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Formalisms and meaning

!  Limits of infinite series

“We describe the behavior of sn by saying that the sum sn approaches the limit 1 as n tends to infinity, and by writing

1 = 1/2 + 1/22 + 1/23 + 1/24 + …” (Courant and Robbins, 1978, p. 64)

and …

!  Continuity

“… since the denominators of these fractions increase without limit, the values of x for which the function sin(1/x) has the values 1, -1, 0, will cluster nearer and nearer to the point x = 0. Between any such point and the origin there will be still an infinite number of oscillations of the function.”

(Courant and Robbins, 1978, p. 283)

Are these just “dead” expressions?

!  the sum sn approaches … !  the hyperbola approaches … !  … as n tends to infinity. !  … oscillations of the function … !  … it never actually reaches …

Continuity

"  A function f is continuous at a number a if the following three conditions are satisfied:

1.  f is defined on an open interval containing a 2.  limx→a f(x) exists, and 3.  limx→a f(x) = f(a).

Where limx→a f(x) is ...

Let a function f be defined on an open interval containing a, except possibly at a itself, and let L be a real number. The statement

limx→a f(x) = L

means that ∀ ε > 0, ∃ δ > 0, such that if 0 < ⏐x - a⏐ < δ,

then ⏐f(x) - L⏐ < ε.

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Axioms of 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 upperbound has a least upper bound.

field

Ordering constraints

Axioms of 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 upperbound has a least upper bound.

field

Ordering constraints

Complete ordered fields

Least upper bounds

!  Upper Bound b is an upper bound for S if x ≤ b, for every x in S.

!  Least Upper Bound b0 is a least upper bound for S if • b0 is an upper bound for S, and • b0 ≤ b for every upper bound b of S.

Pure Math: nothing moves

!  Axiomatic approaches, existential and universal quantifiers, and the Least Upper bound axiom don’t tell us anything about !  Approaching !  Tending to !  Oscillating !  Going farther and farther !  … and so on

So? What moves? …

!  ε-δ formalisms (with existential and universal quantifiers)

!  and other formal definitions don’t give the answer

!  Weierstrass-Dedekind and the Arithmetization of Calculus (19th Century) don’t either,

!  … so, what moves?

Metaphorical Fictive Motion "  Metaphorical Fictive Motion operates on a network of

precise conceptual metaphors (e.g., Numbers Are Locations in Space)

"  providing the inferential structure required to conceive mathematical functions as having motion and directionality

•  sin 1/x oscillates more and more as x approaches zero •  g(x) never goes beyond 1 •  If there exists a number L with the property that f(x) gets closer

and closer to L as x gets larger and larger; limx→∞ f(x) = L

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Metaphorical Fictive Motion

"  Are these mere linguistic expressions (using meaningless and arbitrary notations)?

Metaphorical Fictive Motion

"  Are these mere linguistic expressions (using meaningless and arbitrary notations)?

"  NO. They are actual manifestations of the inherent semantics (meaning), seen through extremely fast, highly efficient, and effortless cognitive mechanisms underlying thought: speech-gesture co-production

!  Mathematics graduate students !  Working in dyads !  Prove the following theorem:

Marghetis, Edwards, and Núñez, 2014

Metaphorical Fictive Motion Co-speech gesture experiment

Marghetis and Núñez, 2010

A (behavioral) experiment

***

*** *

*** p < 0.001 * p <0 .05

Num

ber o

f co-

spee

ch g

estu

res

Marghetis and Núñez, 2010, TopiCS

Metaphorical Fictive Motion

"  The cognitive components of these mathematical ideas (e.g., continuity) "  Dynamic "  Source-path-goal based, etc.,

"  Don’t get captured by the above formalisms and are cognitively incompatible with them "  Static existential and universal quantifiers "  Preservation of closeness "  Gaplessness

Núñez & Lakoff, 1998 Núñez, Edwards, Matos, 1999 Núñez, 2006, 2007

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Metaphorical fictive motion

!  ε-δ definitions neither formalize nor generalize concepts (Núñez & Lakoff, 1998)

!  ε-δ formalisms don’t just symbolize the “intuitive” idea of continuity,

!  They symbolize other—appropriately further metaphorized ideas

Lakoff & Núñez, 2000

f(x) =x sin (1/x) for x ≠ 0

0 for x = 0