Mathematics: Invented, not Discovered?

Mathematics: Invented, not Discovered?

By Basil Gala, Ph.D.
In Search of Meaning

An invention is a device we don't see around us in nature, which we design starting from an idea flashing in our heads. To use the device, we often combine it with materials we find in nature. We don't find an invention by exploring the world outside our minds (though this world may give us hints), but by searching in our inner world of inspiration, fantasy, and dream.

Thus we invented the wheel, the ship, the lightning rod. We did see some stones and spores that looked somewhat like wheels , logs that floated like boats, trees that conducted lightning to the ground; but these things weren't things fashioned to do useful work for us. As we progressed in technology, we invented the watch, the compass, the integrated circuit, the computer--artifacts even further removed from what we can discover in nature (by experimenting, observing, measuring, recording, and fitting data to models). Similarly, item by item, starting with the digits, we invented mathematics with its notions about numbers, quantities, structures, spaces, shapes, relations, changes, and other matters, notions sometimes useful and occasionally important.

Plato taught that mathematical notions were discoveries made in a universe of ideas, a universe more perfect than the material one we live in, an eternal world, unchanging and serene, our own world of the senses being but a shadow show of the truly real and ideal universe of mind. Idealistic or devout thinkers have agreed with Plato for two and a half thousand years. Plato's position is an academic one, not practical, but with profound implications in philosophy I will discuss soon. For now, I simply define an invention as a mental finding, on the inside of our eyes; I define a discovery as a finding in nature, on the outside of our eyes. Still, the separation provided by our eyes and skin is an arbitrary division; our bodies too are part of nature.

We can argue about what is real or ideal, but we can profit more by considering instead what is useful and important among all the different figments of our imagination. A mathematical concept, if true, is like a key that may open a door to the world. If the key works in the keyhole, the door opens; we are able to do some work out there for our survival and progress. If not, the concept may still hold the potential for use in the future, in a different technological environment. It was so with the Fourier transform, not applied to anything for a hundred years after Fourier invented it, until electrical and electronic science emerged with the discoveries of Franklin, Faraday, Galvani, and Marconi. The transform was a good key, mathematically true, eternally true if you like, a key waiting for the for the door of discovery to appear.

The key and lock are ideas fashioned into devices using materials we find in nature, or which we fabricate. Inventions often result from such interractions of the fruitful minds of humans with their natural environment. But many inventions, especially in mathematics, are abstract, not directly involving anything physical. To promote their careers and interests, every year mathematicians publish in journals thousands of papers with new concepts, definitions and theorems that nobody is likely to apply at work; rare are the concepts invented to be both useful and important.






Once in a century we see such an invention as calculus, to which Isaac Newton and Gottfried Leibniz gave birth independently around 1680 AD. Newton invented a practical calculus to explain the motions of objects; a philosopher and mathematician, Leibniz solved a conceptual problem in mathematics, which Archimedes tackled and almost solved in antiquity, the “squaring of the cicle,” involving the constant π. Is π a real part of nature, or a human invention? If we broadcast the first 10 digits of π out into space, can we hope to make ourselves understood by other intelligent races receiving the signal? Since the circle is not found in nature, except in approximation, π is not part of nature either; it was invented. It will mean something to aliens if their intelligence is like our own, having invented the same mathematics.

That is likely, however, because mathematics is about regular patterns, such as the circle or the triangle with exact relations, patterns useful, even essential, to any evolved intelligence. In the SETI program we sent out radio signals into space with the Fibonacci series of numbers, a regular pattern in the signals. Patterns are understood anywhere by any sentient thing, as we see in the reactions of animals towards flowers and fruits. The flower attracts the bee to the center of a pattern of petals, the center with the pistil and stamens.

To decide on whether mathematics is invented or discovered, we must answer the question what is mathematics. It is a language we developed to communicate ideas to each other and to think better in solving problems. It is different from ordinary language in a variety of ways. First, mathematical concepts are not ambiguous, whereas in natural language words often are ambiguous, that is, they have several meanings, often vague meanings. In mathematics each word or symbol means one thing only in a particular subject, which allows mathematicians, scientists, and engineers, to communicate with others in their field better than the public at large and to think more clearly and effectively in their own specialty.

Mathematics fundamentally is the art of patterns. So are the arts of sculpture, painting, culinary, dance, and music, which give us pleasure. Mathematics also gives pleasure to those who appreciate its elegance. A mathematical proof is said to be elegant when it uses few assumptions or previous results; when it is unusually succinct; when it is surprising; when it is based on new insights; and when it can be generalized to solve similar problems. Then there are mathematical values that relate to aesthetics, such as the golden ratio φ (derived from the golden rectangle): φ = (a+b)/a = a/b = (1+√5)/2 = ~1.618.

I find Newton's law of motion F = ma to be beautiful; even more elegant is Einstein's energy equation for the mass of an atom: E = mc². These are actually similar concepts. F is force and E is force over distance. The variable a is the acceleration of an object, which is the change of the change of distance with time. The constant c is the speed of light photons, also a change of distance with time; since c is squared it cancels out the distance unit in E. Photons from the atom are sped by the strong and weak nuclear forces, reversed in direction.

Mathematical structures or patterns have often been compared to those of a musical composition. What is a pattern? The word is more easily interpreted than defined. It is a local regularity, an organization or a structure of elements, which usually involves periodicity, symmetry, repetition. We can define it as a region of low entropy (disorder) in a system, or alternatively a region




of high information content. In sound, musical patterns are distinguished from noise; in vision, paintings are different from graffiti; in taste, we appreciate a chef's dish from hodgepodge; in dance, ballet is not spastic; in every craft we recognize the excellence of the patterns from the master's hands and mind; we recognize the crudeness and inadequacy of the patterns produced by the novice. (The process of how we recognize or classify noisy, distorted patterns has so far largely eluded computer scientists. Some believe pattern recognition will forever be the province of animals, given to them not from the material universe, but from another source. Here we go again to Plato's idealism.) Since physical science also searches and deals with regularities or patterns in nature, mathematics provides powerful tools for understanding and controlling natural events.

Tools relate closely to nature. A pick or a shovel helps to dig up the soil; knife, fork, spoon move food to our mouth; saw, drill, hammer fashion wood and stone. Such tools are common in almost all cultures, though some tools are unusual, like the boomerang or the chopstick. In all cases, tools relate to nature, and so does mathematics; therefore, we shall find aliens in space using much of the same math we have invented.

Mathematics is a bag of tools, but it is also a symbolic language; words are symbols, abstractions, especially so in a jargon like math. An abstraction extracts the essential or characteristic elements, generalizes, factors out details, distancing us from objects. A chair is an abstraction for all pieces of furniture we sit on. A numerical digit, say 3, is an abstraction of any items, oranges, apples, stones, we have learned to pick up and manipulate in the quantity of three. Natural numbers, (0,1,2,3...,) are only the beginning of abstraction. Math continues with negative numbers, real numbers, irrational numbers, complex numbers. Abstraction to algebraic quantities allows for the easier solution of arithmetic problems.

Geometric figures, the triangle, square, circle, are abstractions (ideals) of forms we encounter in nature in less precise shapes. Analytic geometry, introduced by René Descartes, combines algebra with geometry, allowing us to convert algebraic expressions to geometric figures and the reverse. Descartes was also a philosopher who together with Roger Bacon in England laid the foundations of the scientific method. His starting point in this effort was to insist that we accept as true only those beliefs which we know to be true without any doubt, such as that any triangle's angles add up to 180 degrees—true, we would say today, only in Euclidean geometry.

Then came calculus involving abstractions of the changes in functions, using the concepts of the infinitesimal and infinite to solve problems previously thought to be intractable.

Random events, chances, were also intractable, until the abstractions of measure theory, probability and statistics. Initially, the motivation for such inventions was gambling, but we now use abstractions of probability densities, distributions, mean, median, variance, and standard deviation in economics, psychology, sociology, biology, information theory, and quantum mechanics. Mathematics is like a tree, growing branches, and each branch growing more branches indefinitely, with no end in sight.

If abstractions are well conceived and implemented, they give us greater powers of analysis, problem solving, and designing useful systems. Advanced people tend to think in abstract terms, while those less well trained think of particulars and instances. I am not denying the benefits of the concrete



and the particular. Even the greatest idea for a design remains nebulous until someone converts it into concrete terms and materials. From Icarus and Leonardo da Vinci onward, many persons had ideas about flying, but the Wright brothers, bicycle mechanics, built a flying machine that went up in the air with engine on board, turned around and landed. Once the brothers had built the first simple plane, other mechanics, engineers, and scientists could study and improve flying machines culminating with the space shuttle, the most complex artifact to this day.

The first planes had wings, body, controls, and engine that were imprecise, making flying difficult and erratic. (Crashes were frequent, often fatal to pilot and passengers, even with low plane speeds and altitudes. That was the way Will Rogers died in 1935 in Alaska together with aviator Wiley Post in a small plane.) By comparison, the Boeing 787 Dreamliner is built from parts cut with precision, parts produced all over the world and fitted together at the assembly plant in Seattle. A feature of mathematics is precision, exactness, a feature not common in nature; mathematical formulas were applied to the design of the Dreamliner.

A parabola, corresponding to the quadratic formula, has a focus point. If you build a lens or mirror in the shape of a parabola, you concentrate light at the focus. The closest you grind the lens to a mathematical parabola, the more light you collect at the focus. The closer your lens gets to the mathematical form, the more unnatural it becomes. Yet, once built, it's now part of nature. You have projected a concept in your mind out to nature. The lens has approached perfection.

A ball you throw at a basket also follows a parabolic curve. It moves up and away with an initial velocity from the force of your hands, and goes down with the continuous force of gravity. The curve will vary depending on wind resistance, moisture, the basket defender's hands. Control of the curve is difficult, mathematics fails, but the uncanny skill of a champion player will get the ball in the basket most of the time. Albert Einstein: “...as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” Real physical phenomena are far more complex than our mathematical models of them; our models can never match natural events exactly.

Mathematics is not reality, but an abstraction from reality, putting matters in a form simple, noise free, and precise, as we would like matters to be for our control of them. For example, a straight line is the shortest distance between two points, if it is a mathematical (geometric) line; to the extent it deviates from math, it is not the shortest distance, and all physical lines deviate.

Yet, mathematics is a powerful tool for science, especially where we can control events to shape things to our ideal notions, as in lens grinding.

Mathematical notation is also something under our control; we can push around these symbols as we choose, provided we follow the rules of logic, rules we are coming to soon. We can create symbols and define them. The symbols can be very abstract, and can represent many computational steps, such as the sign of integration. Symbols are a great convenience, as are subroutines in computing. There is such a proliferation of symbols and duplication of symbols in mathematics, you need to know the context well to understand an article in a journal, even if you are a mathematician. One specialist can only talk to another specialist in the same field of mathematics. But the notation is not essential, although it can be convenient. The Ancient Greeks did not use it; they had letters instead



of Arabic numerals, and so did the Romans. They produced great math and engineering studies just the same, written out in words. (These people didn't use musical notation also, unfortunately—since music was important in their cultures and was probably as good as their literature, the loss for us of their music was great, unless it survived in the Italian musical tradition.)

Unlike notation, logic is essential to mathematics. Bertrand Russell and other philosopher-mathematicians established that math is founded on just a few logical principles or axioms. To prove a theorem is true, after defining the quantities in the theorem, you follow a chain of reasoning—that chain follows the rules of logic, otherwise we conclude the theorem is false. It is impressive to follow the argument of a mathematician skilled in this reasoning process, who concludes, “it is clear, therefore, that x is equal to z.” Clear? Yes, clear, if you can fill in the missing steps, which the expert mathematician does in his head.

Who invented logic? Aristotle was the first to formalize it; Kant, Frege, Poincaré, Russell and others extended formal logic. But logic is a set of human habits of thought that we developed when we emerged as a fragile species of mammal; logical rules became wired in our brains, because they fostered our survival in a hostile natural environment. Those who could think well with logic, could plan better, control themselves and events better, and survive better. Logic is part of our basic mental equipment, such as our memory and emotional drives. Logic corresponds with the world, the way it is. We can derive other kinds of logic, other math, but they would not help us survive in this universe; we would need to find an alternate universe to match our new logic. In such a place, math could emerge where 2 + 2 = 5. Pick two oranges from a tree; put them in a basket with two other oranges. Five guests come and each gets an orange. What does the fifth guest get to eat? Here on Earth the fifth guest eats nothing but the phantom orange. Burp.

Logic itself is based on axioms, truths we take as self evident. For example, “the whole is greater than the part.” Also, “if equals are added to equals, the wholes are equal.” And another axiom: “things which are equal to the same thing are also equal to one another.” Can you argue against the truth of such axioms? If so, who would listen to you for long?

People have marveled at the capacity of mathematics to solve problems in many disciplines. It appears in any case to work better than rain dances, incantations, mumbo jumbo, or voodoo. Eugene Wigner wrote about “the unreasonable effectiveness of math.” Paul Erdös, a mathematician and atheist, wrote about “finding proofs from God's book of theorems...numbers are beautiful, like Beethoven's Ninth, ineffable.” The effectiveness of math does not seem unreasonable if we consider that it is made of invented devices, which, however, are solidly based on fundamental truths, axioms, wired into our brains during our evolutionary development as a Homo sapiens. Mathematical theorems are inevitably true if they are proven with sufficient rigor. Rigor is the certainty or strength of the link achieved in each step of the chain of reasoning in deriving the proof. If so, we can rely on the result of the theorem and can expect any device that conforms with our equation to perform reliably, for example a rocket reaching its designated orbit, as certainly as the planets revolving around the sun. That's the power of our evolutionary heritage.

One may take a different viewpoint, however, concluding that our reasoning abilities are given to us by God, so that we may be worthy children of the Divine Creator of the universe. If you take that route, you end up again in Plato's academy, deriving mathematics, and other inspirations in musical



melody, fine painting, sculpture, and dance as gifts from our Creator. Thus, mathematical ideas are discoveries in the exploration of Heaven.

God and Heaven, the Madonna, the angels, and all the holy saints (together with Allah, Jehovah, Buddha, Krishna) are in our minds. They exist as long as we project them outward to the world around us. So it is with mathematics. We may thus trek through the heavenly universe in our minds and invent (discover) new mathematical formulas, and put them to good or evil use in the world, building power plants or atomic bombs, laser scalpels or weapons, rockets to explore the galaxy or to fly bombs across the oceans to annihilate people.

Now we enter the inner sanctum of feelings and ethics. Can the tentacles of mathematics reach into emotions, sentiments, beliefs, and motivations? Math can please with its elegant patterns just as music does. Music does at present reach much further into our emotional state than mathematics. So do the other arts of the senses. Music can inspire, it can lift up our mood, it can stir us up to dance and revel, it can motivate (think of a march or the drumbeats, bagpipes, and the trumpets of battle). Can math make us laugh? Only, if a funny person goofs it up. Ordinary language can be molded into jokes, not math. Poetic language, drawn from our common experience of life, can make us cry for pleasure, leap for joy, thrill our inner being with its rhythms and and nuances. Mathematics is cold and austere, a product of our intellect. It can be beautiful; yet it holds no promise, the same as science, for solving our ethical problems, that of war and peace, love and hatred, good and evil.

Richard Dawkins, a vocal atheist, states in “The God Delusion,” that our zeitgeist (spirit of the age) has been getting progressively more ethical, more tolerant towards those different from us, more compassionate--and religion is not involved in this process. He does not explain, however, where he thinks the messages for the evolving zeitgeist are coming from. If not from God, then from what source? Dawkins is a fervent scientist, a preacher for the scientific method and ethos, but science clearly is no more help in ethics than mathematics. Perhaps we are witnessing a better reception of God's message worldwide, excluding certain regions of the Earth where idiocy reigns supreme.

Andros, Greece, June 2009