Where Mathematics Comes From

1. Where Mathematics Comes From George Lakoff Rafael E. Núñez

2. George Lakoff Professor of cognitive linguistics at Berkeley, famous for ideas about metaphor and the embodied mind Rafael E. Núñez Professor of cognitive science at University of California San Diego, known for promoting embodied cognition Authors

3. What is the nature of mathematics? • It exists by virtue of the embodied mind • All mathematical content resides in embodied ideas • Many of the most basic, as well as the most sophisticated, mathematical ideas are metaphorical

4. What are the basic properties of mathematics? • Stable • Precise • Generalizable • Calculable • Consistent within subject matters • Universally available • Effective for precisely conceptualizing a large number of aspects of the world as we experience it

5. The Embodiment of Basic Arithmetic • Number discrimination by babies • Subitizing

6. Schemas • Container schemas (Venn Diagrams, logic) • Source-Path-Goal • Into • Out-of

7. The Grounding Metaphors • Arithmetic as Object Collection • Arithmetic as Object Construction • The Measuring Stick Metaphor • Arithmetic as Motion

8. Numbers are Things • Real objects have these unique properties, so the same should apply to math • Statements about math must be true or false --->Numbers should not be products of minds So, it follows from the empirical study of numbers as a product of the mind that it is natural for people to believe that numbers are not a product of the mind (oh, the irony).

9.  • An empty collection • The lack, absence, or destruction of an object • The ultimate in smallness, lack of any physical measurement • Origin of motion Zero denotes emptiness, nothingness, lack, absence, destruction, ultimate smallness, and origin

10. 1 • Individuality, a lone member • Wholeness, unity • A standard unit • A beginning

11.  • All • The entirety of a set as a whole • Everything countable So… what could possibly be more than an entire set containing everything known?

12. The Embodiment of Infinity • Iterative, imperfective processes • The “number” ∞ From an initial state, produce a resultant state (0->1). Repeat (2, 3, 4, 5… n-1, n), until you get to the “final resultant state”: the “integer” ∞ is unique and larger than every other possible integer So… infinite series of numbers “end” with infinity.

13. Metaphorical Collisions at .999...=1 • Lim n->∞ ∑ni=1 9/10i = 1 if a sequence is infinite, it converges on and is equal to a point L as n->∞. In this case, L=1, so .999…=1

14. Metaphorical Collisions at .999...=1 • Lim n->∞ ∑ni=1 9/10i = 1 if a sequence is infinite, it converges on and is equal to a point L as n->∞. In this case, L=1, so .999…=1 But for every known number n, 1- ∑ni=1 9/10i = 1/10n, so shouldn’t there be a teeny tiny smidgeon of a difference?

15. Infinitesimals & Points • The teeniest tiniest numbers you could ever think of, but smaller • ∆x • Points as disks, with an infinitesimal radius

16. Fractals Fractals

17. What is a fractal? • “a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole” (Mandelbrot)

18. It’s turtles all the way down.