On 2011-05-20 16:05, Simon Wise wrote:
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,
Sorry; that was intended as a joke -- a deliberate ambiguity between zero ("the number zero"), zero ("the set with zero elements"), zero ("false"), and zero ("number of numbers which are 'directly perceivable'"). It was late, I thought it was funny.
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?
"Pair" is a word of English, and a highly ambiguous one at that -- it might be an ordered pair, an unordered pair, a pair of pants, a pair of aces, 'a pair' (aka "couple"), or whatever. Yes, it's semantically and pragmatically complex. The (abstract) number "2" plays a pretty heavy role in all of its sense I can think of at the moment, though.
Or thinking about the distinction between singular and plural forms of words.
What about them? They're usually related by quite simple and obvious rules (e.g. 'add/delete an "s" at the end') except for a very few high frequency lexemes. I agree it's interesting that number (the grammatical feature 'number', i.e. singular/plural) is explicitly encoded in the vast majority of human languages (even in English, which encodes almost nothing else from the known spectrum of grammatical features), and that it usually plays a role not just in morphology (word formation) but also in syntax (sentence structure -- think subject-verb agreement in English); but I'm not sure what you're getting at. Do you mean the semantics usually associated with the feature (singleton vs. non-singleton set) -- it's kinda cool that zero tends to get lumped in with plurals in English (but usually not in German); not sure how other languages go about that one (but I have solicited some references from an acquaintance who worked on numbers and number features pretty intensively a few years ago...)
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.
I think I see what you're getting at, but I'm not sure where it's going. I'll accept the "directly perceivable" term for current purposes, but there's whole heckuvalot more going on in our heads (brains & associated processes) when we look at and identify a small set of like items as a set-of-N than I'm accustomed to calling "direct", and that's just the stuff we know about...
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.
It's a unary predicate, i.e. an intransitive. It takes a single argument. It returns a truth value; albeit in at least one common sense of 'exist' that value depends on the evaluation index (possible world / place and time of utterance / speaker / etc). I'm talking about the kind of existence which is independent of the current index, i.e. __necessary__ existence: existence in every possible world.
Sorry, that was probably annoying. Yes, different people use the word in different ways with different connotations.
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.
They're a different sort of thing for us (humans) certainly: we experience them differently ('direct perception'), quite likely for pretty mundane evolutionary reasons -- there are a lot of sets-of-2, 3, and 4 to deal with in the world, and not so many sets-of-327.
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.
Yup.
But I think that very small counting numbers do have a directly observable nature,
Warning Will Robinson Danger -- I think what's special about small numbers is special to humans, and not to the numbers as such (i.e. as abstracta). I think 2 (e.g. as the cardinality of the set {0,1}) is pretty special from an abstract standpoint as well (binary numbers simulating alphabets of arbitrary finite size, that darned Turing (1937) again), but I'd guess that the ease of small-number recognition is probably just a contigent human-specific brain-related phenomenon along the lines Chris sketched...
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.
Bummer.
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.
Data pending... unfortunately the guy I know who would probably be able to help me out is probably himself wandering around Australia collecting that kind of data at the moment...
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.
There's a thing I feel obliged to point out here which aspiring linguists get to know as the "Sapir-Whorf Hypothesis" (unrelated to Start Trek): basically it states that `if you can't say it, you can't think it', and it's been pretty much totally discredited by now; i.e. just because you don't have a word for it doesn't mean you can't perceive it / think it / know it / talk about it (indirectly).
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.
maybe constructs' = 'referring expressions'? The crux beingreference', i.e. the correspondence (or lack thereof) between a symbol
(word) and the thing-in-the-world? That's a tough one, right enough...
marmosets, Bryan