Where Mathematics Comes From

How the Embodied Mind Brings Mathematics into Being
George Lakoff and Rafael Núñez
Basic Books 2000

At times, dull, slow and repetitive. But fascinating nevertheless. Leaves the impression of covering the whole of mathematics in an elegant way, from a powerful point of view. Puts the concept of infinity at the bottom of the structure.

Mathematics is not built into the universe.
The portrait of mathematics has a human face.

Introduction

Recent Discoveries about the Nature of Mind

p 5
  1. The embodiment of mind
  2. The cognitive unconscious
  3. Metaphorical thought
Insights of the sort we will be giving throughout this book were not even imaginable in the days of the old cognitive science of the disembodied mind, developed in the 1960s and early 1970s. In those days, thought was taken to be the manipulation of purely abstract symbols and all concepts were seen as literal [...] Thought, then, was taken by many to be a form of symbolic logic.

p 6

Conceptual metaphor is a cognitive mechanism for allowing us to reason about one kind of things as if it were another. This means that metaphor is not simply a linguistic phenomenon, a mere figure of speech.

p 6-7

Is zero a point on a line? Or is it the empty set? Or both? Or is it just a number and neither a point nor a set? There is no one answer. Each answer constitutes a choice of metaphor.

Part I — The Embodiment of Basic Arithmetic

Chapter 2 – A Brief Introduction to the Cognitive Science of the Cognitive Mind

pp 27-49
[Image schemas: related to patterns as in A Pattern Language by Christopher Alexander, and to frames as in The Society of Mind by Marvin Minsky]

Conceptual Metaphor

p 45
From the perspective of the embodied mind, spatial logic is primary and the abstract logic of categories is secondarly derived from it via conceptual metaphor. This, of course, is the exact opposite of what formal mathematical logic suggests.

Chapter 3 – Embodied Arithmetic: The Grounding Metaphors

The Metaphorical Meanings of One and Zero

p 75
Four grounding metaphors (4G): Object Collection, Object Construction, Measuring Stick, and Motion Along a Line.

Part II — Algebra, Logic, and Sets

Chapter 7 – Sets and Hypersets

Axiomatic Set Theory and Hypersets

Axiomatic Set Theory

p 145
Here are the classic Zermelo-Fraenkel axioms, including the axiom of choice; together they are commonly called the ZFC axioms:

Part III — The Embodiment of Infinity

Chapter 10 – Transfinite Numbers

Figure 10.1, p 209
Cantor's one-to-one correspondence between the natural numbers and the rational numbers.

Cantor's Diagonalization Proof

p 210

Part IV — Banning Space and Motion:
The Discretization Program That Shaped Modern Mathematics

Chapter 13 – Continuity for Numbers: The Triumph of Dedekind's Metaphors

pp 292-293
The turning point came in the early 1870s, with the classic work of Richard Dedekind and Karl Weierstrass. They successfully launched the movement toward discretization, a reaction against the use of geometric methods via analytic geometry. Descartes had seen the real numbers as points on a naturally continuous line. [...]
Dedekind showed, through a dramatic use of conceptual metaphor, that the real numbers didn't have to be seen as points on a naturally continuous line. And through an implicit use of the Basic Metaphor of Infinity, he showed how to construct the real numbers using sets (infinite sets, of course) of discrete elements. [...]
An important dimension of the discretization program was the concept of "rigor." This meant the use of discrete symbols and precisely defined, systematic algorithmic methods, allowing calculations that were clearly right or wrong. [...]
Geometry involved visualization and spatial intuition.

Part V — Implications for the Philosophy of Mathematics

Chapter 15 – The Theory of Embodied Mathematics

Encounters with the Romance

p 339
The Romance of Mathematics

A Question of Faith: Does Mathematics Exists Outside Us?

Is Mathematics in the Physical World?

p 344
There is a great deal that is wrong with this argument. First, no one observes laws of the universe as such; what are observed empirically are regularities in the universe.

p 345

There are regularities in the universe independent of us.

The Historical Dimension of Embodied Mathematics

Floating-Point Arithmetic: A Case of the Relevance of History in Mathematics

p 360
The standard version of floating-point arithmetic now used is not used because it is objectively true, or even truer than other forms. The version that is now standard was chosen for pragmatic reasons.

Chapter 16 – The Philosophy of Embodied Mathematics

The Point

p 369
The truth is "true" only relative to that metaphorical idea. [...] The same holds for the entity e power pi.i and the truth e power pi.i + 1 = 0.

A Portrait of Mathematics

p 379
The portrait of mathematics has a human face.

Part VI — e πi + 1 = 0

A Case Study of the Cognitive Structure of Classical Mathematics

Case Study 1 – Analytic Geometry and Trigonometry

p 397
Where π was previously only the ratio of the circumference of a circle to its diameter, 2π now becomes a measure of periodicity for recurrent phenomena. [...] This is a new idea: a new meaning for pi.

Case Study 4 – e πi + 1 = 0

How the Fundamental Ideas of Classical Mathematics Fit Together

Mathematical Idea Analysis

p 436
The equation eπi + 1 = 0 is true only by virtue of a large number of profound connections across many fields. It is true because of what it means!

Why eπi = cos y + i sin y

p 444
If you take the function evaluated at one point, 0, and you take all its derivatives at that point, you can determine the entire function! That is, there is enough information in an arbitrarily small neighborhood around that point to determine every value of the function everywhere.

Women, Fire and Dangerous Things,
Philo ToC, Local ToC
Marc Girod
Last modified: Mon Sep 5 13:50:27 EEST 2005