[hist-analytic] The Bride of Kripkenstein
Jlsperanza at aol.com
Jlsperanza at aol.com
Fri Feb 6 20:52:33 EST 2009
Wittgenstein: The Rule ('Gesetz') of the Series ('Reihe')
Who _is_ the Bride of Kripkenstein?
In a message dated 2/6/2009 1:02:06 P.M. Eastern Standard Time,
rbj at rbjones.com writes:
My impression of mathematical usage is that "sequence" is the
"set of terms in succession" (though they need not actually be
"terms") and then a series is a (possible) value expressed as a sequence of
summands (it will only be an actual value if the series "converges").
--- Thanks.
Contrary to the OED, I don't think in either case that the
sequence has to be "lawlike", it might not comply with any rule,
though the ones mathematicians study usually are lawlike since that
kind of sequence is generally more useful.
However, when quantifying over these things you have to take into
account the uninteresting ones too.
---- Exactly. And I think what Wittgenstein is really interested in is what,
in Chomsky's later parlance, we'd have as
*rules
_versus_
* representations
and of course the diverse ideas behind
-- knowing the rule
-- knowing (or 'cognising' as Chomsky irritably prefers) the representation.
----- going down to M. Davies et al. ideas on 'tacit' knowledge,
and Eddington on 'least' (cited in OED under 'least effort') and Grice's
"Principle of the Minimisation of Rational Effort". "The fact," Grice would say,
"that we do not appeal to the rule _explicitly_ should entail that we appeal
to the rule _implicitly_: my ruly thoughts are hardly 'subterranean',
neither are my unruly ones!"
R. B. Jones continues:
For example if a mathematician says something like
"if there exists a sequence" he will very rarely mean law-like, since
that is not a very precise term, and if he did want to say something
like that he would have to stipulate what "lawlike" meant,
e.g. "if there exists a recursively enumerable sequence", where the notion
of law involved is quite definite.
--- I see. Wittgenstein uses, then, 'Reihe' (I've just checked) for
'series', and "Gesatz" for 'rule' or law (I think I consulted the _new_ English
translation!). Then there's 'infinite', for which Wittgenstein uses the rather
poetic, 'endless'. He does speak of 'series' of numbers only (Zahlen) and he
uses 'algebraic formula', but I think it's Algebraisch Einsdruck' in German
which looks more like 'expression' than formula to me.
I forget what he uses for 'pupil'!
---
I don't know much about Wittgenstein's philosophy of mathematics,
but since he is alleged to have been rabidly opposed to set theory,
it is probable that he did not acknowledge the existence of infinite
sequences which do not follow some rule, however in this he differs
from most contemporary mathematicians.
--- Yes, I noted a bit in the relevant passages (that S. Bayne quoted) and
neighbouring ones. He does use 'infinite', as I say, 'endlessly' (but wasn't
the mathematical infinite mainly a progress by the German school of Cantor,
etc.?)
At this point, Wittgenstein was possibly rabidly opposed to Aristotle as
well. So I don't think he would acknowledge anything having to do with
'potential' versus 'actual' (infinites). He seems to suggest, perhaps
common-sensically, that what a pupil (or 'kid' as S. R. Bayne has it) does draw is _finite_.
But students nowadays have perhaps grown wittier:
As in recent "Doubt" with Meryl Streep:
(set in the 1950s middle school in the Bronx):
"You go and write a hundred times, "I should stay silent in class"
Nowadays the student would just provide the 'algebraic' rule -- they are not
even required to _write_ anymore; just type! And with cutting and pasting
it's never clear if they did write the thing '100' times.
If it's a mathematical torture, I would think it _is_ ridiculous for a
teacher to impose on a student an 'infinite' task, but it is notably _not_ unfair
(or ridiculous) to have the student use that fabulous sign of the
mathematicians, for 'infinite'.
the inclined "8"
and so I can explicitly ask my pupil to bring for the next day _five_
statements involving the 'infinite' (i.e. the inclined "8").
Borges wrote, incidentally, "An Abridged Short of the Infinite" (or
eternity) and he was prone to remind that he got it all that from his father playing
with him on Zeno -- the Eleatic -- and his paradoxes!
Cheers,
JL
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