On 2011-05-19 14:01, Simon Wise wrote:
> That is which numbers are directly perceivable, without some more
> abstract mathematical mapping to guide us?
Zero ;-)
> Certainly most people can look at four matches on a table and see that
> there are four, without doing any counting at all.
That's "four matches", not "the number four". If by "number" you mean
the characteristic property of all sets of 4 elements, you're perceiving
something (the matches) which has that property, but you can't directly
perceive the property itself (i.e. its `intension'), because it's a
function (in the mathematical sense) from all possible entities (let's
ignore possible worlds for now) to a truth value indicating whether or
not that entity is a set-of-four. This view is pretty unsatisfying for
a number of reasons (for one thing, it doesn't work well for anything
other than positive integers), but I hope it suffices to show that "the
number four" can't be perceived directly. The same sort of argument
goes for other "simple" qualities like volume, mass, density, color etc
-- this stuff has had epistemologists tearing their hair out for
centuries. There are 2 main camps, and I'm more or less solidly in the
one that says "numbers exist" :-)
> In some languages, where mathematics hasn't become part of the language,
huh? do you happen to know of one specifically?
> and the words for numbers are pre-mathematics, counting goes something
> like "one, two, three, four, many"
... many one, many two, many three, many four, many many,
... many many one, many many two, many many three, many many four,
... LOTS
[courtesy of Terry Pratchett] ;-)
> so I guess that backs up the idea
> that the first few integers are perceived directly,
Again, I take issue with the details, but yes: there's a lot of
empirical evidence that human cognitive/perceptual apparatus does some
specialized handling for small sets, including counting. How we get
those sets to be __sets__ (as opposed to arbitrarily co-occuring random
perceptual data packets) is quite another matter, and im(ns)ho a much
more interesting one.
> but every other
> number - counting numbers past that, zero, negative integers, the rest
> of the rational numbers, the rest of the real numbers, the rest of the
> complex numbers, ... and so forth are all just constructs in the
> language of mathematics which all happen to have some quite useful
> mappings to things we can observe around us. Most integers do not have
> any more 'existence' (however that may be defined) than complex numbers.
I'll agree that integers and complex numbers have the same sort and
degree of existence, but I don't believe they're `constructs'. If
forty-two trees fall in a forest and no one is around to count them,
__forty-two__ trees have still fallen.
marmosets,
Bryan
--
***************************************************
Bryan Jurish
Deutsches Textarchiv
Berlin-Brandenburgische Akademie der Wissenschaften
Jägerstr. 22/23
10117 Berlin
Tel.: +49 (0)30 20370 539
E-Mail: jur...@bbaw.de
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