a crisis of foundation

Originally mathematical ideas were based the physical world, specifically that of geometry. The Greeks, Egyptians, and Pythagoreans measured and used number in a practical and tangible sense. The number 2 'existed' as 2 units of length, maybe the side of a triangle, or the length of a column. Well imagine the pythagoreans disappointment when they stumbled upon the idea of the square root of 2. We know this number to be irrational. The way they discovered the problem was when they split a perfect square into two right angled triangles. If each side length was 1 unit, then the hypotenuse of the triangle should be the square root of 2 units in length, there, they drew it, it must exist. Yet when you try to split the number 1 into even pieces, lets say 5. You can't split the square root of 2 the same way, in fact there is no way to split the square root of 2 evenly. Impossible, but true, such is the case with all irrational numbers. Their values go on and on in the decimal world at infinitum.

Euclids geometry also ran into contradictions. For century's mathematicians have tried to prove that if two lines bisect a third line, and if the angles made add to a number less than 180, then the two bisecting lines must intersect at some other point, in other words, they are not parallel. It seems so obvious it should not be an assumption, but a hard fact that can be proved, yet no one can. Lambert tried, but a proof by contradiction later all he came up with was a famous picture called lamberts rectangle.


So, we can't base mathematics in geometry like we thought, there are too many things left unexplained.

Then the smart guys turned to algebra, and straight numbers for numbers sake. Turns out there's a contradiction here to, but I'll spare you the details.

They tried set theory, and the world of abstract algebra, but there were still mathematical ideas that escaped these axioms and proofs. Like this one... "the set of all sets that don't contain themselves" its a pure paradox.

Then logic, logic must be the answer.... nope even after the book Principa Mathematica was written... a book so complex and precisely defined that it took some 6oo pages for the authors to finally assert that 1 plus 1 does in fact equal 2. Well even this book contained an error, a loop hole that missed some critical information.


So the search for a true foundation for mathematics continues. And while the saga continues, most mathematicians, especially those in the applied realms, turn a blind eye to the predicament, because math still models everyday occurences. Engineers still build bridges, using geometry, actuaries still calculate complex algorithms to predict losses for big businesses, professors still cram the ideas of null sets, convergence theorems, and isomorphic relations into the brains of their pupils, and children still count using their fingers, effectively demonstrating the use of abstraction by the time they are 3 years old. Math still goes on, even when no solid foundation exists.

These thoughts led to others...

What other fields claim to have sure foundations? Science claims to find truth, but only in things tangible and earthly, or at least physical. Yet they model perfect situations that can in no way be tangible... have you ever seen a physics rope in real life? No weight, no friction.... ya, didn't think so. Chemists use imaginary numbers to calculate complex equations, but when it comes down to the actual chemicals, no imaginary numbers are actually used. Then there are the Humanities and Social Sciences. I don't know what they claim to be their foundation, but I'm sure whatever it is changes with the opinion of the professor, or teacher year to year. And we all know the foundation of english is the fact that every rule has an exception, which is a contradictory statement in and of itself! So does any area of study really have a true 'foundation' or building block on which all elements of the field can be derived and explained from? And who really cares anyway? Just the crazed, perfect model seekers called mathematicians probably.

And then I thought about religion. My religion specifically, and the truths I hold to be pure knowledge and wisdom. I remember a time when I was frustrated with the blanket statement that covers everything. If we don't know it, its not because it isn't, it's because we can't, or shouldn't know.... yet. There's always that yet, because with faith and diligence, knowledge of all things must surely be attainable. Faith. Faith is the foundation upon which all truth, light and knowledge builds. That I believe.

A foundation as shaky or as sure as the individual.