Mathematics

Question 1

What is pie?

Answer 1

Circ/diam

Question 2

Galileo quote

Answer 2

“The book of nature is written in the language of mathematics”

Question 3

“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”

Answer 3

Bertrand Russell

Question 4

Math

Answer 4

The science of rigorous proof; deductive reasoning used to derive theorems

Question 5

Axioms

Answer 5

Starting points or basic assumptions believed to be true without proof

Question 6

Euclid’s 5 Axioms

Answer 6

1. 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

Question 7

Proof

Answer 7

A theorem is shown to follow logically the relevant axioms

Question 8

Conjecture

Answer 8

A hypothesis that seems to work, but has not been shown to necessarily be true

Question 9

Goldbach’s conjecture

Answer 9

Every number is the sum of two primes

Question 10

Henri Pointcaré

Answer 10

Stressed the role of intuition

Question 11

Analytic proposition

Answer 11

True by definition

Question 12

Synthetic proposition

Answer 12

Any proposition that is not analytic

Question 13

Math options

Answer 13

Option 1: math is empiricalOption 2: math is analyticOption 3: math as synthetic a priori

Question 14

Euclidean geometry

Answer 14

Self evident, though to be true

Question 15

Baruch Spinoza

Answer 15

Write a book of ethics based on a series of theorems

Question 16

Empiricism

Answer 16

Mathematical truths are empirical generalizations

Question 17

Formalism

Answer 17

Mathematical truth are true by definition Math is invented, only existing in our minds

Question 18

Platonism

Answer 18

Mathematical truths give us a priori insight into the structure of realityPlatonists believe math is discovered and exists “out there” “ideal”

Question 19

Mental fictions

Answer 19

Only exist in our minds

Question 20

Plato’s solution

Answer 20

Mathematical objects exist “out there” but in a unique form of existence

Question 21

Non-Euclidean geometry

Answer 21

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

Question 22

Riemannian geometry axioms

Answer 22

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

Question 23

Gödel’s incompleteness theorem

Answer 23

It is impossible to prove mathematics free of contradiction Kurt gödel

Question 24

Mathematical truths are either

Answer 24

EmpiricalTrue by definitionRational insights to universal truths

Question 25

Analytic

Answer 25

(Def) Characterized by the use of separate words

Question 26

A posteriori

Answer 26

Effect -> causeBased on observation or experience

Question 27

A priori

Answer 27

Cause -> effectGeneral rule to a particular caseIndependent from experience

Question 28

Axioms

Answer 28

Statements taken to be true without proof

Question 29

Conjecture

Answer 29

A conclusion reached by guessing

Question 30

Deduction

Answer 30

Reasoning, inferenceAct of taking away, subtracting

Question 31

Empiricism

Answer 31

The use of methods based on experiment and observation

Question 32

Idealization

Answer 32

Belief that only mental entities are real

Question 33

Theorem

Answer 33

A statement that is to be proved or has been proven

Question 34

What are the four main characteristics of axioms?

Answer 34

1. consistent – something is what it is and is not what it is not! 2. independent – smallest number possible 3. simple – accepted without further proof 4. fruitful – enable proof of theorems