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.
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.
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.
Four grounding metaphors (4G): Object Collection, Object Construction, Measuring Stick, and Motion Along a Line.
Here are the classic Zermelo-Fraenkel axioms, including the axiom of choice; together they are commonly called the ZFC axioms:
- The axiom of Extension: Two sets are equal if and only if they have the same members.
- The axiom of Specification: Given a set A and a one-place predicate P(x) that is either true or false of each member of A, there exists a subset of A whose members are exactly those members of A for which P(x) is true.
- The axiom of Pairing: For any two sets there exists a set that they are both members of.
- The axiom of Union: For every collection of sets, there is a set whose members are exactly the members of the sets of that collection.
- The axiom of Powers: For each set A, there is a set P(A) whose members are exactly the subsets of set A.
- The axiom of Infinity: There exists a set A such that (1) the empty set is a member of A, and (2) if X is a member of A, then the successor of X is a member of A.
- The axiom of Choice: Given a disjoint set S whose members are non-empty sets, there exists a set C which has as its members one and only one element from each member of S.
Figure 10.1, p 209
Cantor's one-to-one correspondence between the natural numbers and
the rational numbers.
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.
- Mathematics is an objective feature of the universe [...]
- What human beings believe [...] has no effect [...]
- Mathematicians are the ultimate scientists [...]
- [...] mathematics characterizes the very nature of rationality.
- [...] Mathematicians are the ultimate experts on [rationality].
- [...] "the book of nature is written in mathematics" [...]
- Mathematics is the queen of the sciences. [...]
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 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.
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.
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.
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!
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.