Math and Truth have an interesting relationship. Mathematics, like truth, is steadfast and immovable and not open to interpretation. 2 plus 2 is equal to 4, period.
Even more interesting is the relationship between Rationality and Truth. Rationality is the process of using reason to arrive at true statements. One then could argue rationality is the vehicle to Truth.
Mathematics then is an artistic expression of rationality; which is why it is always true. However, while mathematics (and reason) certainly leads us towards truth, I would hesitate to say it leads us into Truth. Mathematics can lead us to ideas that are eternally true, but it cannot lead to everything that is True.
This is a controversial statement of sorts. Our society’s collective worldview is ever becoming more secular. By secular, I mean that we will only accept something as true (if there is any Truth) by virtue of the scientific method. We simply will not accept anything unless it has been researched, peer-reviewed, published and endorsed by the scientific community at large. Never mind tradition, if we cannot rationalize it, we cannot accept it as true.
Rationality (by which I include the mathematical arts and the scientific method) is an incredible and important gift that I by no means wish to malign. Yes, we should cherish reason and push scientific discovery as far as we ethically can. However, to operate in a worldview wherein rationality is the sole means to discover Truth is unwise because rationality inevitably produces an interesting phenomenon: irrationality.
Justin E. H. Smith, a professor of philosophy at the University of Paris, writes in a recent book that every time a culture commits to rationality, irrationality always results. And, every effort to thwart irrationality eventually leads to further irrationality.
Perhaps the chief example in the history of irrationality is the Pythagoreans.
Pythagoras was a Pre-Socratic Philosopher whose immortal legacy strikes fear into the hearts of unsuspecting geometry students. He and his followers most notably discovered that the hypotenuse of a right triangle is equal to the square root of the sum of each leg squared. In other words:
a2 + b2 = c2
His influence reaches further than our geometry textbooks. The Pythagoreans were the first to claim that the fundamental building block of reality were non-physical ideals. Where others believed that if one were to take something real and zoom in on it until you find the smallest thing upon which everything is made up, that thing would be something physical. (Here, you and I would say that thing to be the subatomic particles that makeup atoms). However, Pythagoras believed that thing to be something non-physical. He believed that thing to be numbers.
One day Pythagoras was walking through a market and stopped to observe two blacksmiths hammering metal upon an anvil. He took note of the sound each hammer made upon impact. He soon realized that the two sounds, while different, were in harmony because they were the same note in a different octave. He then realized that musical harmony was simply a mathematic ratio. And, if musical harmony could be understood through mathematic ratios, so then could the harmony of the universe.
Pythagoras and his school of followers began to build a worldview upon the idea that numbers were the root of all reality. His followers devoted themselves to mathematic discovery (and provided that Pythagorean Theorem we still use some 2,500 years later). All of their mathematic developments had to have one simple rule: be rational.
In particular, since every number must be rational, every number must be able to be expressed as a ratio.
However, in their commitment to rationality, they discovered irrationality.
Hippasus was a follower of Pythagoras and proved quite simply that the hypotenuse of a right triangle with each leg the length of 1 was impossible to express rationally. In other words, the square root of two cannot be written as a fraction, and its decimal has no discernable pattern continuing indefinitely at random.
Rather than embrace the discovery of irrational numbers, the Pythagoreans dealt with Hippasus their “rational” way: drown him in the ocean.
The Pythagoreans built a worldview on the back of rationality and contributed many important ideas. But in their rationality, they produced irrationality and watched their worldview crumble.
The Pythagoreans should serve as a warning to contemporaries who seek to understand Truth solely as rational.
There is always one definite end to rationality: irrationality.
How then should we live? Abandon rationality altogether?
Absolutely not. We ought to pursue rationality as a means to find truth (not the only means) and embrace irrationality.
G. K. Chesterton, in his book Orthodoxy, describes this balance perfectly. To Chesterton, the line between living rationally and embracing irrationality comes down to freedom. To live simply by rationality is to be, in Chesterton’s words, a scientist; and the one who lives rationality and embraces irrationality lives as a poet. The scientist must live by science and is not free to accept anything but science. However, the poet is free to accept science and fairies.
“ Poets do not go mad; but chess-players do. Mathematicians go mad, and cashiers; but creative artists very seldom. I am not, as will be seen, in any sense attacking logic: I only say that this danger does lie in logic, not imagination… Poetry is sane because it floats easily in an infinite sea; reason seeks to cross the infinite sea, and so make it finite. “G.K. Chesterton, Orthodoxy; Chapter II, “The Maniac”
The wise young men and women should then heed the warning of the Pythagoreans and choose to live as poets in a world of scientists. Anything else would be irrational.