[hist-analytic] A 'Series' of Anscombianisms!
Roger Bishop Jones
rbj at rbjones.com
Fri Feb 6 11:26:14 EST 2009
On Friday 06 February 2009 13:32:05 Jlsperanza at aol.com wrote:
<...>
Is Wittgenstein using the "terms" sequence and series interchangeably?
> I am, typically, in a rush, so I'll end this with (luckily) what I think is
> the relevant 'use', which the OED has as "Math." and defines (I don't
> usually do OED for definitions, but here you are) as
>
> a set of terms in succession
> (finite or infinite in number)
>
> [*not* _pace_ Dummett!]
>
>
> the value of each of which is determined
> by its ordinal position according to a
> definite rule known as the
>
> law
>
> of the series; [Latin 'lex', don't think so. I was
> examining that English 'law' has not really
> Latin cognates. JLS]
> esp. a set of such terms continuously
> added together.
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").
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.
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 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.
Roger Jones
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