A Proof (n.) can ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠be known thoroughly, step by step, without understanding what was proved.

A Proof (n.) is ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠like a stage play that helps me predict how things will go — not an experiment, only midwife to the prediction.

A Proof (n.) is ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠not foundations plus rules of inference but a new building, though in a recognisable style — a new paradigm.

A Proof (n.) is ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠most likely, in its verbal result, to delude us with a myth.

Mathematics (n.) reduced ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠to logic is like tables, chairs and cupboards wrapped in enough paper to look spherical in the end.

Mathematics (n.) forms ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠a network of norms.

Mathematics (n.) cannot ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠simply be a game, or dancing would be mathematics too.

Mathematics (n.) needs ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠its signs to be used in mufti, outside mathematics; that outside use is what makes the sign-game mathematics.

A Calculation (n.) throws ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠brilliant light on its verbal summary, while words cast only a dim glow over the calculation — mistrust the words.

Calculation (n.) that ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠looks like a machine at work makes the calculating human being the machine.

Calculation (n.) cannot ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠be explained by agreement, because we judge identity and agreement by the results of our calculating.

Calculation (n.) rests ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠on the agreement of ratifications — the precondition of our language-game, not something it affirms.

A Rule (n.) that ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠does not compel you is not one you are following.

A Rule (n.) has ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠a hardness easily confused with the hardness of a material.

A Rule (n.) stands ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠detached, alone in its glory, though daily experience gives it its importance — like a king whose dignity must not be explained by his usefulness.

A Rule (n.) conducts ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠you like a gangway with rigid walls — and works on you like an order.

Contradiction (n.) inspires ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠in mathematicians a superstitious dread and veneration.

A Contradiction (n.) is ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠the local symptom of a sickness of the whole calculus — and the body is sick only if we do not know our way about.

A Contradiction (n.) might ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠be taken as the gods hinting that I should act rather than consider.

Contradiction (n.) might ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠be a regular instrument of language: a moving object exists and does not exist in this place, change expressed by contradiction.

A Mathematical proposition (n.) is ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠no more anthropology than a statute book is a report on how a nation treats its thieves; the judge does not use it that way.

A Mathematical proposition (n.) seems ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠to point at a reality outside itself, but only expresses the adoption of a new measure of reality.

A Mathematical proposition (n.) that ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠is never applied is an architrave hanging in the air, supported by nothing and supporting nothing.

A Mathematical proposition (n.) known ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠is not yet knowledge of anything: if we agree on it we have only set our watches, not measured time.

Philosophy (n.) faces ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠this problem: how to tell the truth and pacify these strong prejudices in doing so.

Philosophy (n.) supplies ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠remarks on the natural history of man: not curiosities, but facts no one doubted and no one noticed because they are always before our eyes.

Philosophy (n.) does ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠well to answer a question with a question; an answer may easily be unfair, a counter-question is not.

Philosophy (n.) must ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠find the series of questions that leads through the centre and out into the open — by new examples, never the hackneyed ones.

Logic (n.) presupposes ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠human agreement — not agreement in opinions, much less in opinions about logic.

Logic (n.) is ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠only an auxiliary technique in mathematics; to forget the special techniques is like saying cabinet-making consists in gluing.

Logic (n.) is ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠pictured as a kind of ultra-physics, describing the world's logical structure as perceived by an ultra-experience.

Logic (n.) comes ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠before any agreement between what is said and reality, as a method of measuring comes before any length is right or wrong.

Arithmetic (n.) would ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠end if beans, sticks and fingers kept vanishing or multiplying when counted; 2 + 2 = 4 would become unusable.

An Arithmetic (n.) al ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠proposition (n.) is an empirical proposition hardened into a rule, no longer tested by experience but used to judge it.

Arithmetic (n.) equating ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠625 and 25 × 25 is a grammatical trick that bars one kind of description and steers description into other channels.

An Arithmetic (n.) al ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠proposition (n.) would become useless, not false, if confusion came; its sense, not its truth, rests on the regular working of measurement.

A Inference (n.) is ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠one no experience can contradict without contradicting the premises — a movement within the means of representation.

Inference (n.) uses ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠follows non-temporally — this follows from that — which shows it reports no experiment.

Inference (n.) is ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠a transformation of our expression, like switching a ruler from inches to centimetres; its rightness rests on convention, use, and need.

Inference (n.) cannot ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠be right or wrong, for they help fix the meaning of the signs.