Where do words occur?
Italian astronomer and physicist Galileo Galilei (1564-1642) referred to nature as a book that we are invited by God to study. He realized that even hundreds of the keenest minds could study nature for thousands of years and still not tire of learning what God had revealed “in the open book of the heavens.”[i] He said it was written in the language of mathematics:
Philosophy is written in this grand book the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and to read the alphabet in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.[ii]
As it turns out, we can most effectively define science as the study of patterns in nature and society. For example, chemists study patterns among the elements, leading to the development of tools such as periodic tables. Biologists study patterns among organisms, resulting in charts such as the one for taxonomic rank: domain, kingdom, phylum, class, order, genus, and species. Astronomers study patterns among stars. Economists study patterns in the production, distribution, and consumption of goods and services. Psychologists study patterns in human behavior. Physicists study patterns in matter and energy, discovering such sentences as “Force equals mass times acceleration” and “Energy equals mass times the speed of light squared”. We use the verb equal to indicate the presence of a pattern, where one set of words matches the other.
For that matter, math by itself, apart from its reflection in nature, provides an unending supply of mysteries to be explored. Some equations have taken centuries to solve even after many mathematicians have devoted their careers to finding the solutions. In the past generation alone we have seen the solutions to two such equations: Fermat’s Last theorem, first posed in the 1630s, and the Poincaré Conjecture, first posed in 1904. In each case there has been a long search for what was assumed to be absolute, objective truth.
Mathematicians have often discovered such truths long before scientists discovered the same patterns written, as Galileo said, in nature. Some of the patterns are simple, such as the curve of a seashell or the branching of a tree, both of which follow what we call the golden ratio, 1.618…, which is an irrational number, like pi. Other patterns, such as the changes in a quantum wave function, are so complex that they use imaginary numbers, as discovered by physicist Erwin Schrödinger. Such equations describe and reveal the profound depth of rationality in the cosmos. As Astronomer Royal Sir Martin Rees, Royal Society Research Professor at Cambridge University, put it, “Science advances by discerning patterns and regularities in nature, so that more and more phenomena can be subsumed into general categories and laws.”[iii]
These patterns are not just written in nature. They form the very background and context for nature—even for space itself. In We Have No Idea, Jorge Cham and Daniel Whiteson, professor of experimental particle physics at the University of California, Irvine, talk about how space can only be understood mathematically:
Space is definitely not an empty void and it is definitely not just a relationship between matter. We know this because we have seen space do things that fit neither of those ideas. We have observed space bend and ripple and expand.”[iv]
Space bends and ripples and expands in relation to what? To a nonphysical mathematical grid. The heavens themselves are a medium for profound creative meaning. That’s why so many scientists, such as German Astronomer Johannes Kepler (1571-1630), a friend of Galileo’s, referred to the heavens as a book.
I was merely thinking God’s thoughts after him. Since we astronomers are priests of the highest God in regard to the book of nature, it benefits us to be thoughtful, not of the glory of our minds, but rather, above all else, of the glory of God.[v]
Kepler believed God wanted us to study and understand. “Those laws [of nature] are within the grasp of the human mind; God wanted us to recognize them by creating us after his own image so that we could share in his own thoughts.”[vi]
Similarly, Anglo-Irish chemist Robert Boyle (1627-1691), considered to be one of the founders of the experimental scientific method, found nature to be a source of divine revelation. Perhaps his most famous quote is “Nature abhors a vacuum.” He discovered what chemists call Boyle’s law, which governs the inversely proportional relationship between the volume and pressure of gasses. He looked at chemistry and saw a book available to be translated:
And when with excellent Microscopes I discern in otherwise invisible Objects the Inimitable Subtlety of Nature’s Curious Workmanship; And when, in a word, by the help of Anatomicall Knives, and the light of Chymicall Furnaces, I study the Book of Nature, and consult the Glosses of Aristotle, Epicurus, Paracelsus, Harvey, Helmont, and other learn’d Expositors of that instructive Volumne; I find my self oftentimes reduc’d to exclaim with the Psalmist, How manifold are thy works, O Lord? In wisdom hast thou made them all.[vii]
Like Kepler, he believed that God inspired and invited people to see his work in creation. “If the omniscient author of nature knew that the study of his works tends to make men disbelieve his Being or Attributes, he would not have given them so many invitations to study and contemplate Nature.”[viii]
The more scientists have discovered, the deeper and more profound the mathematical discoveries have come. As one of the most significant physicists of the twentieth century, Englishman Paul Durac, put it:
It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better.[ix]
Even scientists who do not attribute scientific explanations to God still talk about it as language. As Richard Feynman, another one of the greatest physicists of the twentieth century, put it:
To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature … If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.[x]
As far as physicists can tell, the physical laws and constants of nature are uniform across the universe. If they were not uniform, then perhaps the explanations could be called descriptive of nature rather than prescriptive. That is to say that instead of being governing laws they could simply be called generalized observations. Many scientists have tried to find exceptions to this uniformity. But as Richard Muller, professor of physics at the University of California, Berkley, explains, they have always failed.
The equations that we have in physics today—all those that are part of the standard physics, the ones that have been verified experimentally—have the property that they work everywhere. Some people think this is amazing enough that they spend their careers looking for exceptions. They look at things that are very far away, such as distant galaxies or quasars, hoping to find that the laws of physics are a little bit different. So far, no such luck.[xi]
Implications aside, scientists take it for granted that wherever they look in the cosmos, whether they look through a telescope or through a microscope, they will eventually rational explanations. Consider, for example, the following chart.
The point is that all this information that we discover in nature is just as objective and useful as a block of iron ore. It takes rational creative minds at least ten years of study before they can begin to comprehend some of this information. Yet it is simply there, governing the universe, available to be comprehended. Astronomer Royal Martin Rees wrote a book, titled Just Six Numbers, about how the entire universe depends for its existence upon being finely tuned to six numbers. For example, the number called Ω (omega), represents the amount of matter in the universe. Omega equals 1, and Rees says if it were greater than 1 then the universe would have collapsed long ago, but if it were less than 1 no galaxies would have formed.
A few basic physical laws set the ‘rules’; our emergence from a simple Big Bang was sensitive to six ‘cosmic numbers’. Had these numbers not be ‘well tuned’, the gradual unfolding of layer upon layer of complexity would have been quenched.[xii]
Are we saying that rationality governs the universe? Yes, that’s exactly what we’re saying. The one and only reason to deny this conclusion is due to an aversion to spirituality. Yet truth bombards us gently from every direction. “Why do you run around looking for the truth?” asked Laozi. “Be still, and there it is—in the mountain, in the pine, in yourself. Do you imagine the universe is agitated? Go into the desert at night and look at the stars. The practice should answer the question.”
A Single Book With Many Chapters
These mathematical patterns that we discover in nature form a single, all-or-nothing phenomenon. Just as you can’t have only one species of fish, or only one word in a language, or only one baseball player on a team, so also you can’t have just one branch of mathematics or logic or rationality. It is all bound together as a single phenomenon and no one part of it can be taken out of context and isolated. Although we may only study one branch of it at a time, we know that all the branches grow from the same single source. In fact, this is a mathematical fact, proved by German mathematician Georg Canto in 1878.
Cantor used a simple proof to show that, for example, a 2-cm line segment has just as many points on it (just as much mathematical information) as there are points inside a cube the size of the earth—that they both have the same infinite amount of data. “I see it, but I don’t believe it!” he wrote to his friend, Richard Dedekind. On the one hand, it was dazzling to comprehend. On the other hand, it is something that we are all very familiar with. For it is the same reason that children can take in scads of electromagnetic data through their eyes—quadrillions of photons—and then learn to translate it all into short little words like red and blue and yellow. Those small words contain all of that information. It’s the same reason that Isaac Newton was able to compress massive amounts of data into a short sentence like “Force equals mass times acceleration.”
But the implications of Cantor’s discovery go far deeper. To understand them, consider this riddle that uses a geometric figure called the Möbius Strip. In Avengers: Endgame Tony Stark, a.k.a. Ironman, used a Möbius Strip to unlock the secret to time travel. To make one, take a regular piece of printer paper and cut off a one-inch piece lengthwise, so that you have a one by eight-and-a-half ribbon of paper. If you tape the ends together then you have a loop. But if you twist one end of it half a turn before taping it together, then you have a Möbius Strip. Now here is the riddle: How many sides does it have—one or two?
You can draw one line continuously along the entire ribbon without ever having to pick up your pencil. But how can it possible have just one side? Consider not just the paper object but the idea that it represents. Where is the missing side?
It’s a trick question. We’re seeing the “sides” of the Möbius Strip from the wrong context. We asked a question about a two-dimensional representation (the ribbon/plane of the Möbius Strip) using the language and vantage point of three-dimensional space. That does not compute. That is to say that, strictly speaking, the word “side” only applies to three-dimensional objects; two-dimensional concepts do not have any sides at all. For example, if you stack a thousand sheets of paper on each other then you’ll get a 3-D block of paper with six sides. However, if you “stacked” a thousand mathematical planes together, then you would still only have a 2 dimensional plane that doesn’t have any sides at all. But this can be as easy to lose track of as rebates and “tips”.
Now let’s return to Cantor’s discovery about how a one-dimensional line segment has the same number of mathematical points on it as does a two-dimensional plane or a 3-dimensional object. That means that all the information is there, available to be discovered and read, regardless of whether we discover it and read it. It means “book” is a singular phenomenon.
If you don’t believe me, try this out: think again of a two-dimensional mathematical plane. We imagine ourselves looking down on the plane, or looking at it from an angle, or perhaps passing through the plane to look at it from below. Regardless, we can only imagine it from the vantage point of three-dimensional space. Nobody can actually think in two dimensions. We can do a lot of two-dimensional math, but our vantage point will always be 3-D. There is, in fact, an enormous branch of mathematics called non-Euclidean geometry in which two-dimensional math uses three-dimensional space. The Möbius Strip is an example.
Now imagine a one-dimensional line, shooting from infinity to infinity. We can imagine orbiting around the line, or moving along it. Regardless, we can still only imagine it from the vantage point of three-dimensional space. Nobody can actually think in one dimension, even when doing arithmetic.
Thus even though each of the three dimensions is perfectly unique and coherent in and of itself, the three only exist—even in our minds—as a single phenomenon. Just as we recognized that even linguistic vocabulary is only coherent from a multi-dimensional context (such as when we considered the example of the words freedom, knowledge, and determinism), so also mathematics only exists in three dimensions. We might also compare the three dimensions to how white light is composed of red light, green, light, and blue light. (In case you don’t know your optic physics: although blue and yellow paints mix to make green, when it comes to light waves it is green and red light that mix to make yellow. If something is painted yellow, then it will absorb blue light while reflecting red and green light back into our eyes.) Children cannot learn the meaning of the word blue unless they also learn the meaning of red, yellow, white, green, etc.
Furthermore, even though each of the three dimensions is unique and coherent in and of itself, each of the three contains the whole. All mathematical truth can be digitized and translated, for example, into a one-dimensional linear sequence. That is the same reason that a three-dimensional human body can be translated into a linear sequence of about three billion digits (about 262,000 pages in a book) on a DNA molecule.[xiii] And it is why, again, a child can learn to translate massive amounts of three-dimensional light data into small digital color words. And it is why the interactions of the sun and the planets in our solar system can be translated into elegant, linear sentences like “Energy equals mass times the speed of light squared.”
A Dynamic Book
We are only scratching the surface here, but suffice it to say that none of the information that we find in nature is not static. Just as any book is only coherent when the words are read in sequence over a period of time, so also any mathematical or scientific equation reflects movement through time. And so, as Kepler and Boyle said, the book of nature is always inviting and inspiring us. Even its unending constants (such as 3.14159…) confront us with eternity.
Now we have absolutely no idea what time actually is. Einstein showed that it is analogous to the three mathematical dimensions since time passes more slowly for an object moving faster than another object. But calling it a fourth dimension is still just an analogy—a very effective and useful analogy, comparable to how money is a very effective analogy. After all, you cannot see, hear, feel, taste, or smell the value of money. You can only believe by faith what it represents. (Economists usually use the term confidence rather than faith.) Similarly, you can only believe that time is passing. But what exactly is it?
As bizarre as that question may sound, many scientists argue that time is not, in fact, passing and that the impression of an objective flow time illusory. Instead, they say that time is a subjective experience that needs constant adjusting and clarifying. Why do they say this? Well just as it appears to take time for you to read this sentence, so also it takes approximately eight minutes and twenty seconds for light from the sun to reach us and it takes millions of years for light from the stars to reach us. So if you go out at night and stare at the stars, that means you are looking at ancient history. Therefore, there is no such thing as a cosmic, universal now. Therefore, if there is no present, so also there is no past or future. Whatever time is, it does not change and flow. Instead, we are effectively fabricating that idea in order to keep track of things.
One of the leading proponents of this view is Italian theoretical physicist Carlo Rovelli, the founder of what is called the Loop Quantum Gravity theory of physics and the director of the quantum gravity research group at the Centre for Theoretical Physics at Aix-Marseille University. The author of some very popular books, including Reality Is Not What It Seems and The Order of Time, he says that since physicists do not necessarily need a flow of time in their equations, it is likely a subjective phenomenon. Since the illusion of time flow must have emerged in evolution to give us a useful though artificial and blurred perception of the world, he tosses the question to neuroscientists to explain,
I suspect that what we call the “flowing” of time has to be understood by studying the structure of our brain rather than by studying physics: evolution has shaped our brain into a machine that feeds off memory in order to anticipate the future. This is what we are listening to when we listen to the passing of time. Understanding the “flowing” of time is therefore something that may pertain to neuroscience more than to fundamental physics. Searching for the explanation of the feeling of flow in physics might be a mistake.[xiv]
Although it may be disorienting and perhaps even depressing to hear that time is not a fundamental, objective part of life, Rovelli says that such discouragement is easily remedied since the illusion of the flow of time is merely a matter of brain chemistry. “It only takes a few micrograms of LSD to expand our experience of time to an epic and magical scale.”[xv] In fact, he credits the psychedelic drug with inspiring his interest in physics in general and in the nature of time in particular:
It was an extraordinarily strong experience that touched me also intellectually. Among the strange phenomena was the sense of time stopping. Things were happening in my mind but the clock was not going ahead; the flow of time was not passing any more. It was a total subversion of the structure of reality… And I thought: “Well, it’s a chemical that is changing things in my brain. But how do I know that the usual perception is right, and this is wrong? If these two ways of perceiving are so different, what does it mean that one is the correct one?”[xvi]
What does it mean to say that one is the correct one? Well, many scientists have concluded that it means there is a self-evident, objective reality for which they may not necessarily be able to author the explanation. They can know that some things are true even if they cannot necessarily explain them in the laboratory. By contrast, as Professor Muller explains, the only reason not to believe in the flow of time is because one wants to cling to the presuppositions of physicalism (a.k.a. materialism)—something that Einstein himself realized was futile.
Atheists mocked Einstein for drifting away from physics and developing a religious faith in his later years. But they never spoke to his concern that science could not address even these most essential aspects of the world: the flow of time and the meaning of now. Many scientists assume that something that cannot be probed by physics is not part of reality. Is that statement a testable claim, or a religious belief itself? Philosophers give this dogma the name physicalism. Is there a way to test, to prove, a faith that physics encompasses all? Or is such a belief expected for all physicists, just as being Christian has been an informal but effective requirement to qualify as a potential US president? If you challenge physicalism, do you risk being mocked for your drift toward religion, as Einstein was?[xvii]
Speaking of presidents and the flow of time, consider that it takes faith to believe, for example, that George Washington was the first president of the United States—that he actually lived and worked in an earlier time in a place called The United States of America. After all, we cannot actually see him being president. Instead, we can only believe what is written in the historical record. (If that sounds like a silly observation, consider how millions of people doubt the more recent, more detailed historical record of the Holocaust.) Of course, we can also consider the circumstantial evidence for George Washington’s presidency, for there is a direct link between the processes that installed him in office to the processes that installed President Donald Trump.
And just as we believe narratives in American history based upon evidence, so also we believe narratives in natural history based upon evidence. For if science has taught us anything at all over the past 2000 years, it has taught us to have faith that if we study nature carefully enough we will always be able to discover and decipher rational explanations. We will discover information that narrates the unfolding of natural history. And we will inevitably ask who is the author of it all.
[i] As stated by William H. Hobbs, “The Making of Scientific Theories,” Address of the president of Michigan Academy of Science at the Annual Meeting, Ann Arbor (28 Mar 1917) in Science (11 May 1917), N.S. 45, No. 1167, 443.
[ii] Galileo Galilei, The Assayer, (1623).
[iii] Martin Rees, Just Six Numbers: The Deep Forces That Shape the Universe (New York: Basic Books, 2000), 1.
[iv] Jorge Cham and Daniel Whiteson, We Have No Idea (New York: Riverhead Books, 2017), 98.
[vi] Letter (9/10 Apr 1599) to the Bavarian chancellor Herwart von Hohenburg. Collected in Carola Baumgardt and Jamie Callan, Johannes Kepler Life and Letters (1953), 50.
[vii] Robert Boyle, Some Motives and Incentives to the Love of God (1659), 56-7.
[viii] Robert Boyle, “Some considerations touching the usefulness of experimental philosophy” (1663). Quoted in Peter Gay, The Enlightenment (1977), 140.
[ix] Paul Dirac, “The Evolution of the Physicist’s Picture of Nature”. (May 1963). Scientific American. Retrieved 4 April 2013.
[x] Richard Feynman, The Character of Physical Law, Modern Library; Modern Library ed. 1994, chap. 2.
[xi] Richard A. Muller, Now: The Physics of Time. (New York: W. W. Norton & Company, 2016). Kindle Locations 4653-4657.
[xii] Martin Rees, Just Six Numbers: The Deep Forces That Shape the Universe (New York: Basic Books, 2000), 178-179.
[xiv] Carlo Rovelli, “On the Nature of Time”, Financial Times, April 20, 2018. https://www.ft.com/content/ce6ef7b8-429a-11e8-93cf-67ac3a6482fd
[xv] Simon Carnell and Erica Segre, “Interview with Carlo Rovelli”, The Guardian, April 14, 2018. https://www.theguardian.com/books/2018/apr/14/elastic-concept-order-of-time-carlo-rovelli
[xvi] Charlotte Higgins, “’There is no such thing as past or future’: physicist Carlo Rovelli on changing how we think about time”, The Guardian (April 14, 2018). https://www.theguardian.com/books/2018/apr/14/carlo-rovelli-exploding-commonsense-notions-order-of-time-interview
[xvii] Richard A. Muller, Now: The Physics of Time. (W. W. Norton & Company, 2016) Kindle Edition.