R-AESPE-1

The idea of proof

There are different types of proof, legal, scientific/medical (experimental?), mathematical, etc.

The first two proof are judgements based on probability and not absolute statements.

i.e. for a murder case its the most likely case that the accused did the murder

In comparison a proof of Pythaogras' Theorem isn't having just a single right triangle applying or a thousand right triangles applying, as a single case that exists that disproves the theory would mean all these answer do not apply.

In math, it is a reasoned argument that shows such a theory is true that makes a proof! :)

TLDR A mathematical proof is a

reasoned argument, not an accumulation of evidence. Consequently, the truth of mathematical statements can be known with 100% certainty. A mathematical statement which has been proved to be true is called a theorem. A mathematical statement that is yet to be proved is called a conjecture.

Mathematics has a higher standard of proof than most disciplines

Types of True statements

• Axiom: A statement that is self-evidently true; a building block of mathematics
  • e.g. Two straight lines intersect in at most one point.
• Theorem: A statement which requires justification, called a proof.
  • A proof is an argument taking you from something you know to be true, to something you want to show to be true.
  • It demonstrates that what you know implies what you want to show.
  • e.g. Among 3 straight lines, there are at most 3 points where lines intersect.
    • 3 pairs of lines. By axiom 1, each of these pairs intersects in at most 1 point. Hence there are at most three points where 3 straight lines can intersect.

Theorems build upon other theorems; proof of new theorems rarely refer directly to axioms. It's like a brick tower where the foundation is the axioms and the walls are the theorems. You can trace a theorem from the top to the axioms that help prove it.

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