Today I would like to talk about the mathematics of reality. Reality? What is reality? Most of us believe that we are living in reality. We believe that our bodies are real, our minds are real, our family members are real, our friends are real, the tooth brush, paste, water, river, house, car, truck and animals…we believe they are real. We would like to understand these things which we consider real. That is what life is all about..the study of reality. Philosophers call it metaphysics. It is the study of the fundamental nature of reality. Metaphysics is divided into two branches: Ontology and Epistemology.
What is there? is ontology, while ‘How can I know it?’ is epistemology. If I ask you, ‘Are you real?’, you might reply, ‘Off course, I am real?’. Then I ask, ‘What made you real?’ ‘What components of you made you real?’ that is ontology. Then, what tools do I have to prove that you are real? How can I come to the conclusion that you are real? that is epistemology.
What is in the reality? Usually, we come across two kinds of answers. In natural metaphysics, all reality is just natural. It is the Universe plus man with a body. In supernatural metaphysics, reality is made up of fivr things: God plus universe plus Spiritual Beings plus Man with a body and a soul. God + universe + Angels + Demons + Man with a body and a soul.
For a materialist, all reality is made up of matter and its derivatives. Everything in the universe including mathematics, morality, aesthetics, consciousness, love, hate, jealousy..you name it..they are all either matter or derivatives of matter. God, angels, demons, mathematics, morality do not have independent existence. Man created God in his image, it is not ‘God created man in His image’. That is natural metaphysics.
Plato said that mathematics are eternal, unchanging and transcendental with their own existence. That is supernatural metaphysics. People who opposed Plato believed in atomism. They said that mathematics does not have an independent reality. They can be reduced to human experience and experiment. Betrand Russell spent a significant part of his life in reducing all mathematics to logic, logic to mind, mind to brain, brain to body, and body to matter. People who were influenced by Russell started a philosophical movement called Logical Positivism, the atomism of our time. Logical positivism contended that every claim to reality must be empirically verifiable. Later it was shown to be self-defeating because the belief that all reality must be empirically verifiable is not an empirically verifiable belief.
Logical positivist believed that scientific method helps us in the discovery of metaphysics. Many great scientists embraced logical positivism. They said metaphysics must be limited to logical positivism because it is the only road to reality. For example Albert Einstein argued that scientific method leads us in the path of reality. Interestingly, many great scientists rejected such realism. For example, Neils Bohr was a great Danish quantum physicist. He said, ‘Science is great, but it is not about reality’. Science is about practical utility. Is it useful or not?
Who cares about reality? In 1543, Mathematician and astronomer, Nicolaus Copernicus (1473 – 1543) published De revolutionibus, in which he formulated his heliocentric theory. Lutheran theologian Andreas Osiander (1498-1552) wrote the preface to this revolutionary scientific text. He wrote that while heliocentric theory is intelligent, it is not necessarily true, it is not a realist account of the solar system. In the same way, in Stalin’s Russia, science text books on relativity and quantum mechanics had a preface which warned the readers not to take these scientific theories as the realist accounts of nature. They are thoroughly mathematical, intelligent and pragmatic but not accurate descriptions of reality. So, this debate has a long history and is active in our time.
Many atheists in our time are divided into these two camps: Realists and Pragmatists. Realists argue that science leads us to objective reality. Pragmatists argued that science is all about utility, not about reality. The truth for realists is the explanation about the reality; the truth for pragmatists is about its utility. Pragmatism later influenced Postmodernism, which says science is nothing more than a subjective social construction. Steven Weinberg is a Nobel Prize winning physicist. He is an atheist. He told me, ‘I don’t do God’. He takes an objective-realist view of scientific knowledge. He rejects the postmodernist view of science. Welcome to science wars. Mathematician Alan Sokal went so far to publish an intentionally nonsensical paper in a postmodernist journal called Social Text, published by Duke University Press. For the theologian, science is one of the many tools to understand reality. For the atheist realist, science is the only tool to understand reality. But for the atheist pragmatist or postmodernist, science does not even have an access to reality. In the Sokal affair, we see these two groups of atheists strangling each others’ necks. The modern, natural metaphysicians destroy their own tools.
The secular scientific establishment in our time more or less espoused a realist view of logical positivism. They argue that only valid knowledge is scientific knowledge. But, it is not all bad news. Mathematician and philosopher Hilary Putnam (1926-2016) closely followed logical positivism. He studied with Hans Reichenbach, a leading proponent of logical positivism, who was called ‘the greatest empiricist of the 20th century’. Putnam did not agree with Reichenbach. Empiricism is not the only way to knowledge. Putnam taught a course at Harvard University in ‘nonscientific knowledge’. There are real things which are not made up of matter. Putnam argued that knowledge from religion, ethics and aesthetics is also valid knowledge.
Putnam is taking us back to Plato. Wisdom is not just empirical. It is not just experiential and experimental. It is also transcendental. In fact, experiential wisdom is only a shadow of the transcendental wisdom. Mathematics connects us to this transcendental wisdom. Pythagoreans believed that mathematics is the foundation of all reality. According to Plato, Pythagoras said, ‘At its deepest level, reality is mathematical in nature’. Similar sentiment is expressed by Galileo when he said, ‘the universe is written in the language of mathematics’. Isaac Newton titled his most famous work Principia, the Mathematical Principles of Natural Philosophy. For Newton, God is running this universe using unchanging, rational, mathematical laws. Simultaneously with Leibniz,Newton developed calculus to explain how the universe works. Newton and Leibniz had their own conflicts. They quarrelled on the role of God in this universe. But they never doubted that God created the natural reality and it is discoverable by scientific method. They believed in an objective-realist view of scientific knowledge. Alexander Fleming said, “I did not invent penicillin. Nature did that. I only discovered it by accident” Newton would say, “I did not invent mathematics. God did that. I only discovered it by accident”
What is the nature of mathematics? There are different opinions. Plato said, mathematics has its reality independent of this material world; Immanuel Kant said that mathematics is only a tool which helps the human mind in thinking about numbers and shapes. David Hilbert said that mathematics is like the rules of games we choose to play. These great minds did not agree on the nature of mathematics.
When we talk about truth, we consider coherence and correspondence. They are also relevant when we discuss the nature of mathematics. For the realist, mathematics corresponds with reality. For the nonrealists, it does not. The nonrealist is usually satisfied with coherence, ‘It explains my data. That is enough for me’. But truth is more than coherence. Philosopher Brand Blanshard said, “Someone might hold that coherence with a set of beliefs is the test of truth but that truth consists in correspondence to objective facts. If, however, truth consists in correspondence to objective facts, coherence with a set of beliefs will not be a test of truth. This is the case since there is no guarantee that a perfectly coherent set of beliefs matches objective reality”. Objective truth does not satisfy with coherence. It also needs correspondence to reality. Similarly, mathematics does not satisfy with a perfectly coherent set of numbers or theorems. It also needs correspondence to reality.
Mathematics connects us to reality
When Plato ruled the Western philosophy, Mathematics was considered the highway to reality. It’s correspondence to reality is intuitively understood in classical philosophy. From Pythagoras to Putnam, many mathematicians shared the view that mathematics is not just a human invention. It is not an isolated stream. Sooner or later, the stream becomes a river and rushes us towards the ocean of eternal mathematical truth. Some of the greatest mathematicians of our time are taking us back to Plato.
Sir Roger Penrose (b.1931) is a mathematical physicist at the University of Oxford. He was a long time associate of Stephen Hawking. In 2004, Penrose published his famous book, The Road to Reality: A Complete Guide to the Laws of the Universe. In this book, Penrose described three ‘worlds’ and three profound ‘mysteries’. He gave the following figure.
Penrose describes 3 worlds. Physical world, mental world and Platonic mathematical world. He says there are three profound mysteries between these three worlds.
The first mystery is how the Platonic mathematical world organized the physical world.
The second mystery is how our minds interact with the physical world.
The third mystery is how our minds connect us to the Platonic mathematical world.
Penrose wrote these words,
“Does the Platonic mathematical world actually exist, in any meaningful sense? Many people, including philosophers, might regard such a ‘world’ as a complete fiction – a product merely of our unrestrained imaginations. Yet the Platonic viewpoint is indeed an immensely valuable one. It tells us to be careful to distinguish the precise mathematical entities from the approximations that we see around us in the world of physical things. Moreover, it provides us with the blueprint according to which modern science has proceeded ever since….For our individual minds are notoriously imprecise, unreliable, and inconsistent in their judgements. The precision, reliability, and consistency that are required by our scientific theories demand something beyond any one of our individual (untrustworthy) minds. In mathematics, we find a far greater robustness than can be located in any particular mind. Does this not point to something outside ourselves, with a reality that lies beyond what each individual can achieve?”
Does this not point to something outside ourselves? That is the most uncomfortable question Penrose directs to his colleagues at Oxford University such as Richard Dawkins. Penrose says, many people including scientists and philosophers say that the Platonic world is a complete fiction born out of human imagination. But, that is not true. The Platonic world is real and ‘it provides us with the blueprint according to which modern science has proceeded ever since’. He says that our individual minds are notoriously imprecise, unreliable and inconsistent. Our quest for certainty must proceed towards the Platonic mathematical world.
If you look into the history of science, you will see the powerful hand of Platonic mathematics. Let us say, you went to a new town. You checked into the hotel room. On one of the walls of that room, you saw an abstract painting. In the painting, you saw a serene lake, there is a forest of pine trees on the right side of the lake and a hill on the left side of the lake, a motor highway behind the lake. You said in your heart, ‘what a beautiful painting’. That’s an abstract painting. It may or may not represent a particular region of this world. You don’t care. It’s just a painting. But, suppose you walked to the television. In front of the television, there is a folder. Inside the folder, you found a welcome brochure. Inside the brochure, you found a map of the town. When you look at the map, there is a lake. There is a forest of pine trees on the right side of the lake, a hill on the left side of the lake, a motor highway behind the lake. Let us suppose you wanted to take a walk in that town. You would expect a lake. You would expect a forest of pine trees on its right side, a hill on its left side and motor highway behind it. Why? because it is a map. It should represent the reality of that town. The point is, the way you look at a painting is different from the way you look at a map.
Is mathematics a painting or a map? Is it a painting painted by our thoughts or a map given to us by nature? The history of science shows Platonic mathematics is not a painting we draw out of our imagination but a map that guides us to real features of nature. Let me give you some examples from science. Mathematics is a study of patterns and science is also a study of patterns. We have to start with that belief: there are patterns in nature. There are patterns in science and there are patterns in mathematics. We bring them together as we come to the results of a scientific investigation.
Let us start with Copernicus. Nicolas Copernicus (1473-1543) was trained in theology and was working for the Catholic church. He developed a mathematical model of the solar system. He proposed the first ever heliocentric model based on mathematics. Mathematics demanded that planets must revolve around the sun, not around the earth. Later observations established the truth.
Johannes Kepler (1571-1630) believed because this universe was created by God, it must have a mathematical order to it. He derived three empirical laws of planetary motion which provided a mathematical description of solar system orbits. He placed the heliocentric model of Copernicus on a firm mathematical foundation. Later observations established the truthfulness of Kepler’s laws.
Galileo Galilei (1564-1642) realized that empirical mathematical laws not only describe the motions of heavenly objects but also earthly objects. He started to formulate the mathematical laws that describe falling objects. It gets very interesting. He saw the correspondence between the geometrical abstraction of quadratic equations and motions in the natural world. For Galileo, that was not a blind coincidence. He famously said, ‘the universe is written in the language of mathematics’. His famous “rolling ball” experiment showed that the distance traveled by a falling body is proportional to the square of the time of the fall.
Then we have Isaac Newton. He saw the similarities between Kepler’s laws of planetary motions and Galileo’s laws of terrestrial motions. Then, he brought them together to synthesize one set of universal laws of forces and motions. Kepler laid the mathematical foundations for describing celestial mechanics while Galileo laid the mathematical foundations for describing terrestrial mechanics. Newton’s brilliance was to bring them together into one set of mathematical laws which can be used to predict the motion of objects everywhere in the universe. Newton invented a new branch of mathematics called calculus. He established the mathematical relationship among three physical quantities: force, mass, and acceleration.
During 1665-1666, Newton’s most productive years of discovery, he gave us the mathematical description of the universal force of gravity. We all know the famous image of Newton standing with an apple in his hand and looking out into space. He realized that as Galileo said, when it is thrown, the apple follows a parabolic path due to gravity. If thrown with sufficient acceleration, as Kepler said, the same apple will move in an elliptical orbit. Newton established that one force, universal force of gravitation, moves both the apple and the moon and any other object in this universe. That was a great discovery which influenced almost every aspect of human society, from physics to astronomy to chemistry to economics to politics. In 1846, French mathematician Urbain Le Verrier predicted and discovered the planet Neptune using only mathematicians of the laws of Kepler and Newton. That’s the power of mathematics…an unknown planet was predicted and discovered using mathematical calculations. No telescopes or space probes. Just a pencil and a paper. A new planet was discovered.
Thermodynamics is the study of energy. We see different types of energy in nature: thermal energy, chemical energy, motion energy, sound energy etc. Scientists realized that one form of energy can be converted into another form. They reasoned, if nature has a mathematical order, thermodynamics cannot be an exception. If nature has underlying symmetry which gave us – Galileo’s laws of terrestrial motion, Kepler’s laws of planetary motion, and Newton’s laws of universal motion, there must be laws in thermodynamics. There must be a mathematical description of energy.
The search for mathematical order in thermodynamics yielded great results. One by one, four laws of thermodynamics were discovered. James Joule (1818 – 1889) experimentally demonstrated the first law of thermodynamics. He praised God and said, ‘the first law of thermodynamics shows us the beneficence of God’. Then came the second law of thermodynamics. These laws gave us great insights into the nature of the universe.
One of the amazing discoveries of science is the truth that electricity and magnetism are two aspects of the same force. In 1820, Danish physicist Hans Christian Oersted (1777-1851) discovered that electricity can produce magnetic fields. If you believe in mathematical symmetry, you would say, ‘if electricity can produce a magnetic field, then a magnetic field should produce electricity’. That is what Michael Faraday (1791-1867) did. In 1831, he demonstrated that magnetic fields can produce electricity. That discovery turned the world upside down. There is a story, William Gladstone, later to be British Prime minister, heard about Faraday and went to see his laboratory. After meeting Faraday, Gladstone asked him, ‘Mr.Faraday, what is the practical value of all this study into electricity?’. Faraday replied, “Why, sir, there is every probability that you will soon be able to tax it”! Two hundred years later, we can see how Faraday’s prediction was fulfilled.
The mathematical symmetry between electricity and magnetism pervaded all our lives. Every electric motor, every telephone, every television, every computer relies on this symmetry. Scottish physicist James Clerk Maxwell (1831-1879) gave a mathematical form to the connections between electricity and magnetism. He constructed four great equations. Those mathematical equations made wonderful predictions about nature.
Maxwell’s equations predicted the existence of electromagnetic waves, which travel at the speed of light. That shed light on the nature of light, which is understood as an electromagnetic wave – a moving, oscillating disturbance of electric and magnetic fields in empty space. That established the nature of light. His equations predicted there must be other wavelengths of light beside visible light.
Welcome to the era of modern communications. Maxwell’s mathematics predicted the existence of invisible electromagnetic waves. Soon, the beautiful electromagnetic spectrum was unveiled before us. Each region in the spectrum transformed science and technology in breathtaking ways.
Radio waves: Radio waves were predicted by mathematics. In the late 1880s, German physicist Heinrich Hertz (1857-1894) worked on that prediction and discovered radio waves. Later Italian engineer Guglielmo Marconi (1874-1937) demonstrated wireless transmission using radio waves and it brought a revolution in communications. Now, radio waves are being used in radio broadcasting, television, mobile phones, and satellites.
Microwaves: Almost every home has a microwave now. Microwave radars are used in stealth technologies like radar, satellite communication and wireless networking
Infrared waves: we developed infrared vision goggles and film ; infrared imaging in medical diagnostics; infrared satellites tracking forest fires, predicting volcanic eruptions
Visible light: Our visual perception relies on visible light
Ultraviolet light: helped us to identify it as a cause of skin cancer and to take precautions against radiation injury
X-rays: They are useful in diagnostic X-ray imaging in medicine and in X-ray telescopes in astronomy
Gamma rays: They are used in the sterilization of foods, nuclear medicine, and PET
In the electromagnetic spectrum, there are 7 regions and each region led to amazing discoveries and inventions. How did it all start? Mathematics predicted the discovery of this beautiful electromagnetic spectrum.
Coming to Relativity, we all know about Albert Einstein. When Einstein was looking at the mathematical equations developed by Maxwell, he realized one thing: the speed of light must remain constant. Maxwell’s equations demonstrated that the speed of light is a universal constant, at 186,000 miles per second for every observer in the universe. If the speed of light is a constant in all reference frames, then time, length, and mass must be relative.
Einstein developed two theories of relativity: theory of special relativity and the theory of general relativity. Here too, you will see the predictive power of mathematics in these two theories. Some of the greatest mathematicians ever lived such as Bernard Reimann, Hermann Minkowski, David Hilbert, Henri Poincare, George FitzGerald, Hendrik Lorentz influenced Einstein’s thinking. Theory of relativity was positing mind blowing conclusions about the nature of our universe. Often Einstein was reluctant to accept those conclusions. These mathematicians prodded Einstein to see the consequences of his ideas.
Hermann Minkowski showed mathematically how Einstein’s theory could be understood in terms of a four-dimensional spacetime. In 1909 he famously said, “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality”. Minkowski saw space-time in a mathematical vision before it was verified in nature.
Hendrik Lorentz developed transformation equations based on Maxwell’s theory. He developed the equations of special relativity before Einstein figured out the role the equations played in the nature of light. Lorentz drew stunning conclusions from his mathematics such as length contraction and time dilation. Time dilation? What are you talking about? People thought Lorentz was crazy. Yet, he was relentlessly drawing mathematical equations about time dilation. Time dilation is no longer a crazy idea or even futuristic science fiction. Your cellphone does not work well without time dilation. Your GPS does not work without taking ‘time dilation’ into calculations.
Coming to quantum physics, you will see the triumph of abstract mathematics in every inch of quantum physics. In quantum physics, we enter the microworld. Microscopes and telescopes do not help guide us. We must rely on mathematics. Great mathematician John von Neumann titled his famous book on quantum mechanics, ‘Mathematical Foundations of Quantum Mechanics’
What are Feynman diagrams? They are mathematical equations in disguise. There is almost nothing you can do in quantum physics without the guiding hand of mathematics.
Coming to particle physics. Let me give you a couple of examples. In 1928, Paul Dirac mathematically predicted the existence of positron, the antiparticle of the electron. Four years later, in 1932, Carl Anderson discovered the positron and the presence of antimatter. Mathematics predicted antimatter. Paul Dirac discovered half of the universe using mathematics. Murray Gell-Mann predicted the existence of quarks using mathematics and twenty five years later they were discovered.
The pursuit of unity in diversity has been the quest for many great scientific journeys. Mathematics has been a great driving force in this endeavor. Each scientific era unveiled beautiful unifications:
the unification of celestial and terrestrial mechanics in the mathematics of Newton
the unification of electricity and magnetism in the mathematics of Maxwell
the unification of mechanics and electromagnetism in the mathematics of Einstein
the unification of light and matter in the mathematics of quantum physics
These unifications are not fanciful ideas. They led us towards breathtaking discoveries.
the unification of celestial and terrestrial mechanics revealed the nature of gravity
The unification of electricity and magnetism revealed the nature of light
the unification of mechanics and electromagnetism revealed the nature of spacetime
the unification of light and matter revealed the hidden quantum universe
the unification of quantum mechanics and relativity revealed the existence of antimatter
You see, out of each unification, we’ve found a new truth about nature. These mathematical unifications opened new vistas of discovery, shocking unexpected, and even unwanted revelations
Paul Dirac did not want to see antimatter as a consequence of his mathematics
Einstein did not want to see the Big Bang as a consequence of his mathematics
But, they had no choice. That is the power of mathematics!
Darwinism has no justification
We started with the question, ‘Can we know reality?’ That’s a profound question. Evolutionary epistemologists say, ‘we cannot be sure’. Social scientist Donald Campbell (1916-1996) did a lot of research in Darwinism and epistemology. He coined the term ‘evolutionary epistemology’. He believed that there exists an external reality, but he did not believe in direct access to that external reality. He wrote, “all knowing is highly presumptive, involving presumptions not directly or logically justifiable”, “Evolutionary epistemology has in it an unproven assumption of a real world external to the organism, with which the organism is in dialectic interaction”
His favorite slogan was “Cousin to the amoeba, how can we know for certain?” If you believe you are a cousin to ameba, you cannot logically justify that there is an objective external reality and you can understand that reality. Campbell is saying, ‘I am a cousin to the amoeba, how can I be certain that there is a reality and I can know it?’. That was such a humble and honest answer from a great philosopher of science. You don’t find such humility in the likes of Richard Dawkins. They say, ‘I am a cousin to the amoeba, yet I can all reality, and all reality is only material’. They fail to realize that such certainty in evolutionary epistemology is unjustified.
In theism, because God created these three realms together, there is justification to believe that the human mind can interact with the external reality through mathematics. The mathematical world is greater than any individual mathematician or a physicist.
Paul Dirac was looking at Einstein and his mathematics
Einstein was looking at Maxwell and his mathematics.
Maxwell was looking at Faraday and his mathematics.
Faraday was looking at Newton and his mathematics
Newton was looking at Galileo and his mathematics
Galileo was looking at Copernicus and his mathematics
The chain goes on and on. No one is isolated. Mathematics was leading them as a guide to reality.
From Plato to Penrose, most mathematicians think that mathematics has an independent reality. Roger Penrose once said, “There is something absolute and ‘God-given’” about mathematical truth.
Eugene Wigner (1902-1995) was a great mathematical physicist. He was awarded the Nobel Prize in Physics in 1963 for developing a nuclear theory based on mathematical symmetries. He said, “The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it”. In a 1960 essay, he marveled at “the unreasonable effectiveness of mathematics”. Wigner’s brother-in-law, Paul Dirac was also a Nobel Prize winning mathematical physicist. He went further and said, “God is a mathematician of a very high order and He used advanced mathematics in constructing the universe”.
If you take a Christian worldview, you marvel not only at the unreasonable effectiveness of mathematics, but also at the great genius of God, the architect of our universe. I drew this diagram to give you a Christian view of reality. Human mind can interact with the physical world either directly or through mathematics. You don’t need to learn mathematics to become good at fishing but you cannot go anywhere in quantum physics without mathematics.
God created all three realms:
and mental world
God put them in coherence and correspondence. God created them to connect with each other. It is like you bought a new smartphone. When you open the box, open the window, and turn it on, you will see the words, ‘Searching for signal’. When it is found, it connects to Wi-Fi, activates your phone and joins the worldwide web. The designer of the cellphone made it like that. The human mind is like that smartphone. From the day you open the windows of your mind, it starts its journey to metaphysics, in the first step, it says, ‘Searching for signal’. It connects to the mathematical realm which pervades over the universe like a worldwide web. God, the designer of these three realms, put them in coherence and correspondence. God created them to connect to each other and finally to connect to him.
God displayed his glory in the mathematical fabric of this universe. He also revealed to us in the person of Lord Jesus Christ. Today, I ask you to come to Jesus and find meaning and purpose to your life. He loved you so much that He went to the cross to die for your sins and raised from the dead on the third day to give you the promise of eternal life.
Paul Kattupalli MD is a Christian evangelist and a physician. Visit http://www.doctorpaul.org for more of his articles.