Philosophical Remarks

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When in May 1930, the Council of Trinity College, Cambridge, had to decide whether to renew Wittgenstein's research grant, it turned to Bertrand Russell for an assessment of the work Wittgenstein had been doing over the past year. Les mer
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When in May 1930, the Council of Trinity College, Cambridge, had to decide whether to renew Wittgenstein's research grant, it turned to Bertrand Russell for an assessment of the work Wittgenstein had been doing over the past year. His verdict: The theories contained in this new work. . . are novel, very original and indubitably important. Whether they are true, I do not know. As a logician who like simplicity, I should like to think that they are not, but from what I have read of them I am quite sure that he ought to have the opportunity to work them out, since, when completed, they may easily prove to constitute a whole new philosophy.




A proposition is completely logically analysed if its grammar is made clear - in no matter what idiom. All that is possible and necessary is to separate what is essential from what is inessential in our language - which amounts to the construction of a phenomenological language. Phenomenology as the grammar of those facts on which physics builds its theories.


The complexity of philosophy is not in its matter, but in our tangled understanding.


How strange if logic were concerned with an `ideal' language, and not with ours!


If I could describe the point of grammatical conventions by saying they are made necessary by certain properties of the colours (say) - then what would make the conventions superfluous, since in that case I would be able to say precisely that which the conventions exclude my saying.


Might we say: A child must of course learn to speak a particular language, but not to think?


In a certain sense, the use of language is something that cannot be taught.


Grammatical conventions cannot be justified by describing what is represented: any such description already presupposes the grammatical rules.


The kind of co-ordination on the basis of which a heard or seen language functions would be, say: `If you hear a shot or see me wave, run.'


Have philosophers hitherto always spoken nonsense?



Thinking of propositions as instructions for making models. For it to be possible for an expression to guide my hand, it must have the same multiplicity as the action desired. And this must also explain the nature of negative propositions.


How do I know that I can recognize red when I see it? How do I know it is the colour that I meant?


If the image of the colour is not identical with the colour that is really seen, how can a comparison be made?


Language must have the same multiplicity as a control panel that sets off the actions corresponding to its propositions.


Only the application makes a rod into a lever. - Every instruction can be construed as a description, every description as an instruction.


What does it mean, to understand a proposition as a member of a system of propositions? Its complexity is only to be explained by the use for which it is intended.


How do I know that that was what I expected? How do I know that the colour which I now call `white' is the same as the one I saw here yesterday? By recognizing it again.


Should Logic bother itself with the question whether the proposition was merely automatic or thoroughly thought? It is interested in the proposition as part of a language system.


I do not believe that logic can talk about sentences [propositions] in any other than the normal sense in which we say, `There's a sentence written here'.


Agreement of a proposition with reality. We can look at recognition, like memory, in two different ways: as a source of the concepts of the past and of identity, or as a way of checking what happened in the past, and on identity.



If you exclude the element of intention from language, its whole function then collapses.


The essential difference between the picture conception (of intention) and Russell's conception is that the former regards recognition as seeing an internal relation. The causal connection between speech and action is an external relation.


I believe Russell's theory amounts to the following: if I give someone an order and I am happy with what he then does, then he has carried out my order.


If, when learning a language, speech, as it were, is connected up to action, can these connections possibly break down? If so, what means have I for comparing the original arrangement with the subsequent action?


The intention is already expressed in the way I now compare the picture with reality.


Expecting that p will be the case must be the same as expecting the fulfilment of this expectation.


If there were only an external connection, no connection could be described at all, since we only describe the external connection by means of the internal one.


The meaning of a question is the method of answering it. Tell me how you are searching, and I will tell you what you are searching for.


Expecting is connected with looking for. I know what I am looking for, without what I am looking for having to exist. The event that replaces an expectation is the reply to it. That of course implies that the expectation must be in the same space as what is expected.


Expectation is not given an external description by citing what is expected; describing it by means of what is expected is giving an internal description.


If I say `This is the same event as I expected' and `This is the same event as also happened on that occasion', then the word `same' has a different meaning in each case.


Language and intention. If you say, `That's a brake lever, but it doesn't work', you are speaking of intention.


I only use the terms expectation, thought, wish, etc. of something which is articulated.


How you search in one way or another expresses what you expect. Expectation prepares a yardstick for measuring the event. If there were no connection between expectation and reality, you could expect a nonsense.


If I say that the representation must treat of my world, then I cannot say `since otherwise I could not verify it', but `since otherwise it wouldn't even begin to make sense to me'.


The strange thing about expectation is that we know it is an expectation. And that is what shows expectation is immediately connected with reality. We have to be able to give a description comparing expectation with the present.


What I once called `objects' were simply that which we can speak about no matter what may be the case. `I expect three knocks on the door.' What if I replied, `How do you know three knocks exist?'


Is a man who cannot see any red around him at present in the same position as someone incapable of seeing red? If one of them imagines red, that is not a red he sees.


The memory and the reality must be in one space. Also: the image and the reality are in one space.



If I can only see something black and say it isn't red, how do I know that I am not talking nonsense - i.e. that it could be red, that there is red - if red weren't just another graduation mark on the same scale as black?


If there is a valid comparison with a ruler, the word `blue' must give the direction in which I go from black to blue. But how do these different directions find expression in grammar?


A man with red/green colour blindness has a different colour system from a normal man. Is the question then `Can someone who doesn't know what red and green are like really see what we call ``blue'' and ``yellow''?'


Grey must already be conceived as being in lighter/darker space. The yardstick must already be applied: I cannot choose between inner hearing and inner deafness.


For any question there is always a corresponding method of finding. You cannot compare a picture with reality unless you can set it against it as a yardstick.


How is a `formally certified proposition' possible? The application of a yardstick doesn't presuppose any particular length for the object to be measured. That is why I can learn to measure in general.


But are the words in the same space as the object whose length is described? The unit length is part of the symbolism, and it is what contains the specifically spatial element.


A language using a co-ordinate system. The written sign without the co-ordinate system is senseless.



It doesn't strike us at all when we look round us, move about in space, feel our own bodies, etc., etc., because there is nothing that contrasts with the form of our world. The self-evidence of the world expresses itself in the very fact that language can and does only refer to it.


The stream of life, or the stream of the world, flows on and our propositions are so to speak verified only at instants. Then they are commensurable with the present.


Perhaps the difficulty derives from taking the time concept from time in physics and applying it to the course of immediate experience. We don't speak of present, past and future images.


`I do not see the past, only a picture of the past.' But how do I know it's a picture of the past?


On the film strip there is a present picture and past and future pictures: but on the screen there is only the present.


We cannot say `time flows' if by time we mean the possibility of change. - It also appears to us as though memory were a faint picture of what we originally had before us in full clarity. And in the language of physical objects that is so.


But it can also be put differently; and that is important. The phrase `optical illusion', for example, gives the idea of a mistake, even when there is none. One could imagine an absolutely impartial language.


Language can only say those things that we can also imagine otherwise. That everything flows must be expressed in the application of language. And if someone says only the present experience has reality, then the word `present' must be redundant here.


Certain important propositions describing an experience which might have been otherwise: such as the proposition that my visual field is almost incessantly in a state of flux.


If I make a proposition such as `Julius Caesar crossed the Alps', do I merely describe my present mental state? - The proposition states what I believe. If I wish to know what that is, the best thing to do is to ask why I believe it.



One misleading representational technique in our language is the use of the word ``I'', particularly when it is used in representing immediate experience. How would it be if such experience were represented without using the personal pronoun?


Like this, say: If I, L. W., have toothache, that is expressed as `There is toothache'. In other cases: `A is behaving as L. W. does when there is toothache'. Language can have anyone as its centre. That it has me as its centre lies in the application. This privileged status cannot be expressed. Whether I say that what is represented is not one thing among others; or that I cannot express the advantage of my language - both approaches lead to the same result.


It isn't possible to believe something for which you cannot find some kind of verification. In a case where I believe someone is sad I can do this. But I cannot believe that I am sad.


Does it make sense to say two people have the same body?


What distinguishes his toothache from mine?


`When I say he has toothache, I mean he now has what I once had.' But is this a relation toothache once had to me and now has to him?


I could speak of toothache (datum of feeling) in someone else's tooth in the sense that it would be possible to feel pain in a tooth in someone else's mouth.


If I say `A has toothache', I use the image of feeling pain in the same way as, say, the concept of flowing when I talk of an electric current flowing. - The hypotheses that (1) other people have toothache and that (2) they behave just as I do but don't have toothache - possibly have identical senses.


Our language employs the phrases `my pain' and `his pain' and also `I have (or feel) a pain', but `I feel my pain' or `I feel his pain' is nonsense.


What would it be like if I had two bodies, i.e. my body were composed of two separate organisms? - Philosophers who believe you can, in a manner of speaking, extend experience by thinking, ought to remember you can transmit speech over the telephone, but not measles.



Suppose I had such a good memory that I could remember all my sense impressions. I could then describe them, e.g. by representing the visual images plastically, only finishing them so far as I had actually seen them and moving them with a mechanism.


If I describe a language, I am describing something that belongs to physics. But how can a physical language describe the phenomenal?


A phenomenon (specious present) contains time, but isn't in time. Whereas language unwinds in time.


We need a way of speaking with which we can represent the phenomena of visual space, isolated as such.


Visual space is called subjective only in the language of physical space. The essential thing is that the representation of visual space is the representation of an object and contains no suggestion of a subject.


How can I tell that I see the world through the pupil of my eyeball? Surely not in an essentially different way from that of my seeing it through the window.


In visual space there isn't an eye belonging to me and eyes belonging to others. Only the space itself is asymmetrical.


The exceptional position of my body in visual space derives from other feelings, and not from something purely visual.


Is the time of isolated `visual' phenomena the time of our ordinary idioms of physics? I imagine the changes in my visual space are discontinuous and in time with the beats of a metronome. I can then describe them and compare the description with what actually happens. A delusion of memory? No, a delusion that, ex hypothesi, cannot be unmasked isn't a delusion. And here the time of my memory is precisely the time I'm describing.



Incompatible for red and green to be in one place at the same time. What would a mixed colour of red and green be? And different degrees of red are also incompatible with one another. - And yet I can say: `There's an even redder blue than the redder of these two'. That is, from the given I can construct what is not given. - Is a construction possible within the elementary proposition which doesn't work by means of truth functions and also has an effect on one proposition's following logically from another? In that case, two elementary propositions can contradict one another.


This is connected with the idea of a complete description.


That r and g completely occupy the f - that doesn't show itself in our signs. But it must show itself if we look, not at the sign, but at the symbol. For since this includes the form of the objects, then the impossibility of `f(r) · f(g)' must show itself there in this form.


That would imply I can write down two particular propositions, but not their logical product? We can say that the `·' has a different meaning here.


A mixed or intermediate colour of blue and red is such in virtue of an internal relation to the structures of red and blue. But this internal relation is elementary. That is, it doesn't consist in the proposition `a is blue-red' representing a logical product of `a is blue' and `a is red'.


As with colours, so with sounds or electrical charges. It's always a question of the complete description of a certain state at one point or at the same time. But how can I express the fact that e.g. a colour is definitively described? How can I bring it about that a second proposition of the same form contradicts the first? - Two elementary propositions can't contradict one another.


There are rules for the truth functions which also deal with the elementary part of the proposition. In which case propositions become even more like yardsticks. The fact that one measurement is right automatically excludes all others. It isn't a proposition that I put against reality as a yardstick, it's a system of propositions. Equally in the case of negative description: I can't be given the zero point without the yardstick.


The concept of the independent co-ordinates of description. The propositions joined e.g. by `and' are not independent of one another, they form one picture and can be tested for their compatibility or incompatibility.


In that case every assertion would consist in setting a number of scales (yardsticks), and it's impossible to set one scale simultaneously at two graduation marks.


That all propositions contain time appears to be accidental when compared with the fact that the truth functions can be applied to any proposition.


Syntax prohibits a construction such as `a is green and a is red', but for `a is green' the proposition `a is red' is not, so to speak, another proposition, but another form of the same proposition. In this way syntax draws together the propositions that make one determination.



The general proposition `I see a circle on a red background' - a proposition which leaves possibilities open. What would this generality have to do with a totality of objects? Generality in this sense, therefore, enters into the theory of elementary propositions.


If I describe only a part of my visual field, my description must necessarily include the whole visual space. The form (the logical form) of the patch in fact presupposes the whole space.


Can I leave some determination in a proposition open, without at the same time specifying precisely what possibilities are left open? `A red circle is situated in the square.'' How do I know such a proposition? Can I ever know it as an endless disjunction?


Generality and negation. `There is a red circle that is not in the square.' I cannot express the proposition `This circle is not in the square' by placing the `not' at the front of the proposition. That is connected with the fact that it's nonsense to give a circle a name.


`All circles are in the square' can mean either `A certain number of circles are in the square' or: `There is no circle outside it'. But the last proposition is again the negation of a generalization and not the generalization of a negation.


The part of speech is only determined by all the grammatical rules which hold for a word, and seen from this point of view our language contains countless different parts of speech.


The subject-predicate form does not in itself amount to a logical form. The forms of the propositions: `The plate is round', `The man is tall', `The patch is red', have nothing in common. - Concept and object: but that is subject and predicate.


Once you have started doing arithmetic, you don't bother about functions and objects. - The description of an object may not express what would be essential for the existence of the object.


If I give names to three visual circles of equal size - I always name (directly or indirectly) a location. What characterizes propositions of the form `This is...' is only the fact that the reality outside the so-called system of signs somehow enters into the symbol.


What remains in this case, if form and colour alter? For position is part of the form. It is clear that the phrase `bearer of a property' conveys a completely wrong - an impossible - picture.


Roughly speaking, the equation of a circle is the sign for the concept `circle'. So it is as if what corresponds with the objects falling under the concept were here the co-ordinates of the centres. In fact, the number pair that gives the co-ordinates of the centre is not just anything, but characterizes just what in the symbol constitutes the `difference' of the circles.


The specification of the `here' must not prejudge what is here. F(x) must be an external description of x. - but if I now say `Here is a circle' and on another occasion `Here is a sphere', are the two `here's' of the same kind?



Number and concept. Does it make sense to ascribe a number to objects that haven't been brought under a concept? But I can e.g. form the concept, `link between a and b'.


Numbers are pictures of the extensions of concepts. We could regard the extension of a concept as an object whose name has sense only in the context of a proposition. In the symbolism there is an actual correlation, whereas at the level of meaning only the possibility of correlation is at issue.


I can surely always distinguish 3 and 4 in the sign I + I + I + I + I + I + I.


Numbers can only be defined from propositional forms, independently of the question which propositions are true or false. The possibility of grouping these 4 apples into 2 and 2 refers to the sense, not the truth of a proposition.


Can a proposition (A) in PM notation give the sense of 5 + 7 = 12? But how have I obtained the numerical sign in the right-hand bracket if I don't know that it is the result of adding the two left-hand signs?


What tells us that the 5 strokes and the 7 combine precisely to make 12 is always only insight into the internal relations of the structures - not some logical consideration.


An extension is a characteristic of the sense of a proposition.


What A contains apart from the arithmetical schema can only be what is necessary in order to apply it. But nothing at all is necessary.


No investigation of concepts can tell us that 3 + 2 = 5; equally it is not an examination of concepts which tells us that A is a tautology. Numbers must be of a kind with what we use to represent them.


Arithmetic is the grammar of numbers.


Every mathematical calculation is an application of itself and only as such does it have a sense. That is why it isn't necessary to speak about the general form of logical operation here. - Arithmetic is a more general kind of geometry.


It's as if we're surprised that the numerals cut adrift from their definitions function so unerringly; which is connected with the internal consistency of geometry. The general form of the application of arithmetic seems to be represented by the fact that nothing is said about it.


Arithmetical constructions are autonomous, like geometrical ones, and hence they guarantee their own applicability.


If 3 strokes on paper are the sign for the number 3, then you can say the number 3 is to be applied in the way in which 3 strokes can be applied. (Cf. §107)


A statement of number about the extension of a concept is a proposition, but a statement of number about the range of a variable is not, since it can be derived from the variable itself.


Do I know there are 6 permutations of 3 elements in the same way in which I know there are 6 people in this room? No. Therefore the first proposition is of a different kind from the second.



A statement of number doesn't always contain something general or indefinite. For instance, `I see 3 equal circles equidistant from one another'. Something indefinite would be, say: I know that three things have the property φ, but I don't know which. Here it would be nonsense to say I don't know which circles they are.


There is no such concept as `pure colour'. Similarly with permutations. If we say that A B admits of two permutations, it sounds as though we had made a general assertion. But `Two permutations are possible' cannot say any less-i.e. something more general-than the schema A B, B A. They are not the extension of a concept: they are the concept.


There is a mathematical question: `How many permutations of 4 elements are there?' which is the same kind as `What is 25 x 18?' For in both cases there is a general method of solution.


In Russell's theory only an actual correlation can show the ``similarity'' of two classes. Not the possibility of correlation, for this consists precisely in the numerical equality.


What sort of an impossibility is the impossibility of a 1-1 correlation between 3 circles and 2 crosses? - It is nonsense to say of an extension that it has such and such a number, since the number is an internal property of the extension.


Ramsey explains the sign `=' like this: x = x is taut.; x = y is cont. What then is the relation of `Def/=' to `='? - You may compare mathematical equations only with significant propositions, not with tautologies.


An equation is a rule of syntax. You may construe sign-rules as propositions, but you don't have to construe them so. The `heterological' paradox.


The generality of a mathematical assertion is different from the generality of the proposition proved. A mathematical proposition is an allusion to a proof. A generalization only makes sense if it - i.e. all values of its variables - is completely determined.



I grasp an infinite stretch in a different way from an endless one. A proposition about it can't be verified by a putative endless striding, but only in one stride.


It isn't just impossible `for us men' to run through the natural numbers one by one; it's impossible, it means nothing. The totality is only given as a concept.


That, in the case of the logical concept (1, ξ, ξ+1), the existence of its objects is already given with the concept, of itself shows that it determines them. What is fundamental is simply the repetition of an operation. The operation + 1 three times yields and is the number 3.


It looks now as if the quantifiers make no sense for numbers.


If no finite product makes a proposition true, that means no product makes it true. And so it isn't a logical product.


Can I know that a number satisfies the equation without a finite section of the infinite series being marked out as one within which it occurs?


A proposition about all propositions, or all functions, is impossible. Generality in arithmetic is indicated by an induction.


The defect (circle) in Dedekind's explanation of the concept of infinity lies in its application of the concept `all' in the formal implication. What really corresponds to what we mean isn't a proposition at all, it's the inference from φx to ψx, if this inference is permitted - but the inference isn't expressed by a proposition.


Generality in Euclidean geometry. Strange that what holds for one triangle should therefore hold for every other. But once more the construction of a proof is not an experiment; no, a description of the construction must suffice. - What is demonstrated can't be expressed by a proposition.


`The world will eventually come to an end' means nothing at all, for it's compatible with this statement that the world should still exist on any day you care to mention. `How many 9s immediately succeed one another after 3.1415 in the development of π?' If this question is meant to refer to the extension, then it doesn't have the sense of the question which interests us. (`I grasp an infinite stretch in a different way from an endless one.')


The difficulty in applying the simple basic principles shakes our confidence in the principles themselves.


`I saw the ruler move from t1 to t2, therefore I must have seen it at t.' If in such a case I appear to infer a particular case from a general proposition, then the general proposition is never derived from experience, and the proposition isn't a real proposition.


`We only know the infinite by description.' Well then, there's just the description and nothing else.


Does a notation for the infinite presuppose infinite space or infinite time? Then the possibility of such a hypothesis must surely be prefigured somewhere. The problem of the smallest visible distinction.


If I cannot visibly bisect the strip any further, I can't even try to, and so can't see the failure of such an attempt. Continuity in our visual field consists in our not seeing discontinuity.


Experience as experience of the facts gives me the finite; the objects contain the infinite. Of course, not as something rivalling finite experience, but in intension. (Infinite possibility is not a quantity.) Space has no extension, only spatial objects are extended, but infinity is a property of space.


Infinite divisibility: we can conceive of any finite number of parts but not of an infinite number; but that is precisely what constitutes infinite divisibility. - That a patch in visual space can be divided into three parts means that a proposition describing a patch divided in this way makes sense. Whereas infinite divisibility doesn't mean there's a proposition describing a line divided into infinitely many parts. Therefore this possibility is not brought out by any reality of the signs, but by a possibility of a different kind in the signs themselves.


Time contains the possibility of all the future now. The space of human movement is infinite in the same way as time.


The rules for a number system - say, the decimal system - contain everything that is infinite about the numbers. - It all hangs on the syntax of reality and possibility. m = 2n contains the possibility of correlating any number with another, but doesn't correlate all numbers with others.


The propositions `Three things can lie in this direction' and `Infinitely many things can lie in this direction' are only apparently formed in the same way, but are in fact different in structure: the `infinitely many' of the second structure doesn't play the same role as the `three' of the first.


Empty infinite time is only the possibility of facts which alone are the realities. - If there is an infinite reality, then there is also contingency in the infinite. And so, for instance, also an infinite decimal that isn't given by a law. - Infinity lies in the nature of time, it isn't the extension it happens to have.


The infinite number series is only the infinite possibility of finite series of numbers. The signs themselves only contain the possibility and not the reality of their repetition. Mathematics can't even try to speak about their possibility. If it tries to express their possibility, i.e. when it confuses this with their reality, we ought to cut it down to size.


An infinite decimal not given by a rule. `The number that is the result when a man endlessly throws a die', appears to be nonsense. - An infinite row of trees. If there is a law governing the way the trees' heights vary, then the series is defined and can be imagined by means of this law. If I now assume there could be a random series, then that is a series about which, by its very nature, nothing can be known apart from the fact that I can't know it.


The multiplicative axiom. In the case of a finite class of classes we can in fact make a selection. But in the case of infinitely many sub-classes I can only know the law for making a selection. Here the infinity is only in the rule.


What makes us think that perhaps there are infinitely many things is only our confusing the things of physics with the elements of knowledge. `The patch lies somewhere between b and c': the infinite possibility of positions isn't expressed in the analysis of this. - The illusion of an infinite hypothesis in which the parcels of matter are confused with the simple objects. What we can imagine multiplied to infinity are the combinations of the things in accordance with their infinite possibilities, never the things themselves.



While we've as yet no idea how a certain proposition is to be proved, we still ask `Can it be proved or not?' You cannot have a logical plan of search for a sense you don't know. Every proposition teaches us through its sense how we are to convince ourselves whether it is true or false.


A proof of relevance would be a proof which, while yet not proving the proposition, showed the form of a method for testing the proposition.


I can assert the general (algebraic) proposition just as much or as little as the equation 3 x 3 = 9 or 3 x 3 = 11. The general method of solution is in itself a clarification of the nature of the equation. Even in a particular case I see only the rule. `The equation yields a' means: if I transform the equation in accordance with certain rules I get a. But these rules must be given to me before the word `yields' has a meaning and before the question has a sense.


We may only put a question in mathematics where the answer runs: `I must work it out'. The question `How many solutions are there to this equation?' is the holding in readiness of the general method for solving it. And that, in general, is what a question is in mathematics: the holding in readiness of a general method.


I can't ask whether an angle can be trisected until I can see the system `Ruler and Compasses' embedded in a larger one, where this question has a sense. - The system of rules determining a calculus thereby determines the `meaning' of its signs too. If I change the rules, then I change the form, the meaning. - In mathematics, we cannot talk of systems in general, but only within systems.


A mathematical proof is an analysis of a mathematical proposition. It isn't enough to say that p is provable, we have to say: provable according to a particular system. Understanding p means understanding its system.


I can ask `What is the solution of this equation?', but not `Has it a solution?' - It's impossible for us to discover rules of a new type that hold for a form with which we are familiar. - The proposition: `It's possible - though not necessary - that p should hold for all numbers' is nonsense. For `necessary' and `all' belong together in mathematics.


Finding a new system (Sheffer's discovery, for instance). You can't say: I already had all these results, now all I've done is find a better way that leads to all of them. The new way amounts to a new system.


Unravelling knots in mathematics. We may only speak of a genuine attempt at a solution to the extent that the structure of the knot is clearly seen.


You can't write mathematics, you can only do it. - Suppose I hit upon the right way of constructing a regular pentagon by accident. If I don't understand this construction, as far as I'm concerned it doesn't even begin to be the construction of a pentagon. The way I have arrived at it vanishes in what I understand.


Where a connection is now known to exist which was previously unknown, there wasn't a gap before, something incomplete which has now been filled in. - Induction: if I know the law of a spiral, that's in many respects analogous with the case in which I know all the whorls. Yet not completely analogous - and that's all we can say.


But doesn't it still count as a question, whether there is a finite number of primes or not? Once I can write down the general form of primes, e.g. `dividing ... by smaller numbers leaves a remainder' - there is no longer a question of `how many' primes there are. But since it was possible for us to have the phrase `prime number' before we had the strict expression, it was also possible for people to have wrongly formed the question. Only in our verbal language are there in mathematics `as yet unsolved problems'.


A consistency proof can't be essential for the application of the axioms. For these are propositions of syntax.


A polar expedition and a mathematical one. How can there be conjectures in mathematics? Can I make a hypothesis about the distribution of primes? What kind of verification do I then count as valid? I can't conjecture the proof. And if I've got the proof it doesn't prove what was conjectured.


Sheffer's discovery. The systems are certainly not in one space, so that I could say: there are systems with 3 and 2 logical constants, and now I'm trying to reduce the number of constants in the same way. - A mathematical proposition is only the immediately visible surface of a whole body of proof and this surface is the boundary facing us.



A proof for the associative law? As a basic rule of the system it cannot be proved. The usual mistake lies in confusing the extension of its application with what the proof genuinely contains. - Can one prove that by addition of forms ((1 + 1) + 1) etc. numbers of this form would always result? The proof lies in the rule, i.e. in the definition and in nothing else.


A recursive proof is only a general guide to arbitrary special proofs: the general form of continuing along this series. Its generality is not the one we desire but consists in the fact that we can repeat the proof. What we gather from the proof we cannot represent in a proposition at all.


The correct expression for the associative law is not a proposition, but precisely its `proof', which admittedly doesn't state the law. I know the specific equation is correct just as well as if I had given a complete derivation of it. That means it really is proved. The one whorl, in conjunction with the numerical forms of the given equation, is enough.


One says an induction is a sign that such and such holds for all numbers. But an induction isn't a sign for anything but itself. - Compare the generality of genuine propositions with generality in arithmetic. It is differently verified and so is of a different kind.


An induction doesn't prove an algebraic equation, but it justifies the setting up of algebraic equations from the standpoint of their application to arithmetic. That is, it is only through the induction that they gain their sense, not their truth. An induction is related to an algebraic proposition not as proof is to what is proved, but as what is designated to a sign.


If we ask `Does a + (b + c) = (a + c) + c?', what could we be after? - An algebraic proposition doesn't express a generality; this is shown, rather, in the formal relation to the substitution, which proves to be a term of the inductive series.


One can prove any arithmetical equation of the form a x b = c or prove its opposite. A proof of this provability would be the exhibition of an induction from which it could be seen what sort of propositions the ladder leads to.



The theory of aggregates says that you can't grasp the actual infinite by means of arithmetical symbolism at all, it can therefore only be described and not represented. So one could talk about a logical structure without reproducing it in the proposition itself. A method of wrapping a concept up in such a way that its form disappears.


Any proof of the continuity of a function must relate to a number system. The numerical scale, which comes to light when calculating a function, should not be allowed to disappear in the general treatment. - Can the continuum be described? A form cannot be described: it can only be presented.


`The highest point of a curve' doesn't mean `the highest point among all the points of the curve'. In the same way, the maximum of a function isn't the largest value among all its values. No, the highest point is something I construct, i.e. derive from a law.


The expression `(n)...' has a sense if nothing more than the unlimited possibility of going on is presupposed. - Brouwer -. The explanation of the Dedekind cut as if it were clear what was meant by: either R has a last member and L a first, or, etc. In truth none of these cases can be conceived.


Set theory builds on a fictitious symbolism, therefore on nonsense. As if there were something in Logic that could be known, but not by us. If someone says (as Brouwer does) that for (x)· f1x = f2x there is, as well as yes and no, also the case of undecidability, this implies that `(x)...' is meant extensionally and that we may talk of all x happening to have a property.


If one regards the expression `the root of the equation φx = o' as a Russellian description, then a proposition about the root of the equation x + 2 = 6 must have a different sense from one saying the same about 4.


How can a purely internal generality be refuted by the occurrence of a single case (and so by something extensional)? But the particular case refutes the general proposition from within - it attacks the internal proof. - The difference between the two equations x2 = x·x and x2 = 2x isn't one consisting in the extensions of their validity.



That a point in the plane is represented by a number-pair, and in three-dimensional space by a number-triple, is enough to show that the object represented isn't the point at all but the point-network.


Geometry as the syntax of the propositions dealing with objects in space. Whatever is arranged in visual space stands in this sort of order a priori, i.e. in virtue of its logical nature, and geometry here is simply grammar. What the physicist sets into relation with one another in the geometry of physical space are instrument readings, which do not differ in their internal nature whether we live in a linear space or a spherical one.


I can approach any point of an interval indefinitely by always carrying out the bisection prescribed by tossing a coin. Can I divide the rationals into two classes in a similar way, by putting either 0 or 1 in an infinite binary expansion according to the way the coin falls (heads or tails)? No law of succession is described by the instruction to toss a coin; and infinite indefiniteness does not define a number.


Is it possible within the law to abstract from the law and see the extension presented as what is essential? - If I cut at a place where there is no rational number, then there must be approximations to this cut. But closer to what? For the time being I have nothing in the domain of number which I can approach. - All the points of a line can actually be represented by arithmetical rules. In the case of approximation by repeated bisection we approach every point via rational numbers.



What criterion is there for the irrational numbers being complete? Every irrational number runs through a series of rational approximations, and never leaves this series behind. If I have the totality of all irrational numbers except π, and now insert π, I cannot cite a point at which π is really needed; at every point it has a companion agreeing with it. This shows clearly that an irrational number isn't the extension of an infinite decimal fraction, it's a law. If π were an extension, we would never feel the lack of it - it would be impossible for us to detect a gap.


`√2: a rule with an exception. - There must first be the rules for the digits, and then - e.g. - a root is expressed in them. But this expression in a sequence of digits only has significance through being the expression for a real number. If someone subsequently alters it, he has only succeeded in distorting the expression, but not in obtaining a new number.


If `√2 is anything at all, then it is the same as √2, only another expression for it; the expression in another system. It doesn't measure until it is in a system. You would no more say of `√2 that it is a limit towards which the sums of a series are tending than you would of the instruction to throw dice.


That we can apply the law holds also for the law to throw digits like dice. And what distinguishes π' from this can only consist in our knowing that there must be a law governing the occurrences of the digit 7 in π, even if we don't yet know what the law is. π' alludes to a law which is as yet unknown.


Only a law approaches a value.


The letter π stands for a law which has its position in arithmetical space. Whereas π' doesn't use the idioms of arithmetic and so doesn't assign the law a place in this space. For substituting 3 for 7 surely adds absolutely nothing to the law and in this system isn't an arithmetical operation at all.


To determine a real number a rule must be completely intelligible in itself. That is to say, it must not be essentially undecided whether a part of it could be dispensed with. If the extensions of two laws coincide as far as we've gone, and I cannot compare the laws as such, then the numbers defined cannot be compared.


The expansion of π is simultaneously an expression of the nature of π and of the nature of the decimal system. Arithmetical operations only use the decimal system as a means to an end. They can be translated into the language of any other number system, and do not have any of them as their subject matter. - A general rule of operation gets its generality from the generality of the change it effects in the numbers. π' makes the decimal system into its subject matter, and for that reason it is no longer sufficient that we can use the rule to form the extension.


A law where p runs through the series of whole numbers except for those for which Fermat's last theorem doesn't hold. Would this law define a real number? The number F wants to use the spiral ... and choose sections of this spiral according to a principle. But this principle doesn't belong to the spiral. There is admittedly a law there, but it doesn't refer directly to the number. The number is a sort of lawless by-product of the law.



In this context we keep coming up against something that could be called an `arithmetical experiment'. Thus the primes come out from the method for looking for them, as the results of an experiment. I can certainly see a law in the rule, but not in the numbers that result.


A number must measure in and of itself. If it doesn't do that but leaves it to the rationals, we have no need of it. - The true expansion is the one which evokes from the law a comparison with a rational number.


A real number can be compared with the fiction of an infinite spiral, whereas structures like F, P or π' only with finite sections of a spiral.


To compare rational numbers with √2, I have to square them. - They then assume the form √a, where √a is now an arithmetical operation. Written out in this system, they can be compared with √2, and it is for me as if the spiral of the irrational number had shrunk to a point.


Is an arithmetical experiment still possible when a recursive definition has been set up? No, because with the recursion each stage becomes arithmetically comprehensible.


Is it possible to prove a greater than b, without being able to prove at which place the difference will come to light? 1.4 - Is that the square root of 2? No, it's the root of 1.96. That is, I can immediately write it down as an approximation to √2.


If the real number is a rational number a, a comparison of its law with a must show this. That means the law must be so formed as to `click into' the rational number when it comes to the appro-priate place. It wouldn't do, e.g., if we couldn't be sure whether √25 really breaks off at 5.


Can I call a spiral a number if it is one which, for all I know, comes to a stop at a rational point? There is a lack of a method for comparing with the rationals. Expanding indefinitely isn't a method, even when it leads to a result of the comparison.


If the question how F compares with a rational number has no sense, since all expansion still hasn't given us an answer, then this question also had no sense before we tried to settle the matter at random by means of an extension.


It isn't only necessary to be able to say whether a given rational number is the real number: we must also be able to say how close it can possibly come to it. An order of magnitude for the distance apart. Decimal expansion doesn't give me this, since I cannot know e.g. how many 9s will follow a place that has been reached in the expansion. - `e isn't this number' means nothing; we have to say `It is at least this interval away from it'.


Appendix: From F. Waismann's shorthand notes of a conversation on 30 December 1930



It appears to me that negation in arithmetic is interesting only in conjunction with a certain generality. - Indivisibility and inequality. - I don't write `~(5 x 5 = 30)', I write 5 x 5 ≠ 30, since I'm not negating anything but want to establish a relation between 5 x 5 and 30 (and hence something positive). Similarly, when I exclude divisibility, this is equivalent to establishing indivisibility.


There is something recalcitrant to the application of the law of the excluded middle in mathematics - Looking for a law for the distribution of primes. We want to replace the negative criterion for a prime number by a positive one - but this negation isn't what it is in logic, but an indefiniteness. - The negation of an equation is as like and as unlike the denial of a proposition as the affirmation of an equation is as like or unlike the affirmation of a proposition.


Where negation essentially - on logical grounds - corresponds to a disjunction or to the exclusion of one part of a logical series in favour of another - then here it must be one and the same as those logical forms and therefore only apparently a negation.


Yet what is expressed by inequalities is essentially different from what is expressed by equations. And so you can't immediately compare a law yielding places of a decimal expansion which works with inequalities, with one that works with equations. Here we have completely different methods and consequently different kinds of arithmetical structure.


Can you use the prime numbers to define an irrational number? As far as you can foresee the primes, and no further.



Can we say a patch is simpler than a larger one? - It seems as if it is impossible to see a uniformly coloured patch as composite. - The larger geometrical structure isn't composed of smaller geometrical structures. The `pure geometrical figures' are of course only logical possibilities.


Whether it makes sense to say `This part of a red patch is red' depends on whether there is absolute position. It's possible to establish the identity of a position in the visual field, since we would otherwise be unable to distinguish whether a patch always stays in the same place. In visual space there is absolute position, absolute direction, and hence absolute motion. If this were not so, there would be no sense in speaking in this context of the same or different places. This shows the structure of our visual field: for the criterion for its structure is what propositions make sense for it.


Can I say: `The top half of my visual field is red'? - There isn't a relation of `being situated' which would hold between a colour and a position.


It seems to me that the concept of distance is given immediately in the structure of visual space. Measuring in visual space. Equal in length, unequal in parts. Can I be sure that what I count is really the number I see?


But if I can't say there is a definite number of parts in a and b, how in that case am I to describe the visual image? - `Blurred' and `unclear' are relative expressions. - If we were really to see 24 and 25 parts in a and b, we couldn't then see a and b as equal. The word `equal' has a meaning even for visual space which stamps this as a contradiction.


The question is, how to explain certain contradictions that arise when we apply the methods of inference used in Euclidean space to visual space. This happens because we can only see the construction piecemeal and not as a whole: because there's no visual construction that could be composed of these individual visual pieces.


The moment we try to apply exact concepts of measurement to immediate experience, we come up against a peculiar vagueness in this experience. - The words `rough', `approximate', etc. have only a relative sense, but they are still needed and they characterize the nature of our experience. - Problem of the heap of sand. - What corresponds in Euclidean geometry to the visual circle isn't a circle, but a class of figures. - Here it seems as though an exact demarcation of the inexactitude is impossible. We border off a swamp with a wall, and the wall is not the boundary of the swamp.


The correlation between visual space and Euclidean space. If a circle is at all the sort of thing we see, then we must be able to see it and not merely something like it. If I cannot see an exact circle then in this sense neither can I see approximations to one.


We need new concepts and we continually resort to those of the language of physical objects. For instance `precision'. If it is right to say `I do not see a sharp line', then a sharp line is conceivable. If it makes sense to say `I never see an exact circle', then this implies: an exact circle is conceivable in visual space. - The word `equal' used with quite different meanings. - Description of colour patches close to the boundary of the visual field. Clear that the lack of clarity is an internal property of visual space.


What distinctions are there in visual space? The fact that you see a physical hundred-sided polygon as a circle implies nothing as to the possibility of seeing a hundred-sided polygon. Is there a sense in speaking of a visual hundred-sided polygon?


Couldn't I say, `Perhaps I see a perfect circle, but can never know it'? Only if it is established in what cases one calls one measurement more precise than another. It means nothing to say the circle is only an ideal to which reality can only approximate. But it may also be that we call an infinite possibility itself a circle. As with an irrational number. - Now, is the imprecision of measurement the same concept as the imprecision of visual images? Certainly not. - `Seems' and `appears' ambiguous: in one case it is the result of measurement, in another a further appearance.


`Sense datum' contains the idea: if we talk about `the appearance of a tree' we are either taking for a tree something which is one, or something which is not. But this connection isn't there.


Can you try to give `the right model for visual space'? You cannot translate the blurredness of phenomena into an imprecision in the drawing. That visual space isn't Euclidean is already shown by the occurrence of two different kinds of lines and points.



Simple colours - simple as psychological phenomena. I need a purely phenomenological colour theory in which mention is only made of what is actually perceptible and no hypothetical objects - waves, rods, cones and all that - occur. Can I find a metric for colours? Is there a sense in saying, e.g., that with respect to the amount of red in it, one colour is halfway between two other colours?


Orange is a mixture of red and yellow in a sense in which yellow isn't a mixture of red and green although yellow comes between red and green in the colour circle. - If I imagine mixing a blue-green with a yellow-green I see straightaway that it can't happen, that a component part would first have to be `killed'.


I must know what in general is meant by the expression `mixture of colours A and B'. If someone says to me that the colour of a patch lies between violet and red, I understand this and can imagine a redder violet than the one given. But: `The colour lies between this violet and an orange'? The way in which the mixed colour lies between the others is no different here from the way red comes between blue and yellow. - `Red and yellow make orange' doesn't speak of a quantity of components. It means nothing to say this orange and this violet contain the same amount of red. - False comparison between the colour series and a system of two weights on a balance.


The position here is just as it is with the geometry of visual space as compared with Euclidean geometry. There are here quantities of a different sort from that represented by our rational numbers. - If the expression `lie between' on one occasion designates a mixture of two simple colours, and on another a simple component common to two mixed colours, the multiplicity of its application is different in the two cases. - You can also arrange all the shades along a straight line. But then you have to introduce rules to exclude certain transitions, and in the end the representation on the lines has to be given the same kind of topological structure as the octahedron has. Completely analogous to the relation of ordinary language to a `logically purified' mode of expression.


We can't say red has an orange tinge in the same sense as orange has a reddish tinge. `x is composed of y and z' and `x is the common component of y and z' are not interchangeable here.


When we see dots of one colour intermingled with dots of another we seem to have a different sort of colour transition from that on the colour-circle. Not that we establish experimentally that certain colours arise in this way from others. For whether or not such a transition is possible (or conceivable) is an internal property of the colours.


The danger of seeing things as simpler than they really are. - Understanding a Gregorian mode means hearing something new; analogous with suddenly seeing 10 strokes, which I had hitherto only been able to see as twice five strokes, as a characteristic whole.



A proposition, an hypothesis, is coupled with reality - with varying degrees of freedom. All that matters is that the signs in the end still refer to immediate experience and not to an intermediary (a thing in itself). A proposition construed in such a way that it can be uncheckably true or false is completely detached from reality and no longer functions as a proposition.


An hypothesis is a symbol for which certain rules of representation hold. The choice of representation is a process based on so-called induction (not mathematical induction).


We only give up an hypothesis for an even higher gain. The question, how simple a representation is yielded by assuming a particular hypothesis, is connected with the question of probability.


What is essential to an hypothesis is that it arouses an expectation, i.e., its confirmation is never completed. It has a different formal relation to reality from that of verification. - Belief in the uniformity of events. An hypothesis is a law for forming propositions.


The probability of an hypothesis has its measure in how much evidence is needed to make it profitable to throw it out. If I say: I assume the sun will rise again tomorrow, because the opposite is so unlikely, I here mean by `likely' and `unlikely' something completely different from `It's equally likely that I'll throw heads or tails'. - The expectation must make sense now; i.e. I must be able to compare it with how things stand at present.


Describing phenomena by means of the hypothesis of a world of material things compared with a phenomenological description. - Thus the theory of Relativity doesn't represent the logical multiplicity of the phenomena themselves, but that of the regularities observed. This multiplicity corresponds not to one verification, but to a law by verifications.


Hypothesis and postulate. No conceivable experience can refute a postulate, even though it may be extremely inconvenient to hang on to it. Corresponding to the greater or slighter convenience, there is a greater or slighter probability of the postulate. It is senseless to talk of a measure for this probability.


If I say `That will probably occur', this proposition is neither verified by the occurrence nor falsified by its non-occurrence. If we argue about whether it is probable or not, we shall always adduce arguments from the past only. - It's always as if the same state of affairs could be corroborated by experience, whose existence was evident a priori. But that's nonsense. If the experience agrees with the computation, that means my computation is justified by the experience - not its a priori element, but its bases, which are a posteriori: certain natural laws. In the case of throwing a die the natural law takes the form that it is equally likely for any of the six sides to be the side uppermost. It's this law that we test.


Certain possible events must contradict the law if it is to be one at all; and should these occur, they must be explained by a different law. - The prediction that there will be an equal distribution contains an assumption about those natural laws that I don't know precisely.


A man throwing dice every day for a week throws nothing but ones - and not because of any defect in the die. Has he grounds for thinking that there's a natural law at work here which makes him throw nothing but ones? - When an insurance company is guided by probability, it isn't guided by the probability calculus but by a frequency actually observed.


`Straight line with deviations' is only one form of description. If I state `That's the rule', that only has a sense as long as I have determined the maximum number of exceptions I'll allow before knocking down the rule.


It only makes sense to say of the stretch you actually see that it gives the general impression of a straight line, and not of an hypothetical one you assume. An experiment with dice can only give grounds for expecting things to go in the same way.


Any `reasonable' expectation is an expectation that a rule we have observed up to now will continue to hold. But the rule must have been observed and can't, for its part too, be merely expected. - Probability is concerned with the form and a standard of expectation.


A ray of light strikes two different surfaces. The centre of each stretch seems to divide it into equally probable possibilities. This yields apparently incompatible probabilities. But the assumption of the probability of a certain event is verified by a frequency experiment; and, if confirmed, shows itself to be an hypothesis belonging to physics. The geometrical construction merely shows that the equal lengths of the sections was no ground for assuming equal likelihood. I can arbitrarily lay down a law, e.g. that if the lengths of the parts are equal, they are equally likely; but any other law is just as permissible. Similarly with further examples. It is from experience that we determine these possibilities as equally likely. But logic gives this stipulation no precedence.


First Appendix

Complex and Fact


The Concept of Infinity in Mathematics


Second Appendix (from F. Waismann's shorthand notes on Wittgenstein's talks and conversation between December 1929 and September 1931)

Yardstick and Propositional System



Editor's Note347(5)
Translators' Note352(3)
Corrigenda for the German text355

Om forfatteren

Ludwig Wittgenstein - 1889 - 1951 - was an Austrian-British philosopher who taught at the University of Cambridge and is known as one of the most important philosophers of the 20th century. He worked in the areas of logic and the philosophy of mathematics, the mind and language.
The majority of his writing was published after his death Rush Rhees was a pupil of Ludwig Wittgenstein and is one of his literary executors.