Rafael Núñez

This Chilean Professor at the University of California, San Diego, together with George Lakoff from the Berkeley campus of the same University; have looked into the metaphorical character of concepts. Here are two jewels of their thinking:

Time Events are Things in Unidimensional Space.

Time Passing is Motion of an Object.


Núñez  helps me understand what do I do when I teach Mathematics, because he answers the question: What Mathematics Is?

Like Darwin and Galileo before, these Cognitive Scientists are trying to clean all the cobwebs that clutter our minds.

The following are the axioms of the real numbers.

1.  Commutative laws for addition and multiplication.
   
2. Associative laws for addition and multiplication.
3. The distributive law.
4. The existence of identity elements for both addition and multiplication.
5. The existence of additive inverses (i.e., negatives).
6. The existence of multiplicative inverses (i.e. reciprocals)
7. Total ordering
8. If x and y are positive, so is x + y.
9. If x and y are positive, so is x ⋅ y
10. The Least Upper Bound axiom: every nonempty set that has an upper bound has a least upper bound.

The first 6 axioms provide the structure of what is called a field for a set of numbers and two binary operations. Axioms 7 through 9, assure ordering constraints. The first 9 axioms fully characterize ordered fields, such as the rational numbers with  the operations of addition and multiplication. Up to here we already have a lot of structure and complexity. For instance we can characterize and prove theorems about all possible numbers that can be expressed as the division of two whole numbers (i.e., rational numbers). Along a line we can also locate (according to their magnitude) any two different rational numbers and be sure (via proof) that there will always be (infinitely many) more rational numbers between them (a property referred to as density). With the rational numbers we can describe with any given (finite) degree of precision the proportion given by the perimeter of a circle and its diameter (e.g., 3.14;3.1415; etc.). With the rational numbers, however we can not "complete" the points on the line, and we can not express with infinite exactitude the magnitude of the proportion mentioned above (π = 3.14159...). For this we need the full extension of the real numbers. In axiomatic terms, this is accomplished by the tenth axiom: the least upper bound axiom. All ten axioms characterize a complete ordered field.

The concept of motion is in the tenth axiom.

Upper Bound
b is an upper bound for S if
x ≤ b, for every x in S.
Least Upper Bound
b_o is a least upper bound for S if

• b_o is an upper bound for S, and
• b_o  ≤ b for every upper bound b of S.
  

What Nuñez is after in this study is the concept of time. 

Continuity, traced by motion, takes place over time.

The trace of the motion is a static holistic line with no jumps.

These are deep thoughts we should grasp to better teach, and maybe to produce a new paradigm for Physics. Embodied Physics?


Nuñez says:

I believe that mathematics education would benefit tremendously by building on these kinds of findings and by acknowledging, in a deep way, that mathematics is indeed the product of human imagination.