Major Facilities for Mathematical Thinking and Understanding.

(6) Process and Time.

Facility for thinking about processes or sequences of actions that can often be used to good effect in mathematical reasoning.

Consider the composition of the functions f and g,

)).((:)( xgfxgf =o

It is particularly valuable to think of it as a process.

)).(()( xgfxgx →→

A function f (x) can be thought of as a process.

Take for example the process of finding the greatest common divisor (gcd) of the integers 2322 and 654.

Euclid’s Algorithm (Al-Khowarizmi): a and b are positive integers, a > b.If b divides a, then GCD (a, b) = b

If b does not divide a, let

a = kb + r

The key insight is gcd (a, b) = gcd (b, r).

GCD (2322, 654) = GCD (654, 360)

2322 = 654*3 + 360

Hence GCD (2322, 654) = 6.

GCD (294, 66) = GCD (66, 30)

294 = 66*4 + 30

GCD (360, 294) = GCD (294, 66)

360 = 294*1 + 66

GCD (654, 360) = GCD (360, 294)

654 = 360*1 + 294

).mod..( braei ≡

Firstly, let look at the possibility of filling up a space or area with a curve.

We give a construction of such a Peano curve, adapted from David Hilbert's example. The construction is inductive, and is based on replacement rules. We consider building blocks of six shapes,

the length of the straight pieces being twice the radius of the curved ones. A sequence of these patterns end-to-end represents a curve, if we disregard the red and green half-disks.

The replacement rules are the following:

The rules are applied taking into account the way each piece is turned. Here we apply the replacement rules to the initial pattern:

(Rescale the figure on the right so it has the same size as the original.)

Here are the first five steps:

Applying the process repeatedly gives, `in the limit’, a curve that fills up the square. (Note that we still need to work out carefully and critically the idea `limit’.)

Jordan Curve Theorem(Simplified version) Any continuous simple closed curve in the plane, separates the plane into two disjoint regions, the inside and the outside.

`Simple’ here means the curve has no self-crossing (or intersection).

In this figure, it is hard to tell Whether a point is inside or outside the polygonal region.

How about this one?

`Plumb lines in the +ve x-direction.

Aristotle384 - 322 BC

Socrates470-399 BC

427to347BC

Critical Reasoning

The Story of Socrates and the Delphi Oracle

The oracle of Apollo at Delphi

Louis XIV – the Sun King

Archaeological Site of Delphi.

Temple of Apollo at Delphi.

It is proclaimed that Socrates is the wisest among the mortal.

Socrates says that he is wise only in this sense: he knows what he does not know; while as others claim to know what they don’t.

Critical Reasoning – The Method of Conjecture

and Refutation.We already have an idea what are conjectures. Typically, aconjectures is a statement or assertion that is supported (or corroborated) by at least one evident, but is not proven to be true in general. In some rare and extreme cases, when the conjectures are made bypure intuition and are not supported by any evident, we better call them speculations instead.

For instance, the assertion that the equation

has no positive integer solutions

once the index n > 2, was a

conjecture (although it is known

as Fermat’s Last Theorem) before it’s proven by Andrew Wiles. It’s supported by the case n=4 (and others) but was not proven completely.

nnn zyx =+

The asymmetry between proof and refutation.To prove a statement like the Fermat’s Last Theorem, we need to consider ALL the integers n = 3, 4, 5,…. That is, all possible cases must be taken into account.But to disprove or refute a statement, one special case or example is enough.

So there is a huge difference between proof and refutation.

For example, Fermat believed that the numbers

...)3,2,1,0(12: 2 =+= nFn

n

are all prime numbers. To prove the assertion, it is not enough to see that when n = 0, 1, 2, 3, the numbers

,257,1712

,512

,312

3

42

21

0

==+=

=+=

=+=

FF

F

F

are all prime numbers.

But once we find out that forn = 5,

,670041764142949672971212 322

5

5

×==+=+=F

hence it is NOT a prime, we have disproved the statement.

Likewise, consider the `conjecture’ that the equation

1109 22 += yxhas no positive integer solutions.

To prove the statement, it is not enough to try x = 1, 2, 3, …, all The way to, say, 10,000.

Indeed, the first solution occurs

when x=158070671986249 and

y=15140424455100 !

Hence the conjecture is refuted.

The idea on the asymmetry between proof and refutation is developed by Sir Karl Popper (1902-1994).

Poppers

This is what Popper says, “In the empirical sciences…, proofs do not occur, if we mean by `proof’ an argument which establishes once and for ever the truth of a theory. What may occur, however, are refutations of scientific theories.”

Examples are Newton laws of Mechanics, the theory that the Earth is flat, the theory that continents are fixed, etc.

In its simple form, Critical Reasoning can be described as the method of Conjectures and Refutations. Critical Reasoning treats refined statements as conjectures, which can be used and accepted beforefaults are found. Once a fault is found, the statement should be improved (or a completely new statement is propounded). The process of looking for errors an improvement/innovation is continued.

Popper continues, “First, although in science we do our best to find the truth, we are conscious of the fact that we can never be SURE whether we have got it. We have learned in the past, from many disappointments, that we must not expect finality… In other words, we know that our scientific theories must always remain hypotheses… Thus we can say thatin our search for truth, we have replaced scientific certainty by scientific progress.”

1. Formulate a statement (conjecture/hypothesis)

2. Look for faults (seek refutations)

3. Improve the statement / new statement

4. Look for faults

5.Improve the statement again…

It is diametrically opposite to dogmatic or closed thinking, which tends to accept and follow rules and conventions without critical evaluations.

As a dramatic example, consider Euclid’s Fifth Axiom, or equivalently the parallel axiom.

For 2000 years it was accepted as `correct’, intuitive and natural. In 1817, German mathematician Gauss privately expressed criticism toward the fifth axiom. He said,

“I am becoming more and more convinced that the necessity of our geometry cannot be proved, at least by human reason and for human reason.”

He began investigating the sum of the interior angles of very large triangles, and developed curvature in the study of surfaces.

Some time later, Lobachsevskiiand Bolyai expounded the non-Euclidean Geometry.

Let’s examine the critical process. To formulate conjectures, or to improve a statement, often requires imagination and innovation in asking questions.

This point is addressed by Einstein.

Albert Einstein(1879-1955)

‘The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill. To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advance in science.’ – A. Einstein & L. Infeld

The Nobel Laureate/Poet Szymborskadescribe this using metaphors: “And any knowledge that doesn’t lead to new questions quickly dies out: it fails to maintain the temperature required for sustaining life.”

How about the critical process of seeking faults?

It often involves seeing things at different angles.

The Nobel Prize in Physiology or Medicine for 2005 was awarded jointly to

Barry J. Marshall and J. Robin Warrenfor their discovery of

"the bacterium Helicobacter pylori and its role in gastritis ulcer disease"

I was met with constant criticism that my conclusions were premature and not well supported. When the work was presented, my results were disputed and disbelieved, not on the basis of science but because they simply could not be true… I was told that the bacteria were either contaminants or harmless commensals. I realized then that the medical understanding of ulcer disease was akin to a religion. No amount of logical reasoning could budge what people knew in their hearts to be true… To quote historian Daniel Boorstin: “The greatest obstacle to knowledge is not ignorance; it is the illusion of knowledge”.

On looking back to this event, I am impressed by the great limitations of the human mind. How quick are we to learn, that is, to imitate what others have done or thought before.

And how slow to understand, that is, to see the deeper connections.

Slowest of all, however, are we in inventing new connections or even in applying old ideas in a new field. (Frits Zernike , Nobel Lecture, 1953.)

Conspiracy Theory (1997)

A widely held attitude that seeks to explain a event by finding out the schemer(s) behind – the man or group who have plotted and conspired to bring it about.

Unintended Consequences.

Reflexivity

Risk management, or hedging.

On Black Wednesday (September 16, 1992), Soros became instantly famous when he sold short more than $10bn worth of pounds, profiting from the Bank of England's excessively rigid currency policy.

George Soros

Quantum Fund.

Finally, the Bank of England was forced to withdraw the currency out of the European Exchange Rate Mechanism and to devaluate the Pound Sterling.

Soros reportedly earned US$ 1.1 billion in the process. He was dubbed "the man who broke the Bank of England.”

The event was repeated 6 years later for Thai Baht.

• 15 years ago, hedge funds, those private, lightly regulated investment vehicles aimed at the ultrawealthy, managed less than $40 billion.

• Today, the figure is approaching $1 trillion

In 2003, the 25 highest-paid hedge fund managers earned more than $200 million, on average.

The top-ranked manager, George Soros, took home $750 million that year.

International Herald Tribune.