Mathematics Flashcards
What is pie?
Circ/diam
Galileo quote
“The book of nature is written in the language of mathematics”
“This is one of the greatest events of my life, as dazzling as my first love. I had not imagined that there was anything so delicious in this world”
Bertrand Russell
Math
The science of rigorous proof; deductive reasoning used to derive theorems
Axioms
Starting points or basic assumptions believed to be true without proof
Euclid’s 5 Axioms
- It shall be possible to draw a straight line to join any two points 2. A finite line may be extended without the limit in either direction3. 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 another5. There is just one straight line through a given point which is parallel to a given line
Proof
A theorem is shown to follow logically the relevant axioms
Conjecture
A hypothesis that seems to work, but has not been shown to necessarily be true
Goldbach’s conjecture
Every number is the sum of two primes
Henri Pointcaré
Stressed the role of intuition
Analytic proposition
True by definition
Synthetic proposition
Any proposition that is not analytic
Math options
Option 1: math is empiricalOption 2: math is analyticOption 3: math as synthetic a priori
Euclidean geometry
Self evident, though to be true
Baruch Spinoza
Write a book of ethics based on a series of theorems
Empiricism
Mathematical truths are empirical generalizations
Formalism
Mathematical truth are true by definition Math is invented, only existing in our minds
Platonism
Mathematical truths give us a priori insight into the structure of realityPlatonists believe math is discovered and exists “out there” “ideal”
Mental fictions
Only exist in our minds
Plato’s solution
Mathematical objects exist “out there” but in a unique form of existence
Non-Euclidean geometry
Georg f b Riemann Replaced Euclid’s axioms with their contraries Two points may determine more than one lineAll lines are finite in length but endlessThere are no parallel lines
Riemannian geometry axioms
All perpendiculars to a straight line meet at one pointTwo straight lines enclose an areaThe sum of the angles of any triangle are greater than 180 degrees Never proved system was free from contradiction = problem of consistency
Gödel’s incompleteness theorem
It is impossible to prove mathematics free of contradiction Kurt gödel
Mathematical truths are either
EmpiricalTrue by definitionRational insights to universal truths
Analytic
(Def) Characterized by the use of separate words
A posteriori
Effect -> causeBased on observation or experience
A priori
Cause -> effectGeneral rule to a particular caseIndependent from experience
Axioms
Statements taken to be true without proof
Conjecture
A conclusion reached by guessing
Deduction
Reasoning, inferenceAct of taking away, subtracting
Empiricism
The use of methods based on experiment and observation
Idealization
Belief that only mental entities are real
Theorem
A statement that is to be proved or has been proven
What are the four main characteristics of axioms?
- consistent – something is what it is and is not what it is not!
- independent – smallest number possible
- simple – accepted without further proof
- fruitful – enable proof of theorems
