In a message dated 9/16/2015 5:27:43 A.M. Eastern Daylight Time,
"[E]ven if we could "observe" an even number, we cannot "observe" every
even number in an infinite sequence, and Goldbach's conjecture covers
even number: and moving from a finite sample to a generalization, about
'every' in a larger sample, raises issues of "induction"".
In my previous post, I mentioned a substitutional reading for Pears's
'every' that solves the problem that McEvoy detected.
I also mentioned INTUITIONISM (since Grice was aware that empiricists were
rather narrow in what they counted as 'experience' -- qua intentionalist,
he knew that having an intention is, for example, an 'experience' --, and
mutatis mutandis, having an intuition.
But McEvoy uses an English word that logicians have formalised as
as when McEvoy writes:
"Goldbach's conjecture", that D. F. Pears, but not Russell, quotes, "covers
EVERY" (now substitutionally interpreted) "even number: and moving from a
FINITE sample to a generalisation about 'every' in a larger [IMPLICATURE:
INFINITE, ∞] SAMPLE, raises issues."
Which is just as well: as Grice says, if philosophy (Italian 'filosofia',
f.) generated no 'issues' she would be dead.
Grice once joked about
i. As far as I know, there are infinitely many stars; and every one
This is so difficult to symbolise as far as the logical form is complex
that Grice thought the implicatures would be 'otiose' -- in fact, he ranks (i)
as a "silly" thing to say.
The existence of the natural numbers is given by a first act of
intuitionism, that is by the perception of the movement of time and the falling
of a life moment into two distinct things: what was, 1, and what is
together with what was, 2, and from there to 3, 4, ...
In contrast to classical mathematics such as Russell post-Bradley
developed, in intuitionism all infinity
is considered to be potential infinity.
(And while, as Grice notes, 'what is actual is possible', surely what is
potential is not actual).
In particular this is the case for the infinity
of the natural numbers -- a class of which was the object of a rather odd
conjecture by Goldbach (He was so excited about it that on the SAME day he
wrote to Euler for advice: "Do you conjecture my conjecture is a mere
Therefore, statements that QUANTIFY (as when we use 'every', as Pears and
Goldbach do, but Russell doesn't) over this set have to be treated with
caution -- as Pears implicates.
On the other hand, the principle of induction is fully acceptable from an
intuitionistic point of view, and Pears having had conversations with
Dummett was well aware of this.
Because of the finiteness of a natural number in contrast to, for example,
a real number, many arithmetical statements of a finite nature that are
true in classical mathematics are so in intuitionism as well.
And some which are only 'provable' in classical mathematics are similarly
merely 'provable' in intuitionism.
For example, in intuitionism every natural number has a prime
There exist computably enumerable sets that are not computable.
A∨¬A) holds for all quantifier free statements A.
For more complex statements, such as van der Waerden's theorem or Kruskal's
theorem, intuitionistic validity is not so straightforward.
In fact, the intuitionistic proofs of both statements are complex and
deviate from the classical proofs.
Thus in the context of the natural numbers, intuitionism and classical
mathematics have a lot in common.
It is only when other infinite sets such as the real numbers are considered
that intuitionism starts to differ more dramatically from classical
mathematics, and from most other forms of constructivism as well.
And Pears knew this. Still he is merely making a general point that an
empiricist approach to Russell's logical atomism is consistent with a
rationalist approach (in terms of absolute atomicity) according to which the
attainment of the logical residues that Russell calls "logical atoms"
as they are, and thus better termed corpuscules) may be achieved and
_proved_, hopefully (Pears writes, putting in Russell's shoes) "one day" --
I say, he intelligently leaves the date open...
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