1. Philosophy of Mathematics: a sneak peek

2. Mathematics exists independently of the human mind.
REALISM
Mathematics exists independently of the human mind.
Humans and “aliens” did not invent mathematics. Math is discovered.
Math is physically real.

3. REALISM
One type is Mathematical Platonism: mathematics is abstract, eternal and unchanging.

4. Mathematics is “invented” by the human mind (e.g., Formalism).
ANTI-REALISM
Mathematics is “invented” by the human mind (e.g., Formalism).
Math is not physically real.

5. NON-PLATONISTIC SCHOOLS OF THOUGHT
Logicism
Intuitionism
Formalism

6. reduce mathematics to logic
1. LOGICISM
reduce mathematics to logic

7. mathematical knowledge is acquired by deduction from basic principles
DEDUCTIVE LOGIC
mathematical knowledge is acquired by deduction from basic principles

8. 2. INTUITIONISM mathematics is an activity of construction
“The real numbers are mental constructions, proofs and theorems are mental constructions, mathematical meaning is a mental construction”

9. 3. FORMALISM “higher mathematics is no more than a formal game”
proving mathematical statements is a game in which symbols (abstract objects) are manipulated according to fixed rules

10. 3. FORMALISM
a necessary requirement of a formal system (e.g., axiomatic system) is that it should be at least consistent [Hilbert]
An axiomatic system is consistent if there is no statement such that both the statement and its negation are axioms or theorems of the axiomatic system. Contradictory axioms or theorems are not desired in an axiomatic system.

11. AXIOMATIC SYSTEM
MATH 10

12. AXIOMATIC (POSTULATE) SYSTEM
consists of some undefined terms and a list of statements, called axioms, concerning the undefined terms
one obtains a mathematical theory by proving new statements, called theorems, using only the axioms, logic system, and previous theorems
definitions are made in the process in order to be more concise

13. INGREDIENTS Undefined terms/primitive terms Defined terms
Axioms/postulates - accepted unproved statements
Theorems - proved statements

14. A theory consists of an axiomatic system and all its derived theorems.

15. “THEORY” Examples: Approximation theory Automata theory Chaos theory
Coding theory
Game theory
Graph theory
Group theory
Information theory
Knot theory
Number theory
Probability theory
Queueing theory
Set theory

16. ENGLISH LANGUAGE
Here, we will use the English language as an auxiliary language. Note: “2+2=6.” is a sentence.

17. ENGLISH LANGUAGE
Here, we will use the English language to as an auxiliary language.

18. UNDEFINED TERMS
Undefined terms are of two types:
objects or elements (e.g., point, line, plane)
relationships between objects, called relations (e.g., on, between)
“A point is on a line.”

19. UNDEFINED TERMS
Undefined terms are not anymore defined to eliminate infinite regress of defining terms.

20. UNDEFINED TERMS
A model of an axiomatic system is obtained if we can assign meaning to the undefined terms of the axiomatic system which convert the axioms into true statements about the assigned concepts.

21. UNDEFINED TERMS Two types of models:
Concrete model: meanings assigned to the undefined terms are objects and relations adapted from the real world.
Abstract model: meanings assigned to the undefined terms are objects and relations adapted from another axiomatic development.

22. THEOREMS
As we prove a “new” statement to be true, it becomes a theorem. Proof should be based only on the axioms, logic system, and previous theorems.

23. NOTE ON MATHEMATICAL PROOF
WARNING!
The word “obvious” is not acceptable as a proof.
Do not make an additional assumption outside the system being studied.
Do not depend on any preconceived idea or picture. Pictures should only be used as an intuitive aid in developing the proof.

24. Conjecture
A statement formed without proof (not yet proved or disproved) Fermat’s Last Theorem (already proven by Andrew Wiles after 350 years death of Fermat): “No three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2."

25. Example 1 Axiom 1. Every dog has at least two human owners.
Axiom 1. Every dog has at least two human owners.
Axiom 2. Every human owner has at least two dogs.
Axiom 3. There exists at least one dog.

26. Example 1 Suppose the real number system holds.
What are the undefined terms in this axiom set?
Elements: dog; human owner
Relation: has

27. Example 1
Axiom 3 guarantees the existence of a dog, but no axiom explicitly states that there exists a human owner. Theorem 1. There exists at least one human owner. Proof: By Axiom 3, there exists a dog. Now since each dog must have at least two human owners by Axiom 1, there exists at least one human owner. QED

28. IDEAL AXIOMATIC SYSTEM
Consistent: the system lacks contradiction, that is, we cannot derive both a statement and its negation from the system's axioms (a necessary requirement)
Independent: an axiom is independent if it is not a theorem that follows from the other axioms (not a necessary requirement)
Complete: every statement containing the undefined and defined terms of the system can be proved valid or invalid

29. NOTE ON CONSISTENCY
“Inconsistent theories prove everything, including their consistency.”

30. NOTE ON INDEPENDENT AXIOMS
In some high school geometry courses, theorems which are long and difficult to prove are usually taken as axioms/postulates. Hence in most high school geometry courses, the axiom sets are usually not independent.

31. Example 1 Axiom 1. Every dog has at least two human owners.
Axiom 2. Every human owner has at least two dogs.
Axiom 3. There exists at least one dog.
Axiom 4. There exists at least one human owner. (dependent)

32. Mathematician's worst nightmare
Kurt Gödel with his Incompleteness Theorem (1931) demonstrated that even in elementary parts of arithmetic there exist propositions which cannot be proved or disproved within the system.

33. Mathematician's worst nightmare
The liar paradox: "This sentence is false."
A Gödel sentence G for a system M (truth replaced by provability): "G is not provable in the system M."
Note: It is not possible to replace "not provable" with "false”.

34. Mathematician's worst nightmare
“A complete and consistent list of axioms can never be formulated.”
Each time an additional consistent statement is added as an axiom, there are other true statements that still cannot be proved.
If an axiom is added that makes the system complete, it does so at the cost of making the system inconsistent.
Gödel’s Incompleteness Theorem says that there will always be statements that are eternally doomed to be proven neither true or false.

35. Mathematician's worst nightmare
“At the bottom of any logic system (such as science) are statements that must be taken on faith alone.” However, this ”incompleteness” provides opportunity for growth and improvement of mathematics/science.

36. Mathematician's worst nightmare
Generally, an axiomatic system does not stand alone.

37. EXAMPLES
MATH 10

38. Example 1 Axiom 1. Every dog has at least two human owners.
Axiom 2. Every human owner has at least two dogs.
Axiom 3. There exists at least one dog.

39. Example 1
Prove that the minimum number of dogs is two. Proof: By Theorem 1, there exists a human owner, call it H1. Then by Axiom 2, H1 must have two dogs call them D1 and D2. Hence, there are at least two dogs. By Axiom 1, D1 must have a human owner other than H1, call it H2. We form a model where H1 and H2 both are assigned to D1 and D2, then we have exactly two dogs, which demonstrates that the minimum number of dogs is two. QED

40. Example 1
This axiomatic system is consistent. But not complete. For example, we cannot prove or disprove: “There exist at least four dogs.”

41. Example 2
Axiom 1. Every heart has at least two paths. Axiom 2. Every path has at least two hearts. Axiom 3. There exists at least one heart.

42. Example 2
Example 1 and Example 2 are abstractly the same, only the notation is different. 

43. Example 2
Axiom 1. Every heart has at least two paths. Axiom 2. Every path has at least two hearts. Axiom 3. There exists at least one heart. Definition: A path P is said to connect two hearts, say Puso1 and Puso2, iff P has Puso1 and Puso2.

44. Example 2
Axiom 1. Every heart has at least two paths. Axiom 2. Every path has at least two hearts. Axiom 3. There exists at least one heart. Theorem: Every path connects two hearts.

45. Example 2
Suppose there are exactly 2 hearts.

46. Example 2
Suppose there are exactly 3 hearts.

47. Example 2
Suppose there are exactly 4 hearts.

48. Example 2
Suppose there are exactly 4 hearts.

49. Example 2
Suppose there are exactly 4 hearts.

50. Example 3
Axiom 1. Every heart has at least two paths. Axiom 2. Every path has at least two hearts. Axiom 3. There exists exactly two hearts. Axiom 4. Any two paths have at most one heart in common.

51. Example 3
This axiomatic system is inconsistent.

52. Example 4
Axiom 1. Every hive is a collection of bees. Axiom 2. Any two distinct hives have one and only one bee in common. Axiom 3. Every bee belongs to two and only two hives. Axiom 4. There are exactly four hives.

53. Example 4
Deduce the following theorems: Theorem 1. There are exactly six bees. Theorem 2. There are exactly three bees in each hive. Theorem 3. For each bee there is exactly one other bee not in the same hive with it.

54. Example 4
Hive 1: {A,B,C} Hive 2: {A,D,E} Hive 3: {B,D,F} Hive 4: {C,E,F}

55. Example 5: Peano’s Axiom
Axiom 1: N is a set and 1 ∈ N.
Axiom 2: Each element x of N has a unique successor in N denoted x’.
Axiom 3: 1 is not the successor of any element of N.
Axiom 4: If x' = y' then x = y.
Axiom 5: If M ⊂ N satisfies both
1 ∈ M
x ∈ M implies x' ∈ M
then M = N.

56. Example 6: Euclid’s Postulates (restated)
Axiom 1 (ruler): Given any two points, you can draw a straight line between them (making what's called a line segment). Axiom 2: Any line segment can be extended indefinitely. Axiom 3 (compass): A circle may be drawn with any given radius and an arbitrary center. Axiom 4: All right angles are equal to each other. Axiom 5: Through a given point, only one line can be drawn parallel to a given line. [Playfair]

57. EXERCISE
Axiom 1. Every heart has at least one connection. Axiom 2. Every connection has exactly two hearts. Axiom 3. There exists at least one connection.

58. Prove the following theorems: Theorem 1
Prove the following theorems: Theorem 1. There exist at least two hearts.
EXERCISE
Theorem 2. The minimum number of hearts is two and minimum number of connections is one.

59. Prove the following theorems:
EXERCISE
Theorem 3. Suppose there are exactly three hearts. There are only two possibilities: there are two connections or there are three connections.

60. https://plato.stanford.edu/entries/philosophy-mathematics/