“Mathematics, rightly viewed, possesses not only truth, but

supreme beauty – a beauty cold and austere, like that of a sculpture.”

Bertrand Russell(1872 – 1970)

The vain search for absolute truth

1. What in your view is Mathematics?2. Choose three words that best describe Maths.

Post your words at http://www.wallwisher.com/wall/MrsVTOK1http://www.wallwisher.com/wall/MrsVTOK2

Four questions for you to answer:

1.Primes were invented (discovered?) by:a) The Ancient Greeksb) Insectsc) The Ancient Chinese

2. The length of coastline in Britain is:a) 18 000 kmb) 36 000 kmc) infinited) all of the above

3. The sum of the angles in a triangle is:a) 180 degressb) less than 180 degreesc) more than 180 degreesd) all of the above

4. The main purpose of mathematics is to,

a) be a tool for predicting real world events and real world problemsb) develop critical thinking skillsc) to create new mathematicsd) provide a challengee) provide a qualification for employmentf) be abstractg) give something for geeky dudes to do

New answers to old questions:

A Mathematician’s Answers

1. b) The primes were invented by insects (we will discuss later whether the word invented is correctly used)

2. d) Wikipedia gives 18000km for the coastline of Britain, but a Mathematician has a different answer – it depends on the accuracy of your measurement (How do culture and geography have an impact on the development of Mathematics Mathematics?)

3. d) All of the above!

A Mathematician’s Answers

4. c) The main purpose of Mathematics is to create new Mathematics

Formation of Mathematical Knowledge

What is the average of the numbers below?

1 , 1 , 1 , 1 , 3 , 4 , 4 , 4 , 5 , 5 , 1200

“Philosophy is a game with objectives and no rules. Mathematics is a game with rules and no objectives.” Anon

Euclid’s axiomatic system:

• 1. Any two points can be joined by a straight line.

• 2. Any straight line segment can be extended indefinitely in a straight line.

• 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

• 4. All right angles are congruent.

• 5. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitable must intersect each other on that side if extended enough. (Parallel Postulate)

• 6. Things that equal the same thing also equal one other.

• 7. If equals are added to equals, then the wholes are equal.

• 8. If equals are subtracted from equals, then the remainders are equal.

• 9. Things that coincide with one other equal one other.

• 10. The whole is greater than a part.

• 5. Through a point not on a given straight line, one and only one line can be drawn that never meets the given line. (Playfair’s Parallel Postulate XVIII)

Girolamo Saccheri (1667-1733) – proof by contradiction

5. Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.

5a. There exists no parallel to a given line through a point not on that line.

5b. There exist more than one parallel to a given line through a point not on that line.

This would lead to a

contradiction with the rest of

the axioms

This alternative did not lead to

any contradiction

Carl Friederich Gauss (1777 – 1855)

• The fifth postulate 5. There exist more than one parallel to a given line through a point not on that line

Hyperbolic geometry

Janos Bolyai (1802 – 1860)

Nicolai Ivanovitch Lobachevsky (1793 – 1856)

Bernhard Riemann (1826 – 1866)

• The fifth postulate 5. There exists no parallel to a given line through a point not on that line

Elliptic geometry

• The second postulate 2. Any straight line segment can be extended indefinitely into a straight line 2. A straight line is unbounded

Consider the Euclidian geometry: what is the area of a triangle? Can you justify your answer?

How did your teacher presented this claim?

Proof in Mathematics

Mathematics as a form of artA mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. (G.H. Hardy, 1877-1947)

What is that mathematicians do? If the world would be divided into “poetic dreamers” and “rational thinker”, in which category would you place mathematicians?

Mathematics is the purest of the arts, as well as the most misunderstood

It allows more freedom of expression than poetry, art of music – which depend on the the properties of the physical universe.

Aesthetic principle in Mathematics: simple is beautiful.

Exercise:1. Draw a circle and then indicate a point on this circle. How

many regions do you see inside the circle? Answer: 12. Consider one more point on this circle. Connect the two

points with a line segment. How many regions do you see?

3. Consider one more point on the same circle. Connect this last point with any of the two points. How many regions do you see?

4. Repeat the procedure. You will have now 5 points. How many regions do you expect to see? Check your answer.

5. Add another point. You have 6 points of the circle, Join the last point to each of the previous 5. How many regions do you expect to see?

Any mathematical conjecture faces the judgment of a

FORMAL PROOF

“I remember the time when I was kidnapped, and they sent a piece of my finger to my father. He said he wanted more proof.” Rodney Dangerfield, American actor

Theorem. The sum of any two even numbers is even.Proof: (try to produce a convincing argument)

Definition. An even number is a number that is a multiple of 2.

Let a and b be the two even numbers.

a is a multiple of two, therefore a can be written as 2k, where k is a positive integer.

b is a multiple of two, therefore b can be written as 2l, where l is a positive integer.

Therefore a+b=2k+2l.

Factorising the expression in the right hand side, a+b=2(k+l)

Therefore a+b is a multiple of 2, so it is an even number. I am done!

Mathematical proof makes mathematics a unique form of knowledge.

The formation of a proof is based on axioms, language and a strong system of formal reasoning.

Characteristics of formal proof:- is non-observational. Does not rely on our perception on nature- is a deductive process- proof doesn’t give only certitude, but also understanding and very often new knowledge- conclusions that come through mathematical proof are certain within the system of axioms that it is built upon

Pierre de Fermat (1601 – 1665)

Andrew Wiles (born 1953)

Back to the formation of mathematical knowledge:

Forming knowledge claims the existence of a set of axioms

consistent

complete

(independent)axiomatic system

An axiomatic system is called consistent if it lacks contradiction, i.e. within the system cannot be proven both a statement and its negation.The system is called to be complete if any statement can be proven or disproven within the system.Independence is not a requirement for the axiomatic system. One axiom is called independent if cannot be proven from the other axioms. In an independent system all axioms are independent.

The vain search for absolute truth

At the end of 19th century the critics of mathematics reached a shocking conclusion

Mathematical knowledge has limitations. Before the development of non-Euclidian geometries, mathematicians thought that they possessed the absolute truth.

The battle to restore the status of mathematics as the area of knowledge providing the eternal truth has begun

“We must know, we will know” (Hilbert, 1900)

Could Mathematics reach the state of completeness?

Can all statements be proved or disproved from the axioms within the system?

Can the consistency of axioms be proved?

“Many people would sooner die than think. In fact, they do so” Bertrand Russell

“Not ignorance, but ignorance of ignorance is the death of knowledge.” Alfred North Whitehead

Attempt to construct the entire field of mathematics from the principles of logic alone.

Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves.Under this scenario, we can ask the following question: Does the barber shave himself?

Kurt Godel (1906 – 1978)

Within any consistent axiomatic system there are true propositions that cannot be proven to be true.

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

http://www.youtube.com/watch?v=dK_narib4do

Mathematics and Natural Sciences