[lit-ideas] Grice and Popper on Intuitionism and Finitism
- From: Jlsperanza@xxxxxxx
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Sat, 8 Jun 2013 10:50:54 -0400 (EDT)
Or something like that. I'm looking for the right kind of keyword. (Even if
not finding it).
In "Re: Grice's Infinity, Popper's Infinity", in a message dated 6/8/2013
5:35:56 A.M. UTC-02, donalmcevoyuk@xxxxxxxxxxx writes:
OMG. How did JLS know of this "footnote"? Has JLS actually looked at the
book? [There is such a footnote but it hasn't been mentioned on THIS LIST I
think.]
Perhaps it wasn't but I thought it was an interesting use, by Popper, of
the idea of 'meaning' -- as in "meaningless".
For:
Mathematics is a SPECIAL branch of stuff -- and it shouldn't provide much
of a parameter when dealing with ontology. For, indeed, there is nothing
ILLEGAL about Euclid's Theorem becoming "meaningless" in a finitist model. If
Popper were really serious about this, he should provide a total reductio
ad absurdum of a finitism, which he can't (or Kant).
---- In other words, mathematics (including Euclid's theorem) is a set of
tautologies. So it becomes 'meaningful' or "meaningless" according to the
postulates one chooses to adhere to.
---- The fact that Popper needs to focus on mathematical tautologies to
give 'existence' to what he calls World 3 will hardly impress the empiricist
who is a finitist and an intuitionistic to boot.
---
McEvoy goes on:
"This World 3.3 terminology, or even W3.3 to be briefer, was mentioned in
my posts."
I'm glad to learn. Sorry I missed that.
I thought it was a good terminology.
"Popper uses it in his Schilpp volumes. Sooner or later it is best to use
it, because there is otherwise the temptation to think that when W3 content
is embodied it becomes physical content i.e. that W3 content when embodied
becomes W1 content. This is not Popper's view afaiunderstand. W3 content
may be embodied but does not lose its character as W3 content: the W3 content
never becomes W1 content."
I see. Of course one could take more seriously the idea of a Venn diagramme
here. For W3.3 is that bit of a triadic Venn set, and Popper should
provide perhaps a more reasoned (or general) proof of the existence of
non-intersected bits -- of W3 -- rather than _by example_ (casuistic). Note
that his
alleged proof of W3.3 relies on psychological terms, as when Popper notably
uses "intuitive grasp", in the segment cited by McEvoy:
"Euclid solved this problem. Neither the formulation of the problem or the
solution of the problem was based on, or could be read off from, encoded
World 3 material. They were based directly on an intuitive grasp of the World
3 situation: of the infinite sequence of natural numbers."
McEvoy goes on to refer to the link:
In this connection the urled piece says two things:-
"It is recognizable that World 3.2 cannot be reduced to World 2: the
subjective thinking processes (which belong to World 2) and the objective
contents of such processes (which belong to World 3) are nonequal, and it is
widely believed that people cannot grasp the objective contents as such,
without any subjective interpretation"
and comments, rightly:
"We should truncate this to remove the idea that we "cannot grasp the
objective contents as such, without any subjective interpretation": for I think
Popper's example of Euclid's theorem is meant to show a case where Euclid's
own subjectivity manages to "grasp the objective contents as such" - and
Popper does not see any diametric opposition between a 'grasp' being at once
subjective in some sense and yet of objective contents as such."
Good McEvoy focuses on this, since it relates to the segment cited above of
the "intuitive grasp" -- Note that Popper unfortunately uses 'intuitive'
which is the KEY term for INTUITIONISM, which can develop into a finitist
model where
"∞"
or for that matter,
"ℵ" (aleph).
Should Popper have cared to symbolise his claims using these handy symbols
for different aspects of infinity would have helped, for, as Hilbert
points out, it all 'boils down' (metaphorically) to whether one allows
quantification over this kind of stuff -- something Euclid (but not Russell)
was
unaware of -- as he had what Popper dignifies with the name of an 'intuitive
grasp' that attempts to prove his triadic nonmonism).
Indeed, for David Hilbert (whom Popper knew), 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 )whom Popper knew) 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.
Note that, by Harvey Friedman's grand conjecture most mathematical results
should be provable using finitistic means (as Euclid would be pleased to
learn).
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.
Today most classical mathematicians (except those who've heard of Grice)
are considered Platonist and believe in the existence of infinite
mathematical objects and a set-theoretical universe.
Grice speaks of 'deem' as the KEYWORD here.
For Euclid may have had some sort of a 'grasp' of
"∞"
or
"ℵ" (aleph).
--- Similarly, the utterer who says, sillily,
"As far as I know, there are infinitely many stars"
may THINK he has a sort of an intuitive grasp of infinity.
Note that if we turn Grice's example of a 'silly thing silly people say'
("As far as I know, there are infinitely many stars") into a mathematical
example, it turns out that Euclid _was_ silly:
"As far as I know, there are infinitely many prime numbers".
---- ("It's better to stick with _stars_, since, as Frege knew, we hardly
_know_ what a plain number is").
----- Popper plays with this when he goes on to contradict the well-known
adage, "Natural numbers were created by God" and propose an anthropocentric
variant of it: "Quite the contrary: numbers are the product of language".
In a phrase that is bound to offend (some) anthropologists, he goes on to
mention that according to some 'primitive languages', people cannot count
higher than '3', and divide numbers into "1, 2, and many".
Some languages have a very limited set of numerals, and in some cases they
arguably do not have any numerals at all, but instead use more generic
quantifiers or number words, such as 'pair' or 'many'. However, by now most
such languages have borrowed the numeral system or part of the numeral system
of a national or colonial language, though in a few cases (such as
Guarani), a numeral system has been invented internally rather than borrowed.
Other
languages had an indigenous system but borrowed a second set of numerals
anyway. An example is Japanese, which uses either native or Chinese-derived
numerals depending on what is being counted.
Not all languages have numeral systems. Specifically, there is not much
need for numeral systems among hunter-gatherers who do not engage in commerce.
Many languages around the world have no numerals above two to four—or at
least did not before contact with the colonial societies—and speakers of
these languages may have no tradition of using the numerals they did have for
counting. Indeed, several languages from the Amazon have been independently
reported to have no specific number words other than 'one'. These include
Nadëb, pre-contact Mocoví and Pilagá, Culina and pre-contact Jarawara,
Jabutí, Canela-Krahô, Botocudo (Krenák), Chiquitano, the Campa languages,
Arabela, and Achuar. Some languages of Australia, such as Warlpiri, do not
have
words for quantities above two, as did many Khoisan languages at the time
of European contact. Such languages do not have a word class of 'numeral'.
--- end of NUMERAL interlude -- discussed briefly by Levinson in his book
on "Implicature".
McEvoy goes on:
"Nevertheless, it is interesting and on the right lines that the author
notes "that World 3.2 cannot be reduced to World 2: the subjective thinking
processes (which belong to World 2) and the objective contents of such
processes (which belong to World 3) are nonequal""
and comments:
"In particular, for Popper, they are non equal because the possibility of
critical revision of such processes depends on converting merely "subjective
thinking processes" into "objective contents of such processes", for in
this way the "objective contents" become inter-subjectively criticizable in
the light of standards like truth or truthlikeness."
However, Popper's prose tends to be too informal to some, as when he
dismisses empiricist philosophers for claiming that such 'objective content' is
a mere 'fancy of the brain' (quotation needed). Popper is using a strawman
and simplifying the view of his opponent, since 'fancy of the brain' is his
own, not his opponent's actual way of approaching these complex issues.
McEvoy goes on to quote from the link:
""The unmaterialized objects of World 3 can affect on World 1 only, if a
human mind or some machine (computer) recognizes them. (Question: can a
computer recognize such object, without any human intervention?)"
and writes:
"Just yesterday I posted on this point: computers can never recognise W3
content as such, only humans with a W2 can."
Well, one thing that Popper should address, and Grice did, is that terms
like 'soul' do not JUST belong in animals like MAN. Surely there is some sort
of 'sequence' as Aristotle saw it (never mind 'evolutionary'). Just as
'number' gets meaningful when considered in a sequence, so, Aristotle (and
Grice) argued, does 'soul'.
Yet, as we do not need to postulate a W3.3 in, say, a cat's perception of
milk, the typical empiricist approach runs as to avoid the postulation of an
independent or autonomous W3.3 in the case of just one species amongst
many: homo sapiens sapiens.
(Why would abstractions from homo-sapiens-sapiens' contents of
psychological attitudes require a different realm of reality?)
McEvoy goes on:
"The apparent recognition by a computer of W3 content, such as "2 + 2 = 4",
is not actual recognition by the computer of such content in its W3 terms
but our being able to interpret what the computer has processed in W3 terms
- i.e. to convert what the computer has processed merely at a W1 level
into the terms of its W3 significance. Say we put a computer to task of
sorting data to find whether there is any contradiction in the data and the
computer reports it has found a contradiction: the computer is simply
following
its encoded instructions and does not grasp what a contradiction is in W3
terms - it simply has a programme which it uses to detect contradiction by
surveying data put in W1 code and then alerting us when it locates data of
both an 'x' and 'non-x' sort i.e. a contradiction. It only recognises this
'contradiction' in this sense that it has a W1 programme to detect and
report when it comes across date of this contradictory sort; but it does not
recognise contradiction at all in W3 terms for it lacks any W2 grasp of the W3
idea of contradiction."
Examples from computers while lovely pose a Searlean type of illustration
to Grice.
As we know, Grice distinguished between 'mean' as in "I know what you mean"
and a broader use of 'mean' (improper, strictly) as when we say, "Black
clouds mean rain".
It may be argued that there are things which the computer "means" (but
McEvoy and Popper denies) in this 'sense'.
Or not.
Cheers,
Speranza
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