• Mathematics as empirical

• Mathematics as analytic

• Mathematics as synthetic a priori

What is mathematics as empirical?

It basically means that all math statements are based on verifiable observations or experience, not just random theories.

For example: we are more certain that 3 x 3 = 9 than all basketball players are tall.

The Mathematical paradigm

Math is "the science of rigorous proof"

Introduction

Mathematics as the search for abstract patterns

Mathematics not only seems to give us certainty, but is also of enormous practical value.

The certainty and usefulness of mathematics may help to explain its enduring appeal.

**AOK - Math**

Formal system

Axioms

Deductive reasoning

Theorems

intuition & Mathematics

Creative imagination and intuition also play a key role in mathematics to come up with a new theorem.

Everything that can be counted does not count. Everything that counts cannot be counted.

- Albert Einstein

AxiOms

Deductive reasoning

theorems

Beauty, elegance and intuition

Mathematics and certainty

Option 1: Mathematics as empirical

option 2: Mathematics as analytic

option 3: Mathematics as synthetic a priori

The first person to organize geometry into a rigorous body of knowledge

His ideas have had an enduring influence on civilization.

Euclid - The Father of Geometry

The model of reasoning developed by Euclid

Basic assumptions

- self-evident truths which provided firm foundation for mathematical knowledge.

4 traditional requirements for a set of axioms:

Consistent

Independent

Simple

Fruitful

5 axioms postulated by Euclid:

It shall be possible to draw a straight line joining any two points.

A finite straight line may be extended without limit in either direction.

It shall possible to draw a circle with a given center and through a given point.

All right angles are equal to one another.

There is just one straight line through a given point which is parallel to a given line.

There are two premises and one conclusion. If the two premises are true, then the conclusion must be true.

In mathematics axioms are like premises and theorems are like conclusions.

Examples:

All even numbers can be divided by 2,

28 is an even number,

then 28 can be divided by 2.

There are odd numbers in prime numbers,

15 is an odd number,

then 15 is a prime number.

Using the five axioms and deductive reasoning, Euclid derived various simple theorems:

Lines perpendicular to the same line are parallel.

Two straight lines do not enclose an area.

The sum of the angles of a triangle is 180 degrees.

The angles on a straight line sum to 180 degrees.

These theorems are then used to construct more complex proofs.

When a + d = 180

then we prove that a = c

"I think you should be more explicit here in step two"

How to Prove A Mathematical Theory

Moodle Questions

Is Math invented or discovered?

Is Math the universal truth?

Since any logical sequence of statements which leads to a theorem counts as a proof in mathematics, so there can be many different proofs of a theorem.

Example:

Standard Approach: 1+2=3, 3+3=6, 6+4=10, etc.

Elegant Approach:

1 2 3 4 5... 46 47 48 49 50

100 99 98 97 96 ...55 54 53 52 51

Finding the sum of 1-100

However, mathematicians generally seek the proof that is the most clear, economical and elegant.

What WOK(s) do you think the mathematicians need in order to come up with an elegant approach?

However, some mathematicians, such as Henri Poincare (1854-1912), stress the role played by intuition in creative mathematical work.

Here is an example requires intuition:

If you tie a string tightly around the 'equator' of a football, and you then want to make it go all way round the ball one inch from its surface - as in the diagram - it turns out that you will need about 6 extra inches to your original place of string.

Now, Imagine that you are going to do the same thing for the Earth, how much string do you think you would need to add? Make an intelligent guess.

The answer for the earth is the same as the answer for the football - roughly 6 inches.

This might go against most people's natural intuitions.

Even though some people's intuitions are the truth, sometimes such intuitions may not be accepted by the mathematical community until they have been proven.

There are two distinctions about the nature of mathematical certainty

1. The nature of propositions

a. Analytic proposition – proposition that is true by definition

b. Synthetic proposition – any proposition that is not analytic

Therefore: all propositions are either analytic or synthetic

2. How we are able to know if a proposition is true

a. It is knowable it a priori if it can be true independent of experience

b. It is a posteriori if it cannot be known to be true independent of experience

Therefore: all true propositions are either priori or posteriori

A group of random numbers isn’t enough to prove that math is empirical.

Numbers are unable to explain everything, there are many cases we are able to surpass the game of numbers and be creative.

Is mathematics really empirical though?

What is 4-3?

What is mathematics as analytic?

It is referring that math is based on concepts that are well defined in order to prevent denials, leaving the denials contradictory.

We claim that the proposition 2+2=4 is true through definition. It is exactly the same as saying (1+1) + (1+1) = (1+1+1+1). Math is like unpacking the truth.

Unwrapping Math

Is Math Analytical?

despite math being plausible, there are some problems when saying math is a string of definitional truths

if math was easy and can be solved by explained definitions, why will math even be hard?

Is Math Analytical?

mathematical truth isn't always trivially true

claiming that math is analytical is less plausible when we think of Goldbach's conjecture, which isn't proved right or wrong yet.

if mathematical propositions are true by definition, we are left with the question that everything seems to fit in well

math gives non-trivial, substantial knowledge about reality

knowledge can be known to be true independent of experience

it also means that human beings are able to discover truths about the nature of reality

The time of euclid

father of geometry

popular during the 19th century

active in Alexandria

lived under the reign of Ptolemy I

Euclid's 2 beliefs

if you begin with self-evident axioms and use deductive reasons, you obtain true theorems

math gave substantial knowledge about the nature of physical space

it is quite true since we use geometry to estimate the circumference of earth, building churches, etc.

Euclidean geometry

it was a new model of knowledge

became many philosophers' dream to achieve what Euclid achieved in the area of geometry

even Descartes wrote a book on ethics proving that various theorems is what he believed to be ethical axioms

"To speak freely, I am convinced that it (mathematics) is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency, as being the source of all others"

-Descartes

if math is indeed synthetic a priori...

how is the human mind able to discover truths about the world on the basis of reason alone?

God created a "pre-established harmony" between human mind and universe

if math is indeed synthetic a priori...

hard to see why mathematical ability would be an evolutionary advantage

nature had no way of knowing that deep math will be useful to humans one day

therefore, mathematical ability is most probably a by-product of other abilities that has actual values

conclusion

There are limits to certainty.

We can never prove that mathematics is free from contradiction.

Mathematics is applied to the real world and we can test axioms against reality.

Although mathematics cannot give us absolute certainty, it continues to play a key role in a wide varitey of subjects.

We cannot ca[tire everything om tje abstract map of mathematics.

There is no reason to believe that it is the only, or always the best, tool for making sense of reality.

So is intuition really important in Maths?