The Axiomatic Method

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The axiomatic method I: Mathematical Proofs

Why do we need to prove things?

How do we resolve paradoxes?

Eudlid’s “Elements”

arranged in order many of Eudoxus's theorems, perfected many of Theaetetus's, and brought to irrefutable demonstration only loosely proved by his predecessors Ptolemy once asked him if there were a shorted way to study geometry than the Elements, … In his aim he was a Platonist, being in sympathy with this philosophy, whence he made the end of the whole "Elements" the construction of the so-called Platonic figures.

Elements begins with 5 postulates. Also … axioms Euclid calls 'common notions’… general assumptions that allow mathematics to proceed deductively. For example: “Things which are equal to the same thing are equal to each other.””

Euclid's Postulates 1. A straight line segment can be drawn joining any two points.

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 are drawn which 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 inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

Euclid's fifth postulate cannot be proven from others, though attempted by many people. Euclid used only 1—4 for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th.

In 1823, Bolyai and Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold.

Occam’s Razor

• the explanation of any phenomenon should make as few assumptions as possible, eliminating those that make no difference in the observable predictions of the explanatory hypothesis or theory. The principle is often expressed in Latin as the lexparsimoniae ("law of parsimony”

Turtles all the way down

• • • • A well-known scientist … gave public lecture on astronomy… described how earth orbits around sun a sun orbits around center of galaxy.

At the end of the lecture, a little old lady at the back of the room got up and said: "What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise.” Scientist replies, "What is tortoise standing on?" "You're very clever, young man" said the old lady. "But it's turtles all the way down!”

Mathematics vs Pure Reason

• • • • • What distinguishes mathematical reasoning from other reasoning?

Exs: If I study hard I will get a good grade I think, therefore I am The area of a triangle is ½*b*h

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Descartes: Scientific Method

Descartes’ major work, published in 1637 “Discourse on the Method for Rightly Directing One’s Reason and Searching for Truth in the Sciences.” Descartes proposed that all science become demonstrative in the way Euclid made geometry demonstrative, namely as a series of valid deductions from self evident truths, rather than as something rooted in observation and experiment.

Geometry, without the parallel postulate

Other geometries

Hyperbolic geometry

axiomatic method

• reduce a “coherent body” of mathematical to a minimal set of “axioms.

• In their Principia Mathematica, Whitehead and Bertrand Russell attempted to show that all mathematical theory could be reduced to some collection of axioms. • The explication of the particular axioms can help to clarify a suitable level of mathematical abstraction (is the set of assumptions too many or not enough – compare this to Occam’s razor • The Zermelo-Fraenkel axioms, the result of the axiomatic method applied to set theory, allowed the proper formulation of set theory problems and helped avoided the paradoxes of naïve set theory. One such problem was the Continuum hypothesis.

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Peano axioms

Peano axioms define properties natural numbers, N. 1. For every natural number x, x = x. (reflexive) 2. For all xand y, if x = y, then y = x (symmetric).

3. For all x, y and z, if x = y and y = z, then x = z (transitive).

4. For all a and b, if a is a natural number and a = b, then b is also a natural number.

the naturals are assumed to be closed under a "successor" function S.

5. 0 is a natural number.

6. For every natural number n, S(n) is a natural number.

properties of S.

7. For every natural number n, S(n) ≠ 0.

8. For all m and n, if S(m) = S(n), then m = n.

…the set of natural numbers is infinite, 9. If K is a set such that: * 0 is in K, and * for every natural number n, if n is in K, then S(n) is in K, then K contains every natural number.