[lit-ideas] Re: Grice's Realm
- From: Jlsperanza@xxxxxxx
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Mon, 10 Jun 2013 01:58:46 -0400 (EDT)
In a message dated 6/10/2013 2:30:57 A.M. UTC-02, R. Paul, writing for the
Cantor Institute, writes about
"∞"
and "א"
>you've left out 'Aleph null,' or
'Aleph sub-naught,'
Indeed. Sorry about that.
This should do (or _could_ do, seeing that the 'zero' is not yet quite
subscripted):
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.
Note that 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
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’
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)
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 the existence of the infinite.
From the geometrical source of knowledge flows the infinite in the genuine
and strictest sense of this
word . . . 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:
It is evident that sense perception can yield nothing infinite. 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 sucient 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).
Note that Frege, like Popper, must otiosely distinguish between (where
'between' is misused, since it strictly applies to TWO realms only, not three),
or 'among':
-- 'infinite' as it applies to what Frege calls the second realm (of
material objects) and Popper calls the first world of material objects -- as in
Grice's example: "As far as I know, there are infinitely many stars -- in
the sky".
-- 'infinite' as it applies to what Frege calls the first realm
(psychological processes) and Popper calls the second world. "This
mathematician knew
infinity".
-- 'infinite' as it applies to a 'third realm' (Frege) or 'third world'.
Since Grice's motto is: do not multiply realms (or worlds) beyond
necessity, the Fregean difficulties are solved.
Cheers,
Speranza
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