It has been said that the whole Western philosophy is a footnote to Plato. He advocated the transcendental nature of mathematics. In this message, Dr.Paul Kattupalli explains how the three modern, atheistic philosophies of mathematics – Logicism, formalism and intuitionism fail to explain the unreasonable effectiveness of mathematics and point us to God, the Geometer of the universe. Visit us http://www.doctorpaul.org
For Greek philosopher Plato, mathematics has a reality of their own. It cannot be reduced to human thought. Mathematical laws – they have their own existence. They have their own reality. But such a view supports theism. Such a view points us to a Creator God. So, in the modern age, three philosophies of mathematics were put forward by materialist thinkers. Their aim is to reject the immaterial reality of mathematics. Their aim is to refute Plato. We can call them Anti-Platonic views of mathematics.
They are called Logicism, formalism, and intuitionism. First, Logicism: A prominent advocate of logicism was Gottlob Frege (1848 – 1925), a German philosopher and mathematician. Frege started a grand project to reduce arithmetic to logic. Frege wanted to reduce mathematics to logic, thus eliminating the independent existence of mathematics. Once all the mathematics are explained in terms of logic, mathematics will lose its independent existence.
But the whole project came to a screeching halt when Frege received a letter from a young mathematician named Bertrand Russell. Frege was asked how his system could account for a self-referential paradox. Russel put it this way: “Who shaves the barber in a town where barber shaves everybody who does not shave themselves?” Frege realized that his system is self-referential and self-contradictory. But Bertrand Russell did not give up. With the assistance of Alfred North Whitehead he took upon himself a much larger project, Principia Mathematica : to reduce all of mathematics to logic. Not just arithmetic, but all of mathematics. Why would you do that? Because they have no choice.
In naturalism, everything must be reduced to matter. If your friend fails, you should pick up the pieces and try again. It’s like evolutionists believing in abiogenesis, original life arose from inanimate substances. First, they believed that life spontaneously results from putrefaction. In a series of experiments, Louis Pasteur disproved it. They said, ‘well, he only prove that life did not come from dead meat, so, we have to look somewhere else,may be in a primordial pond, or may be in a mid-ocean ridge, may be in some extraterrestrial space.’ In the same way, the mathematician who does not accept Plato’s immaterial mathematical world has no choice. Thus,Bertrand Russel took upon himself the very project he disrupted when it was in the hands of Frege. Needless to say, it also failed. Russell’s system is also self-referential and self-contradictory. Mathematics uses logic, but it is not logic. Mathematics cannot be reduced to logic. It is its own branch of knowledge.
Formalism: The second school of thought is called formalism. It’s prominent advocate was David Hilbert (1862-1943), a German mathematician.
Hilbert was baptized and raised in a Prussian Evangelical Church. But later he became an agnostic. He argued that mathematical truth was independent of the existence of God. He said mathematics is a kind of logic game. It is a logic structure but cannot be reduced to logic. We invented mathematics for our utilitarian purposes. For example, we invented basketball. We created the NBA – National Basketball Association, which formulates all rules and regulations for playing basketball. There is nothing transcendental or eternal about those laws of basketball. Similarly, we created mathematics for our own pleasure. We invented such terms as number, line, point, function, theorem, proof for our purposes. Mathematics has no connection at all to anything outside itself. It has no meaning outside of itself. Hilbert argued mathematics can attain consistency and completeness within itself.
In the year 1900, speaking at a mathematics conference in Prais, Hilbert enumerated 23 mathematical problems that must be solved. Ironically, those problems he wanted to solve destroyed his philosophy of mathematics.Two of those problems were, 1. to show that mathematics is consistent and complete, and 2. to show that mathematics provides an effective decision procedure for solving any problem. Mathematics is consistent, complete, and also a perfect problem solver. Now let me introduce you to two great mathematicians, Kurt Gödel and Alan Turing.
Kurt Gödel (1906-1978) was a great mathematician. In 1931, when he was just 25 years old, he threw a bomb shell on Hilbert’s formalism.Gödel’s incompleteness theorem showed that the logical consistency and completeness of mathematics cannot be proven. In the entire history of mathematics, Gödel’s proof was a big sensation. It showed that the formalist interpretation of mathematics was not possible.
Then, we have Alan Turing (1912-1954). He was a highly influential English mathematician. In 1936, Alan Turing proved that no algorithm or procedure could exist, even in principle, that would guarantee a solution to every mathematical problem.This eventually led to the foundation of modern computer science. Thus, Kurt Gödel and Alan Turing became the two morticians who put Hilbert’s formalism to rest.
Intuitionism: The third school was advocated by Dutch mathematician Luitzen Brouwer. He was influenced by Immanuel Kant and Arthur Schopenhauer. He argued there is no connection between mathematics and reality. Our minds impose order on experience using mathematics. Our minds do not discover the mathematical truth. They only invent it. Mathematical truths are truths we experience. If you cannot experience it, it is not true. Our intuition creates the illusion of order in mathematics. Mathematical truths are just mental projections.
Intuitionism did not gain much ground among mathematicians because the scientists were discovering groundbreaking fruitfulness of mathematics in almost every branch of science. Nobel laureate Eugene Wigner called it ‘the unreasonable effectiveness of mathematics’. When great physicists were discovering hitherto unknown, and often unexpected features of nature using mathematics, it would not be convincing to argue, ‘Mr.Wigner, it’s all in your mind, it has nothing to do with reality’.
Intuitionism also destroys the certainty we put in science and mathematics. Would you believe a scientist who says, ‘What I just described is only my mental construction, it has nothing to with reality’? Would you put certainty in a mathematician who says, ‘What I just described is only my mental construction. It has nothing to do with reality’. The very basis of science will be jeopardized. No wonder these theories provided fertile soil for post-modernists who lost faith in science, mathematics and logic.
Both formalists and intuitionists denied any connection between mathematical truths and reality. But, intuitionists went further. They denied that the law of non-contradiction is a valid logical law. They argued that the law of non-contradiction is symptomatic of faith in the transcendental existence and truth of mathematical statements. They are right. If the mathematical order is what we impose on nature, then there is no validity to the law of non-contradiction. To put it in other words, the law of non-contradiction is valid only if mathematics is connected to reality. Because humanity started with the belief that mathematics is connected to reality, it embraced the law of non-contradiction which is foundational to the birth of logic and science.
Today most mathematicians will agree, at least privately, that mathematical objects do exist independently of the mind. We discover the mathematical truths, not invent them. Why were logicism, formalism and intuitionism born in the first place? Because, like their cousins in biology, secular mathematicians were not willing to allow any Divine Foot at their door steps.
Logical positivist Otto Neurath once said, “We are like sailors who have to rebuild their ship on the open sea, without being able to dismantle it in dry dock and reconstruct it from the best components. Only metaphysics can disappear without trace”
For over two thousand years, the ship of mathematics has been driven by the winds of Platonic philosophy. But that must change. Neurath said, we must rebuild this ship. We must dismantle it and reconstruct it without a trace of Platonic metaphysics. But the new discoveries in all branches of science only deepened our appreciation for the role of mathematics. The law of non-contradiction does not cease to exist even if humanity ceases to exist. The laws of gravity, the laws of chemistry, the laws of quantum mechanics…they are all mathematical laws which will exist even if we disappear. Mathematical reasoning has to be true even if humanity ceases to exist.
Those mathematical truths were here long before we came and they will be here after we are gone. They are independent of our existence. They are independent of our mind. The secular ship logical positivists have built has been sinking by the ever intensifying winds of Platonism.
Plato believed in the transcendental, spiritual, eternal basis for mathematical truths. The secular philosophers came out with Logicism, Formalism and Intuitionism.
All three schools of philosophy were shown to be fallacious and today most mathematicians are going back to Platonic spiritual philosophy. This takes us back to God, the geometer, who designed this world on mathematical truths.
Christians can identify with Plato’s philosophy to some extent. Plato argued the material objects are imperfect representations of spiritual entities. Christians believe that our physical bodies are imperfect habitations of our eternal souls. The tabernacle in the Old Testament is a replica of the Tabernacle which is in heaven.
In Exodus 25:9, God says to Moses, “According to all that I shew thee, after the pattern of the tabernacle,and the pattern of all the instruments thereof, even so shall ye make it”
Mathematics is the study of patterns. Moses was commanded to follow the heavenly patterns for the construction of his earthly objects. He was told to build the tabernacle and all its instruments.
He climbed Mt.Sinai and looked into heaven. He saw the heavenly tabernacle and all its instruments. Then he went down, set to build the earthly tabernacle. The earthly tabernacle is just a replica of the heavenly tabernacle.
Hebrews 8:5 says “Who serve unto the example and shadow of heavenly things, as Moses was admonished of God when he was about to make the tabernacle: for, See, saith he, that thou make all things according to the pattern shwed to thee in the mount”
Moses followed a heavenly pattern. In 1 Corinthians 13:12, Apostle Paul says, For we know in part, and we prophesy in part. But when that which is perfect is come, then that which is in part shall be done away. When I was a child, I spake as a child, I understood as a child, I thought as a child; but when I became a man, I put away childish things. For now we see through a glass, darkly; but then face to face; now I know in part; but then shall I know even as also I am known. And now abideth faith, hope, charity, these three; but the greatest of these is charity.
When that which is perfect is come….this is not a perfect world. This is a corrupted version of a perfect world. The perfect world is yet to come.
When I was a child, I spake as a child, I understood as a child, I thought as a child: This is not the perfect wisdom. What we have today is only limited wisdom. It is only a slice of God’s eternal wisdom.
‘For now we see through a glass, darkly; but then face to face’: Our current vision of God is imperfect. We are only seeing through a glass. Soon, we will see our Savior face to face.
‘But the greatest of these is charity’: When we love a child, when we love our spouse, when we love a friend, when we love a stranger, it is only a reflection of God’s eternal love for us.
There are exceptions to Platonism. Our Lord Jesus Christ is not an imperfect copy of God. He is the perfect, real substance of God. He was God in human flesh. He is the perfect embodiment of divine glory. He is the glory of heaven.
In Revelation 21:22, we read “I saw no temple in the city, for its temple is the Lord God the Almighty and the Lamb”
There is no temple in heaven. The temple is the Lord God the Almighty and the Lamb. In heaven, God himself is the temple. Jesus himself is
the temple. Every temple we build in this world is a reflection of that heavenly temple. I found Platonic realism to be more logical, more empirical, more satisfying and more enduring. The three materialistic
philosophies of mathematics proved to be self-defeating. That is why atheism makes no sense to me.
Let us pray, Lord Jesus, we see your great wisdom in the mathematics of the universe. Yet, like individuals who were imprisoned in Plato’s Cave,
many people in our time are blinded by materialism and atheism. We pray
that you open their eyes to see your love, beauty and great wisdom. In your
precious name we pray, Amen.