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Axiom vs. Theorem

Euclid's Geometry

Where it all began....

  • Geometry originated in Egypt as an art of Earth measurement
  • Euclid (325 BCE-265 BCE): The Father of Geometry
  • The first Egyptian mathematician who initiated a new way of thinking the study of geometry.
  • He introduced the method of proving a geometrical result by deductive reasoning based upon previously proved result & some self evident specific assumptions called axioms.

Definition

~An axiom is a finite statement that is accepted to be true without proof.

~A starting point for deducing and inferring other (theory dependent) truths.

Axiom

A Postulate

To demand or claim without proof

Also known as...

Peano Arithmetic Axiom

~System of natural numbers

~Accessibly infinite formal system.

A type of axiom

Axioms will not lead to contradiction.

Often used when speaking of mathematics or logic.

Different beliefs in Mathematics can lead to useful real-life applications, previously unthought of.

Axioms must be consistent to base assumptions in mathematics.

Examples

~Everything that is, is.

~What isn’t not is more like is than is not.

~What is not false is sometimes true.

~What is not true is sometimes false.

~What is always sometimes true is at least a little bit true.

~What is always sometimes false is at least a little bit false.

~If two lines intersect, then they intersect at exactly one point.

~All planets revolve around the sun.

~The earth turns 360 degrees everyday.

~Probability lies between 0 & 1

Examples

Definition

~A theorem is a theoretical proposition, statement, or formula proved by a chain of reasoning; a truth established by means of accepted truths.

Theorem

Pythagorean

Theorem

Pythagorean Theorem

Pythagorean equation: a2 + b2 = c2. Pythagorean theorem states that the area of the square formed by the longest side of the right triangle (the hypotenuse) is equal to the sum of the area of the squares formed by the other two sides of the right triangle:

Kuratowski's Theorem

Kuratowski's Theorem

In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski.

Prime Number Theorem

In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs.

Prime Number Theorem

Relationship between Axioms & Theorems

How are they related?

An axiom is a statement that is considered to be true, based on logic; however, it cannot be proven or demonstrated because it is simply considered as self-evident. ... A theorem, by definition, is a statement proven based on axioms, other theorems, and some set of logical connectives.

The difference....

How are they different?

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