Grice's Changing Conception of Number

Speranza

Incidentally, re: Jones's commentary, about the

"the revelations about how far Frege by 1924 had further weakened his logicism",

the reference was to:


http://people.umass.edu/klement/fnum.pdf

-- an essay by Klement, in "Theoria" -- "Frege's changing conception of number", which I facetiously retitled as per title to this post.

The references to Klement's interesting essay below.

Of course, while Grice did change (views) he did not change _AS MUCH AS FREGE did_: I was surprised to note that some very late manuscripts by Grice revert to his very initial concern with topics like 'personal identity' (the meaning of "I" statements) in terms of 'logical constructions', for example.

Note that Klement also quotes from Sluga, who loved Grice (and talked on his Oxonian cricket experiences during the Grice bench-warming ceremony on Berkeley campus -- Moses Hall) (Indeed Grice quotes from Sluga in "Presupposition and conversational implicature", even if the original footnote was cleared when the essay was reprinted in WoW).

Cheers,

REFERENCES

Cocchiarella, Nino (1987). Logical Studies in Early Analytic Philosophy. Columbus:
Ohio State University Press.

Frege, Gottlob (1891). “Function and concept.” In Frege (1984), pp. 137–56.
———(1892a). “Comments on sense and meaning.” In Frege (1979), pp. 118–25.
———(1892b). “On concept and object.” In Frege (1984), pp. 182–94.
———(1892c). “On concept and object [draft].” In Frege (1979), pp. 87–117.
——— (1895). “A critical elucidation of some points in E. Schröder’s Lectures
on the Algebra of Logic.” In Frege (1984), pp. 211–28.
———(1897). “Logic.” In Frege (1979), pp. 126–151.
———(1906). “On Schoenflies: Die logischen Paradoxien der Mengenlehre.” In
Frege (1979), pp. 176–83.
———(1914). “Logic in mathematics.” In Frege (1979), pp. 203–250.
———(1919). “Notes for Ludwig Darmstädter.” In Frege (1979), pp. 253–57.
——— (1924a). “Diary entries on the concept of numbers.” In Frege (1979), pp.
263–64.
——— (1924b). “A new attempt at a foundation for arithmetic.” In Frege (1979),
pp. 278–81.
———(1924c). “Number.” In Frege (1979), pp. 265–66.
———(1924d). “Numbers and arithmetic.” In Frege (1979), pp. 275–77.
———(1924e). “Sources of knowledge of mathematics and natural sciences.” In
Frege (1979), pp. 267–74.
——— (1950). The Foundations of Arithmetic. Evanston: Northwestern University
Press. Translated by J. L. Austin; first published in 1884 as Die Grundlagen
der Arithmetik (Breslau: W. Köbner).c
——— (1964). Basic Laws of Arithmetic. Berkeley: University of California
Press. Translated by M. Furth; originally published in 1893–1902 as Grundgesetze
der Arithmetik, 2 vols. (Jena: H. Pohle).
——— (1979). Posthumous Writings. Chicago: University of Chicago Press.
Translated by P. Long and R. White.
———(1980). Philosophical and Mathematical Correspondence. Chicago: University
of Chicago Press.
——— (1984). Collected Papers on Mathematics, Logic and Philosophy. New
York: Basil Blackwell. Edited by B. McGuinness.
——— (1997). The Frege Reader. Oxford: Blackwell.

GRICE, H. P. The Grice Papers. Bancroft Library, UCalifornia/Berkeley.

Klement, Kevin C. (2001). “Russell’s paradox in appendix B of the Principles
of Mathematics: Was Frege’s response adequate?” History and Philosophy of
Logic 22, 13–28.
——— (2002). Frege and the Logic of Sense and Reference. New York: Routledge.
———(2004). “Putting form before function: Logical grammar in Frege, Russell
and Wittgenstein.” Philosopher’s Imprint 4, 1–47.

Landini, Gregory (2006). “Frege’s cardinal numbers as concept-correlates.” Erkenntnis
65, 207–43.

Quine, W. V. (1955). “On Frege’s way out.” Mind 64, 145–59.

Russell, Bertrand (1919). Introduction to Mathematical Philosophy. London:
George Allen & Unwin.

Sluga, Hans (1980). Gottlob Frege. London: Routledge and Kegan Paul.

"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.

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

Speranza

Somewhat out of the blue, Grice gives this example in an early essay in WoW (Way of Words -- coined after Locke's "Way of Ideas, Way of Words":

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

-- which may lead us to reconsider what Grice -- and for that matter, Carnap, since we love him, too -- said or thought about "∞" (for surely we need to symbolise all this (*)) and "ℵ1".

--- (* "If you can't put in in symbols, it's not worth saying" -- Grice to Strawson -- with Strawson's impolite retort).

Jones comments in post to GRICE CLUB:

"This is one of the few philosophical issues in relation to which relatively recent developments in mathematics make a real difference, and threaten to enter decisively into a matter one might have thought the province of Philosophy."

Which is good to know. Grice wrote this example early enough, so we cannot say he was aware of recent developments in mathematics. He was concerned with an argument with Malcolm. I never knew WHY Grice developed such an interest in replying a rather minor point in Malcolm's writings.

Jones goes on:

"Locke here [in section "Infinity", as per "Essay concerning Human Understanding", as edited by R. B. Jones himself] seems to be taking the Aristotelian path of allowing potential but rejecting actual infinity, as well as a more specific rejection of infinite numbers."

Indeed. Of course, Locke is into this trichotomy, shall we call it:

way of words: "infinitely many" (stars, numbers)

way of ideas: the NEGATIVE idea of 'infinity' (be it potential or actual)

way of THINGS: the infinity itself!

---

Note that the motto for Locke (allegedly refuted by Mill, System of Logic) is that words signify MEDIATELY things -- what they IMMEDIATELY signify is _ideas_. Mill tried to get away from this psychologism, and his 'semantics' is pre-Fregean in that words _refer_ (denote) and signify (connote) things directly.

---

Jones goes on:

"Later Bishop Berkeley was to take aim at the use by mathematicians in the differential and integral calculus of infintary quantities or numbers (those which are infinitely small)."

Great reference. Indeed, it may occupy the pages of a whole book. R. Paul was referring to a book on this, with the rather hyperbolic (and metaphorical) title, "The man who knew infinity" -- NOT Bishop Berkeley or Locke.

I always loved and admired Berkeley, if only because he gave the motto to the town or village (not incidentally called Berkeley) where Grice settled for _years_:


And there's a painting, too:

	Emanuel Leutze, "Westward the course of empire takes its way".

The painting takes its inspiration from the closing lines of George Berkeley's Verses on the Prospect of Planting Arts and Learning in America:

Westward the course of empire takes its way;
The first four Acts already past,
A fifth shall close the Drama with the day;
Time's noblest offspring is the last.

--- end of Berkeleyan interlude.

Jones goes on:

"Since then [i.e. Locke and Berkeley] a number of developments in Mathematics and Logic have made a significant difference."

-- which Grice should be aware of -- as far as logic is concerned. I treasure the gem of a reference in the Bartlett dictionary: "Grice: British logician".

Jones:

"The first was the "rigorisation" of analysis, taking place in the first half of the 19th century, which put right what Berkeley complained of by re-casting analysis without the use of infinitesimals.
From there on in the news is bad for philosophical sceptics about infinity (at least in mathematics)."

----

I like the epithet: 'sceptic about infinity'. For strictly, a sceptic rejects he can KNOW. But perhaps Locke is not so much sceptic (or skeptic as the students at UC/Berkeley must now spell the thing -- but not Grice) as a psychologist, in thinking that only an intuitively constructed notion of 'infinity' is all we need to account for what is after all a rather 'negative' concept: IN-finite, with IN- being that never otiose negative affix (cfr. IN-DE-finite).

Jones:

"First Cantor came up with a coherent account of what an infinite number might be, in his theory of cardinal numbers, and these became a standard aspect of set theoretical foundations for "classical" mathematics."

Indeed,

hence the "ℵ0"

which should read

ℵ0

-- and cfr. Borges's story, "The aleph".

And as we wonder if the use of

ℵ

without a subscript (since Grice, as we know, loved them -- cfr. Jones on "Vacuous Names", online, pdf document) makes or should make sense.

--- I was marvelled to learn that Frege's idea of number (including infinite numbers, as it were) was influenced by Cantor.

Jones goes on:

"These are perhaps not the kinds of infinity of which Locke and Berkeley spoke."

Well, Locke does refer to mathematicians. In the closing bit of "Infinity" he says something along the lines: "What I say is what an ordinary English speaker may think about this stuff. Never mind what an obscure mathematician may!"

Since that above sounds rude, I should find the lovely charming passage in Locke's never improvable prose:

Indeed, it's the four last paragraphs in the section:

The first reads:

"I pretend NOT to treat of them [ideas of infinity] in their  full latitude."

The second paragraph reads:

"It suffices to my design to show how the mind receives them, such  as they are, from sensation and reflection; and how even the idea we have of infinity, how remote soever it may seem to be from
any object of sense, or  operation of our mind, has, nevertheless, as all our other ideas, its original  there."

It's the THIRD paragraph that is a caveat against claims by this or that mathematician (and we may wonder, alla Yolton, who has studied these things, or R. Hall, who edits the Locke Newsletter) whom Locke is thinking of.

Locke:

"Some mathematicians perhaps, of advanced speculations, may have other  ways to introduce into their minds ideas of infinity."

This I think is charming, for Locke, like Grice, are into 'ways of words' and 'ways of ideas'. And Locke is ALLOWING that the "idea" of infinity "in some mathematician" (he uses the plural) may be, out of the result of what Locke charmingly calls an "advanced speculation" (again he uses the plural) -- where 'advanced' need to be read as "advanced to me" [Have you noted the horrible implicature here: Person A meets person B: "You ARE tall" (rude remark). "Tall" is never to be understood _simpliciter_ but "tall to me (who am rather short)" or something].

Locke concludes his treatment in the last paragraph:

"But this hinders not but  that they themselves, as well as all
other men, got the first ideas which they  had of infinity from sensation and
reflection, in the method we have here set  down."

which again is charming in noting that after all, mathematicians of the advanced speculation sort are indeed, after all, 'men' ("like us").

I think the keyword there is "first" ideas as different from 'second', shall we say, ideas. He is saying that Cantor (or Robert Paul, who writes for the Cantor Institute) must, as a baby [sic] have gotten [sic] his idea of 'infinitely many stars' from beholding the deep blue sky -- and come up with

ℵ0, ℵ1, ℵ2, ℵn

as a result. Or not!

Jones goes on:

"Later a Robinson invented "non-standard" analysis"

Loved the 'a' -- of course to distinguish him from the ('the') Robinson that counts and that Grice adored. The author of "Definition".

IN THE SELDOM QUOTED "ACTIONS AND EVENTS" (PPQ 1986) Grice quotes on the very first page ("p. 1", that is)  the well-known Oriel Fellow, Richard Robinson, "You name it".

This was, Grice tells us, Robinson's reply to what kind of ontology his metaphysics committed him.

In other words, you name an entity, I commit to it. Very ingenious of this Oriel Fellow!
Jones goes on about this 'a Robinson':

"in which infinitesimal and infinite numbers are re-instated, retrospectively justifying the Leibnizian methods which Berkeley had criticised (though to make a bit of mathematics respectable retrospectively is odd, since one expects proofs to be complete in the first place)."

Indeed. But which may lead us to reconsider where Leibniz got his 'advanced speculations' from originally? Nicholas of Cusa?

---

Jones goes on:

"To this one may then add the usurpation of metaphysics by physicists, who since Einstein have thought that observation and experiment can tell us about the structure of space, including whether space is or is not in fact infinite in extent."

Indeed. I would think this is what Locke would call,

"way of things": "∞" and "ℵ0" as they apply to REALITY, as it were.

"way of ideas" -- Locke's negative idea of infinity

"way of words" -- Grice on the silly implicatures of things like "As far as I know, there are infinitely many stars" (why not the shorter, "[as far as I know] there are infinite stars".

Incidentally, a point may be made about the alleged tautological character (unlike the contingent synthetic a posteriori?) of mathematical claims, as opposed to physical claims. I would thus distinguish between Grice's physical claim:

As far as I know, there are infinitely many stars [in the infinite universe, out there].

versus the mathematical (tautological?) claim:

As far as I know, there are infintely many numbers.

(Indeed, this whole point about infinity as misused by philosophers was motivated by a reference by Donal McEvoy, elsewhere, to Popper's alleged proof of the causal efficacy of world-3 ideas like 'infinity' onto this or that specific mathematical process (seen psychologically) as per Euclid's mind when he 'saw' the theorem ("there are infinitely many prime numbers")).

Jones:

"I believed that received opinion now is that space is infinite in extent, but that the amount of matter in the universe is nevertheless finite."

Would be nice to symbolise this. I propose the existential quantifier:

(∃x∞) Ux ("space is infinite in extent")

&

~(∃x∞) Mx ("the amount of matter [in the universe is not infinite" (but rather finite).

Or something!

The idea is to develop something like a numerical quantifier alla Quine, Methods of Logic,

There are twelve apostles

(∃12x) Ax

-- Neither more nor less -- cfr. Grice on "numerous meanings").

Jones concludes:

"How all this fits into Grice's philosophising I don't know, but I thought I would throw it in."

Thanks.

Further to the Frege connection, which may relate, I have found a rather neat document, excerpt of which I append.

-- INTERLUDE on Cantor --

The crucial condition was suggested by the problem of proving the  Cantor-Bendixson theorem.

On that basis, Cantor could establish the results that the cardinality of  the “second number class” is greater than that of N; and that no intermediate  cardinality exists.

Thus, if you write card(N) = ℵ0

his theorems justified calling the cardinality of the “second number class”  ℵ1.

After the second number class comes a “third number class” (all transfinite  ordinals whose set of predecessors has cardinality ℵ1).

The cardinality of this new number class can be proved to be ℵ2.

And so on.

The first function of the transfinite ordinals was, thus, to establish a  well-defined scale of increasing transfinite cardinalities.

The aleph notation used above was introduced by Cantor only in 1895.

This made it possible to formulate much more precisely the problem of the continuum.

Cantor's conjecture became the hypothesis that card(R) = ℵ1.

-- end of interlude on Cantor.

Begin of Frege connection:


Frege's views on the nature of cardinality were in part indeed  anticipated by Georg Cantor.


With this understanding of the direction of Frege’s thought, we are finally, this online source states, in a position to understand what led him to the conclusion that  geometrical sources of knowledge were necessary in arithmetic.

In certain entries in his diary from 1924, Frege resigns himself to having 
failed in his “efforts to become clear about what is meant by number” (Frege,  1924a, p. 263).

However, in particular, he accuses himself of having been misled  by language into thinking that numbers are objects:

The sentences

‘Six is an even number’

‘Four is a square number’

‘Five  is a prime number’

appear analogous to the sentences

‘Sirius is a fixed  star’

‘Europe is a continent’

– sentences whose function
is to represent an  object as falling under a concept.

Thus the words ‘six’
, ‘four’ and ‘five’ look  like
proper names of objects . . .

"But . . . when one has been occupied  with these questions for a long time one comes to suspect that
our way of  using language is misleading, that number-words are not proper names of objects  at all
. . . and that consequently a sentence like

‘Four is a square number’ 

[or "Caesar is a prime number", to use Carnap's example?]

simply does not express that an object is subsumed under a concept and so  just cannot be construed like  the sentence ‘Sirius is a fixed
star’. But how  then is it to be construed?"

(Frege, 1924a, p. 263)

The online source goes on to note that Frege does not answer this question in this context; indeed, nowhere in this final period does he give a worked out view about how to understand such 
sentences.

However, it seems fairly clear that the natural thing for him to have said  is that the proper construal of statements about numbers is in terms of  second-level concepts, and when we claim that a certain number has a certain  feature, we are in effect claiming that a certain third-level concept applies  to a second-level concept.

That it is this understanding of numbers Frege wishes to preserve in these 
writings is further attested by the precise role and importance he seems to  assign
to the geometrical source of knowledge.

It is through it that we are able to come to recognize, to put it not mildly,

 the  Existence of the Infinite.


"From the geometrical source of knowledge flows "the infinite ", in the  genuine and strictest sense of this word", Frege writes.

"We have infinitely many points  on every interval of a straight
line, on every circle, and infinitely many  lines through any point"
(Frege, 1924e, p. 273)

Since, according to Frege, points in space are, logically considered, 
objects, the geometrical source of knowledge a ords us knowledge of the  existence of infinitely
many objects.

Frege is explicit that this sort of knowledge has both spatial  and geometrical aspects, and that it is a priori and independent of sense  perception.

Frege writes:

"It is evident that sense perception can yield nothing infinite." (_pace_ dear old Locke).

"However  many starts we may include in our inventories, there will never be infinitely  many, and the same goes for us with the grains of sand on the seashore."

"And  so, where we may legitimately claim to recognize the infinite, we have  not
obtained it from sense perception."

"For this we need a special source of 
knowledge, and one such is
the geometrical."

"Besides the spatial, the  temporal must also be recognized."

"A source of
knowledge corresponds to  this
too, and from this also we derive the infinite."

"Time stretching to 
infinity in both directions is like a
line stretching to infinity in both  directions."

(Frege, 1924e, p. 274)

A guarantee of infinitely many objects  is precisely what is needed in
order to guarantee that the sequence of  natural numbers, when construed as
second-level concepts, does not come to an  end.

That this is Frege’s reason for appealing to the geometrical source of  knowledge comes out rather explicitly in discussing the failure of his former  views:

"I myself at one time held it to be possible to conquer the entire  number
domain, continuing along
a purely logical path from the  kindergarten-numbers."

"I have seen the
mistake in this."

"I was right  in
thinking that you cannot do this if you take an empirical route."

"I may 
have arrived at this conviction
as a result of the following consideration:  that the series of whole
numbers should eventually come
to an end, that there  should be a greatest whole number, is manifestly
absurd."

"This shows  that
arithmetic cannot be based on sense perception."

"For if it could be so 
based, we should have to
reconcile ourselves to the brute fact of the series  of whole numbers
coming to an end, as we may
one day have to reconcile  ourselves to there being no stars above a
certain size."

"But here surely
the  position is di erent: that the series of whole numbers should
eventually come to  an end is not
just false."

"We  find the idea absurd."

"So an a priori mode of  cognition must
be involved here."

"But this cognition does not have to flow  from purely logical principles, as I
originally assumed."

"There
is the further  possibility that it has a geometrical source."
(Frege, 1924d, pp. 276–277; Cf.  1924b,
p. 279).

The upshot of the appeal to geometry seems precisely to a ord us  knowledge
of the existence of objects, which Frege is now explicit that he  thinks
cannot be yielded by the logical source of knowledge alone (Frege,  1924b, p. 279).
Once the existence of a su cient number of objects is  guaranteed in an a priori
way, we are free to continue to understand a  statement of number as containing an
assertion about a concept: indeed, Frege  is explicit in these final manuscripts that
this is a
thesis of his earlier  work he still regards as true (Frege, 1924d, pp. 275–
76; 1924b,
p.  278).

And so on, as we say (hyperbolically * -- cfr. Grice on hyperbole as conversational implicature, WoW -- "Every nice girl loves a sailor"), 'ad infinitum'.

The Implicatures of Plasticarian

Speranza

From today's World Wide Words,

World Wide Words © Michael Quinion 2013 http://www.worldwidewords.org.


"When Thomas Smith, a chemistry PhD
student from Manchester, was given a plastic
lid for his takeaway tea by the staff at his university
café, he had a novel comeback. “I can’t take that,”
he said. “I’m a plasticarian.”


Quinion notes about the suffix, -arian:

"it’s rare in asserting opposition
rather than acceptance."

and provides an earlier quote:

"Becoming a plasticarian will affect my life and my diet..."

Quinion compares 'plasticarian' to:

humanitarian
libertarian
vegetarian

and as he notes, it is rare to have '-arian' IMPLICATING opposition (or 
rejection) rather than support. But since now, after T. Smith, everybody
(that's  a hyperbole) seems to be using 'plasticarian', it may do to play with
the three  other items in terms of 'opposition':

"I'm a humanitarian: I avoid humans". (alla "I'm a plasticarian: I avoid 
plastics")
"I'm a libertarian: I avoid liberty" -- cfr. the Statue of Liberty in 
Regent Street, London (Sir Arthur Lasenby Liberty (13 August 1843 – 11 May 
1917)).

"I'm a vegetarian; I avoid vegetals."

THE PEDANTIC Manchester TEA provider:

"When Thomas Smith, a chemistry PhD
student from Manchester, was given  a plastic
lid for his takeaway tea by the staff at his university
café,  he had a novel comeback. “I can’t take that,”
he said. “I’m a  plasticarian.”

THOMAS SMITH: I can't take that [plastic lid]. I'm a plasticarian.

STAFF: You mean an anti-plasticarian. Strictly.

THOMAS SMITH. Hmpf.

Cheers,

Speranza

As Far As I Know, There Are Infinitely Many Stars -- Grice on "∞" and "ℵ"

Speranza

--- is an example by Grice in "Way in the Way of Words" -- his argument against Malcolm.

Let us reconsider what we mean, 'infinity', or

"∞"


or

"ℵ" (aleph).

We can start with the excellent job done by R. B. Jones at

http://www.rbjones.com/rbjpub/philos/classics/locke/ctb2c17.htm

 

where Locke's Essay is converted to HTML for RBJones.com by first edition 1994/10/29 last
modified 2009/3/30.

Chapter XVII

 

"Of Infinity"

 Locke writes:

 

"[Let us speak of] "infinity", in its original intention [i.e. signification], attributed to 
space, duration, and number."

"He that would know what kind of idea it is to which we give the name of 
"infinity", cannot do it better than by considering to what infinity is by the 
mind more immediately attributed."

"And then we cannot do it better than by considering how the mind comes to frame it."

""Finite" and "infinite" seem to me to be looked upon by the mind as the modes 
of quantity, and to be attributed primarily in their first designation only
to  those things which have parts, and are capable of increase or
diminution by the  addition or subtraction of any the least part."

"And such are the ideas of space, duration, and number, which we have 
considered in the foregoing chapters."

"It is true, that we cannot but be assured, that the great God, of whom and
from whom are all things, is incomprehensibly infinite."

"But yet, when we apply to that first and supreme Being our idea of 
"infinite", in our weak and narrow thoughts, we do it primarily in respect to his 
duration and ubiquity."

"And, I think, more figuratively to his power, wisdom, and goodness, and 
other attributes, which are properly inexhaustible and incomprehensible,  &c."

"For, when we call them "infinite", we have no other idea of this infinity 
but what carries with it some reflection on, and imitation of, that number or
extent of the acts or objects of God's power, wisdom, and goodness, which
can  never be supposed so great, or so many, which these attributes will not
always  surmount and exceed, let us multiply them in our thoughts as far as
we can, with  all the infinity of endless number."

"I do not pretend to say
how these attributes  are in God, who is infinitely beyond the reach of our
narrow capacities."

"They  do, without doubt, contain in them all possible
perfection."

"But this, I say, is  our way of conceiving them, and these our ideas
of their infinity."


"The idea of "finite" is easily got."

"Finite then, and infinite, being by the 
mind looked on as modifications of expansion and duration, the next thing to
be  considered, is,- How the mind comes by them.

As for the idea of finite,
there is  no great difficulty.

The obvious portions of extension that affect
our senses,  carry with them into the mind the idea of finite: and the
ordinary periods of  succession, whereby we measure time and duration, as hours,
days, and years, are  bounded lengths.

The difficulty is, how we come by
those boundless ideas of  eternity and immensity; since the objects we converse
with come so much short of  any approach or proportion to that largeness."

"How we come by the idea of infinity.

Every one that has any idea of  any
stated lengths of space, as a foot, finds that he can repeat that idea; and 
joining it to the former, make the idea of two feet.

And by the addition of
a  third, three feet; and so on, without ever coming to an end of his
additions,  whether of the same idea of a foot, or, if he pleases, of doubling it,
or any  other idea he has of any length, as a mile, or diameter of the
earth, or of the  orbis magnus.

For whichever of these he takes, and how often
soever he doubles,  or any otherwise multiplies it, he finds, that, after he
has continued his  doubling in his thoughts, and enlarged his idea as much
as he pleases, he has no  more reason to stop, nor is one jot nearer the end
of such addition, than he was  at first setting out: the power of enlarging
his idea of space by further  additions remaining still the same, he hence
takes the idea of infinite  space."

"For let us consider our idea of space as boundless."

"This, I think, is the way whereby the mind gets the idea of infinite 
space.

It is a quite different consideration, to examine whether the mind has 
the idea of such a boundless space actually existing; since our ideas are not
always proofs of the existence of things: but yet, since this comes here
in our  way, I suppose I may say, that we are apt to think that space in
itself is  actually boundless, to which imagination the idea of space or
expansion of  itself naturally leads us.

For, it being considered by us, either as
the  extension of body, or as existing by itself, without any solid matter
taking it  up, (for of such a void space we have not only the idea, but I
have proved, as I  think, from the motion of body, its necessary existence), it
is impossible the  mind should be ever able to find or suppose any end of
it, or be stopped  anywhere in its progress in this space, how far soever it
extends its thoughts. 

Any bounds made with body, even adamantine walls, are
so far from putting a stop  to the mind in its further progress in space
and extension that it rather  facilitates and enlarges it.

For so far as that
body reaches, so far no one can  doubt of extension; and when we are come to
the utmost extremity of body, what  is there that can there put a stop, and
satisfy the mind that it is at the end  of space, when it perceives that it
is not; nay, when it is satisfied that body  itself can move into it?

For,
if it be necessary for the motion of body, that  there should be an empty
space, though ever so little, here amongst bodies; and  if it be possible for
body to move in or through that empty space;- nay, it is  impossible for any
particle of matter to move but into an empty space; the same  possibility
of a body's moving into a void space, beyond the utmost bounds of  body, as
well as into a void space interspersed amongst bodies, will always  remain
clear and evident: the idea of empty pure space, whether within or beyond  the
confines of all bodies, being exactly the same, differing not in nature, 
though in bulk; and there being nothing to hinder body from moving into it.

So  that wherever the mind places itself by any thought, either amongst, or
remote  from all bodies, it can, in this uniform idea of space, nowhere find
any bounds,  any end; and so must necessarily conclude it, by the very
nature and idea of  each part of it, to be actually infinite."


"And so of duration."

"As, by the power we find in ourselves of repeating, as often as we will, 
any idea of space, we get the idea of immensity; so, by being able to repeat
the  idea of any length of duration we have in our minds, with all the
endless  addition of number, we come by the idea of eternity.

For we find in
ourselves,  we can no more come to an end of such repeated ideas than we can
come to the end  of number; which every one perceives he cannot.

But here
again it is another  question, quite different from our having an idea of
eternity, to know whether  there were any real being, whose duration has been
eternal.

And as to this, I  say, he that considers something now existing, must
necessarily come to  Something eternal.

But having spoke of this in another
place, I shall say here  no more of it, but proceed on to some other
considerations of our idea of  infinity."

"Why other ideas are not capable of infinity."

"If it be so, that our idea of infinity be got from the power we observe in
ourselves of repeating, without end, our own ideas, it may be demanded,-

Why we  do not attribute infinity to other ideas, as well as those of space
and  duration."

"Since they may be as easily, and as often, repeated in our minds as the 
other: and yet nobody ever thinks of infinite sweetness, or infinite
whiteness,  though he can repeat the idea of sweet or white, as frequently as those
of a  yard or a day?

"To which I answer as follows.

All the ideas that are considered as having parts, and
are capable of increase by the addition of any equal or less parts, afford
us,  by their repetition, the idea of infinity; because, with this endless 
repetition, there is continued an enlargement of which there can be no end.

But  in other ideas it is not so.

For to the largest idea of extension or
duration  that I at present have, the addition of any the least part makes an
increase;  but to the perfectest idea I have of the whitest whiteness, if I
add another of  a less or equal whiteness, (and of a whiter than I have, I
cannot add the idea),  it makes no increase, and enlarges not my idea at
all; and therefore the  different ideas of whiteness, &c. are called degrees."

"For those ideas that consist of parts are capable of being augmented by 
every addition of the least part; but if you take the idea of white, which
one  parcel of snow yielded yesterday to our sight, and another idea of white
from  another parcel of snow you see to-day, and put them together in your
mind, they  embody, as it were, and run into one, and the idea of whiteness
is not at all  increased; and if we add a less degree of whiteness to a
greater, we are so far  from increasing, that we diminish it.

Those ideas that
consist not of parts  cannot be augmented to what proportion men please, or be
stretched beyond what  they have received by their senses; but space,
duration, and number, being  capable of increase by repetition, leave in the mind
an idea of endless room for  more; nor can we conceive anywhere a stop to a
further addition or progression:  and so those ideas alone lead our minds
towards the thought of infinity."

"There is a Difference between infinity of space, and space  infinite."

"Though our idea of infinity arise from the contemplation of quantity, and 
the endless increase the mind is able to make in quantity, by the repeated 
additions of what portions thereof it pleases; yet I guess we cause great 
confusion in our thoughts, when we join infinity to any supposed idea of 
quantity the mind can be thought to have, and so discourse or reason about an 
infinite quantity, as an infinite space, or an infinite duration.

For, as
our  idea of infinity being, as I think, an endless growing idea, but the
idea of any  quantity the mind has, being at that time terminated in that idea,
(for be it as  great as it will, it can be no greater than it is,)- to join
infinity to it, is  to adjust a standing measure to a growing bulk; and
therefore I think it is not  an insignificant subtilty, if I say, that we are
carefully to distinguish  between the idea of the infinity of space, and the
idea of a space infinite.

The  first is nothing but a supposed endless
progression of the mind, over what  repeated ideas of space it pleases; but to
have actually in the mind the idea of  a space infinite, is to suppose the
mind already passed over, and actually to  have a view of all those repeated
ideas of space which an endless repetition can  never totally represent to it;
which carries in it a plain  contradiction.

"We have NO idea of infinite space."

"This, perhaps, will be a little plainer, if we consider it in  numbers."

-------------------------

"The infinity of numbers, to the end of whose addition every one perceives 
there is no approach, easily appears to any one that reflects on it."

"But, how clear soever this idea of the infinity of number be, there is 
nothing yet more evident than the absurdity of the actual idea of an infinite 
number."

"Whatsoever positive ideas we have in our minds of any space, duration, or 
number, let them be ever so great, they are still finite."

"But when we suppose an inexhaustible remainder, from which we remove all 
bounds, and wherein we allow the mind an endless progression of thought,
without  ever completing the idea, there we have our idea of infinity: which,
though it  seems to be pretty clear when we consider nothing else in it but
the negation of  an end, yet, when we would frame in our minds the idea of an
infinite space or  duration, that idea is very obscure and confused,
because it is made up of two  parts, very different, if not inconsistent."

"For, let a man frame in his mind an idea of any space or number, as great 
as he will; it is plain the mind rests and terminates in that idea, which
is  contrary to the idea of infinity, which consists in a supposed endless 
progression.

And therefore I think it is that we are so easily confounded,
when  we come to argue and reason about infinite space or duration, &c.

Because  the parts of such an idea not being perceived to be, as they are,
inconsistent,  the one side or other always perplexes, whatever consequences we
draw from the  other; as an idea of motion not passing on would perplex any one
who should  argue from such an idea, which is not better than an idea of
motion at rest.

And  such another seems to me to be the idea of a space, or
(which is the same thing)  a number infinite, i.e. of a space or number which
the mind actually has, and so  views and terminates in; and of a space or
number, which, in a constant and  endless enlarging and progression, it can in
thought never attain to.

For, how  large soever an idea of space I have in
my mind, it is no larger than it is that  instant that I have it, though I
be capable the next instant to double it, and  so on in infinitum; for that
alone is infinite which has no bounds; and that the  idea of infinity, in
which our thoughts can find none."

"Number affords us the clearest idea of infinity.


But of all other ideas, 
it is number, as I have said, which I think furnishes us with the clearest
and  most distinct idea of infinity we are capable of.

For, even in space and
duration, when the mind pursues the idea of infinity, it there makes use
of the  ideas and repetitions of numbers, as of millions and millions of
miles, or  years, which are so many distinct ideas,- kept best by number from
running into  a confused heap, wherein the mind loses itself; and when it has
added together  as many millions, &c., as it pleases, of known lengths of
space or duration,  the clearest idea it can get of infinity, is the confused
incomprehensible  remainder of endless addible numbers, which affords no
prospect of stop or  boundary."

"Our different conceptions of the infinity of number can be contrasted with those 
of duration and expansion.

It will, perhaps, give us a little further light
into  the idea we have of infinity, and discover to us, that it is nothing
but the  infinity of number applied to determinate parts, of which we have
in our minds  the distinct ideas, if we consider that number is not generally
thought by us  infinite, whereas duration and extension are apt to be so;
which arises from  hence,- that in number we are at one end, as it were: for
there being in number  nothing less than an unit, we there stop, and are at
an end; but in addition, or  increase of number, we can set no bounds: and
so it is like a line, whereof one  end terminating with us, the other is
extended still forwards, beyond all that  we can conceive. But in space and
duration it is otherwise.

For in duration we  consider it as if this line of
number were extended both ways- to an  unconceivable, undeterminate, and
infinite length; which is evident to any one  that will but reflect on what
consideration he hath of Eternity; which, I  suppose, will find to be nothing else
but the turning this infinity of number  both ways, a parte ante, and a
parte post, as they speak.

For, when we would  consider eternity, a parte ante,
what do we but, beginning from ourselves and  the present time we are in,
repeat in our minds the ideas of years, or ages, or  any other assignable
portion of duration past, with a prospect of proceeding in  such addition with
all the infinity of number: and when we would consider  eternity, a parte
post, we just after the same rate begin from ourselves, and  reckon by
multiplied periods yet to come, still extending that line of number as  before.

And
these two being put together, are that infinite duration we call  Eternity:
which, as we turn our view either way, forwards or backwards, appears 
infinite, because we still turn that way the infinite end of number, i.e. the 
power still of adding more.

"How we conceive the infinity of space.

The same happens also in space, 
wherein, conceiving ourselves to be, as it were, in the centre, we do on all 
sides pursue those indeterminable lines of number; and reckoning any way
from  ourselves, a yard, mile, diameter of the earth, or orbis magnus,- by the 
infinity of number, we add others to them, as often as we will.

And having
no  more reason to set bounds to those repeated ideas than we have to set
bounds to  number, we have that indeterminable idea of immensity."

"Let us speak of Infinite divisibility."

"And since in any bulk of matter our thoughts can never arrive at the 
utmost divisibility, therefore there is an apparent infinity to us also in that,
which has the infinity also of number; but with this difference,- that, in
the  former considerations of the infinity of space and duration, we only
use  addition of numbers; whereas this is like the division of an unit into
its  fractions, wherein the mind also can proceed in infinitum, as well as in
the  former additions; it being indeed but the addition still of new
numbers: though  in the addition of the one, we can have no more the positive idea
of a space  infinitely great, than, in the division of the other, we can
have the [positive]  idea of a body infinitely little;- our idea of infinity
being, as I may say, a  growing or fugitive idea, still in a boundless
progression, that can stop  nowhere."

"There is No positive idea of infinity."

"Though it be hard, I think, to find anyone so absurd as to say he has the 
positive idea of an actual infinite number;- the infinity whereof lies only
in a  power still of adding any combination of units to any former number,
and that as  long and as much as one will; the like also being in the
infinity of space and  duration, which power leaves always to the mind room for
endless additions;- yet  there be those who imagine they have positive ideas
of infinite duration and  space.

It would, I think, be enough to destroy any
such positive idea of  infinite, to ask him that has it,- whether he could
add to it or no; which would  easily show the mistake of such a positive
idea.

We can, I think, have no  positive idea of any space or duration which is
not made up of, and commensurate  to, repeated numbers of feet or yards, or
days and years; which are the common  measures, whereof we have the ideas in
our minds, and whereby we judge of the  greatness of this sort of
quantities.

And therefore, since an infinite idea of  space or duration must needs be
made up of infinite parts, it can have no other  infinity than that of
number capable still of further addition; but not an  actual positive idea of a
number infinite.

For, I think it is evident, that the  addition of finite
things together (as are all lengths whereof we have the  positive ideas) can
never otherwise produce the idea of infinite than as number  does; which,
consisting of additions of finite units one to another, suggests  the idea of
infinite, only by a power we find we have of still increasing the  sum, and
adding more of the same kind; without coming one jot nearer the end of  such
progression."

"Let us show how we cannot have a positive idea of infinity in  quantity."

"They who would prove their idea of infinite to be positive, seem to me to 
do it by a pleasant argument, taken from the negation of an end; which
being  negative, the negation of it is positive.

He that considers that the end
is, in  body, but the extremity or superficies of that body, will not
perhaps be forward  to grant that the end is a bare negative: and he that
perceives the end of his  pen is black or white, will be apt to think that the end
is something more than  a pure negation.

Nor is it, when applied to duration,
the bare negation of  existence, but more properly the last moment of it.

But if they will have the  end to be nothing but the bare negation of
existence, I am sure they cannot deny  but the beginning is the first instant of
being, and is not by any body  conceived to be a bare negation; and therefore,
by their own argument, the idea  of eternal, a parte ante, or of a duration
without a beginning, is but a  negative idea."

"Let us consider what is positive, what negative, in our idea of  infinite."

"The idea of infinite has, I confess, something of positive in all those 
things we apply to it."

"When we would think of infinite space or duration, we at first step 
usually make some very large idea, as perhaps of millions of ages, or miles, 
which possibly we double and multiply several times.

All that we thus amass 
together in our thoughts is positive, and the assemblage of a great number of 
positive ideas of space or duration.

But what still remains beyond this we
have  no more a positive distinct notion of than a mariner has of the depth
of the  sea; where, having let down a large portion of his sounding-line, he
reaches no  bottom.

Whereby he knows the depth to be so many fathoms, and
more; but how much  the more is, he hath no distinct notion at all: and could
he always supply new  line, and find the plummet always sink, without ever
stopping, he would be  something in the posture of the mind reaching after a
complete and positive idea  of infinity.

In which case, let this line be
ten, or ten thousand fathoms long,  it equally discovers what is beyond it,
and gives only this confused and  comparative idea, that this is not all, but
one may yet go farther.

So much as  the mind comprehends of any space, it
has a positive idea of: but in  endeavouring to make it infinite,- it being
always enlarging, always advancing,-  the idea is still imperfect and
incomplete.

So much space as the mind takes a  view of in its contemplation of
greatness, is a clear picture, and positive in  the understanding: but infinite
is still greater. 1.

Then the idea of so much is  positive and clear. 2.

The
idea of greater is also clear; but it is but a  comparative idea, the idea
of so much greater as cannot be comprehended. 3.

And  this is plainly
negative: not positive.

For he has no positive clear idea of the  largeness of any
extension, (which is that sought for in the idea of infinite),  that has not
a comprehensive idea of the dimensions of it: and such, nobody, I  think,
pretends to in what is infinite.

For to say a man has a positive clear  idea
of any quantity, without knowing how great it is, is as reasonable as to 
say, he has the positive clear idea of the number of the sands on the
sea-shore,  who knows not how many there be, but only that they are more than
twenty.

For  just such a perfect and positive idea has he of an infinite space or
duration,  who says it is larger than the extent or duration of ten, one
hundred, one  thousand, or any other number of miles, or years, whereof he has
or can have a  positive idea; which is all the idea, I think, we have of
infinite.

So that what  lies beyond our positive idea towards infinity, lies in
obscurity, and has the  indeterminate confusion of a negative idea, wherein
I know I neither do nor can  comprehend all I would, it being too large for
a finite and narrow capacity.

And  that cannot but be very far from a
positive complete idea, wherein the greatest  part of what I would comprehend is
left out, under the undeterminate intimation  of being still greater.

For to
say, that, having in any quantity measured so  much, or gone so far, you
are not yet at the end, is only to say that that  quantity is greater.

So that
the negation of an end in any quantity is, in other  words, only to say
that it is bigger; and a total negation of an end is but  carrying this bigger
still with you, in all the progressions of your thoughts  shall make in
quantity; and adding this idea of still greater to all the ideas  you have, or
can be supposed to have, of quantity.

Now, whether such an idea as  that be
positive, I leave any one to consider."

"We have no positive idea of an infinite duration.

I ask those who say they
have a positive idea of eternity, whether their idea of duration includes
in it  succession, or not?

If it does not, they ought to show the difference
of their  notion of duration, when applied to an eternal Being, and to a
finite; since,  perhaps, there may be others as well as I, who will own to
them their weakness  of understanding in this point, and acknowledge that the
notion they have of  duration forces them to conceive, that whatever has
duration, is of a longer  continuance to-day than it was yesterday.

If, to avoid
succession in external  existence, they return to the punctum stans of the
schools, I suppose they will  thereby very little mend the matter, or help
us to a more clear and positive  idea of infinite duration; there being
nothing more inconceivable to me than  duration without succession.

Besides, that
punctum stans, if it signify  anything, being not quantum, finite or
infinite cannot belong to it.

But, if our  weak apprehensions cannot separate
succession from any duration whatsoever, our  idea of eternity can be nothing
but of infinite succession of moments of  duration wherein anything does
exist; and whether any one has, or can have, a  positive idea of an actual
infinite number, I leave him to consider, till his  infinite number be so great
that he himself can add no more to it; and as long  as he can increase it, I
doubt he himself will think the idea he hath of it a  little too scanty for
positive infinity."

"No complete idea of eternal being.

I think it unavoidable for every 
considering, rational creature, that will but examine his own or any other 
existence, to have the notion of an eternal, wise Being, who had no beginning: 
and such an idea of infinite duration I am sure I have.

But this negation of
a  beginning, being but the negation of a positive thing, scarce gives me a 
positive idea of infinity; which, whenever I endeavour to extend my
thoughts to,  I confess myself at a loss, and I find I cannot attain any clear
comprehension  of it."

"There is No positive idea of infinite space."

"He that thinks he has a positive idea of infinite space, will, when he 
considers it, find that he can no more have a positive idea of the greatest, 
than he has of the least space.

For in this latter, which seems the easier
of  the two, and more within our comprehension, we are capable only of a
comparative  idea of smallness, which will always be less than any one whereof
we have the  positive idea.

All our positive ideas of any quantity, whether
great or little,  have always bounds, though our comparative idea, whereby we
can always add to  the one, and take from the other, hath no bounds.

For
that which remains, either  great or little, not being comprehended in that
positive idea which we have,  lies in obscurity; and we have no other idea of
it, but of the power of  enlarging the one and diminishing the other,
without ceasing.

A pestle and  mortar will as soon bring any particle of matter to
indivisibility, as the  acutest thought of a mathematician; and a surveyor
may as soon with his chain  measure out infinite space, as a philosopher by
the quickest flight of mind  reach it, or by thinking comprehend it; which
is to have a positive idea of it. 

He that thinks on a cube of an inch
diameter, has a clear and positive idea of  it in his mind, and so can frame one
of 1/2, 1/4, 1/8, and so on, till he has  the idea in his thoughts of
something very little; but yet reaches not the idea  of that incomprehensible
littleness which division can produce.

What remains of  smallness is as far from
his thoughts as when he first began; and therefore he  never comes at all to
have a clear and positive idea of that smallness which is  consequent to
infinite divisibility."

"What is positive, what negative, in our idea of infinite.

Every one that 
looks towards infinity does, as I have said, at first glance make some very 
large idea of that which he applies it to, let it be space or duration; and
possibly he wearies his thoughts, by multiplying in his mind that first
large  idea: but yet by that he comes no nearer to the having a positive clear
idea of  what remains to make up a positive infinite, than the country
fellow had of the  water which was yet to come, and pass the channel of the
river where he  stood."

Rusticus expectat dum defluat amnis, at ille
Labitur, et labetur in  omne volubilis oevum.

"Some think they have a positive idea of eternity, and not of infinite 
space."

"There are some I have met that put so much difference between infinite 
duration and infinite space, that they persuade themselves that they have a 
positive idea of eternity, but that they have not, nor can have any idea of 
infinite space.

The reason of which mistake I suppose to be this- that
finding,  by a due contemplation of causes and effects, that it is necessary to
admit some  Eternal Being, and so to consider the real existence of that
Being as taken up  and commensurate to their idea of eternity; but, on the other
side, not finding  it necessary, but, on the contrary, apparently absurd,
that body should be  infinite, they forwardly conclude that they can have no
idea of infinite space,  because they can have no idea of infinite matter.

Which consequence, I conceive,  is very ill collected, because the existence
of matter is no ways necessary to  the existence of space, no more than the
existence of motion, or the sun, is  necessary to duration, though duration
used to be measured by it.

And I doubt  not but that a man may have the idea
of ten thousand miles square, without any  body so big, as well as the idea
of ten thousand years, without any body so old. 

It seems as easy to me to
have the idea of space empty of body, as to think of  the capacity of a
bushel without corn, or the hollow of a nut-shell without a  kernel in it: it
being no more necessary that there should be existing a solid  body,
infinitely extended, because we have an idea of the infinity of space,  than it is
necessary that the world should be eternal, because we have an idea  of
infinite duration.

And why should we think our idea of infinite space  requires
the real existence of matter to support it, when we find that we have  as
clear an idea of an infinite duration to come, as we have of infinite  duration
past?

Though I suppose nobody thinks it conceivable that anything does  or
has existed in that future duration.

Nor is it possible to join our idea of 
future duration with present or past existence, any more than it is possible
to  make the ideas of yesterday, to-day, and to-morrow to be the same; or
bring ages  past and future together, and make them contemporary. But if
these men are of  the mind, that they have clearer ideas of infinite duration
than of infinite  space, because it is past doubt that God has existed from
all eternity, but  there is no real matter co-extended with infinite space;
yet those philosophers  who are of opinion that infinite space is possessed by
God's infinite  omnipresence, as well as infinite duration by his eternal
existence, must be  allowed to have as clear an idea of infinite space as of
infinite duration;  though neither of them, I think, has any positive idea
of infinity in either  case.

For whatsoever positive ideas a man has in his
mind of any quantity, he  can repeat it, and add it to the former, as easy as
he can add together the  ideas of two days, or two paces, which are
positive ideas of lengths he has in  his mind, and so on as long as he pleases:
whereby, if a man had a positive idea  of infinite, either duration or space,
he could add two infinities together;  nay, make one infinite infinitely
bigger than another- absurdities too gross to  be confuted."

"The Supposed positive ideas of infinity is the cause of mistakes."

"But yet if after all this, there be men who persuade themselves that they 
have clear positive comprehensive ideas of infinity, it is fit they enjoy
their  privilege: and I should be very glad (with some others that I know,
who  acknowledge they have none such) to be better informed by their
communication. 

For I have been hitherto apt to think that the great and inextricable
difficulties which perpetually involve all discourses concerning
infinity,-  whether of space, duration, or divisibility, have been the certain marks
of a  defect in our ideas of infinity, and the disproportion the nature
thereof has to  the comprehension of our narrow capacities.

For, whilst men talk
and dispute of  infinite space or duration, as if they had as complete and
positive ideas of  them as they have of the names they use for them, or as
they have of a yard, or  an hour, or any other determinate quantity; it is no
wonder if the  incomprehensible nature of the thing they discourse of, or
reason about, leads  them into perplexities and contradictions, and their
minds be overlaid by an  object too large and mighty to be surveyed and managed
by them."

"All these are modes of ideas got from sensation and reflection.

If I have 
dwelt pretty long on the consideration of duration, space, and number, and
what  arises from the contemplation of them,- Infinity, it is possibly no
more than  the matter requires; there being few simple ideas whose modes give
more exercise  to the thoughts of men than those do.

 I pretend not to treat
of them in their  full latitude. It suffices to my design to show how the
mind receives them, such  as they are, from sensation and reflection; and how
even the idea we have of  infinity, how remote soever it may seem to be from
any object of sense, or  operation of our mind, has, nevertheless, as all
our other ideas, its original  there.

Some mathematicians perhaps, of
advanced speculations, may have other  ways to introduce into their minds ideas of
infinity.

But this hinders not but  that they themselves, as well as all
other men, got the first ideas which they  had of infinity from sensation and
reflection, in the method we have here set  down."

Earl Grey never drank Earl Grey

Speranza

From today's

From today's "World Wide Words" (© Michael Quinion 2013 --
http://www.worldwidewords.org):

Earl Grey never drank Earl Grey.

-- which Grice might find interesting to analyse implicaturally. ("We should proceed by symbolising the claim", he might add.

Quinion concludes his piece:

"As always, there are loose ends. But the stories that connect Earl Grey tea to a nineteenth-century aristocrat have been debunked. Earl Grey never drank Earl Grey."

How did it all start?

 

Quinion writes:

"For the most part, etymology isn’t a flashy subject. It needs care and patience rather than academic brilliance and is rarely rewarded by moments of breathtaking insight. But at times a search for the provenance of a term turns into an intriguing detective story with an unexpected dénouement.
In October 2012, the Oxford English Dictionary issued an appeal for information about the term Earl Grey tea."

"This is a blend of black China teas flavoured with bergamot, an oil derived from a citrus fruit native to the Far East but widely grown in Italy. Various stories link it to the second Earl Grey, who was British prime minister between 1830 and 1834 and largely responsible for the Great Reform Act of 1832 as well as removing the monopoly of the East India Company on importing tea from China. One legend says that the tea was a reward for his (or an envoy of his) rescuing the son of a Chinese mandarin; another, that a Chinese diplomat gave him a gift of it while he was prime minister. The website of the family home, Howick Hall, says that it was specially blended by a Chinese mandarin to offset the lime taste of the water from the local well and that Lady Grey used it when she was entertaining in London. (A version of Earl Grey tea called Lady Grey tea with a less pungent flavour, created in the early 1990s by the tea merchants Twinings, is named after her.)

"The etymological problem for the OED was that the first example of the term Earl Grey tea it had on record was dated 1929, though they knew of Earl Grey’s mixture from 1891."

"Various contributors progressively took the story back. An advert from about 1928 by Jacksons of Piccadilly claimed to have introduced it at the request of Earl Grey in 1836. A tale appeared in several versions in the decade after 1891 claiming that Earl Grey’s mixture was so named because the earl had introduced it to Her Majesty. But he had retired to Howick after leaving office, aged 70, and may not even have met Queen Victoria, who ascended the throne in 1837. A further advertisement for Earl Grey’s mixture, in the Morning Post in 1884, announced that “this choice Tea can only be obtained of the Introducers and Sole Proprietors, Charlton and Co” of Piccadilly."

"The story took a surprising twist when researchers on the Foods of England site found that Charlton and Co had advertised a tea in 1867 as the rather expensive “celebrated Grey mixture”, with no reference to any aristocratic connection, though it did boast of its “most distinguished patronage”. Might the business have added a noble association later on as a marketing ploy, one that was to be copied by others? It could well have done. Victorian advertisers weren’t renowned for their strict adherence to truth."

"The search for the name runs into the sand at this point. But it’s not the end of the story. The use of bergamot as a flavouring and scent long predates any connection with Earl Grey — for example, it was added to snuff early in the eighteenth century. But its early associations with tea are disreputable. A newspaper report in 1824 was ominously headed, “To render Tea at 5s a Pound equal to Tea at 12s”. It explained."

"If we can discover any fine-flavoured substance, and add it to the tea in a proper manner, so as to make it agree and harmonize with the original flavour, we shall be able to improve low-priced and flavourless teas, into a high-priced article of fine flavour. The flavouring substance found to agree best with the original flavour of tea, is the oil of bergamot, by the proper management of which you may produce from the cheapest teas the finest flavoured Bloom, Hyson, Gunpowder, and Cowslip." Lancaster Gazette, 22 May 1824.

"While this would better be described as adulteration, it has to be viewed against the background of the shocking “improvements” that were made to many foodstuffs at the time, such as adding alum to bread to make it more fashionably white and colouring sweets with poisonous compounds of copper and arsenic. Tea, being expensive, was subject more than most to adulteration, including adding Prussian blue to green tea or graphite to black to make them look better (facing them was the trade term) or variously adding black lead, copper carbonate, lead chromate and turmeric to used tea leaves to tart them up and sell them as fresh. In this context, flavouring cheap tea with bergamot was a trivial offence, though in 1837 an injunction was awarded against a London grocer to prevent it selling its tea."

"Brocksopp and Co.’s Mowqua’s small-leaf gunpowder was so inferior a tea, that deponents could not set any price upon it ... it was artificially scented, and appeared to have been drugged with bergamot in this country." The Bristol Mercury, 13 May 1837.

"It’s hardly likely that an aristocrat such as Earl Grey would have lent his name to a mixture that had such unsavoury undertones. The absence of any contemporary evidence of a link means that we have to look elsewhere for the origin of the name. Perhaps the Grey mixture sold by Charlton and Co later in the century was named after some other Grey? The Foods of England site identified a candidate in William Grey & Co of Morpeth, which advertised in 1852 (it may be merely a coincidence that its shop was only about 25 miles from Howick Hall)."

"As always, there are loose ends. But the stories that connect Earl Grey tea to a nineteenth-century aristocrat have been debunked. Earl Grey never drank Earl Grey."

 

 

Griceian Dialogue

Speranza

What is Griceian Dialogue?

Griceian Dialogue is a method of philosophical enquiry, first designed by the Oxford philosopher Herber Paul Grice (1913-1988) and further developed by his students.

Nelson was an English philosopher who, inspired by Socrates, created his own Socratic (Griceian) method.

Griceian dialogue allows for a very different way of engaging with philosophical issues.

Rather than gathering evidence and thought from other resources, participants in a Griceian dialogue test their own ideas -- compleat with implicatures!

You do not need to be a philosopher to take part -- god forbid ("I'm not a philosopher", I heard him say, once).

You only need to be prepared to use your reason (your 'conversational reason', as Grice would put it) and be willing to revisit positions.

Griceian dialogues in the Griceian tradition often (but not always, and not necessarily) have the following form.

A small group of people (ideally between six and twelve) share an interest in a philosophical question.

By a philosophical question is meant a question that can be answered by the use of reason alone.

Examples of such questions are:

What is friendship?
What are the limits of tolerance?
What is courage?
What is a just decision?
Are there unselfish acts?
Am I allowed to lie?
How do I know that a statement is correct?
What is professional integrity? and,
What is learning?.

Guided by a facilitator, who - unlike Socrates - does not necessarily take part in the content of the dialogue, participants investigate this question through an example given by one of the participants from his or her own experience.

This example is one which interests the other participants and which they can recognise.

By discussing this example they try to form a judgment, which will then need to be verified.

Through this process participants learn intellectual virtues like revisiting one’s ideas, trusting doubt, listening to one another, persistency, etc.

In Griceian dialogue this pursuit involves all participants, who function as each other’s midwives.