Pure math does not exist. No matter how abstract the concept, there is always some application to the real world, even if it’s not immediately apparent. Knot theory (the science of telling knots apart) started out as nothing more than an intellectual challenge in the late 19th century, and is now used to help understand how DNA combines and splits apart. Mathematical breakthroughs have long anticipated discoveries in the physical world. From the speed of light, to the normal distribution of probabilities and Einstein’s theory of relativity, the universe is built on math. In Is God A Mathematician?, astrophysicist Mario Livio tries to decipher what that means.
Despite the title, the book isn’t about theology. Livio is more interested in math than God. What he wants to figure out is whether math was invented or discovered. In other words, are numbers, and the relationships between them, something humans cooked up out of thin air, or did they always exist, just waiting to be figured out? Math is the universal language among humans, but would the same hold true for alien species? Would intelligent squids who lived on the ocean floor have any knowledge of prime numbers? It may seem like an absurd hypothetical, but looking for an answer is an interesting window into the long and tangled history of math, which is Livio’s real passion.
The greatest mathematicians all united seemingly different fields of study. Descartes combined geometry and algebra with the invention of graphing and the coordinate system. Newton used math and physics together to create calculus. Einstein’s theories found a way to treat space and time as one. The holy grail of modern scientific research is finding a theory of everything, a single unifying framework which reconciles the effects on gravity on incredibly large objects like galaxies (general relativity) and incredibly small ones like electrons (quantum field theory). Neither theory can explain what happens when a huge amount of mass is confined to a microscopic area, like the center of a black hole or the universe in the second after the Big Bang. The progression of mathematical history seems to point towards one underlying series of equations that determine the very nature of our reality.
The people who think math was discovered point to these natural laws as proof. It doesn’t matter whether or not we know about the speed of light, objects still can’t go faster than it. Just because we can’t perceive the platonic world of mathematical forms doesn’t mean it doesn’t exist. That’s the idea behind Plato’s famous allegory of The Cave, where the ancient Greek philosopher imagined a group of human beings chained to the floor, unable to move their heads. The only thing they can see is the wall in front of them, and any shadows cast on the wall by the fire behind them. They see the reflections on the wall rather than the objects that are actually being reflected, in the same way that the objects we see in the world are merely shadows of their ideal forms.
The other school of thought thinks the shadows are all that there is, that there is no ideal world. Our perception is what creates reality. One of the most fascinating discoveries in quantum mechanics is the observer effect: the mere act of being observed will change the ways in which objects behave. When scientists shoot an electron at a wall through a piece of paper with two slits cut in it, which slit the electron goes through (or whether it goes through both) depends on whether or not a measuring device is placed in front of the paper to see. If there’s no device, the electron creates an interference pattern on the wall, as if two ripples of water were intersecting over the surface of the lake. However, if there is a device, the electron is forced to choose one of the two slits, and no interference pattern is created. No one knows why this happens.
The debate goes back to the very beginning of math. Euclidean geometry has been around for thousands of years, and it was long seen as proof that mathematical knowledge was based on a solid foundation of universal truth. That changed in the 19th century, when people realized one of its primary axioms could be altered. Axioms are the building blocks of Euclidean geometry, the things assumed to be true without needing to be proven: there’s only one straight line between two points, two parallel lines don’t intersect etc. However, not all of them hold true in every situation. The sum of the angles in a triangle only adds up to 180 degrees on a flat plane. They add up to less than 180 on a curved structure, and more than 180 on a spherical one. Euclidean geometry only works in two-dimensional space. Modifying even one axiom results in dramatically different conclusions about the world.
This became important as mathematicians tried to formalize logic to create a rational framework to analyze the truth of any statement. Here it is as its most basic form: if P or Q, and not P, then Q. What could we prove about the world given these tools? Was math invented or discovered? And if we discovered it, then who invented it? Livio dances around the question throughout the book, but he shows his hand here, when he quotes George Boole, one of the architects of formalized logic:
To show the power of his methods, Boole attempted to use his logical symbols for everything he deemed important. For instance, he even analyzed the arguments of the philosophers Samuel Clarke and Baruch Spinoza for the existence and attributes of God: “It is not possible, I think, to rise from the perusal of arguments of Clarke and Spinoza without a deep conviction of the futility of all endeavors to establish, entirely a priori, the existence of an Infinite Being, His attributes, and His relation to the universe.” In spite of the soundness of Boole’s conclusion, apparently not everybody was convinced of the futility of such endeavors, since updated versions of the ontological arguments for God’s existence continue to emerge today.The main logical arguments for God fall under three main categories. There is the cosmological argument, which says that since everything in this world is created by a creator, the universe itself must have been created. The teleological argument says God must exist since the universe itself was not created by accident. Finally, the Cartesian circle says that God must exist because without an ultimate source of truth there is no way to trust in the validity or human reasoning. However, as Livio points out, theists don’t need these arguments to be persuaded in the existence of God while atheists are not persuaded by them.
He is more right than he knows. That’s what the apostle Paul is referring to when he talks about “the mystery of faith”. There’s no way to reason backwards to faith because reason is ultimately built on faith. Every set of logical beliefs about the world is built on first principles. Geometry doesn't work without axioms. You can’t prove anything without assuming something else to be true. Descartes and Newton were devout Christians whose discoveries stemmed from their faith in God. Livio has the search backwards. You can’t reason your way to first principles. You can only judge them by their results:
Beware of false prophets, who come to you in sheep’s clothing but inwardly are ravenous wolves. You will recognize them by their fruits. Are grapes gathered from thorn bushes, or figs from thistles? So, every healthy tree bears good fruit, but the diseased tree bears bad fruit. A healthy tree cannot bear bad fruit, nor can a diseased tree bear good fruit. Every tree that does not bear good fruit is cut down and thrown into the fire. Thus you will recognize them by their fruits.
- Matthew 7:15-20