The Basics PURE MATHEMATICS FORMAL SYSTEMS Did you

The Basics PURE MATHEMATICS

FORMAL SYSTEMS Did you understand the Introduction to Formal Systems? Essentially, it is when you have a set of basic premises (axioms) that are used to develop ideas (theorems) and verification of those ideas (proofs) What is the link between Math and Logic? It is sometimes said that mathematical reasoning is a process of logical deduction. If this is true, and if the conclusion of a proof must always be implied by (contained in) its premises, how can there ever be new mathematical knowledge?

AXIOMS So, what is an axiom? Let’s take a moment and examine Euclid’s Axioms for Geometry: 1. It shall be possible to draw a straight line joining any two points. 2. A finite straight line may be extended without limit in either direction. 3. It shall be possible to draw a circle with a given center and through a given point. 4. All right angles are equal to one another. 5. There is just one straight line through a given point which is parallel to a given line.

AXIOMS So, where/how did Euclid come up with these Axioms? Axioms are examples of a priori knowledge. “A ‘self-evident’ truth that requires no proof” “A universally accepted principle or rule” “A proposition that is assumed without proof for the sake of studying the consequences that follow from it” So, where is the “proof” for these axioms?

AXIOMATIC IMPLICATIONS How do we choose the axioms underlying mathematics? Is this an act of faith? Axioms are just “accepted as true” in order to use them to build on and search out their greater implications, otherwise we would be dealing with the issue of infinite regress. What does it mean to say that mathematics is an axiomatic system? “Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. ” Bertrand Russell (1917)

THEOREMS & PROOFS What is a theorem? What is a proof (in a mathematical sense)? How important are these to math? How are they different from theories and laws in science? What is a conjecture and how is it different from a theorem? How is it different from a proof? Why would someone even pose a conjecture if they didn’t have a pretty good idea it was true? Are all mathematical statements either true or false? Can a mathematical statement be true before it has been proven?

SET THEORY & RUSSELL’S PARADOX A ‘set’ is any defined group of objects. This is used in multiple ways, but the simplest we learn as young children – counting ‘things’ that are alike – “How many apples? ” A paradox is a statement or set of statements that cause contradiction or defies intuition. Example – An old lady that refuses to get rid of a box full of string labeled “string too short to be useful. ”

SET THEORY & RUSSELL’S PARADOX Bertrand Russell was a mathematician and philosopher from the early 20 th century which found a problem theory. Here goes: Set of allwith andsetonly those There is a regimental himself a members of thebarber, regiment who of thethe regiment, shaves allwho and only barberwho shaves and do those membersnot of the regiment who do not shave themselves. Does the barber shave himself? Let’s take a moment to look at the set that has been defined. So, does the barber shave himself?

FERMAT’S THEOREM Do you understand Fermat’s Theorem that it is not possible to have xn + yn = zn for n>2? Can you believe that someone spent 30 YEARS of his life trying to prove this? To quote Mrs. Snow, “What a waste of time. ” Do you agree with her? Did you watch the documentary about Dr. Wiles? What are the roles of empirical evidence and inductive reasoning in establishing a mathematical claim? What is the role of the mathematical community in determining the validity of a mathematical proof?

a=b a - b = 0 (subtract both sides by b) (a - b)/(a - b) = 0 / (a-b) (divide both sides by a - b) 1 = 0 (simplify)

A CURIOUS INCIDENT What are prime numbers? Why are they “special? ” Why does Christopher see them as special? Other people believe that mathematics is a formal game devoid of intrinsic meaning? What do they mean by this? If this is the case, how can there be such a wealth of applications in the real world? Do you think math could hold the keys to unlocking the secrets of life?

HOMEWORK READ THE FOLLOWING: “A Father and Son “Discuss” Math” “WHAT ARE NUMBERS, REALLY? ” “What happens when you can't count past four? ” They are a little long, but you REALLY need to read them! DON’T SKIM!