[lit-ideas] Re: Principia Mathematica
- From: Jlsperanza@xxxxxxx
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Sun, 26 Jan 2014 07:04:11 -0500 (EST)
In a message dated 1/26/2014 6:15:25 A.M. Eastern Standard Time,
Palma@xxxxxxxxxx writes:
>the demon has no way to change necessary truths
in reply to O. K.'s interesting bringing in Descartes's demon (who is
always 'malign', as I recall, as opposed to the Greek 'eudaimon' that makes us
happy).
It may do to go back to McCreery's original quote then:
>I am, oddly enough, reading a very long book:
>Isabelle Stengers. Thinking with Whitehead: A Free and Wild Creation of
Concepts. Harvard University Press, 2011
So the keyword is "Whitehead", and I guess the rhetorical answer to
McCreery's rhetorical question, "Shall we consider another philosopher?" is:
"Yes, A. N. Whitehead.
McCreery notes:
>According to Stengers, Whitehead's reputation almost disappeared during
the triumph of analytic >philosophy following World War II.
And I'm not even sure Whitehead was considered a PHILOSOPHER while at
Cambridge. I think he attained an official philosophical post only in Harvard.
But I should re-read Wikipedia:
"In 1880, Whitehead began attending Trinity College, Cambridge, and studied
mathematics. He earned his BA from Trinity in 1884, and graduated as
fourth wrangler. Elected a fellow of Trinity in 1884, Whitehead would teach
and
write on mathematics and physics at the college until 1910, spending the
1890s writing his Treatise on Universal Algebra (1898), and the 1900s
collaborating with his former pupil, Bertrand Russell, on the first edition of
Principia Mathematica. In 1890, Whitehead married Evelyn Wade, an Irish woman
raised in France; they had a daughter, Jessie Whitehead, and two sons,
Thomas North Whitehead and Eric Whitehead. Eric Whitehead died in action while
serving in the Royal Flying Corps during World War I at the age of 19. ...
The period between 1910 and 1926 was spent mostly at University College
London and Imperial College London, where he taught and wrote on physics, the
philosophy of science, and the theory and practice of education.[58] Toward
the end of his time in England, Whitehead turned his attention to
philosophy. [... He] had no advanced training in philosophy..."
So perhaps McCreery could implicate:
"Shall we consider another mathematician?" :)
McCreery continues:
"For years, the only people who took him seriously were American
theologians."
Note the emphasis on "American". Oddly, Quine sort of resurrected Whitehead
with his 'Gavagai' examples. He was saying that we cannot understand, alla
Sapir-Whorf thesis, ideas about other cultures (or that HE could not
understand). Quine goes on to argue that "Gavagai" may refer not to a
rabbit-object, but to a rabbit-process. And at this point, I think Whitehead's
EVENT
or PROCESS philosophy, perhaps influenced by Native American cosmologies,
came into view. Or not. So, he ended up being seriously taken by
COSMOLOGISTS and metaphysicians, at last?
McCreery:
>Stengers herself is an interesting figure...
>she ... has many provocative things to say.
At this point McCreery provides the quote that seems to implicate again,
"Shall we consider a mathematician?" -- and not a philosopher.
Stengers:
"Whitehead was a mathematician, and it
is no doubt because he was a mathematician, because
he knew and loved the way mathematics forces mathematicians to think, but
all knew the rigorous constraints to which every mathematical definition
must respond,
that he never thought that mathematics could constitute a model that was
generalizable.
The kind of necessity proper to mathematical demonstrations
cannot be transferred to philosophy."
In my "Principia Mathematica" first post I empahsised the programme behind
"Principia Mathematica". It is generally regarded as a PHILOSOPHICAL,
rather than mathematical, work (Grice loved it) that aimed at reducing
mathematical concepts (notably algebra) to FIRST-order and second-order
predicate
calculus of the type that PHILOSOPHERS were for long concerned with.
And I was wondering about the irony of the quote:
For it is only a PHILOSOPHER (who issues a philosophical claim) who can
state that the kind of necessity proper to mathematical demonstrations cannot
be transferred to philosophy."
The paradoxical character of this can be made explicit by expanding on the
quote:
"The kind of necessity proper to mathematical demonstrations [sic] cannot
[sic -- cfr. 'should not'] be transferred to philosophy, including the
philosophy of mathematics one of whose corollaries is that the kind of
necessity
proper to mathematical demonstration is unique."
In other words, it is the PHILOSOPHY of mathematics, and one type of the
philosophy of mathematics, that comes up with such a statement. Variants
include:
According to some philosophies of mathematics, there is a lot of
non-deductive reasoning in mathematics" or "non-deductive mathematical
reasoning".
According to some philosophers, the kind of necessity TYPICAL of
mathematical demonstration MAY And SHOULD _show_ in areas other than algebra;
notably 'logic', on which algebra depends -- according to Whitehead/Russell,
authors -- in that order -- of "Principia Mathematica" -- and morality --
Spinoza 'more geometrico --. And cfr., to echo O. K., Descartes's work on the
coordinates.
---- Let us be reminded of Stenger's book title:
>Thinking with Whitehead: A Free and Wild Creation of Concepts.
Note that she wrote in French, and so some plurals in French seem best
expressed in the singular in English:
>"The kind of necessity proper to mathematical demonstration"
[I would say, rather than what she and her translator has, "demonstrations"
in the plural]
"cannot" [but then perhaps she meant 'MAY not' -- there is no such
distinction between 'may' and 'can' in French]
be transferred to philosophy."
---- as if somebody asked that it should!? :)
Creery goes on to quote from the translation to Stenger's book:
"Philosophical reasoning that tries to be demonstrative in this sense could
only produce an imitation unworthy of the adventure that, for
mathematicians, is constituted by the production of a demonstration."
This seems rather circular. It could be turned into a non-circular
statement by using 'pseudo-demonstrative'. But I won't!
Stengers:
"What is more, in order to conform to the logical-mathematical model, such
reasoning would require the goodwill of the readers, their submission to
definitions that are simplistic compared with the extraordinary subtlety both
of the situations and the usages of natural language as it confronts these
situations."
This seems to obviate Grice and I'm surprised that Stengers comes from the
Belgium (and shall I say near the Dutch-speaking area), because it is a
linguist in that area who wrote a whole book about Griceian concepts of number:
ABSTRACT: INTERLUDE ON A BOOK ON THE IMPLICATURES OF MATHEMATICAL DISCOURSE
-- which may well apply to what Stengers may have in mind when she talks
of the 'extraordinary' (why not simply ordinary, alla 'ordinary language
philosophy') subtlety behind, say, the use of 'two' in "two apples":
Outlandish as it may seem to the uninitiated, the meaning of English
cardinal numbers has been the object of many heated and fascinating debates.
Notwithstanding the numerous important objections that have been formulated
in the last three decades, the Gricean, scalar account is still the
standard semantic description of numerals.
Bultinck writes the history of this implicature-driven approach.
Bultinck's "Numerous Meanings" is a unique contribution to the semantics
and pragmatics of cardinal numbers, taking Grice's theory of implicatures to
its limits, and raising numerous and original questions about meaning along
the way."
----- END OF INTERLUDE.
McCreery continues with Stengers:
Stengers writes:
"Such simplistic definitions, which mutilate questions, would be the price
to pay for an approach that would finally be rational. As a
mathematician-cum-philosopher, Whitehead transferred from mathematics to
philosophy not
the authority produced by demonstration, but the adventure and commitment to
and for a question, the “bad faith” with regard to every “as is well know,
” all consensual plausibilities."
Well, no wonder he was taken seriously by philosophers. Let's rewrite the
above trying to reconstruct the original French!
"Simplistic definitions" is the simple term in 'semantic analysis' for
Carnap's meaning postulates, as when we say,
"a bachelor is an unmarried male".
Or when we look for nomologico-deductive proofs alla Reichenbach,
"All crows are black".
It's via semantics (and pragmatics) that philosophers proceed. Simplistic
is better than overtly complicated, at THIS point.
It's not clear what Whitehead transferred to HIS TYPE of philosophy:
"the adventure and commitment"
(apples and pears?)
"to and for a question"
"the 'bad faith' -- who coined this phrase? Can it please theologians? :)
"with regard to every "as is well known""
-- this should sound better in French, since "as is well known" is NOT a
motto used by philosophers A LOT.
And here she (or her translator) adds, "all consensual plausibilites",
which seems like an overt defense (or defence, if you must) of relativism.
In his second post on this McCreery grants the rightness of O. K's and R.
P.'s observations and adds:
"What Robert and Omar said. Let us agree that we understand what Aristotle
meant and that he was talking about ethical/political debate, where
demanding scientific proofs from a rhetorician is as foolish as accepting only
probably proof from a mathematician. What, then, of Whitehead's stronger
claim, that the kind of necessity proper to mathematical demonstrations cannot
be transferred to philosophy? And if mathematical demonstration cannot be
expected of philosophy, what alternatives do we have for judging the quality
of a philosopher's work?"
As I note, Stenger's prose is best left in French! I would not speak of
'mathematical demonstrations' but 'demonstration' in the singular, and would
think overpresumptuous of a mathematician-cum-philosopher or
philosopher-cum-mathematician ("neither fish nor fowl"?) to speak of ALL
mathematical
demonstration. It may be that probability theory, a branch of mathematics
never, by definition, displays that 'kind of necessity'. Mereology, either.
And
of course 'intuitionism' and the mathematics and logics that it produces
is still a different animal.
McCreery:
"If mathematical demonstration cannot be expected of philosophy, what
alternatives do we have for judging the quality of a philosopher's work?"
Well, if that philosopher's work is a work on the philosophy of
mathematics, like Whitehead's and Russell's "Principia Mathematica" is, then we
may
end doubting as to whether we can use as a premise in our condemnation of
philosophy the outcome of a philosophical kind that advises as to what kind of
'necessity' is proper to this and that.
If Aristotle was a master of ethics and politics, he was also the master of
what Tom Weller (and Geach?) called the Sillygistics!
All mathematical reasoning is demonstrative.
This type of necessity can not be transferred to philosophy.
But it should.
----
So let's get a copy of Spinoza, "More Geometrico".
Or not!
>"what alternatives do we have for judging the quality of a philosopher's
work?"
There are zillions. But let us stick with Whitehead's misexpressed one:
"the adventure and commitment to and for a question, the “bad faith” with
regard to every “as is well know,” all consensual plausibilities."
Surely there are other outcomes to Whitehead's alleged challenge:
Stenger:
"I]n order to conform to the logical-mathematical model, such reasoning
would require the goodwill of the readers, their submission to definitions
that are simplistic compared with the extraordinary subtlety both of the
situations and the usages of natural language as it confronts these
situations."
Not if we are armed with Grice's idea of implicature. The idea of
implicature is there to help the philosopher in these cases.
A CALCULUS of the mathematical type (e.g. for the use of 'and', 'or',
'not' and 'if' -- alla 'natural deduction' all Gentzen (vide Wikipedia) SHOULD
be adopted, and ANY DIVERGENCE between 'this' mathematical 'usage' of
'and', 'or', 'if' and 'not', 'all', 'some', 'the', etc. is explained in terms
of
implicature. Take the concept of number:
"He showed me two apples"
---- Therefore, he showed me one apple."
Bultinck notes that there ARE complications with Grice's or a Griceian
(since this was not a topic that Grice particularly explored) idea of 'two' (or
'three' for that matter) -- but the IDEA is _there_. Cfr. the colloquial:
I should get back to you, "in a minute OR TWO". Or not.
Cheers,
Speranza
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