[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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