God and Math

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.

God and Math: Greek Worship of the Infinity of Natural Numbers

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.

The Modern Worship of Logic

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.

Gödel’s Theorem

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.

1. Validity . . . all conclusions are reached by valid reasoning.
2. Consistency . . . no conclusions contradict any other conclusions.
3. Completeness . . . all statements made in the system are either true or
   false.

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.

Conclusion: God and Math

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).

SourceThis article originally appeared on answersingenesis.org

Views: 0