TaK

“This was one of the great events of my life, as dazzling as first love. I had not imagined that there was anything so delicious in the world”

Bertrand Russell When he recalled his first study of Geometry

TaK - Mathematics

• Choose three words that for you best describe the essence of mathematical knowledge.

• Share them with classmates

• Do you find others have the same words and ideas?

• Do you think that the group impression of Mathematics is a good general picture – or a stereotype?

TaK - Mathematics

What makes Mathematics distinct and different from other Areas of Knowledge?

TaK - Mathematics

Why do we do Mathematics?

TaK - Mathematics

How do we do Mathematics?

TaK - Mathematics

What could be less ambiguous, more clearly defined than a mathematical problem?

A mathematical problem may not be easy to resolve, but there will be a right answer – and

little room for debate.

TaK - Mathematics

“I think you should be more explicit here in step 2…”

TaK - Mathematics

Maths gives relative certainty. If it presents us with reliable knowledge then we can learn

precisely how it does that, and see if we can apply the techniques elsewhere.

The techniques of Mathematics may provide us with a tool that will be central in our search

for reliable knowledge.

TaK - Mathematics

“A mathematician is cautious in the presence

of the obvious” Kip Sumner

TaK - Mathematics

Maths is 35% formulae, 35% proofs and

35% imagination

TaK - Mathematics

Fence the largest possible areaWith the least amount of fencing.

TaK - Mathematics

Maths: invented or discovered?

TaK - Mathematics

• Axioms• Considered by Euclid to be derived from

experience, requiring no proof and true• We now consider them to be

assumptions, premises, “givens”

• Deductive reasoning• Theorems

TaK - Mathematics

If you are to be labelled ‘a knower’, then you must be prepared to be concerned with:

Objectivity

Reliability

Validity

Precision

TaK - Mathematics

Problem &Algorithm

Data &Examples

Conjecture

Theorem

New Algorithm

New Data & New Examples

Further Application

ImplementationProof

Observation

…and so on

TaK - Mathematics

Mathematics

PerceptionDoes perception play any role in mathematics?

ReasonCan maths be

reduced to logic?

EthicsAre basic ethical truths as certain

as basic mathematical

truths?

ArtsHow important is

beauty in mathematics?

LanguageHow is

mathematics like a language?

HistoryWhat role can

mathematics play in history?

EmotionHow important is

intuition in maths?

Natural SciencesIs the book of

nature written in the language of

maths?

TaK - Mathematics

Some key points:

• Mathematics, which can be defined as the ‘science of rigorous proof’, begins with axioms and uses deductive reason to derive theorems

• Although proof is the logical matter of deriving theorems from axioms, mathematicians consider some proofs to be more beautiful than others

• According to three different views about the nature of mathematical truths they are either: 1) empirical, 2) true by definition, or 3) rational insights into universal truths

• While some people believe that mathematics is discovered, others claim it is invented; but neither view seems to be entirely satisfactory

• Mathematicians and philosophers are still perplexed by the extraordinary usefulness of mathematics