The Unreasonable Effectiveness of (High School) Mathematics - - PowerPoint PPT Presentation

The Unreasonable Effectiveness of (High School) Mathematics

5th Annual Jesus College Graduate Conference

April 27th 2012

Dominic Orchard

2 + 2 = 1 + 0 = 0 + 3 = 0 + 0 =

Some sums...

(elementary school!)

4 1 3

x + 0 = 0 + x =

... with variables

(high school)

x x

3 + 4 + 2 = 6 + 1 + 4 =

... with three numbers

How did you do it?

9 11

6 + 1 + 4

Reducing “pairs”

( ) 6 + 1 + 4 ( ) = 11 = 7 + 4 = 6 + 5 = 11

Some axioms of +

x + 0 = x 0 + x = x (x + y) + z = x + (y + z)

0 does nothing (with respect to +) grouping into pairs doesn’t change result

2 × 2 = 3 × 4 = 1 × 3 = 4 × 1 =

Some more sums...

4 12 3 4

x × 1 = 1 × x =

... with variables

x x

2 × 3 × 4

Reducing “pairs”

( ) 2 × 3 × 4 ( ) = 6 × 4 = 24 = 2 × 12 = 24

Some axioms of ×

x × 1 = x 1 × x = x (x × y) × z = x × (y × z)

1 does nothing (with respect to ×) grouping into pairs doesn’t change result

Axioms of × and +

x × 1 = x x + 0 = x

(x × y) × z = x × (y × z) (x + y) + z = x + (y + z)

1 × x = x 0 + x = x x × y = y × x x + y = y + x ( )

Common structure... monoids

Not the Dr. Who aliens with one eye.....

Monoids

• A collection of things X

x ⊕ n = x {right unit} n ⊕ x = x {left unit} (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) {associativity}

⊕

• A special X, call it n, that does “nothing” with

⊕

• An operation that turns two Xs into one X

e.g. (whole) numbers e.g. + or × e.g. 0 (for +) or 1 (for ×)

Unreasonably effective

• Extremely ubiquitous
• Maths
• Physics
• Topology
• Computer Science
• Logic
• Linguistics (semantics)
• Everyday phenomena

* The Unreasonable Effectiveness of Mathematics (R. W. HAMMING, 1980) *

• Monoids are a simple (abstract) concept

(cf. counting*)

Not trivially effective

• Plenty of things are not monoids

3 - 0 = 3 ✔ ✘ 0 - 3 = -3 (2 - 3) - 4 = -5 2 - (3 - 4) = 3 ✘

• Interesting to study the things that are not

New example: Paint mixing!

⊕

• Collection of “things” X

Acrylic paints

Paint-mixing

• Operation that turns two Xs into one X

⊕

• Special X that does “nothing” with

n = “Extender” base

⊕ =

Monoids of “containment”

• means take two containers where one

is inside the other, and flatten into one:

⊕

1 2

⊕

=

1 2

Monoids of “containment”

x

⊕

=

x x

⊕

• “Nothing” (trivial) container, n =

x

=

x

Dreams as containers...

(idea originally due to Dan Piponi)

(“Containment monoids” usually called monads)

Dreams as container monoid...

• Anything can be put into a (trivial) dream

n

y

Dreams as container monoid...

• A dream inside a dream is just a dream (collapse)

x

⊕

x y

⊕

Computations (functions)

A B

f

B C

g

⊕

A C

f g

⊕

=

Computations (functions)

A B

f

⊕

=

x x

n

A B

f

My work....

⊕

=

A B

f g

C B A

f g

⊕

C

• Special kinds of monoids for and

Importance of monoidality

• Underlying equational theory

... + x + 5 + (-5) + y + ..... Additional property ... + x + 0 + y + ..... ... + x + y + ..... Monoidality

Thanks!

x ⊕ n = x {right unit} n ⊕ x = x {left unit} (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) {associativity}

dorchard.co.uk

• Monoids pervasive (many more examples)
• Unreasonably effective but very simple
• My work: more complex models of computation

with underlying monoid properties:

Back-up slides

Computations (functions)

A B

f

B C

g

⊕

A C

f g

⊕

=

C D

h

⊕

C D

h

⊕

=

( )

g

A C

f ⊕ h

⊕

Computations (functions)

A B

f

B C

g

⊕

B C

g h

⊕

=

C D

h

⊕ ⊕

=

( )

g f

A C

⊕

h

⊕

A B

f

Non-deterministic Computations

A

f

B* A B

f

• Previously, output single result
• Non-deterministic: output many possible

results

g

Non-deterministic Computations

⊕

A

f

B* B C*

g ?

=

A C*

f ⊕ *

g

Non-deterministic Computations

⊕

A

f

B* B* C*

g

=

A C*

f ⊕

*

*

Non-deterministic Computations

• Need to design the * operation and a “nothing”

computation id* to satisfy monoid axioms e.g.: ?

id*

?*

⊕

A*

f

B*

*

=

A

f

B*