Axioms and Proofs

In the 19th century, there was a crisis among mathematicians once they realized there are fundamental assumptions their field is built on that they had not yet clarified. Disregarding certain statements formerly considered axioms creates internally consistent yet alien frameworks that nevertheless hold use within scientific fields. To demonstrate, we'll have to reiterate high school geometry.

Math is about proofs. Your school may have lead you to believe that math is about computation. This is mostly false. Anything that can be computed is by definition a solved problem, disregarding the resources it takes to compute. The frontiers of mathematics are pushed by proving new results from prior results and axioms. My personal favorite fields of math prove the existence of more efficient methods of computation; faster algorithms give faster results. The Greek philosopher and mathematician Euclid attempted to prove results about geometry from four axioms:

A straight line segment can be drawn joining any two points.

Any straight line segment can be extended indefinitely in a straight line.

Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

All right angles are congruent.

The fifth axiom, the "parallel postulate", is significantly more complex.

If a straight line intersects two other straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

The converse of a statement "A implies B" is "B implies A". This isn't always true; "someone living in Paris implying they also live in France" doesn't imply "someone living in France implies they also live in Paris". Let's take the converse of the parallel postulate.

If a straight line intersects two other straight lines that intersect on some side of first line, the two interior angles on this side add to less than two right angles.

The parallel postulate is assumed as an axiom in Euclidean geometry, the geometry you learned in school. The converse is proven from the parallel postulate and prior results. This is not the only possible geometry.

To construct an alternate geometry, let's replace the parallel postulate with something a bit simpler to understand in my opinion, Playfair's axiom.

Given a line, and a point not on it, at most one parallel to the given line can be drawn through the given point.

If I just told you that we're replacing one axiom with another, it wouldn't be equivalent. We'll have to prove that first. In order to prove two statements A and B are equivalent, one must first prove A implies B, then prove B implies A. Then if A is true, B is true, and if A is false, B is false. They're "paired", in a sense, and equivalent. Let's begin.

Parallel implies Playfair

  • Assume two lines, one drawn through point A and one drawn through point B intersect another line drawn through both A and B, and their interior angles on one side sum to <180 degrees.
  • The first two lines must meet, due to the parallel postulate.
  • Draw another line through point B such that the interior angles sum to >180 degrees on one side.
  • Then the interior angles must sum to <180 degrees on the other side, and the two lines meet on that side.
  • Therefore, in order for the lines to not meet on either side, the interior angles must sum to 180 on both sides.
  • This line through point B is unique.
  • Therefore, the parallel postulate implies Playfair's axiom; given a line, the line through point A, and a point not on it, point B, at most one parallel to the given line can be drawn through the given point.

Playfair implies Parallel

Proving "if B, then A" is the same as proving "if not A, then not B". It turns out that direction is easier for this example.

  • Assume two lines, one drawn through point A and one drawn through point B intersect another line drawn through both A and B, and their interior angles on one side sum to <180 degrees.
  • Since we're not assuming the parallel postulate, the first two lines don't have to meet.
  • Draw another line through point B such that the interior angles between A and B are still less than 180 degrees.
  • This new line and the line through point A don't have to meet either.
  • Playfair's axiom states that given a line and a point not on it, at most one parallel to the given line can be drawn through the point.
  • Therefore, if the parallel postulate is false, then Playfair's axiom is also false, so Playfair's axiom implies the parallel postulate.

Therefore, we can swap the parallel postulate for Playfair's axiom and get the exact same geometry. However, we don't really need either; the first 28 proofs in Euclid's Elements are proven without the parallel postulate, in a so-called "neutral geometry" that says nothing about the number of parallel lines that can be drawn to another line through a given point. What if we replaced the axiom?

Given a line, and a point not on it, at least two parallels to the given line can be drawn through the given point.

This forms hyperbolic geometry, the subject of various games and Youtube videos due to its ease of visualization. It's an internally consistent geometry full of its own proofs that found later applications in physics. Every time you see the words "non-Euclidean geometry", think of lines, not Lovecraft, and what may be beyond the horror protagonist's comprehension will be within yours.

The 19th century mathematicians didn't take it as easily. The parallel postulate was an assumption thousands of years old, upended by the rigorous work of geometers showing supposed "proofs" of the axiom from the prior four postulates were flawed. What else were they assuming? Their naive set theory didn't hold up to such paradoxes as "does a set containing all sets that do not contain itselves, contain itself", commonly stated as "does a (presumed male) barber who shaves any man who does not shave himself, shave himself". Bertrand Russell attempted to prove the foundations of mathematics in the Principia Mathematica, taking hundreds of pages to eventually prove 1 + 1 = 2, but Kurt Gödel proved Russell's attempt was futile with his Incompleteness Theorems.

Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e. there are statements of the language of F which can neither be proved nor disproved in F.

The canonical consistency statement Cons(F), stating that there exist no contradictions within F, will not be provable in F.

For any system of logic you construct, there are statements that can be assumed true or false within the system with no contradiction, and the consistency of the system cannot be proven within the system itself. It's up to mathematicians to agree on the ssytem they wish to use, and so they mostly did; Zermelo-Fraenkel set theory with an additional axiom, the Axiom of Choice, sprung from this debate. This is a good-enough definition of set theory with common minimal assumptions to prove most results we already took for granted, and it's still used to this day.

There exist fields of mathematics that change the axioms used and test which statements are still provable, or which alternate statements are provable instead. These constitute entire parallel universes of logic accessible to the mind and the computer, hand-chosen to describe the properties of the desired object of study.