[PD] CVs

[Simon Wise]

On 19/05/11 23:12, Bryan Jurish wrote:
> 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 ;-)

My point is that it is not zero, that looking at a pile of things and saying 
that it is a pile of one or two or three is something that can be observed 
without a mathematical model, without even counting or any more elaborate 
language than these few words for these small quantities. But they have become 
so wrapped in the rest of our words for numbers that it is difficult to make the 
distinction. Think about of what words like pair mean, is pair a number? is it a 
synonym for two? or is it a directly observable quality which is quite different 
from either a single thing or a few things? Or thinking about the distinction 
between singular and plural forms of words. This is why the languages I 
mentioned are so interesting in this context.

>
>> 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

this is the core of what I am saying - that three or four are something other 
than  the result of counting the members of a set, and that for some unusual 
people quite surprisingly large numbers are perceived directly, independently of 
the process of counting. Occasionally the different status of these 'numbers' in 
language can be seen, they can be seen as words for some observable quality 
rather than as the first few of an infinite series of integers, used to describe 
a characteristic of sets of things.

> 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" :-)

I am also in this camp, models do 'exist' in the way I use the word exist, but 
there are other ways to use this word, and so discussion gets tricky. I was 
suggesting that small counting numbers are a different kind thing to the other 
quantities listed here. They are observable in a different way, without the 
constructs that other measures require. They fit into a much narrower definition 
of exist than the others.

What I am saying about numbers is referring to your earlier remarks about pi and 
sqrt(2) in the context of discrete v continuous. Sqrt(-1) has a very practical 
and useful place in talking about physical spaces, it exists in exactly the same 
way the 1,734,834 exists. If you leave irrational numbers like sqrt(2) out of 
your model for describing lengths, and limit the non-integer numbers you talk 
about to those you can construct by divide two integers, then you get into 
trouble because those 'rational' numbers are not continuous, there are gaps 
between each one. Whether the possible values of 'length' is a continuous thing 
or a series of discrete possible values depends hugely on context, but models 
using continuous values are very useful all the same.

But I think that very small counting numbers do have a directly observable 
nature, the quantity four is recognisable without using some way of counting in 
a way that larger numbers are not. This becomes much more noticeable with the 
rare person who can just see a pile of 51 things as the same as another pile of 
51 things, and different from a pile of 52 things, without counting them or 
calculating or constructing the number in some way. The numbers here are not 
derived from counting the objects, but are some quality recognised directly in 
the pile as a whole. The words for numbers in the languages mentioned seem to 
suggest that those words may be referring to this observable quality of a group, 
rather than being part of a counting system and a way of talking about numbers 
more generally. I think I read about the person who could 'see' 52 in one of 
Oliver Sacks books, and maybe elsewhere as well.

>
>> In some languages, where mathematics hasn't become part of the language,
>
> huh?  do you happen to know of one specifically?

I can't recall the details of the examples given, but there were a number of 
languages with this kind of counting, that is with no words to quantify sets of 
things with more than a few elements, the biggest number before 'many' varied - 
I recall something like 3 and 5 in the ones described. Unfortunately this was a 
book I read a long time ago, and my books are thousands of kilometres away in 
Sydney. Obviously they were languages spoken by people without the kind of 
accounting and writing of records which some suggest were the motivation for 
some of our earlier numbering systems. There are many many languages spoken in 
the world, and in places like New Guinea there are groups who first met or heard 
of anything outside their very local area in the last 80 years. I have a 
remarkable set of oral history tapes, a series produced 20 or 30 years ago by 
the ABC here, called Time Belong Masta I think, with the recollections of many 
people who where the first outsiders to visit these places and some of these 
language and similar issues are discussed.

For a sense of how isolated a group can be in this kind of countryside one story 
was about a person who had travelled for the first time outside their valley, 
when the group reached a river not very far away this person had never seen or 
heard of water existing in such quantities, he had no words for it and it was 
completely beyond anything that his society had any knowledge of. The examples 
above may have been from New Guinea, but certainly somewhere as isolated as this.

>
>> 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] ;-)

The ideas we are talking about are interesting - the language used for numbers 
crops up in lots of speculative fiction - sci-fi, fantasy etc  for exactly this 
reason.


>
> 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.

yes, they have, and the number that fell is not dependent on someone counting 
them, but as the numbers get higher or more sophisticated then the ways to 
assign numbers become more abstracted, more dependent on the mathematical 
language. The quantities exist, the numbers that we use to describe them exist, 
the connections between these things often requires a model, this is what I mean 
by 'constructs'. There may well be better words to use for this.

Simon