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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

What are Sets?

! Container-schemas

! Path-schemas

What are sets? !  On the formalist view of the axiomatic method a

“set” is any mathematical structure that “satisfies” the axioms of set theory

!  A classic group of axioms is the Zermelo-Fraenkel axioms (ZFC)

!  Often they are taught as been about sets conceptualized as containers (e.g., through Venn diagrams)

Are hypersets, sets?

What are sets?: Sets Are Containers

Source Domain Container schemas

Target Domain Sets

Interior of Containers schemas → Sets

Objects in interiors → Set members

Being an object in an interior → The membership relation

An interior of one container schema within a larger one

→ A subset in a larger one

The overlap of the interiors of two Container schemas

→ The intersection of two sets

The totality of the interiors of two Container schemas

→ The union of two sets

The exterior of a Container schema

→ The complement of a set

B A

A intersection B ZFC axioms

!  axiom of extension !  axiom of specification !  axiom of pairing !  axiom of union

!  axiom of powers !  axiom of infinity !  axiom of choice

There is absolutely nothing in these axioms that explicitly requires sets to be containers

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The point is that ...

!  Within formal mathematics “sets” are not technically conceptualized through Container Schemas

!  “Sets” don’t have interiors, boundaries, and exteriors

!  “Sets” are simply entities that “fit” the axioms

The point is that ...

!  But, empirical data show that educated people (incl. mathematicians) often DO conceptualize sets in terms of Container Schemas

!  and that is perfectly consistent with the ZFC axioms

!  However ...

Self-containment?

!  When we conceptualize sets as Container Schemas, a particular entailment follows:

!  Sets cannot be members of themselves! (Since containers cannot be inside themselves)

!  Strictly speaking, this entailment doesn’t follow from the ZFC axioms, but from the inferential structure of the container schema (i.e., metaphorical understanding)

An extra axiom is needed!

!  The ZFC axioms don’t rule out sets that “contain” themselves

!  Indeed, an extra axiom was proposed by Von Neumann to rule out this possibility!

!  The axiom of foundation: !  There are no infinite descending sequences

of sets under the membership relationship. !  That is, . . . Si+1 ∈ Si ∈ . . . ∈ S is ruled out.

An extra axiom is needed!

!  Since allowing sets to be members of themselves would result in such a sequence, . . . Si+1 ∈ Si ∈ . . . ∈ S

!  the axiom of foundation has the indirect effect of ruling out-self membership

But, sometimes we may need “self-containment” ...

...111

11

++

+=x

xx 11+=

… when modeling recursive structures like

which can be seen as

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Hypersets !  Some set-theorists realized that a new

non-container metaphor was needed to think about sets

!  and they explicitly constructed one (Barwise and Moss, 1991)

!  The idea is to use Graphs, not containers, for characterizing sets

!  Accessible Pointed Graphs (APGs) !  … A new conceptual metaphor:

Sets Are Graphs!!

“The box metaphor is the motivating intuition that gives rise to the Axiom of Foundation”

Barwise and Moss, 1991, p. 36 Hypersets: Sets As APGs !  Accessible, indicates that there is a single node which

is linked to all other nodes in the graph !  Pointed, indicates an asymmetric relation between

nodes in the graph (arrows)

...

What are sets?: Sets Are Graphs

Source Domain Accesible Pointed

Graphs

Target Domain Sets

An APG → The membership structure of a set

An arrow → The membership relation

Nodes that are tails of arrows → Sets

Decorations on nodes that are heads of arrows

→ Members

APG’s with no loops → Classical sets with the Foundation Axiom

APG’s with or without loops → Hypersets with the Anti-foundation Axiom

Mathematics, a Human conceptual system

!  What is ultimately a set, then? !  Like construals of time, both concepts of sets are

internally consistent but mutually inconsistent !  Truth in these purely imaginary conceptual systems

is relative to the very human mappings that made them possible

B A

A intersection B

...

So, what are sets, really?

!  Mathematics has (at least) two internally consistent but mutually inconsistent metaphorical conceptions of sets !  one metaphorized in terms of container schemas !  and another one in terms of graphs