"As Far As I Know, There Are Infintely Many Stars" -- Grice on "∞" and "ℵ0"

Speranza

I will provide some commentary on Jones's remarks on the other post, with my thanks for them.

Jones writes:

"Among the many points of interest in your essay here Speranza, are the revelations about how far Frege by 1924 had further weakened his logicism.

Thanks. And indeed. I should provide the specific references, when I find them. I find something particularly interesting about Frege, perhaps because Grice's favourite colleague, J. L. Austin, cared to translate his Arithmetic!

Jones:

"The contrast between Frege and Carnap in this is stark. The ease with which Carnap the arch anti-metaphysician reconciles himself to liberal ontologies is the more remarkable when one sees the difficulties into which both Frege and Russell fell with infinity."

Indeed. Of course, it's best to see Grice as a Russellian of sorts (surely an anti-Strawsonian in matters logical), so I'm glad Russell fell into difficulties with infinity. What is philosophy without a big fall? On the other hand, there's Grice's Bootstraps, or how to pull oneself up by one's own bootstraps, as he called it (in the matter at hand: do not introduce "infinity" in the object-language unless you want to deal with it in the meta-language, too).

Jones goes on:

"Probably the source of Carnap's strength here (if I may call it that) is Hilbert."

Intersting that you should mention this.

As I was telling, my current interest in this (rather momentary) was due to a commentary, elsewhere, by Donal McEvoy regarding Popper (McEvoy is a Popper expert) and his use of Euclid's theorem (regarding the infinite series of primes). And we found that Hilbert indeed endorsed a total FINITISM. I will see if I find the nice wiki reference, in the infinity entry.

Actually, it is in the "FINITISM" Wikipedia entry.

BEGIN WIKIPEDIA QUOTE:

"[A different] position was endorsed by David Hilbert."

"Finite mathematical objects are concrete objects, infinite mathematical objects are ideal objects, and accepting ideal mathematical objects does not cause a problem regarding finite mathematical objects."

"More formally, Hilbert believed that it is possible to show that any theorem about finite mathematical objects that can be obtained using ideal infinite objects can be also obtained without them."

"Therefore allowing infinite mathematical objects would not cause a problem regarding finite objects."

"This lead to Hilbert's program of proving consistency of set theory using finitistic means as this would imply that adding ideal mathematical objects is conservative over the finitistic part."

"Hilbert's views are also associated with formalist philosophy of mathematics."

"Hilbert's goal of proving the consistency of set theory or even arithmetic through finitistic means turned out to be an impossible task due to Kurt Gödel's incompleteness theorems."

"However, by Harvey Friedman's grand conjecture most mathematical results should be provable using finitistic means."

[Must say I love the use of 'conjecture' here -- cfr. Fermat's former conjecture now a theorem. Although cfr. Speranza's motto: once a conjecture, always a conjecture -- which got Speranza an argument with Adriano Palma].

"Hilbert did not give a rigorous explanation of what he considered finitistic and refer to as elementary."

"However, based on his work with Paul Bernays some experts like William Tait have argued that the primitive recursive arithmetic can be considered as an upper bound on what Hilbert considered as finitistic mathematics."

"In the years following Gödel's theorems, as it became clear that there is no hope of proving consistency of mathematics, and with development of axiomatic set theories like Zermelo–Fraenkel set theory and the lack of any evidence against its consistency, most mathematicians lost interest in the topic."

"Today most classical mathematicians are considered Platonist and believe in the existence of infinite mathematical objects and a set-theoretical universe."

--- END OF WIKIPEDIA quote. Although the last paragraph relates to Jones's query at the end of his post. More below.


Jones goes on:

"In moving from the logical ontology necessary for mathematics to formal treatment of the more concrete ontologies required by empirical science, Carnap moved from the use of explicit definitions (used in the systems of Frege and Russell once the foundation is in place) which could only yield abstract entities from an abstract starting point to the more liberal implicit definitions found in Hilbert's axiom systems, in which any needed ontology can be obtained by citing the properties which characterise it."

Again, intersting reference to Russell, which was Grice's favourite (There is a mimeo by Grice on this, on "Definite descriptions" (but surely he could have generalised over any logical construction)"in Russell and in the vernacular", as Grice puts it!

Jones:

"This is Carnap's move from universalist to pluralist."

"The ("external") question of whether these entities exist is for Carnap meaningless, subject to the pragmatic constraint that the chosen axioms are logically coherent. We have discussed briefly before how this connects with Grice, in that Carnap and Grice share a liberal attitude to ontology which confounds an expectation of conflict."

Indeed.

"On the matter of infinity, in which we are told the interest of Locke and Grice alike (but not likely Carnap) is in ordinary rather than expert usage, what do they make of the inevitable conflict, what do they say to the Mathematician about his ontology?"

Well, as I was saying, I would think Grice would endorse to that commentary from Wikipedia:

"Today most classical mathematicians are considered Platonist and believe in the existence of infinite mathematical objects and a set-theoretical universe."

-- where the reference to Plato (after all, as Whitehead used to say, all philosophy is footnotes to Plato, and then some) should NOT be gratuitous.

Grice deals in various bits with what he considers a crucial concept in his 'eschatology': that of 'deeming'. (His illustration: a dog deemed a cat in Oxford, as per a resolution of the governing body of a college).

Grice wants to say that some concepts are 'ideal' (not unlike Locke on the negative idea of 'infinity'). Grice gives two examples: 'circle' and "know" -- and wants to extend that to "mean" (or "meaning"). This he does in "Meaning Revisited".

But his point is general. And so he may say that while, alla Hilbert, only an INTUITIONISTIC finitist approach is logically constructible, this should not lead us to think that the concept, thus constructed, may NOT be _deemed_ to stand for the 'ideal' concept ('infinity' in this case).

--- The reference to intuitionism should not be gratuitious either, in seeing that Grice became more and more associated with colleagues who had felt the influence of M. A. E. Dummett in Oxford, who crucially brought intuitionism to the fore -- if that's the expression.

Jones:

"On the matter of infinity, in which we are told the interest of Locke and Grice alike (but not likely Carnap) is in ordinary rather than expert usage, what do they make of the inevitable conflict, what do they say to the Mathematician about his ontology?"

Well, Grice would I think distinguish between:

"As far as I know, there are infinitely many stars" -- a claim in physics.

"As far as I know, there are infinitely many prime numbers" -- a claim in mathematics.

---- It is true that a few mathematical concepts have a correlate in the 'vernacular' (or 'non-expert' use, to use Jones's phrase). So Grice may need to provide an adequate answer to THAT.

Grice has a mimeo entitled, "The learned and the vulgar". This is a late mimeo. He became interested in the ways a scientist approach ('learned') may conflict with 'the vulgar' ordinary approach to things.

But again, it may do to revise Grice on the infinitely many stars.

The sentence, as per header,

"As far as I know, there are infinitely many stars."

occurs on p. 163 of "Way of Words."

Grice gives this as an example which

"would seem to  involve nothing but an ordinary use of
language by any standard but that of  freedom from absurdity."

It is "not, so far as I can see, technical,  philosophical,
figurative, or strained".

I.e. it is not a claim in advanced physics.

Similarly, he would consider that

"As far as I know, there are infinitely many primes"

is a corresponding non-technical mathematical claim, rather than one belonging to what Locke calls the 'advanced speculation' of your average mathematician.

The sentence about the stars, according to Grice, is an example "of the sorts of  things which have been said and meant by numbers of actual persons."

"Yet," and this is Grice's crucial point against Malcolm, it is "open, I think, at least to the
suspicion of self-contradictoriness, absurdity,  or some other kind of
meaninglessness."

--- And this may relate to the claim of meaninglessness as advocated by Carnap on this or that.

Grice's view, at this point, may be compared with that of Wittgenstein, whom the Wikipedia entry also has as an adherent of FINITISM -- the later Witters?

Indeed, the talk of 'meaningless' compares to Popper (whom Carnap knew). For Popper notes in a footnote to the book where he discusses Euclid, that naturally, in a finitist model 
of mathematics, Euclid's theorem becomes, rightly, 'meaningless'.

The idea behind all this is that mathematics is tautological and so, indeed, the choice of axioms dictate internal questions only. What is meaningless in a system may well be meaningful in another.

We may end this post with a reference to the first bit of Grice's favourite philosopher: Kantotle.

For Aristotle may be characterized as a strict finitist, along with Hilbert, Locke, and, perhaps Grice.

Aristotle especially promoted the potential infinity as a middle option  between strict finitism and actual infinity.

Aristotle's actual infinity means simply an actualization of  something never-ending in nature, when in contrast the Cantorist actual infinity  means the transfinite cardinal and ordinal numbers, that have nothing to do with  the things in nature.

Specifically, Aristotle wrote in Book 3, chapter 6, of "Physics" (that Grice would have studied in Greek at Corpus Christi).

"But on the other hand,"

writes Aristotle,

"to suppose that the infinite does not exist in  any
way leads obviously to many impossible consequences."

"There will be a beginning and end of time, a magnitude will not be 
divisible into magnitudes, number will not be infinite."

"If, then, in view of the above considerations, neither alternative seems 
possible, an arbiter must be called in."

"Allow me to be him," as Grice would say.

Interstingly, according to some historians of mathematics, the Greeks lacked a real concept of the 'infinity', which should make us reconsider if the above is the right translation of Aristotle's "Physics".

It has been noted that Euclid never used the Greek equivalent to 'infinite'. Rather, what he thought he proved is a bit of a roundabout 'ta legomena'.

Euclid did NOT have a word for 'infinity'. 
On top of that, seeing that Grice LOVED Hardy (the mathematician), who favoured 'complete proofs', Euclid wrote his "proofs" in a style which would be unacceptable today.

Euclid woulg give an example rather
than handling the general case. 

It may be obvious that  he did understand
the general case.

Perhaps, crucially, he just did not have the notation to express it -- or in other words, lacked the 'vulgar' to counterpart the 'learned' speculation he was after.

Euclid's
proof of his theorem about prime numbers is one of those cases.

What Euclid actually wrote was:


"Prime numbers are more than any
assigned multitude of prime  numbers."

which contrasts with the rather more perspicuous:

"As far as I know, there are infinitely many prime numbers."'

----- [INCIDENTALLY, and to go back to Grice's 'star' illustration, "As far as I know, there are infinitely many stars", seeing that, as Jones remarks, cosmologists assume that matter is FINITE, the claim selected by Grice ends up a falsehood, rather than a meaningless claim].

Note that Euclid's phrasing is still _vaguer_ than one which uses  'indefinite' as an alternate to 'infinite'.

Euclid proceeds to deal with his claim,
"Prime numbers are more than any
assigned multitude of prime  numbers."

in a constructivist way (alla Grice and Hardy):

"Let A, B, and C be the assigned prime numbers."

"I say that there  are more
prime numbers than A, B, and C.
Take the least number DE  measured by A, B, and C. Add the unit DF to DE. 
Then EF is either prime or  not."

"First, let it be prime. Then the prime
numbers A, B, C, and EF have been  found which are more than A, B, and C. Next,
let EF not be prime."

"Therefore it  is measured by some prime number. Let it
be measured by the prime number G.  VII.31  I say that G is not the same
with any of the numbers A, B, and C. 
If possible, let it be so. Now A, B, and C measure DE, therefore G also 
measures DE. But it also measures EF. Therefore G, being a number, measures
the  remainder, the unit DF, which is absurd.  Therefore G is not the same
with  any one of the numbers A, B, and C. And by hypothesis it is prime.
Therefore the  prime numbers A, B, C, and G have been found which are more than
the assigned  multitude of A, B, and C. Therefore, prime numbers are more than
any assigned  multitude of prime numbers."

----

Incidentally, while Hardy did believe that what Euclid engages himself in is a 'reductio ad absurdum', the claim has been questioned elsewhere.

Etc. Or, "and so on, ad infinitum."

Cheers.