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 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
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
• 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