Mathematical Hikes

This page is a sub-page of our page on First Class Mathematics.

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Related pages:

Math Rehab
Mathematics is Representation

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Story-scripts/maps for three different hikes:

1) From Numbers, via Music, to Vectors and Linear Transformations.

2) From Numbers, via Music and Vectors, to Category Theory.

3) From Conics and Quadrics, via Projective Geometry, Projective Metrics
and Non-Euclidean Geometry, to Einstein’s Special Theory of Relativity.

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Hike 2) is an expansion of hike 1). However, a “birds-eye view” of Category Theory should be introduced also in hike 1) as a kind of “vista point” since Category Theory provides a structural framework that unifies an amazingly large number of different mathematical concepts and constructions (including Numbers, Vectors, and Musical Notation).

The overall idea is that structures, constructions, concepts, … are difficult to understand unless one is able to modify them in a controlled way. A point of entry into the subject of changing the background of a representation can be found at our page on Shift of Basis.

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Eleven stories designed to raise interest in mathematics

1. The story of the people who thought the world was understandable.
From Thales and Pythagoras to Demokritos and Aristarkos.

2. The story of the people that wanted to escape the realm of the senses.
From Plato via Augustine and Aquino to the “scholastic age.”

3. The mathematics of the eye: The development of true perspective.
From Pappus via della Franchesca and da Vinci to Desargues, Pascal, Poncelét, Plücker, Grassmann and Klein.

4. Einstein for Flatlanders: Two-dimensional relativity theory.
The story about the flatlanders that lived on a sphere and the flatlanders that lived on a torus (“dough-nut”).

5. The story of the people that disregarded almost everything.
The evolution of abstract thinking: From induction and abduction to abstraction and deduction. “The power of thinking is knowing what not to think about.”

6. On the difficulties of overcoming psychological complexes.
The story about the development of the concept of number: From “positive” or “natural” to “negative”, from “rational” to “irrational”, from “real” to “imaginary” and “complex”.

7. What is there between the atoms?
Does the world consist of particles or waves – or maybe something else? The historical debate from Thales versus Pythagoras via Newton versus Huygens to Einstein versus Bohr and Heisenberg and the break-up of the particle concept through the emergence of string theory and superstring theory.

8. The mysterious law about the degradation of work:
The principles of energy and entropy. The development of the energy concept from Leibniz via Count Rumford and Carnot to Mayer, Joule and Boltzmann.

9. The story of the long-lived demon that was unable to forget.
Maxwell’s demon and the deep connections between information theory and thermodynamics.

10. The evolution of geometric arithmetic.

11. Linear Algebra: The story of the linear war
between the planets Vectoria and Vectoria’.

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