Foundations and Philosophy

Question 1

Is mathematics a science?

Answer 1

If mathematics is a science, then it is a very different science from others we know (biology, physics, etc.)

Question 2

Do mathematical objects exist?

Answer 2

In biology, for example, you can physically see objects you study.
In maths, you cannot see everything

Question 3

Is mathematical knowledge absolutely certain?

Any examples?

Answer 3

On the surface, it would appear so. However, when you look deeper it is more ambiguous.
For example, if you agree with Euclid’s 5th postulate (parallel line postulate) then you can construct Euclidean geometry. But if you disagree, then you can construct other forms of geometry.

Question 4

What does the “Foundations of Mathematics” refer to?

Answer 4

Questions about what mathematics is built on, and what it actually is

This is more philosphy than mathematics

Question 5

What does philosophy include?

Answer 5

1. Ontology - The nature of existence
2. Epistemology - The nature of knowledge

Question 6

Explain Ontology

Answer 6

Ontology - the nature of existence
- What kinds of things exist?
- In what sense do they exist?

Question 7

Explain Epistemology

Answer 7

Epistemology - the nature of knowledge
- How can we know about things?
- Is the knowledge certain, provisional, open to correction, etc.

Question 8

What is Goldbach’s conjecture?

Answer 8

Every even integer >= 4 can be written as the sum of 2 prime numers

A very famous unsolved problem in number theory

Question 9

What is Platonism?

Answer 9

A follower of the philosophy of Plato, particularly his idea that mathematical objects exist outside the material world (“World of Ideals”)

Question 10

Describe a Platonist view of Golbach’s conjecture.

Answer 10

1. The infinite sequence of natural numbers is “just there”
2. In the sequence, there is either a counterexample (so GC is false) or there isn’t (so GC is true)
3. If it’s true, then it always has been true regardless of what we know about it. It would still be true even if no-one had ever asked the question

Question 11

Describe a Platonist view of mathematics.

Answer 11

• The view that mathematical objects have a real, but non-material, existence (independent of humans, but accesible to thought)
• Mathematics is already decided and exist in a “world of ideals”, and just needs to be discovered.
• Mathematics would still be mathematics, even if humans never existed.

Question 12

What are some objections to the Platonist view (of the Goldbach conjecture)?

Answer 12

• What sort of thing are these numbers? Just ideas?
• Where is “just there”?
• How can you know that they exist, or know anything about them?

Question 13

What is intuitionism?

Answer 13

Mathematics is considered to be purely the result of the constructive mental activity of humans, rather than the discovery of fundamental principles claimed to exist in an objective reality.

Question 14

Describe an Intuitionist view of Goldbach’s conjecture.

Answer 14

1. No one can understand infinitely many numbers all at once, so it doesn’t make sense to consider the whole sequence of natural numbers as one single object
2. To say that GC is true means nothing more than having a method which guarantees that, for any given even number, it can return two primes that add to that number
3. To say that GC is false means nothing more than having a way to see any such method would be contradictory

It could be that GC is neither true nor false, since we could have no way to do either step 2 or 3

Question 15

Describe an Intuitionist view of mathematics.

Answer 15

• Can accept statements as neither true nor false
• Based on intuition and what can be achieved by construction
• For example, since we live in a finite universe, we have no way to construct an infinite quantity (it could be very large, but always finite), so intuitionist might reject the idea of infinites

Question 16

What are some objections to the Intuitionist view (of the Goldbach conjecture)?

Answer 16

• The version of true and false is too individualistic (something which is not true for you today might be true for me today or true for you tomorrow)
• Deciding true or false based on human construction goes against classical logic
• How can something be neither true nor false?
• If there are statements which are neither true nor false, then it isn’t legitimate to use Proof by Contradiction

Question 17

What is Formalism?

Answer 17

The view that (mathematical and logical) statements can be considered to be statements about the consequences of the manipulation of strings using established rules.

Question 18

Describe a Formalist view of Goldbach’s conjecture.

Answer 18

• If, within the rules of the “game”, we can get to a string of symbols which says that GC is true/false then we have proved/disproved it
• It doesn’t make sense to ask the “meaning” or the “truth” of the strings of symbols

Question 19

Describe the Formalist view of mathematics.

Answer 19

That mathematics is really just like a game with strings of symbols, and rules for manipulation. There are certain strings we can start with (axioms), and rules to combine/transform strings already written down (Rules of Inference)

A formalist is not concerned with asking the “meaning” of the symbols of strings

Question 20

What are some objections to the Formalist view (of the Goldbach conjecture)?

Answer 20

• If maths is “just a game”, then why should it have anything to do with real-life, science, engineering, etc?
• Which rules do we choose to work with? Why have you chosen certain axioms and rules of classical logic, not other types?
• What if neither the string saying that GC is true, nor the string saying it is false, can be reached within the rules of the game?
• What if both of the strings can be reached?

Question 21

What is Fictionalism?

Answer 21

The idea that mathematics is nothing more than a fiction (or novel).
Used to “tell stories” about mathematical ideas, acknowledging their utility without requiring belief in their literal truth.

Question 22

Which view of mathematics is the “most naive”? Why?

Answer 22

Formalism, since it seems to have flawed reasoning for foundations (its choice of rules and axioms, etc)

Interestingly though, this is probably the most popular view adopted by mathematicians

Question 23

What is a Syllogism? What is the general form? Give a valid and invalid example.

Answer 23

Patterns of valid logical deduction, considered by Aristotle
- General form: “Every A is a B. x is an A. Therefore x is a B”
- Valid: “Socrates is a man. All men are mortal. Therefore, Socrates is mortal.”
- Invalid: “All cats have four legs. Fido has four legs. Therefore Fido is a cat.”

Question 24

What is the difference between “actually infinite” and “potentially infinite”? Give an example.

Answer 24

• Actually Infinite - an object which is infinite (e.g., a straight line extending indefinitely in both directions)
• Potentially Infinite - an object you start with, which you can always make bigger (e.g., a line segment, which can be extended but we are unsure how far it needs to be extended)

Euclid considers potential infinites in The Elements (postulates 2 and 5)

Question 25

Describe the Arabic mathematic approach to the Parallel Postulate

Answer 25

- ca. 1000-1200 CE - Felt that the Parallel Postulate was unsatifactory - Gave (incorrect) proofs that it followed from the other axioms

Question 26

Describe Saccheri's approach to the Parallel Postulate.

Answer 26

1. Let ABCD be a quadrilateral in which A, B are right angles, and AD=BC 2. Showed that angles C, D are equal 3. Call the probability that C,D are acute/obtuse the "hypothesis of the acute/obtuse angle" 4. He believed that he had proved that each the hypothesis of the acute and obtuse angles led to contradictions, meaning the Parallel Postulate must be true ## Footnote (This is not right)

Question 27

Describe Lambert's approach to the Parallel Postulate.

Answer 27

Lambert knew that Saccheri's hypotheses of acute/obtuse angles is true for spherical geometry (as opposed to a plane)

Question 28

Describe Kant's way to divide statements.

Answer 28

Two ways to divide statements - Method 1: 1. Analytic - "All bachelors are unmarried" 2. Synthetic - "It was raining yesterday" - Method 2: 1. Empirical - "The sky is blue today" 2. A Priori - "2 + 2 = 4"

Question 29

Describe Kant's approach to Euclidean geometry.

Answer 29

- According to Kant: "The angles of a triangle add up to two right angles" is a synthetic statement, but also a priori - This is possible because we can't help but percieve things as located in time and space - Our minds seem to be constitued with intuitive notions of time/space *built in* so Kant believed Euclidean geometry must be true because it was impossible (at least for him) to conceive space in any other way.

Question 30

Give two examples of non-Euclidean geometry. Why did they arise? ## Footnote Dates?

Answer 30

1. Spherical Geometry (1800s) 2. Hyperbolic Geometry (1800s) Both originally aimed to find a contradiction, but cam to believe that their theories were valid in some possible form of geometry

Question 31

What were David Hilbert's grouping of the axioms?

Answer 31

- Axioms of Connection/Incidence - Axioms of Order - Axioms of Parallels - Axioms of Congruence - Axioms of Continuity/Completeness

Question 32

What was Cantor's fundamental sequence of rational numbers?

Answer 32

- Georg Cantor reagarded real numers as associated to fundamental sequences of rational numbers - "{a_n} for n ∈ N is a fundamental sequence if, given any ε>0, there is some N with |a_n - a_m|<ε whenever m,n>N"

Question 33

What was Euclid's proof of no largest prime number?

Answer 33

1. Assume you have a list containing all prime numbers in increasing order: p_1, p_2, p_3, ..., p_n (where p_n is the largest prime) 2. Compose a new number n=p_1 x p_2 x p_3 x ... x p_n 3. Then consider n+1, which must have a prime factorisation 4. n+1 is not divisible by any prime in the original list, so must be prime itself 5. n+1>p_n so a contradiction is found

Question 34

What are some problems of infinity?

Answer 34

- What are statements of infinity about, if we never "get there"? - Actual infinity vs Potential infinity - Countability (being able to write out all elements of a set) vs uncountability - Limits and infinite series - Infinitesimals and calculus - Two different infinities not being equal

Question 35

What is Cantor's Paradise?

Answer 35

- A termed used by David Hilbert to describe Cantor's work on comparing the sizes of infinite sets. - Cantor showed there were as many rational numbers as positive integers, but more real numbers than positive integers

Question 36

Were Cantor's ideas well received?

Answer 36

At first, no they were not: - Some mathematicians were unwilling to accept the actual (and uncountable) infinities which were essential in Cantor's work. - Others felt that only things which could be constructed should be allowed

Question 37

What did Cantor's work indicate?

Answer 37

That the concept of a set, rather than a number, should be the starting point of mathematics.

Question 38

What is Peano's axiomatic description of the natural numbers?

Answer 38

He described the natural numbers (with 0), N_0={0,1,2,...} in terms of the "successor" functions S: 1. 0 ∈ N_0 2. S(a) ∈ N_0 for all a 3. There is no a∈N_0 with S(a) = 0 4. If S(a)=S(b) then a=b 5. If T ⊆ N_0 satisfies: (1) 0∈T and (2) for all a∈N_0, if a∈T implies S(a)∈T then T=N_0

Question 39

How did Peano define addition and multiplication?

Answer 39

Addition and multiplication can be defined recursively: - a + 0 = a - a + S(b) = S(a+b) and: - a x 0 = 0 - a x S(b) = (a x b) + a

Question 40

What was Frege's idea of set theory? How was it scuppered?

Answer 40

- In Frege's set theory, a set can be any collection of objects: we can form sets of sets, and the set of all sets - It was devastated by Russell's paradox: "What about the set S of all those sets which do not belong to themselves? Does S belong to itself? Either of the possible answers 'Yes' or 'No' leads to a contradiction."

Question 41

What was an outcome of Russell's critique of Frege's set theory?

Answer 41

- We needed axioms which specify how we can create new sets from old ones - This must be restrictive enough to prevent contradictions, but free enough to construct mathematical objects

Question 42

What is logicism?

Answer 42

- That everything follows from unpacking the definitions following the laws of logic - Since it seeks to make mathematics true, independently of human understanding, logicism can be seen as a form of Platonism

Question 43

What is the Law of the Excluded middle? Who rejects the law?

Answer 43

- That for any statement P, either P or its negation ¬P must be true - Rejected by Brouwer (and other intuitionists)

Question 44

What is Brouwer's Fixed Point Theorem? How was it proved? What is an interesting observation?

Answer 44

- Theorem: "Let D be the closed unit disc with boundary the circle S. Let f:D->D be any continuous function, then f has a fixed point (this is some a∈D with f(a)=a)" - Proof: "Suppose f has no fixed point. Then we can define a continuous function g:d->S as follows: for each z∈D draw a straight line from f(z) to z and let g(z) be the point where this line meets S. If z∈S then g(z)=z which contradicts the lemma" - Interesting observation: this is a non-constructive proof that uses contradiction (both are disregarded by intuitionists)

Question 45

What was Hilbert's program?

Answer 45

- He wanted to put mathematics on a solid foundation by starting from set theory axioms and formalising the laws of inference - This meant that proofs could be checked for validity

Question 46

What is an interesting observation about Hilbert's program, in relation to Formalism?

Answer 46

- To show that the Hilbert's system is consistent and/or complete requires reasoning about the system - However, Formalists consider the "meaning" of symbols to be irrelevant

Question 47

How did Godel approach Hilbert's program?

Answer 47

- Kurt Godel considered a Hilbert system of mathematics, powerful enough for integer arithmetic - In any such system, the strings of symbols could be encoded as numbers, and the laws of inference interpreted as operations on the numbers - This means that assertations abouot provability in the system can be turned into arithmetic statements with the system

Question 48

What are Godel's 2 Incompleteness Theorems?

Answer 48

1. In any formal mathematical system S, there are always true statements within S that cannot be proved within S 2. Such a formal system cannot prove itself to be consistent (assuming that it is indeed consistent).

Question 49

What are some potential crises in Logicism, Formalism, and Intuitionism?

Answer 49

- Logicism gives no guarantee that mathematics has solid foundations - Formalism - Godels theorems show that formal approaches cannot provide a consistent and complete system from any finite set of axioms - Intuitionism rejects non-constructable infinite sets (and much of classical mathematics)

Question 50

List some mathematical revolutions since 1900.

Answer 50

1. Gender in mathematics: particularly the accessibility and advancement in training and careers to women 2. A two-way interaction between development of computers and mathematics 3. The intensity and availability of mathematical training in industrialised nations

Question 51

Describe Russell and Whitehead’s set theory system.

Answer 51

- Russell and Whitehead produced a system entirely built on set theory, which avoids Russell's paradox - However, there is no guarantee that their axiosm are consistent (a paradox could be waiting to be uncovered)

Question 52

What is the 'unreasonable effectiveness of mathematics'?

Answer 52

- Eugene Wigner used the phrase in his 1960 paper to argue that it is a mystery that mathematics could be so useful - It is a miracle/gift which we do not understand nor deserve - We should be grateful for its existence and hope it remains in future - Aligns with the Platonist idea that maths exists in the "world of ideals" - Intuitionism and Formalism would not align with this idea since they argue that mathematics is purely a human construction.