The beauty and power of mathematical truths have impressed and astounded mankind since they were first discovered. At one point, mathematics was almost revered as supreme truth . . . until a mathematician stumbled upon a proof that pointed even higher.
Ever since Adam and Eve rejected God’s authority, humankind has searched for a source of ultimate truth apart from Him. This quest—to replace God with another truth—has taken many strange twists over the millennia. But one of the most fascinating has been the effort to replace God with “pure reason,” or logic.
“Your faith should not be in the wisdom of men but in the power of God.”
—1 Corinthians 2:5
The effort reached its first pinnacle at the time of the ancient Greeks, but it raised its head again in the last few centuries. At one point, some of the world’s leading minds seemed to be close to reaching their goal. Ironically, another mathematician stepped in to prove that they would never reach it! This mathematician proved that there must be true statements in any given mathematical system that cannot be proved within that system. Thus, math cannot be the ultimate foundation for truth; it must appeal to something beyond itself.
The lesson for Christians is exciting. No matter how hard people try to disprove or sideline God as the foundation for all truth and life, His eternal power and nature shine forth even more brightly. The very effort to destroy Him merely reminds fallible humans, by their own efforts, that God gets the ultimate glory . . . even in the mental world of mathematics and logic.
Throughout history, mathematics has offered glimpses of infinity that point man’s attention to God.
Throughout history, mathematics has offered glimpses of infinity that point man’s attention to God. However, humans in their rebellious state do not want even a glimpse of God because they suppress the truth in unrighteousness (Romans 1:18).
In ancient Greece, Pythagoras (572–492 BC) chose to worship the infinity of natural (counting) numbers instead of God.
Pythagoras is best known for proving the mathematical theorem now named after him: a2 + b2 = c2, where a, b, and c are the sides of a right triangle. He was especially enamored with natural numbers (1, 2, 3, etc.), which can be described as lengths of sides on right triangles. Some natural numbers satisfy his theorem, such as 3, 4, 5 and 5, 12, 13. (These triples are now called Pythagorean triples in his honor.) Plato (429–348 BC) continued this fascination with the properties of natural numbers. He saw clear contrasts between the imperfect, physical world around us and the perfect, abstract world of ideas. His form of abstract thought-worship came to be called Platonism.
Yet Pythagoras and Plato both stumbled over the limits of their god. Using the Pythagorean Theorem, they recognized that when a = 1 and b = 1, then c (the square root of 2) isn’t a natural number and can’t even be written as a fraction (a ratio of two natural numbers). Natural numbers aren’t the ultimate truth. This mystified and angered them but did not change their minds about numbers.
Similarly, Europe’s Enlightenment spawned a bevy of philosopher-mathematicians in the late 1800s and early 1900s who worshipped logic and reason as the ultimate source of truth. Gottlob Frege (1848–1925), Bertrand Russell (1872–1970), and Alfred North Whitehead (1861–1947) promoted logic as the ultimate foundation for mathematics. Their philosophy of mathematics was called Logicism because it attempted to prove every mathematical fact on the basis of logic alone.
In Principia Mathematica (1910–1913), Russell and Whitehead proved using only logic that 1 + 1 = 2. From here, they hoped to prove every other mathematical fact. By 1920 they thought they were getting close.
David Hilbert (1862–1943) went a step further in the 1920s. Since he considered logic to be a branch of mathematics, he claimed that mathematics was self-dependent. In other words, it did not need to refer to any authority outside itself in order to prove any of its truth claims. This supposedly made mathematics autonomous (its own final authority, independent of all outside authorities), like God Himself. Hilbert’s philosophy of math, called Formalism, promoted mathematics as its own foundation and set as its goal absolute knowledge.1
Few modern readers realize how influential these thoughts were and are. Math was considered completely knowable. These men believed that someday every last theorem would be proved, and then all math would be proved and known. This self-confidence was paralleled in the sciences where many scientists thought they would eventually learn everything and mankind would make every last imaginable discovery.
In 1931 these false philosophies of math crumbled into dust when Gödel proved his Undecideability Theorem. Kurt Gödel (1906–1978) proved that no logical systems (if they include the counting numbers) can have all three of the following properties.
The details filled a book, but the basic concept was simple and elegant. He summed it up this way: “Anything you can draw a circle around cannot explain itself without referring to something outside the circle—something you have to assume but cannot prove.” For this reason, his proof is also called the Incompleteness Theorem.
Kurt Gödel had dropped a bomb on the foundations of mathematics. Math could not play the role of God as infinite and autonomous.
Kurt Gödel had dropped a bomb on the foundations of mathematics. Math could not play the role of God as infinite and autonomous. It was shocking, though, that logic could prove that mathematics could not be its own ultimate foundation.
Christians should not have been surprised. The first two conditions are true about math: it is valid and consistent. But only God fulfills the third condition. Only He is complete and therefore self-dependent (autonomous). God alone is “all in all” (1 Corinthians 15:28), “the beginning and the end” (Revelation 22:13). God is the ultimate authority (Hebrews 6:13), and in Christ are hidden all the treasures of wisdom and knowledge (Colossians 2:3).
There will always be a statement in any system that can’t be shown to be true or false. From a Christian perspective, Gödel proved that complete knowledge is unattainable. There will always be a question to confound the greatest minds; there will always be an unsolvable problem. Gödel’s proof shows that neither math nor logic can be the foundation for math.
An effort to recover from the fallout of this atomic blast continues today. Luitzen Brouwer (1882–1966) turned to the human mind as the foundation of math. Instead of giving God His rightful place, Brouwer redefined the second condition, consistency. He proposed a third category for truth values. Besides true and false, he added a possibility, which he called “indeterminate.”
His philosophy, called Intuitionism, makes human intuition the foundation of mathematics. He rejected the idea that math is discovered, and he promoted instead the view that math is invented by men. In his view, the human mind is the foundation of math instead of God.2
Many secular mathematicians now embrace Intuitionism. However, many others see insoluble problems with it. If math is an invention of many human minds, then why should all these minds agree on what is correct? This is nonsensical, if math is only an art. Do all artists agree on how to paint and what should be painted?
Second, why should it be useful in so many realms of knowledge, from biology to psychology, from engineering to medicine, from chemistry to business? Did our minds create the universe, too?
Third, why has the same thought occurred to different thinkers independently so many times? Since no two artists have ever conceived of the same painting independently, the invention of the very same mathematical concepts by mere Intuitionism seems ridiculous. How did both Newton and Leibniz come up with calculus separately? How did Gauss, Riemann, and Lobachevsky all come up with non-Euclidean geometry independently, as a response to past mathematicians’ failures—during hundreds of years of fruitless labor—to prove Euclid’s Parallel Postulate?
The Christian philosphy of math begins with God, who numbered the days of creation as recorded in Genesis 1.
These problems are death knells for Intuitionism. Secular mathematicians who understand this failure frequently fall back on Logicism or Formalism, even though it has already been proven impossible. They have nowhere else to go except to God.
The Christian philosophy of math, in contrast, begins with God, who numbered the days of creation as recorded in Genesis 1. The founder of the true philosophy of math is Jesus Christ, the source of math is the Bible, and the purpose of math is the glory of God.3 “For no other foundation can anyone lay than that which is laid, which is Jesus Christ” (1 Corinthians 3:11).