"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'.