Start Introduction Definition: Basic assumptions or starting points. Lagemaat says that they are self-evident truths that provide a foundation for mathematical knowledge.

Infinite regress: endless chain of reasoning (proving A in terms of B and B in terms of C, and so on forever).

Four requirements for axioms:

•Consistent- If you can deduce two contradicting ideas from the set of axioms, then they are not consistent. When inconsistency occurs, anything can be proven wrong.

•Independent- Begin with the smallest possible number of axioms. You should not be able to deduce on of the axioms from the other; that would be a theorem.

•Simple- Axioms have to be clear and simple enough to be accepted without further proof.

•Fruitful- Should be able to prove as many theorems as possible with the fewest amount of axioms.

Examples that Euclid came up with:

•It shall be possible to draw a straight line joining any two points.

•A finite straight line may be extended without limit in either direction.

•It shall be possible to draw a circle with a given center through a given point.

•All right angles are equal to one another.

•There is just one straight line through a given point which is parallel to a given line. Deductive Reasoning Although math is defined as ‘the science of rigorous proof’ and is typically thought of as a subject with a lot of certainty, this certainty is limited. Although math can’t give us complete certainty, it is essential in many subjects such as physics and economics. It’s important to remember that math is not the only way to make sense of the world. Claire Cecil and Nanavale Toma Math! :] Lagemaat suggests that some people feel threatened by the fact that you can’t hide behind your mistakes in math. It’s either correct or incorrect. He also suggests that this comforts some people because there is actual proof of whether or not they have the correct answer.

Definition: the science of rigorous proof.

Galileo said that the book of nature is written in the language of mathematics.

Bertrand Russell began studying geometry when he was 11. The book quotes him saying that geometry “was one of the great events of [his] life, as dazzling as first love”. Euclid

Most well-known Greek mathematician: Euclid. He lived in Egypt somewhere near 300 BCE and did a lot of work to organize geometry into one area of knowledge. The geometry taught in most middle/high schools today is what Euclid worked with.

Euclid’s model of reasoning:

Axioms

Deductive reasoning

Theorems Axioms Syllogisms

Example:

All human beings are mortal (1)

Socrates is a human being (2)

Therefore Socrates is mortal (3)

1 and 2 are the premises and 3 is the conclusion of the argument. If 1 and 2are true, then 3 is necessarily true. In math, axioms are like premises and theorems are like conclusions. Theorems Using five axioms and deductive reasoning, Euclid came up with the following simple theorems:

1.)Lines perpendicular to the same line are parallel.

2.)Two straight lines do not enclose an area.

3.)The sum of the angles of a triangle is 180 degrees.

4.)The angles on a straight line sum to 180 degrees.

These can be used to make proofs.

Inductive reasoning: reasoning from particular to general. It is useful but cannot really give us certainty.

Goldbach’s conjecture: a famous mathematical conjecture according to which every even number is the sum of two primes. The conjecture still works through hundreds, thousands, and even ten thousands. However, there is uncertainty of how far you have to go before something is actually proven. Even if a conjecture is proven through a million, the next number could disprove it. A million may seem like a lot to us, but it’s nothing compared to infinity. Beauty, Elegance, and Intuition Sometimes a proof that is considered more elegant than others may be described as “beautiful”. Lagemaat says that in order for a proof to be “beautiful”, they have to be clear, economical, and elegant.

Questions:

There are 1,024 people in a knock-out tennis tournament. What is the total number of games that must be played before a champion can be declared?

What is the sum of the integers from 1 to 100? More Questions More questions:

1.)Do you think that mathematical insight can be taught, or would you say that it is something inborn and either you’ve got it or you haven’t?

2.)We sometimes use calculators and computers to help us solve mathematical problems. Does it follow that machines understand mathematics? Mathematics and Uncertainty Analytic: proposition that is true by definition.

Synthetic: any proposition that is not analytic.

a priori: A proposition that can be known to be true independent of experience.

A posteriori: A proposition that cannot be known to be true independent of experience. Discovered or Invented? Platonists- people who believe mathematical entities are discovered and exist ‘out there’.

Formalists- people who argue that they are invented and only exist in the mind.

Questions:

1.)What is the difference between saying that something has been ‘discovered’ and saying that it has been ‘invented’? What sorts of things do we usually say are discovered, and what sorts of things invented?

2.)Do you think that intelligent aliens would come up with the same mathematics as us, or might they develop a completely different kind of mathematics? Continued We like to say that math really exists because we have rules such as what defines a circle. However, don’t circles exist in everyday life? Wheels, coins, etc. No, apparently they do not, in a strict mathematical sense.

Since objects in math do not really exist in the real world, they must be “mental fictions”.

At the same time, since it is possible for us to discover them, they can’t be “mental fictions”.

Plato’s argument for the superior reality of mathematical over physical objects can be reduced o two key claims:

1.Mathematics is more certain than perception

2.Mathematics is timelessly true.

Questions:

1.)‘In order for something to exist, it must be possible to observe it’. Do you agree or disagree with this statement? Give reasons.

2.)Do you think that numbers have always existed? Did they exist at the time of the Big Bang? If they exist, where do you suppose they exist?

3.)Do you think that the full expansion of pi, which goes on forever, exists ‘out there’, and that we are gradually discovering more and more about it? Non-Euclidean Geometry For a long time, Euclidean geometry was considered to be a model of knowledge because it seemed to be certain and informative.

Riemannian geometry

In the 19th century, Georg Friedrich Bernard Riemann (1822-66) came up with the idea to replace some of Euclid’s axioms with their opposites. A lot of people thought the system would collapse because of contradictions. However, no contradictions were found in Riemann’s system.

Riemann’s axioms differed from Euclid’s as followed:

A.Two points may determine more than one line (instead of axiom 1).

B.All lines are finite in length but endless, like circles (instead of axiom 2);

C.There are no parallel lines (instead of axiom 5).

Among the theorems that can be deduced from these axioms are:

1.All perpendiculars to a straight line meet at one point.

2.Two straight lines enclose an area.

3.The sum of the angles of any triangle is greater than 180 degrees.

Gödel’s incompleteness Theorem

This theorem proved that it is impossible to prove that a formal mathematical system is free from contradiction. He did not prove that math actually contains contradictions, buy that we cannot be certain it doesn’t. Applied Mathematics Applied Mathematics

Definition: Math that is used to model and solve problems in the real world.

Mathematical ideas that are developed as a purely intellectual exercise sometimes turn out to be applicable to the real world. An example is in the 3rd century BCE when the Greeks became interested in the geometry of ellipses and a mathematician, Apollonius of Perga, wrote 8 volumes about them. The knowledge was useless until Johannes Kepler was studying planetary motion and discovered that the orbits of planets are not circular, but elliptical.

Question:

To what extent do you think governments should fund ‘useless’ research in pure mathematics? Conclusion 2 =1+1 4 =2+2 6 =3+3 8 =5+3 10 =5+5 12 =7+5 14 =7+7 16 =13+3 18 =13+5 20 =17+3

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# ToK: Math Presentation

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