#### Transcript of Geometric Paradigm

**Mathematics**

Geometric Paradigm

Paradigm = Pattern

Examples:

DaVinci's Virtuvian Man (Golden Ratio)

Flowers, Spider Webs

Important for TOK because it helps increase our knowledge of the world with a more logical way of thinking.

For example, with the mathematical explanation of a geometric paradigm, we can understand better why spider webs are so symmetrical.

DaVinci's Vitruvian Man

Conclusion

Proofs

A proof should logically follow from its axioms and are examples of deductive reasoning.

Example: In an equilateral triangle ABC, a point P is found such that AP = 3, BP = 4, and CP = 5.

Empiricism

The theory that all knowledge is derived from sense-experience.

Mathematical truths are empirical generalizations based on a vast number of experiences that are no different from scientific statements.

Example: We feel certain that 2 + 2 = 4 because we've seen more confirming instances, also known as empiricism.

Theorems and Deductive Reasoning

Theorems and deductive reasoning are sub groups of axioms because they are derived from specific axioms.

Deductive reasoning is from general to specific

Out of the two (inductive and deductive), deductive reasoning is more certain but less informative.

If we think of axioms, theorems and deductive reasoning in terms of syllogisms.

Axioms are the premises and theorems are the conclusions.

Multiple theorems can be pulled to gather to form a proof.

Conjecture

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

Ex: Goldbach's Conjecture

A famous mathematical conjecture according to which every even number is the sum of two primes.

1 + 1 = 2

2 + 2 = 4

3 + 3 = 6

5 + 3 = 8

Axioms

They must have these four requirements:

Consistent

Independent

Simple

Fruitful

When you reason formally, you begin with Axioms.

Axioms are self-evident truths which provide foundations for mathematical knowledge.

Example:

I

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

All right angles are equal to one another.

Mathematics is an important area of knowledge because it helps us learn about the world through a logical and objective manner.

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