Posted by: Alexandre Borovik | August 30, 2008

## Viva La Diférence!

I continue my hunt for paradoxes discovered by children in their personal experiences of  learning of mathematics. This one comes from a compendum of responses to What had astounded your? question accumulated at avva’s Live Journal and reproduced in my previous post. I quote:

“В дошкольном детстве помню поразило объяснение относительности временной шкалы, т.е. что нулевой год – это произвольная отметка на линии времени, а где у этой линии начало никто не знает.”

Translation: “When I was a preschooler, I remember I was struck by an expanation of a relative nature of the time scale, that is, that Year Zero was just an arbitrary mark on the timeline, and no one knew where the time had started.”

Just today I read a paper by David Carraher, Bárbara M. Brizuela and Darrell Earnest, The Reification of Additive Differences in Early Algebra: Viva La Diférence! [In H. Chick, K. Stacey, J. Vincent, & J. Vicent (Eds.), Proceedings of the 12th ICMI study conference: The future of the teaching and learning of algebra (pp. 163-170). Melbourne, Australia: The University of Melbourne.] I quote from the paper:

The problem talks about the differences in heights among three characters without revealing their actual heights. This problem seemed appropriate for introducing students to additive functions. The heights could be thought to vary insofar as they could take on a set of possible values. Of course that was our view. The point of researching the issue was to see what sense the students made of such a problem.

One can easily agree that both timeline without Year Zero and relative height are instances of the same structure: one-dimensional affine space. Here is a general definition from Wikipedia:

an affine space is a set S, together with a vector space V, and a map

$(a, b) \mapsto \Theta(a, b).$

The image Θ(a,b) is written as ab and can be thought of as the vector from b to a. The map has the properties that:

1. for every b in S the map
$a \mapsto a - b\,$
is a bijection, and
2. for every a, b and c in S we have
$(a-b) + (b-c) = a-c.\,$

Formal axioms for affine spaces (and, on the way, for vector spaces) were introduced by Hermann Weyl in his famous book of 1918 Space Time Matter. He clearly explains the underlying philosophy:

Time is homogeneous, i.e. a single point of time can only be given by being specified individually. There is no inherent property arising from the general nature of time which may be ascribed to any one point but not to any other ; or, every property logically derivable from these two fundamental relations belongs either to all points or to none.

All beginnings are obscure. Inasmuch as the mathematician operates with his conceptions along strict and formal lines, he, above all, must be reminded from time to time that the origins of things lie in greater depths than those to which his methods enable him to descend. Beyond the knowledge gained from the individual sciences, there remains the task of comprehending. In spite of the fact that the views of philosophy sway from one system to another, we cannot dispense with it unless we are to convert knowledge into a meaningless chaos.