Re: [xmca] Why "Natural" Numbers?

[Anna Sfard <annasfar@math.msu.edu>]

Hi David,

I cannot say when the adjective "natural" was first used to describe numbers such as 1, 2, 3, etc., but it seems that the reason for the name is the widespread conviction that these numbers are objects-in-the-world, just like stars, rocks, trees, and elephants. That is, they are product of nature rather than of human mind. This sentiment has been best expressed by Leopold Kronecker, a distinguished German mathematician (1823 - 1891) in his famous statement: "God created integers [and what he probably meant were natural numbers], all else is the work of man". He was trying to discredit the then newly invented set theory, whose author, Georg Cantor (1845 -1918), claimed that natural numbers are constructed and defined them as such. Kroneker's opposition is often seen as the main reason for Cantor's mental illness - natural, it seems, is not always healthy! 

anna

----- Original Message ----- 
From: David Kellogg 
To: xmca 
Sent: Sunday, November 15, 2009 9:03 AM
Subject: [xmca] Why "Natural" Numbers?


Can any of the marvelous mathematics educators on this list help me out?

I have a lecture to give tomorrow on Davydov and Schmittau, and I just realized that I have not the faintest idea why the "natural" numbers are called that. I know why the reals are real and the rationals are rational and why the irrationals are irrational. But I haven't a clue what is "natural" about the naturals. A quick trip through Wikipedia is even more confusing: it turns out that the canonical set of naturals used in mathematics includes zero! Now what the devil is "natural" about the number zero?

I suppose a stupid answer is that "natural" numbers are connected to "natural" meaning, the sort of meaning that Vygotsky talks about when he speaks of theories of meaning that do not distinguish between language meanings and "meaning" that we find in nature (just as there are acoustic theories of phonetics that do not distinguish between language sounds and those that we find in nature). 

But somehow that suggests that counting is somehow more characteristic of animal behavior than measuring. That can't be right. It's easy to think of examples of animals eyeballing a distance before they attempt to leap it, and Vygotsky himself talks about how a dog can distinguish between two heaps of treats without counting them. 

(Interestingly, if you look at Paula's video of the eight year old subject doing the Vygotsky blocks test you'll find lots of examples of measuring but no examples of counting!)

David Kellogg
Seoul National University of Education


      
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