Mathematics versus Truth

HS #35 2018.6.5

Mathematics versus Truth

Mathematics versus Truth – what does that mean? Isn’t mathematics the prime example of truth? Not so fast. The story begins 2300 years ago with Euclid. He was the first to ask, “Is geometry true? How do we know?” Geometry had been used successfully for thousands of years to build the Parthenon and pyramids, but that wasn’t enough for Euclid. He wanted certainty.

Euclid realized that he didn’t have to accept all 470 geometry facts (theorems) as true. Instead, beginning with just five (axioms) as the foundation, the other 465 could be proved via mathematical argument. 

With that accomplishment, which you likely studied in high school, geometry was considered the prime example of truth for the next two millennia. The great philosopher Immanuel Kant based his argument for the existence of truth on Euclid’s geometry.  

But just as Elijah’s servant noticed a little cloud no larger than a man’s hand, trouble loomed on the horizon.  Mathematicians had always been reluctant to accept one axiom: that parallel lines (lines that never intersect) exist. No one doubted its truth – they just weren’t comfortable assuming it as an axiom. So they tried to prove it as a theorem.  Their method was to assume it was false and show that geometry falls apart as a result.

That’s when the world changed.  Geometry DIDN’T fall apart. Instead, they just invented other types of geometry. As with Euclid’s, the new geometries were fully self-consistent; they just followed different rules and described different worlds. It’s much like changing the rules of a card game. As long as no contradictions follow, the new rules give rise to perfectly fine new games. 

Ironically, one new geometry that was discovered is the one we live on. Just as one gets straight lines on a plane by pulling a string tight, so also a taut string joining any two points on a globe gives a straight line, called a great circle, in spherical geometry. The equator and longitude lines are great circles. You can form many others. Notice that ALL of these lines intersect. There are no parallel lines. 

Then in the 1600’s Rene Descartes took Euclid’s question a step further. How do we know that ANYTHING is true? His own attempt at a foundational axiom (I think, therefore I am) got mathematicians wondering about the truth of all mathematics. 

The logician Frege spent ten years trying to prove the truth of mathematics by using set theory - the study of collections of things – as its foundation. Seems simple enough. It’s so simple that after Sputnik was launched, the shocked U.S. started teaching set theory in elementary school as part of the new math.  

But as with the geometry axioms, there was a hidden crack in the foundation, and Bertrand Russell found it. Russell, a great polymath, found a logical contradiction similar to the self-referencing statement, “This sentence is false.” (Think about it – is that sentence true or false? See the problem?) 

Russell took no joy in his discovery. He was a famous atheist (authored “Why I am not a Christian”) who wanted certainty in life and thought it only found in mathematics. He, together with Alfred Whitehead, spent fifteen years constructing a foundation for mathematics based on logic. But the foundation he sought proved elusive. It was much deeper and harder to attain than he realized. It was only on page 360 of Volume 2 of his unreadable tome, Principia Mathematica, where he finally proved that 1+1=2. (No kidding!) 

Kurt Gdel, a friend of Einstein, dealt the final blow to an easy relationship between truth and mathematics by proving that not all mathematical truths are provable. Starting to sound insane? Well, he likely was – Gdel died of starvation because he was afraid of being poisoned. 

Is all of this just a story of eggheads squabbling over nonsense? Hardly.  These discoveries have raised anew Pilate’s question, “What is truth?” The resulting waves have spread through other disciplines and cultures to our daily lives.  It is most certainly a contributing factor to the rise of post modernism that calls into question all assertions of truth. One personal example: In college I was active in Campus Crusade for Christ. We brought the CCC author and Christian apologist Josh McDowell to campus to speak. I met him again a few years ago when he spoke at Central Wesleyan. He said that in the 1970’s students would object to his truth claims of Christianity with, “That’s not true!” Now the objection is, “You’re not being tolerant.”