Re: The modern mathematical concept of infinity is ...

Jesse F. Hughes wrote:
Ralf Bader <bader@xxxxxxxxxx> writes:

Jesse F. Hughes wrote:

Perhaps Isles has a mathematical theory
(I wouldn't know),

I wouldn't call it a theory (theories, for my feeling, are book-size, not
paper-size). [....]

Thanks much for the summary of Isles and its (non-)relation to WM's
ramblings.

Hmm. Here's a paragraph from Isles' paper (page 2):

--- QUOTE ---
It is true that the notation 2^65536 is of the same kind as the term ||
^|| or
||^(||^A||) for which the analogous computations do terminate but what
does that
prove? In other areas of life we know that parallelism of form, and
even par-
allelism with respect to forms of processes, does not imply similar
behavior (un-
less the parallel is "exact"; but the appropriateness of the adjective
in the present
situation is what is being challenged here). After all, the activity
of swimming
is the same whether it is done in a pool or in the ocean. Yet from the
fact that
I can swim across a pool it hardly follows that I can swim across an
ocean.
--- END QUOTE ---

This seems to be a denial of mathematical reasoning. "Induction might
not
work when we get to Truly Enormous numbers, because we've never been
there"...

But is he really discussing linguistics? He appears to claim not,
citing John Locke (1632 - 1704) as an authority to convince us that
numeral notation is what mathematics is really about. Perhaps that
_is_ what "mathematics" was about in the 17th century? (This is all
rather reminiscent of WM's modus operandi, isn't it?)

One reason this interests me is that I think the reverse confusion
also happens. Linguistics is "the scientific study of natural
language" (Wikipedia), and natural language is a facet of the natural
world. Thus linguistics is a natural science, affected by evidence: Do
Japanese verbs agree with anything in number? Do Bantu languages have
a case system? Natural sciences don't (and can't) start from axioms,
and don't really support extended chains of logical inference -- in
particular, reduction ad absurdum is not reliable unless every
inference is absolute (as it would be in maths). If we develop a
branch of mathematics called "Formal language theory" (or similar), we
can use this as a mathematical model of natural language, up to the
point at which it is useful, but we cannot use this mathematical
theory to make empirically unsupported (or unsupporatable) claims
which are genuinely about natural language. But commonly this sort of
claim occurs, leading to discussions with titles like "Is Finnish
morphology infinite?"

(Or equivalently, "Is Japanese morphology infinite?", which I can do.
Japanese, like Finnish, has a plug-n-play system of verb suffixes,
including a causative one we'll call 'sase'. Thus:

taberu : means 'eat'
tabe-sase-ru : means 'cause to eat', as in "feed the children"
tabe-sase-sase-ru : means 'cause to cause to eat', as in "get the maid
to feed the children"
.... ["and so on", or not!]

It is normally possible to persuade a native speaker to agree that
tabesasesaseru "is Japanese". But the putative argument is whether
tabe-sase^n-ru "is Japanese" for any natural n. The formal model you
get by writing the obvious grammar in Prolog obviously allows any n,
because (a) it's much easier to do and (b) any limit would be quite
arbitrary. But clearly once the value of n reaches double figures, the
result is something with no relation to the actual Japanese language
as spoken. In this case, to claim that tabe-sase^i-ru (where i is
"Isles's number") is not a Japanese word seems pretty reasonable.

Nonetheless, in a spectacular culmination to this "speculative
extrapolation", there is a whole book (surely makes it a "theory"?) by
Langendoen and Postal (two respected linguists) titled "The Vastness
of Natural Languages", which purports to show that the cardinality[?]
of the set of sentences of Japanese is actually larger than any
cardinal. They are careful to repeat the term "NL", for "natural
language", to make it clear their claim relates not to some formal
mathematical objects, but to, well, natural languages. This totally
batty thesis is supported by lots of arm waving, appeals to Occam's
razor, and the oldest trick in the book: "We can't actually prove P,
but we assert that you can't disprove it either, so meeurrgh..."

Well, there seems to me to be some basic confusion going on here.

Brian Chandler
.

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