Grice and Witters on 'infinitely many'

Speranza

Some [--people] like Witters, but Moore's MY man" (Austin, to Grice).

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From:

Rodych, Victor, "Wittgenstein's Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Summer 2011 Edition), Edward N. Zalta (ed.), URL = .

http://plato.stanford.edu/entries/wittgenstein-mathematics/

"As in his intermediate position, the later [Witters] claims that

‘ℵ₀’

and “infinite series” get their mathematical uses from the use of ‘infinity’ in ordinary language (RFM II, §60).

-- where by ordinary language (or lingo) we mean things like German, or English.

"Although, in ordinary language, we often use ‘infinite’ and “infinitely many” as answers to the question “how many?,” and though we associate infinity with the enormously large, the principal use we make of ‘infinite’ and ‘infinity’ is to speak of the unlimited (RFM V, §14) and unlimited techniques (RFM II, §45; PI §218).

PUPIL: How many stars are there?

GRICE: There are infinitely many stars.

PUPIL: Infinitely many?

GRICE: As far as I know, yes.

"This fact is brought out by the fact “that the technique of learning ℵ₀ numerals is different from the technique of learning 100,000 numerals” (LFM 31).

"When we say, e.g.,

There are an infinite number of even numbers

or

There are infinitely many stars

we MEAN, as it were, that we have a mathematical technique or rule for generating even numbers [or stars?] which is limitless, which is markedly different from a limited technique or rule for generating a finite number of numbers, such as 1–100,000,000.

“We learn an endless technique,” says Wittgenstein (RFM V, §19), “but what is in question here is not some gigantic extension.”

"An infinite sequence, for example, is not a gigantic extension because it is not an extension, and ‘ℵ₀’ is not a cardinal number, for “how is this picture connected with the calculus,” given that “its connexion is not that of the picture | | | | with 4” (i.e., given that ‘ℵ₀’ is not connected to a (finite) extension)?"

 

"This shows, says Wittgenstein (RFM II, §58), that we ought to AVOID the word ‘infinite’ in mathematics wherever it seems to give a meaning to the calculus, rather than acquiring its meaning from the calculus and its use in the calculus."

 

IMPORTANT point above. Grice's discussion is other. He is concerned with the claim by Malcolm that ordinary language is sacred and free from self-contradictoriness. His example of the 'infinitely many stars' is one among many (four or five).

"Once we see that the calculus contains nothing infinite, we should not be ‘disappointed’ (RFM II, §60), but simply note (RFM II, §59) that it is not “really necessary… to conjure up the picture of the infinite (of the enormously big).”"

"A second strong indication that the later Wittgenstein maintains his finitism is his continued and consistent treatment of ‘propositions’ of the type “There are three consecutive 7s in the decimal expansion of π” (hereafter ‘PIC’).^"

 

^"n the middle period, PIC (and its putative negation, ¬PIC, namely, “It is not the case that there are three consecutive 7s in the decimal expansion of π”) is not a meaningful mathematical “statement at all” (WVC 81–82: Footnote #1)."

 

"On Wittgenstein's intermediate view, PIC—like FLT, GC, and the Fundamental Theorem of Algebra—is not a mathematical proposition because we do not have in hand an applicable decision procedure by which we can decide it in a particular calculus. For this reason, we can only meaningfully state finitistic propositions regarding the expansion of π, such as “There exist three consecutive 7s in the first 10,000 places of the expansion of π” (WVC 71; 81–82, Footnote #1)."

 

"The later Wittgenstein maintains this position in various passages in RFM (Bernays 1959 [1986, 176]). For example, to someone who says that since “the rule of expansion determine[s] the series completely,” “it must implicitly determine all questions about the structure of the series,” Wittgenstein replies: “Here you are thinking of finite series” (RFM V, §11). If PIC were a mathematical question (or problem)—if it were finitistically restricted—it would be algorithmically decidable, which it is not [(RFM V, §21), (LFM 31–32, 111, 170), (WVC 102–03)]."

 

"As Wittgenstein says at (RFM V, §9): “The question… changes its status, when it becomes decidable,” “[f]or a connexion is made then, which formerly was not there.”  And if, moreover, one invokes the Law of the Excluded Middle to establish that PIC is a mathematical proposition—i.e., by saying that one of these “two pictures… must correspond to the fact” (RFM V, §10)—one simply begs the question (RFM V, §12), for if we have doubts about the mathematical status of PIC, we will not be swayed by a person who asserts “PIC ∨ ¬PIC” (RFM VII, §41; V, §13)."

"Wittgenstein's finitism, constructivism, and conception of mathematical decidability are interestingly connected at (RFM VII, §41, par. 2–5)."

Witters:

"What harm is done e.g. by saying that God knows all irrational numbers?  Or: that they are already there, even though we only know certain of them?  Why are these pictures not harmless?"

 

For one thing, they hide certain problems.— (MS 124, p. 139; March 16, 1944)

"Suppose that people go on and on calculating the expansion of π."

 

"So God, who knows everything, knows whether they will have reached ‘777’ by the end of the world."

 

"But can his omniscience decide whether they would have reached it after the end of the world?"

 

"It cannot. I want to say: Even God can determine something mathematical only by mathematics.  Even for him the mere rule of expansion cannot decide anything that it does not decide for us."

"We might put it like this: if the rule for the expansion has been given us, a calculation can tell us that there is a ‘2’ at the fifth place."

 

"Could God have known this, without the calculation, purely from the rule of expansion?  I want to say: No. (MS 124, pp. 175–176; March 23–24, 1944)."

 

"What Wittgenstein means here is that God's omniscience might, by calculation, find that ‘777’ occurs at the interval [n,n+2], but, on the other hand, God might go on calculating forever without ‘777’ ever turning up."

"Since π is not a completed infinite extension that can be completely surveyed by an omniscient being (i.e., it is not a fact that can be known by an omniscient mind), even God has only the rule, and so God's omniscience is no advantage in this case [(LFM 103–04); cf. (Weyl, 1921 [1998, 97])]. "

"Like us, with our modest minds, an omniscient mind (i.e., God) can only calculate the expansion of π to some n^th decimal place—where our n is minute and God's n is (relatively) enormous—and at no n^th decimal place could any mind rightly conclude that because ‘777’ has not turned up, it, therefore, will never turn up."

Or not, i.e., or it will.