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“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.

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**“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**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**TaK - Mathematics**• Why do we do Mathematics?**TaK - Mathematics**• How do we do 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**TaK - Mathematics**“I think you should be more explicit here in step 2…”**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 area**• With 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**New Data & New Examples**Further Application TaK - Mathematics …and so on Data & Examples Observation Problem & Algorithm New Algorithm Conjecture Theorem Implementation Proof**Reason**Can maths be reduced to logic? Emotion How important is intuition in maths? Perception Does perception play any role in mathematics? TaK - Mathematics Natural Sciences Is the book of nature written in the language of maths? Ethics Are basic ethical truths as certain as basic mathematical truths? Mathematics History What role can mathematics play in history? Arts How important is beauty in mathematics? Language How is mathematics like a language?**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 TaK - Mathematics

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