Remarks on the Foundations of Mathematics, revised edition

by Wittgenstein, Ludwig; Von Wright, G. H.; Rhees, R.; Anscombe, G. E. M.

• ISBN13: 9780262730679
• ISBN10: 0262730677
• Edition: Revised
• Format: Paperback
• Copyright: 1983-05-10
• Publisher: The MIT Press

Summary

This analyzes in depth such topics logical compulsion and mathematical conviction; calculation as experiment; mathematical surprise, discovery, and invention; Russell's logic, Godel's theorem, cantor's diagonal procedure, Dedekind's cuts; the nature of proof and contradiction; and the role of mathematical propositions in the forming of concepts.

Table of Contents

Part I
Following a rule (Cf. Philosphical Investigations § 189-90 ff.).---Steps determined by a formula (1-2). Continuation of a series (3). Inexorability of mathematics; mathematics and truth (4-5). Remark on measuring (5).	35	(4)
Logical inference.---The word ``all''; inference from `(x) .fx' to `fa' (10-16). Inference and truth (17-23).	39	(7)
Proof.---Proof as pattern or model (paradigm). Example: hand and pentacle (25 ff.). Proof as picture of an experiment (36). Example: 100 marbles (36 ff.). Construction of figures out of their parts (42-72). Mathematical surprise. Proof and conviction. Mathematics and essence (32, 73, 74). The depth of the essence: the deep need for the convention (74).	46	(19)
Calculation and experiment.---The `unfolding' of mathematical properties. Example of 100 marbles (75, 86, 88). Unfolding the properties of a polygon (76), and of a chain (79, 80, 91, 92). Measuring (93, 94). Geometrical examples (96-98). Internal properties and relations (102-105); examples from the logic of colour.	65	(11)
Mathematical belief.	76	(3)
Logical compulsion.---In what sense does logical argument compel? (113-117). The inexorability of logic compared with that of the law (118). The `logical machine' and the kinematics of rigid bodies' (119-125). `The hardness of the logical must' (121). The machine as symbolizing its way of working (122). The employment of a word grasped in a flash (123-130). Possibility as a shadow of reality (125). The employment of a word misunderstood and interpreted as a queer process (127). (For 122-130 see also Philosophical Investigations § 191-197.) The laws of logic as `laws of thought' (131-133). Logic interprets `proposition' and `language' (134). Going wrong in a calculation (136-137). Remark on measuring (140). Logical impossibility (141). ``What we are supplying is really remarks on the natural history of man'' (142).	79	(13)
Foundation of a calculating procedure and of a logical inference.---Calculating without propositions (143-145). Example: sale of timber (143-152). ``Are our laws of inference eternal and unalterable''? (155). Logic precedes truth (156).	92	(4)
Mathematics, logic and experience.---Proof and experiment (157-169). What about mathematics is logic: it moves in the rules of our language (165). The mathematician is an inventor, not a discoverer (168). The grounds of the logical must (169-171).	96	(6)
Appendix I
Two sorts of negation. One cancels out when repeated, the other is still a negation. How we mean the one or the other sort in double negation. Meaning tested by the expression of meaning (3). Negation compared with rotation through 180° in geometry (1, 6, 7).	102	(3)
We can imagine a `more primitive' logic, in which only sentences containing no negation can be negated (8). The question whether negation has the same meaning for these people as for us (9). Does the figure one mean the same both times in ``This rod is 1 metre long'' and ``Here is 1 soldier'' (10)?	105	(1)
Two systems for measuring length. 1 foot = 1W, but 2W=4 foot, etc. Do ``W'' and ``foot'' mean the same?	106	(1)
Meaning as the function of a word in a sentence.	107	(1)
Use of the word ``is'' as copula and as sign of identity. The personal union by the same word a mere accident. Essential and inessential features of a notation. Comparison with the role of a piece in a game. A game does not just have rules, it has a point (20).	108	(3)
Appendix II
The surprising may play completely different roles in mathematics in two different ways.---A state of affairs may appear astonishing because of the kind of way it is represented. Or the emergence of a particular result may be seen as in itself surprising. The second kind of surprise not legitimate. Criticism of the conception that mathematical demonstration brings something hidden to light (2). ``No mystery here'' means: Look around (4). A calculation compared with a way of turning up a card (5). When we don't command a view over what we have been doing, it strikes us as mysterious. We adopt a particular form of expression and become dominated by it in acting and thinking (8).	111	(5)
Appendix III
Kinds of proposition.---Arithmetic done without propositions (4).	116	(1)
Truth and provability in the system of Principia Mathematica.	117	(1)
Discussion of a proposition `P' which asserts its own unprovability in the system of Principia Mathematica.---The role of contradiction in the language-game (11-14, 17).	118	(5)
The propositions of logic. `Proposition' and `proposition-like formation'.	123	(2)
Part II
(``Additions'').---The diagonal procedure. What use can be made of the diagonal number? (3) ``The motto here is: Take a wider look round'' (6). We should regard with suspicion the result of a calculation expressed in words (7). The concept `nondenumerable' (10-13). Comparison of the concepts `real number' and `cardinal number' from the point of view of serial ordering (16-22).	125	(7)
The sickness of a time.	132	(1)
Discussion of the proposition ``There is no greatest cardinal number.'' ``We say of a licence that it does not terminate'' (26).	133	(1)
Irrational numbers. There is no system of irrational numbers (33). But Cantor defines a higher order difference, namely a difference between a development and a system of developments (34).	133	(2)
N0 From our having a use for a kind of numeral that gives the number of terms of an infinite series it doesn't follow that it also makes any sense to talk about the number of the concept `infinite series'. There is no grammatical technique suggesting the use of such an expression. Such a use is not still to be discovered, but takes inventing (38).	135	(2)
Discussion of the proposition ``Fractions cannot be arranged in a series in order of magnitude.''	137	(2)
How do we compare games?	139	(1)
Discussion of the proposition that fractions (number-pairs) can be ordered in an infinite series.	139	(2)
``Is the word `infinite' to be avoided in mathematics?''	141	(1)
Finitism and Behaviourism are similar trends. Both deny the existence of something with a view to getting out of a muddle.	142	(1)
``What I am doing is not showing up calculations as wrong, but subjecting the interest of calculations to test.''	142	(1)
Part III
Proof.---Mathematical proof must be perspicuous. Role of definitions (2).	143
Russell's logic and the idea of the reduction of arithmetic to symbolic logic.---The application of the calculation must take care of itself (4). Proof in the Russellian calculus, in the decimal calculus, and in the stroke calculus.	114	(35)
Proof.---Proof as a memorable picture (9). The reproduction of the pattern of a proof (10-11).	149	(2)
Russell's logic and the problem of the mutual relation of different calculating techniques.---What is the invention of the decimal system? (12.) Proof in the Russellian calculus and in the decimal system (13). Signs for numbers that can, and that cannot, be taken in (16). Relation of abbreviated and unabbreviated calculating techniques to one another (17-20).	151	(7)
Proof.---Identity and reproducibility of a proof (21). The proof as model. Proof and experiment (22-24). Proof and mathematical conviction (25-26). In the proof we have won through to a decision (27). The proved proposition as a rule. It serves to shew us what it makes sense to say (28). The propositions of mathematics as `instruments of language' (29). The mathematical must correspond to a track which I lay down in language (30). The proof introduces a new concept (31). What concept does `p⊃p' produce? (32). `p⊃p' as pivot of the linguistic method of representation (33). The proof as part of an institution (36). Importance of the distinction between determining and using a sense (37). Acceptance of a proof; the `geometrical' conception of proof (38-40). The proof as adoption of a particular employment of signs (41). ``The proof must be a procedure plain to view'' (42). ``Logic as foundation of mathematics does not work if only because the cogency of the logical proof stands and falls with its geometrical cogency'' (43). In mathematics we can get away from logical proofs (44).	158	(17)
Russell's logic.---Relation between the ordinary and the Russellian technique of proof (45). Criticism of the conception of logic as the `foundation' of mathematics. Mathematics is a motley of calculating techniques. The abbreviated technique as a new aspect of the unabbreviated (46-48). Remark on trigonometry (50). The decimal notation is independent of calculation with unit strokes (51). Why Russell's logic does not teach us to divide (52). Why mathematics is not logic (53). Recursive proof (54). Proof and experiment (55). The correspondence of different calculi; stroke notation and decimal notation (56-57). Several proofs of one and the same proposition; proof and the sense of a mathematical proposition (58-62). The exact correspondence of a convincing transition in music and in mathematics (63).	175	(17)
Calculation and experiment.---Are the propositions of mathematics anthropological propositions? (65). Mathematical propositions conceived as prophecies of concordant results of calculating (66). Agreement is part of the phenomenon of calculating (67). If a calculation is an experiment, what in that case is a mistake in calculation? (68). The calculation as an experiment and as a path (69). A proof subserves mutual understanding. An experiment presupposes it (71). Mathematics and the science of conditioned calculating reflexes (72). The concept of calculation excludes confusion (75-76).	192	(10)
Contradiction.---A game in which the one who moves first must always win (77). Calculating with (a - a). The chasms in a calculus are not there if I do not see them (78). Discussion of the heterological paradox (79). Contradiction regarded from the point of view of the language-game. Contradiction as a `hidden sickness' of the calculus (80). Contradiction and the usability of a calculus (81). The consistency proof and the misuse of the idea of mechanical insurance against contradiction (82-89). ``My aim is to change the attitude towards contradiction and the consistency proof.'' (82) The role of the proposition: ``I must have miscalculated''---the key to understanding the `foundations' of mathematics (90).	202	(21)
Part IV
On axioms.---The self-evidence of axioms (1-3). Self-evidence and use (2-3). Axiom and empirical proposition (4-5). The negation of an axiom (5). The mathematical proposition stands on four legs, not on three (7).	223	(4)
Following a rule.---Description by means of a rule (8).	227	(2)
The arithmetical assumption is not tied to experience.	229	(1)
The conception of arithmetic as the natural history of numbers.---Judging experience by means of the picture (12).	229	(2)
External relation of the logical (mathematical) proposition.	231	(1)
The possibility of doing applied mathematics without pure mathematics.---Mathematics need not be done in propositions; the centre of gravity may lie in action (15). The commutative law as an example (16-17).	232	(2)
Calculation as a mechanical activity.	234	(1)
The picture as a proof.	235	(1)
Intuition.	235	(1)
What is the difference between not calculating and calculating wrong?	236	(1)
Proof and mathematical concept formation.---Proof alters concept formation. Concept formation as the limit of the empirical (29). Proof does not compel, but guides (30). Proof conducts our experience into definite channels (31, 33). Proof and prediction (33).	237	(5)
The philosophical problem is: how can we tell the truth and at the same time pacify these strong prejudices?	242	(1)
The mathematical proposition.---We acknowledge it by turning our back on it (35). The effect of the proof: one plunges into the new rule (36).	243	(2)
Synthetic character of mathematical propositions.---The distribution of primes as an example (42).	245	(1)
The result set up as equivalent to the operation.	245	(1)
That proof must be perspicuous means that causality plays no part in proof.	246	(1)
Intuition in mathematics.	246	(2)
The mathematical proposition as determination of a concept, following upon the discovery of a new form.	248	(1)
The working of the mathematical machine is only the picture of the working of a machine.	249	(1)
The picture as a proof.	249	(1)
Reversal of a word.	250	(3)
Mathematical and empirical propositions.---The assumption of a mathematical concept expresses the confident expectation of certain experiences; but the establishment of this measure is not equivalent to the expression of the expectations (53).	253	(1)
Contradiction.---The liar (58). Contradiction conceived as something supra-propositional, as a monument with a Janus head enthroned above the propositions of logic (59).	254	(3)
Part V
Mathematics as a game and as a machine-like activity.---Does the calculating machine calculate? (2). How far is it necessary to have a concept of `proposition' in order to understand Russell's mathematical logic? (4).	257	(2)
Is a misunderstanding about the possible application of mathematics any objection to a calculation as part of mathematics?---Set theory (7).	259	(7)
The law of excluded middle in mathematics.---Where there is nothing to base a decision on, we must invent something in order to give the application of the law of excluded middle a sense.	266	(6)
`Alchemy' of the concept of infinity and of other mathematical concepts whose application is not understood.---Infinite predictions (23).	272	(3)
The law of excluded middle. The mathematical proposition as a commandment. Mathematical existence.	275	(6)
Existence proofs in mathematics.---``The harmful invasion of mathematics by logic'' (24; see also 46 and 48). The mathematically general does not stand to the mathematically particular in the same relation as does the general to the particular elsewhere (25). Existence proofs which do not admit of the construction of what exists (26-27).	281	(4)
Proof by reductio ad absurdum.	285	(1)
Extensional and intensional in mathematics; the Dedekind cut---geometrical illustration of analysis (2q). Dedekind's theorem without irrational numbers (30). How does this theorem come by its deep content? (31) The picture of the number-line (32, 37). Discussion of the concept ``cut'' (33, 34). The totality of functions is an unordered totality (38). Discussion of the mathematical concept of a function; extension and intension in analysis (39-40).	285	(10)
Concepts occurring in `necessary' propositions must also have a meaning in non-necessary ones.	295	(1)
On proof and understanding of a mathematical proposition.---The proof conceived as a movement from one concept to another (42). Understanding a mathematical proposition (45-46). The proof introduces a new concept. The proof serves to convince one of something (45). Existence proof and construction (46).	295	(4)
A concept is not essentially a predicate.	299	(1)
`Mathematical logic' has completely blinded the thinking of mathematicians and philosophers.	300	(1)
The numerical sign goes along with the sign for a concept and only together with this is it a measure.	300	(1)
On the concept of generality.	300	(1)
The proof shews how the result is yielded.	301	(1)
General remarks.---The philosopher is the man who has to cure himself of many sicknesses of the understanding before he can reach the notions of the healthy human understanding.	301	(2)
Part VI
Proofs give propositions an order.	303	(1)
Only within a technique of transformation is a proof a formal test. The addition of certain numbers is called a formal test of the numerals, but only because adding is a practised technique. The proof also hangs together with the application.	303	(2)
When the proposition in application does not seem to be right, the proof must shew one why and how it must be right.	305	(1)
The proof shews how, and thus why the rule---e.g., that 8 x 9 makes 72---can be used.	305	(1)
The queer thing is that the picture, not the reality, is supposed to prove a proposition.	306	(1)
The Euclidean proof teaches us a technique of finding a prime number between p and p! And we become convinced that this technique must always lead to a prime number > p.	307	(1)
The spectator sees the impressive procedure and judges ``I realise that it must be like that.''---This ``must'' shews what kind of instruction he has drawn from the scene.	308	(1)
This must shew that he has adopted a concept, i.e., a method; as opposed to the application of the method.	309	(1)
``Shew me how 3 and 2 make 5.'' Child and abacus.---One would ask someone ``how?'' if one wanted to get him to shew that he understands what is in question here.	310	(2)
``Shew me how ...'' in distinction from ``Shew me that...''	312	(1)
The proof of a proposition doesn't mention the whole system of calculation behind the proposition, which gives it sense.	313	(1)
The domain of the tasks of philosophy.	314	(1)
Ought we to say that mathematicians don't understand Fermat's last theorem?	314	(1)
What would it be to ``shew how there are infinitely many prime numbers''?	315	(1)
The process of copying. The process of construction according to a rule. Must one always have a clear idea whether his prediction is intended mathematically or otherwise?	316	(2)
The inexorable proposition is that by this rule this number follows after that one. The result of the operation here turns into the criterion for this operation's having been carried out. So we are able to judge in a new sense whether someone has followed the rule.	318	(2)
The learning of a rule.---``It's supposed to keep on as I have shewn him.''---What I understand by `uniform' I would explain by means of examples.	320	(1)
Here definitions are no help to me.	321	(1)
Paraphrasing the rule only makes it more intelligible to someone who can already obey the paraphrase.	321	(1)
Teaching someone to multiply: we reject different patterns of multiplication with the same initial segment.	322	(1)
It would be nonsense to say: just once in the history of the world, someone followed a rule.---Disputes do not break out about whether a proceeding has been according to a rule or not.---This belongs to the structure from which our language goes out to give a description, for example.	322	(2)
As if we had hardened an empirical proposition into a rule and now experience is compared with it and it is used to judge experience.	324	(1)
In calculating everything depends on whether one calculates right or wrong.---The arithmetical proposition is withdrawn from experiential checking.	324	(2)
``If I follow the rule, then this is the only number I can get to from that one.'' That is how I proceed; don't ask for a reason!	326	(1)
Suppose someone were to work out the multiplication tables, log tables, etc., because he didn't trust them?---Does it make any difference whether we utter a sentence of a calculus as a proposition of arithmetic or an empirical proposition?	327	(1)
``According to the rule that I see in this sequence, this is how it continues.'' Not: according to experience! Rather, that just is the sense of the rule.	327	(1)
When I see a rule in the series---can this consist in my seeing an algebraic expression before me? Must it not belong to a language?	328	(1)
The certainty with which I call the colour ``red'' is not to be called in question when I give the description. For this characterises what we call describing.---Following according to a rule is the bottom of our language-game. Because this (e.g., 252 = 625) is the proceeding upon which we build all judging.	329	(1)
Law, and the empirical proposition that we give this law.---``If I obey the order, I do this!'' does not mean: If I obey the order, I obey the order. So I must have a different way of identifying the ``this.''	330	(2)
If humans who have been educated in this fashion calculate like this anyway, then what do we need the rule for? ``252 = 625'' does not mean that human beings calculate like this, for 252 ≠ 625 would not be the proposition that humans get not this but a different result.	332	(1)
``Apply the rule to those numbers.''---If I want to follow it, have I any choice left?
How far is it possible to describe the function of a rule?	333	(1)
If you aren't master of any rule, all I can do is train you.
But how can I explain the nature of the rule to myself?
What surrounding is requisite for someone to be able to invent chess (e.g.)?	334	(1)
Is regularity possible without repetition?
Not: how often must he have calculated right to prove to others that he can calculate, but: how often, to prove it to himself?
Can we imagine someone's knowing that he can calculate although he has never done any calculating?	335	(1)
In order to describe the phenomenon of language, one has to describe a practice.	335	(1)
A country that exists for two minutes, and is a projection of part of England, with everything that is going on in two minutes. Someone is doing the same as a mathematician in England, who is calculating. In this two-minute-man calculating?
How do I know that the name of this colour is ``green''? If I asked other people and they did not agree with me, I should become totally confused and take them or myself for crazy. Here I react with this word (``green''); and that is also how I know how I have to follow the rule now.	336	(2)
How a polygon of forces gets drawn according to the given arrows.
``The word OBEN has four sounds.''---Is someone who counts the letters making an experiment? It may be one.	338	(2)
Compare: (1) The word there has 7 sounds
(2) The sound-picture ``Dædalus'' has 7 sounds.
The second proposition is tenseless. The employment of the two propositions must be different.
``By counting off the sounds one may obtain an empirical proposition---or again, a rule''.
Definitions---new ways of belonging together.	340	(1)
``How can one follow a rule?''---Here we misunderstand the facts that stare us in the face.	341	(1)
It is important that the enormous majority of us agree in certain things.	342	(1)
Languages of different tribes who all had the same vocabulary, but the meanings of the words were different.---In order to communicate the people would have to agree about meanings---they would have to agree not merely about definitions, but also in their judgments.
The temptation to say ``I can't understand language-game (2) because the explanation consists only in examples of the application.''	343	(1)
A cave-man who produces regular sequences of signs for himself. We do not say that he is acting according to a rule because we can form the general expression of such a rule.	344	(1)
Under what kind of circumstances should we say: someone is giving a rule by drawing this figure? Under what kind of circumstances: that someone is following this rule, when he draws a series of such figures?	345	(1)
Only within a particular technique of acting, speaking, thinking can anyone resolve upon something. (This ``can'' is the ``can'' of grammar.)	345	(2)
Two procedures: (1) deriving number after number in sequence according to an algebraic expression; (2) this procedure: as he looks at a certain sign, someone has a digit occur to him; when he looks again at the sign and the digit, again a digit occurs to him, and so on. Acting according to a rule presupposes some kind of uniformity.	347	(1)
Instruction in acting according to a rule. The effect of ``and so on'' will be that we almost all count and calculate the same. It only makes sense to say ``and so on'' when the other continues in the same way as I.	348	(2)
If something must come out, then it is a ground of judgment that I do not assail.	350	(1)
Is it not enough that this certainty exists? Why should I seek a source for it?	350	(1)
As we use the word ``order'' and ``obey'', gestures in less than words are engulfed in a set of many relationships. In a strange tribe, is the man whom the rest obey unconditionally the chief?---What is the difference between making a wrong inference and not inferring at all?	351	(1)
`` `That logic belongs to the natural history of mankind'---is not combinable with the logical `must'.'' The agreement of human beings which is a presupposition of the phenomenon of logic, is not an agreement in opinions, much less in opinions about questions of logic.	352	(3)
Part VII
The role of propositions that treat of measures and are not empirical propositions. Such a proposition (e.g. 12 inches = 1 foot) is embedded in a technique, and so in the conditions of this technique; but it is not a statement of those conditions.	355	(1)
The role of a rule. It can also be used to make predictions. This depends on properties of the measuring rods and of the people who use them.	355	(2)
A mathematical proposition---a transformation of the expression. The rule considered from the point of view of usefulness and from that of dignity. How are two arithmetical expressions supposed to say the same thing? They are made equivalent in arithmetic.	357	(2)
Someone who learns arithmetic simply by following my examples. If I say, ``If you do with these numbers what I did for you with the others, you will get such-and-such a result''---this seems to be both a prediction and a mathematical proposition.	359	(2)
The difference between being surprised that the figures on paper seem to behave like this and being surprised that this is what comes out.	361	(2)
Isn't the contrast between rules of representation and descriptive propositions one which falls away on every side?	363	(1)
What is common to a mathematical proposition and a mathematical proof, that they should both be called ``mathematical''?	364	(1)
If we regard proving as the transformation of a proposition by an experiemnt, and the man who does the calculation as an apparatus, then the intermediate steps are an uninteresting by-product.	364	(1)
Proof as a picture. It is not approval alone which makes this picture into a calculation, but the consensus of approvals.	365	(1)
Does the sense of the proposition change when a proof has been found? The new proof gives the proposition a place in a new system.	366	(3)
Let us say we have got some of our results because of a hidden contradiction. Does that make them illegitimate?	369	(2)
Might we not let a contradiction stand?
``A method for avoiding a contradiction mechanically.'' It is not bad mathematics that is amended here, but a new bit of mathematics is invented.	371	(1)
Must logical axioms always be convincing?	372	(1)
The people who sometimes cancel out by expressions of value 0.	373	(1)
If the calculation has lost its point for me, as soon as I know that I can get any arbitrary result from it---did it have no point as long as I did not know this?	374	(4)
One thinks that contradiction has to be senseless.
What does mathematics need a foundation for?	378	(1)
A good angel will always be necessary.
The practical value of calculating. Calculation and experiment. A calculation as part of the technique of an experiment. The activity of calculating may also be an experiment.	379	(2)
Is mathematics supposed to bring facts to light? Does it not take mathematics to determine the character of what we call a `fact'? Doesn't it teach us how to enquire after facts?	381	(2)
In mathematics there are no causal connexions, only the connexions of the pattern.
Remarks.	383	(1)
The network of joins in a wall. Why do we call this a mathematical problem?	384	(1)
Does mathematics make experiments with units?
``The proposition that says of itself that it is unprovable''---how is this to be understood?	385	(2)
The construction of a propositional sign from axioms according to rules; it appears that we have demonstrated the actual sense of the proposition to be false, and at the same time proved it.	387	(2)
The question ``How many?''	389	(1)
Measurement and unit.
What we call the mathematical conception of a proposition belongs together with the special place calculation has among our other activities.	389	(2)
What is the criterion that here I have followed the paradigm?	391	(1)
Anyone who describes learning how to `proceed according to the rule' for me, will himself be applying the expression of a rule in his description and will presuppose that I understand it.	392	(1)
Not allowing contradiction to stand characterizes the technique of our application of truth-functions---a man's understanding of ``and so on'' is shown by his saying this and acting thus in certain cases.	393	(2)
Does the ``heterological'' contradiction shew a logical property of this concept?	395	(1)
A game. And after a certain move any attempt to go on playing proves to be against the rules.	396	(1)
Logical inference is part of a language game.	397	(1)
Logical inference and non-logical inference.
The rules of logical inference cannot be either wrong or right.
They determine the meaning of the signs.
A reasonable procedure with numerals need not be what we call ``calculating''.	398	(1)
Is not a mathematics with an application that is sheer fantasy, still mathematics?	399	(1)
The formation of concepts may be essential to a great part of mathematics; and have no role in other parts.	399	(1)
A people who do not notice a contradiction, and draw conclusions from it.	400	(1)
Can it be a mathematical task to make mathematics into mathematics?
If a contradiction were actually found in arithmetic, this would shew that an arithmetic with such a contradiction can serve us very well.	401	(1)
``The class of lions is not a lion, but the class of classes is a class.''	401	(3)
``I always lie.'' What part might this sentence play in human life?	404	(1)
Logical inference. Is not a rule something arbitrary?	404	(1)
``It is impossible for human beings to recognize an object as different from itself.''
``Correct---i.e. it conforms to the rule.''	405	(1)
``Bringing the same''---how can I explain this to someone?	406	(1)
When should we speak of a proof of the existence of `777' in an expansion?	407	(1)
``Concept formation'' may mean various things.	408	(1)
The concept of a rule for forming an infinite decimal.
Is it essential to the concept of calculating, that people generally reach this result?	409	(1)
If I ask, e.g., whether a certain body moves according to the equation of a parabola---what does mathematics do in this case?	410	(1)
Questions about the way in which mathematics forms concepts.	411	(2)
Can one not make mathematical experiments after all?	413	(1)
The pupil has got hold of the rule when he reacts to it in such-and-such a way. This reaction presupposes a surrounding of particular circumstances, forms of life and of language.	413	(1)
``The line intimates to me how I am to go.''	414	(1)
In some circumstances: the line seems to intimate to him how he is to go, but it isn't a rule.	415	(1)
How are we to decide if he is always doing the same thing?	415	(1)
Whether he is doing the same, or a different thing every time---does not yet determine whether he is obeying a rule.	416	(1)
If you enumerate examples and then say ``and so on'', this last expression is not explained in the same way as the examples.	416	(1)
Suppose an inner voice, a sort of inspiration, tells me how to follow the line. What is the difference between this procedure and following a rule?	417	(1)
Doing the same thing is linked with following a rule.	418	(1)
Can I play the language-game if I don't have such and such spontaneous reactions?	419	(1)
What we do when following a rule, we see from the point of view of ``Always the same thing.''	419	(1)
Calculating prodigies who get the right result but can't say how---are we to say they don't calculate?	419	(1)
``Thinking you are following a rule.''	420	(1)
How can I explain the word ``same''---how do I get the feeling that something lies in my understanding beyond what I can say?	420	(1)
What is impalpable about intimation: there is nothing between the rule and my action.	421	(1)
Adding shapes. Possibilities in folding a piece of paper. Suppose we did not separate geometrical and physical possibility?	422	(3)
Might not people in certain circumstances calculate with figures, without a particular result's having to come out?
If the calculation shews you a causal connexion, you are not calculating.
Mathematics is normative.
The introduction of a new rule of inference as a transition to a new language game.	425	(1)
Observation that a surface is red and blue, but not that it is red. Inferences from this.	425	(1)
Can logic tell us what we must observe?
A surface with stripes of changing colours.	426	(2)
Could implications be observed?
Someone says he sees a red and yellow star, but not anything yellow.	428	(1)
``I hold to a rule.''	429	(1)
The mathematical must---the expression of an attitude towards the technique of calculating.	430	(1)
The expression of the fact that mathematics forms concepts.
The case of seeing the complex formed from A and B, but seeing neither A nor B.	431	(1)
Can I see A and B, but only observe A ∨ B?
And vice versa.
Experiences and timeless propositions.	432	(1)
In what sense can a proposition of arithmetic be said to give us a concept?	432	(1)
Not every language-game contains something that we want to call a ``concept''.	433	(1)
Proof and picture.	434	(1)
A language-game in which there are axioms, proofs and proved propositions.	435	(1)
Any proof in applied mathematics can be regarded as a proof in pure mathematics proving that this proposition can be got from these propositions by such and such operations---Any empirical proposition can serve as a rule if it is made immovable and becomes a part of the system of coordinates.	436