if his utterances had the form of assertions, they would only have the status of interjections--of expressions of mental states or processes, between which and such states or processes in another person there could be no contradiction. And in that case his assertion that something was true only for us and through being recognized by us as such would have this status too. If this view were true, it would be impossible to daim that any of his own opinions was more justified in the eyes of others than the opposite opinion. A view that made such a claim would be unjustified; this would mean, however, that every opinion would be unjustified in the usual sense of the word, and so also those opinions to which we were opposed. There would be no science, no error and no ~:orrection of error; properly speaking, there would be nothing true in the uormal sense of the word. For this is so closely bound up with that independence of being recognized as true, which we are emphasizing here, that it cannot be separated from it. If anyone seriously and sincerely defended the view we are here attacking, we should have no recourse but to ussume that he was attaching a different sense to the word 'true'.
We can go a step further. In order to be true, thoughts--e. g. laws of nature-not only do not need to be recognized by us as true: they do not have to have been thought by us at all. A law of nature is not invented by us, hut discovered, and just as a desolate island in the Arctic Ocean was there long before anyone had set eyes on it, so the laws of nature, and likewise those of mathematics, have held good at all times and not just since they were discovered. This shows us that these thoughts, if true, are not only true independently of our recognizing them to be so, but that they are indepen- dent of our thinking as such. A thought does not belong specially to the person who thinks it, as does an idea to the person who has it: everyone who grasps it encounters it in the same way, as the same thought. Otherwise two people would never attach the same thought to the same sentence, but cuch would have his own thought; and if, say, one man put 2 ? 2 = 4 forward ns true whilst another denied it, there would be no contradiction, because what was asserted by one would be different from what was rejected by the other. It would be quite impossible for the assertions of different people to contradict one another, for a contradiction occurs only when it is the very Nume thought that one person is asserting to be true and another to be false. So a dispute about the truth of something would be futile. There would Nimply be no common ground to fight on; each thought would be enclosed in its own private world and a contradiction between the thoughts of ditTerent people would be like a war between ourselves and the inhabitants of Mars. Nor must we say that one person might communicate his thought to another and a conflict would then flare up in the latter's private world. It would be quite impossible for a thought to be so communicated that it
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should pass out of the private world of one person into that of another. The thought that entered the latter's mind as a result of the communication would be different from the thought in the former's mind; and the slightest alteration can transform a truth into a falsehood. If we wanted to regard a thought as something psychological, as a structure of ideas, without, however, adopting a wholly subjective standpoint, we should have to explain the assertion that 2 + 3 = 5 on something like the following lines 'It has been observed that with many people certain ideas form themselves in association with the sentence "2 + 3 = 5". We call a formation of this kind the sense of the sentence "2 + 3 = 5". So far as we have observed hitherto these formations are always true; we may therefore make the provisional statement "Going by the observations made hitherto, the sense of the sentence '2 + 3 = 5' is true". ' But it is obvious that this explanation would not work at all. And it would leave us where we were, for the sense of the sentence 'It has been observed that with many people certain ideas form themselves etc. ' would of course be a formation of ideas too and the whole thing would begin over again. A soup that tastes pleasant to one person, may be nauseous to another. In such a case each person is really making a judgement about his own sensation of taste, and this is different from the other's. The same would hold for thoughts if a thought were related to a sentence in the same kind of way as sensations of taste are related to the chemical stimuli that excite them.
If a thought, like an idea, were something private and mental, then the truth of a thought could surely only consist in a relation to something that was not private or mental. So if we wanted to know whether a thought was true, we should have to ask whether the relation in question obtained and thus whether the thought that this relation obtained was true. And so we should be in the position of a man on a treadmill who makes a step forwards and upwards, but the step he treads on keeps giving way and he falls back to where he was before.
A thought is something impersonal. If we see the sentence '2 + 3 = 5' written on a wall, we have no difficulty at all in recognizing the thought expressed by it, and we do not need to know who has written it there in order to understand it.
A sentence like 'I am cold' may seem to be a counter-example to our thesis that a thought is independent of the person thinking it, in so far as it can be true for one person and false for another, and thus not true in itself. The reason for this is that the sentence expresses a different thought in the mouth of one person from what it expresses in the mouth of another. In this case the mere words do not contain the entire sense: we have in addition to take into account who utters it. There are many cases like this in which the spoken word has to be supplemented by the speaker's gesture and expression, and the accompanying circumstances. The word 'I' simply designates a different person in the mouths of different people. It is not necessary that the person who feels cold should himself give utterance to the
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thought that he feels cold. Another person can do this by using a name to designate the one who feels cold.
In this way a thought can be clothed in a sentence that is more in keeping with its being independent of the person thinking it. The possibility of doing this distinguishes it from a mental state expressed by an interjection. Words like 'here' and 'now' only acquire their full sense through the circumstances in which they are used. If someone says 'it is raining' the time and place of utterance has to be supplied. If such a sentence is written down it often no longer has a complete sense because there is nothing to indicate who uttered it, and where and when. As regards a sentence containing a judgement of taste like 'This rose is beautiful', the identity of the speaker is essential to the sense, even though the word 'I' does not occur in it. So the explanation for all these apparent exceptions is that the same sentence does not always express the same thought, because the words need to be supplemented in order to get a complete sense, and how this is done can vary according to the circumstances.
Whereas ideas (in the psychological sense of the word) have no fixed boundaries, but are constantly changing and, Proteus-like, assume different forms, thoughts always remain the same. It is of the essence of a thought to he non-temporal and non-spatial. In the case of the thought that 3 + 4 = 7 and the laws of nature there is hardly any need to support this statement. If it should turn out that the law of gravitation ceased to be true from a certain moment onwards, we should conclude that it was not true at all, and put ourselves out to discover a new law: the new one would differ in containing a condition which would be satisfied at one time but not at another. It is the same with place. If it should transpire that the law of gravitation was not valid in the neighbourhood of Sirius, we should search for another law which contained a condition that was satisfied in our solar system but not in the neighbourhood of Sirius. If someone wished to cite, say, 'The total number of inhabitants of the German Empire is 52 000 000', as a counter-example to the timelessness of thoughts, I should reply: This sentence is not a com- plete expression of a thought at all, since it lacks a time-determination. If we add such a determination, for example, 'at noon on l January 1897 by central European time', then the thought is either true, in which case it is always, or better, timelessly, true, or it is false and in that case it is false without qualification. This holds of any particular historical fact: if it is true, it is true independently of the time at which it is judged to be true. It is no objection that a sentence may acquire a different sense in the course of time; for what changes in such a case is of course the language, not the thought. In another language this shift need not take place. It is true of course that we speak of men's thoughts as being liable to change. However it is not the thoughts which are true at one time and false at another: it is only that they arc held to be true at one time and false at another.
What if it is objected that I am attaching to the word 'thought' a sense that it does not ordinarily have, and that other people understand by it an
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act of thinking, which is obviously private and mental? Well, the important thing is that I remain true to my way of using it; whether this agrees with the ordinary use is of less importance. It may well be the case that people sometimes understand by the word 'thought' an act of thinking-in any case this is not always so*-and such an act cannot be true.
In logic, as in other sciences, it is open to us to coin technical terms,
? Dedekind, in theor. 66 of his book Was sind und was sol/en die Zahlen? , uses this word as I do. For he is attempting there to prove that the totality of things that can be objects of his thinking is infinite. Let s be such an object; then Dedekind calls rp(s) the thought that s can be an object of his thinking. And this thought can now itself be an object of his thinking. Thus rp(rp(s)) is the thought that the thought that s can be an object of his thinking can be an object of his thinking. We can see from this what 'rp(rp(rp(s)))', 'rp(rp(rp(rp(s))))' and so on, are supposed to mean. It is essential to the proof that the sentence 's can be an object of Dedekind's thinking' always expresses a thought when the letter 's' designates such an object. Now if, as Dedekind wishes to prove, there are infinitely many such objects s, there must also be
infinitely many such thoughts rp(s). Now presumably we shall not hurt Dedekind's feelings if we assume that he has not thought infinitely many thoughts. Equally he should not assume that others have already thought infinitely many thoughts which could be the objects of his thinking; for this would be to assume what was to be proved. Now if infinitely many thoughts have not yet been thought, the infinitely many thoughts rp(s) must comprise infinitely many thoughts that are not thought, in which case it cannot be essential to a thought that it should be thought. And this is precisely what I am maintaining. If there were only thoughts that are thought, the sign 'rp(s)' would not always have a meaning; to ensure that it did have, it is not sufficient for 's' to mean something that could be an object of Dedekind's thinking: it would also have to have been thought by someone in order to be a possible object of Dedekind's thinking. If this were not the case, then the sign 'rp(s)' would have no meaning for the given s. The sun (0) can be an object of Dedekind's thinking; hence the first two members and perhaps a few successive members of the series '0, 'rp(0)', rp(rp(0))' . . . have a meaning. But as we progressed along the series we would be bound eventually to reach a member that was meaningless, because the thought which it was meant to designate had not been thought, and so was not to hand. In that case 'rp(s)' would resemble a power series which did not converge for every value of the argument. The fact that the series diverged would correspond to the sign 'rp(s)' becoming meaningless. If we assume a power series with radius of convergence 4 and if we assume, further, that the series has the value 2 for l as argument and the value 5 for 2 as argument, then the corresponding series of numbers l, 2, 5 comes to an end at this point and does not go on to infinity. In the same way the series 0, rp(0), rp(rp(0)), does not go on to infinity if there are only thoughts that are thought. So the validity of Dedekind's proofs rests on the assumption that thoughts obtain independently of our thinking. We can see how this use is one to which the word 'thought' naturally lends itself.
? Logic 137
regardless of whether the words are always used in precisely that way in l'vcryday life. It doesn't matter if the meaning we fix on is not altogether in line with everyday use or doesn't accord with the word's etymology; what does matter is to make it as appropriate a vehicle as possible for use in npressing laws. Provided there is no loss of rigour, the more compendious the formulation of the complete system of laws, the more felicitous is the apparatus of technical terms.
Now we cannot regard thinking as a process which generates thoughts. It would be just as wrong to identify a thought with an act of thinking, so that a thought is related to thinking as a leap is to leaping. This view is in harmony with many of our ways of talking. For do we not say that the same thought ~~grasped by this person and by that person? And that each person has the same thought over and over again? Now if thoughts only came into txistence as a result of thinking or if they were constituted by thinking, then the same thought could come into existence, cease to exist, a! )d then come 111to existence again, which is absurd. As I do not create a tree by looking at 1t or cause a pencil to come into existence by taking hold of it, neither do I 1/,Cnerate a thought by thinking. And still less does the brain secrete thoughts, as the liver does gall.
The metaphors that underlie the expressions we use when we speak of 11. rasping a thought, of conceiving, laying hold of, seizing, understanding, of mpere, percipere, comprehendere, intelligere, put the matter in essentially the right perspective. What is grasped, taken hold of, is already there and all we do is take possession of it. Likewise, what we see into or single out from nmongst other things is already there and does not come into existence as a result of these activities. Of course all metaphors go lame at some point. We nrc inclined to regard what is independent of our mental processes as Nornething spatial or material, and the words that we have just listed make it look as if this is what a thought actually is. But this is not where the point of lhc comparison lies. What is independent of our mental processes, what is ohjcctive, does not have to be spatial or material or actual. If we were to disregard this we should easily slip into a kind of mythology.