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Does a kid, which learns the meaning of the term "distance" (or any other expressions which might be thought of as physical quantities) automatically also develope a concept of numbers?

If I know that my desk is longer than my pen, does that imply I know that the one can be thought of as being multiple times as long as the other and do I know this in terms of numbers, i.e. "Oh, it's about seven times as long!".

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The origin of the notion of number is a matter of some debate, and a fairly large literature exists on this (and related problems, such as whether numbers actually exist in a Platonic sense, or are just conventions, like names.)

I don't know of any philosophers who argue that the notion of number follows necessarily from the notion of extension (i.e., distance)-- most link it, rather, to the act of seeing two of something (like two hands), and abstracting from there.

So, I think that most philosophers would argue that it is possible to see that the desk is longer than the pen, without necessarily quantifying that relation in terms of units.

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    But if you argue that one automatically comes up with numbers just by "seeing two things", i.e. if you "discover numbers" so easily, then I think it's even harder to argue that one can have a concept of comparing distances without (immediately associatng it with) numbers.
    – Nikolaj-K
    May 8 '12 at 11:37
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    Note: I did't say it was "automatic" or "easy"-- the concept of number is a fairly sophisticated abstraction. May 8 '12 at 11:47
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I think if anything this question is an experimental one. There are some studies about how well animals are dealing with numbers. I have read something about pigeons being able to count up to some small number. Here is some link supporting that.

From this one can argue that some concept of numbers seem to come pretty natural. I have also heard of some studies that show some insight of how numbers are perceived by humans or what people with some specific brain damage can still do in mathematics and what not. Also, I am pretty sure I have read that people after some accidents couldn't understand most words, but numerical words as "seven, one, two-billion-and-five-hundred-twenty-four" were understood. The author then suggested, that it seems that we deal totally different with "numerical words" than with other words. Numbers seem to be memorized somewhere else in the brain than say names of animals. I am sorry that I totally forgot where I read that, but anyhow, I am sure you find a lot of similar studies after googeling for a while. The/One related branch is explained here.

So from what I have seen I would say one can confidently claim that naive mathematical understanding (counting small numbers, comparing numbers) evolves rather soon and (all?) somewhat sophisticated animals seem to share that. - Oh and the same goes for children, see also in the Wikipedia link.

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