Topic: number and arithmetic as part of language

topics > philosophy > Group: philosophy of mathematics

analytic truth
mathematical proof as a social process
number as a named set of numbers
private language argument for skepticism about meaning
problems with analytic truth
what is a number


Wittgenstein argues that the meaning of number and arithmetic arises entirely from its use in language. It is absurd to discuss the truth of "12*12=144" because it is true by the implicit agreement of using a common language. Just imagine the difficulties if we all computed different values!

How we see number appears to be arbitrary but much of this flexibility occurs in assigning concepts. Many mathematicians believe in the ultimate reality of mathematical objects. As Erdos says, the best theorems are written in "God's book". (cbb 3/94)

Subtopic: math/logic started with language -- Wittgenstein up

Quote: there is no such thing as a logical machinery behind our symbols; where logical machinery is like a clock plus necessity [»wittL_1939]
Quote: the calculi of mathematics was invented to suit experience and then made independent of experience [»wittL_1939]
Quote: mathematics and logic are part of the apparatus of language, not part of its application; allows us to use "900" in our daily life [»wittL_1939]
Quote: a child has got to the bottom of arithmetic by learning to use it; that's all there is to it [»wittL_1939]

Subtopic: math rules part of language up

Quote: words such as "two" and "and", and rules such as "2+2=4" have many uses but not much of a meaning [»wittL_1939]
Quote: can always affix "by definition" to a mathematical proposition; this refers to a picture, e.g., "12*12=144", in a standards archive [»wittL_1939]
Quote: 1+1=2 is not know by calculation or intuition, but because we all were taught it and know it [»wittL_1939]
Quote: it is absurd to say that "12*12=144 may be wrong" because agreement is the justification for this technique; calculation is based on agreement [»wittL_1939]
Quote: we fix our technique that 13 follows 12; the only discovery is that this is a valuable thing to do [»wittL_1939]
Quote: that we mean addition by '+' is part of a 'language game' that sustains itself by the brute fact that we generally agree [»kripSA_1982]

Subtopic: no distinction between right and wrong rules up

Quote: there is no objective fact about '+' that explains our agreement in particular cases
Quote: there is no fact about my internal mental history or external behavior that distinguishes quus from plus; no foundation for language [»kripSA_1982]
Quote: consider quus which is 5 if either summand > 57; it could be '+' if we hadn't seen any summands > 57 [»kripSA_1982]
Quote: if define 'plus' by a machine or program, it could be 'quus' due to instruction definition, finiteness, or machine malfunction [»kripSA_1982]
Quote: suppose someone says "I know what series '1 4 9 16'" is but then writes 20,000 for the 100'th element; may still be following a rule [»wittL_1939]
Quote: how does a series define a rule? can always interpret the rule otherwise, e.g., adding two [»pitkHF_1972]

Subtopic: rules by agreement up

Quote: a person is an adder if the community agrees about his additions and procedures; those who deviate are corrected [»kripSA_1982]
Quote: someone who is an incorrigible deviant in enough respects simply cannot participate in the life of the community and in communication
Quote: a customer when dealing with the grocer expects the grocer to count and add as he does [»kripSA_1982]
Quote: our entire lives depend on the 'game' of attributing to others the mastery of certain rules; e.g., the rule of addition [»kripSA_1982]
Quote: each person who claims to be following a rule can be checked by others in the community; a primitive part of the language game [»kripSA_1982]
Quote: the relation of meaning and intention to future action is normative, not descriptive; e.g., 68+57 should be 125 [»kripSA_1982]

Subtopic: number as language up

Quote: saying that number is a one-one correlation between classes is just substituting another expression for number [»wittL_1939]
Quote: are all classes of the same size, the same 'number'? if so, when do names refer to the same thing (e.g., overlapping shadows) [»wittL_1939]
Quote: two strokes is as good a definition of "two" as any others [»wittL_1939]
Quote: a natural language for very large numbers [»cbb_1990, OK]

Subtopic: number as abstract up

Quote: number is entirely the creature of the mind; 1 house has many windows, 1 city has many houses [»fregG_1884]
Quote: number applies to concepts that are abstracted from things; this explains number's wide range of applicability [»fregG_1884]
Quote: classical mathematics requires and assumes universals as values of its bound variables; e.g., numbers [»quinWV2_1947]

Subtopic: identity of self is unambiguous up

Quote: saying two pieces of chalk are the same could mean different things; but saying everything is the same as itself seems utterly unambiguous [»wittL_1939]

Related Topics up

Topic: analytic truth (51 items)
Topic: mathematical proof as a social process (14 items)
Topic: number as a named set of numbers (15 items)
Topic: private language argument for skepticism about meaning (34 items)
Topic: problems with analytic truth (20 items)
Topic: rules (43 items)
Topic: what is a number
(55 items)

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