Does rationality saturate the cosmos?
A very simple, effective way to define science is to say that it is the study of patterns in nature and society. 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 side matches the other.
Everywhere we look in nature we find similar such rational, creative patterns that we can translate into English, etc. 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 order within nature. 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.”[i]
Such patterns are just as objective and useful as a chunk of iron ore.
These Patterns Are Objective and Useful
Galileo Galilei, the father of modern science, discovered that the distances traveled by falling bodies are proportional to the succession of odd numbers. As he put it, such useful information saturates the universe:
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.” (The Assayer, published in 1623)
Or as Richard Feynman, 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.[ii]
Or as Physicist 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.[iii]
Notice how easily Dirac said it seems to point to a Cosmic Mathematician. The scientific establishment is acutely aware of this. Indeed, they have one and only one reason to question the objectivity of mathematics: it completely defies their presupposition of materialism. As mathematician Reuben Hersh famously put it:
Most writers on the subject seem to agree that the typical ‘working mathematician’ is a Platonist on weekdays and a formalist on Sundays. That is, when he is doing mathematics, he is convinced that he is dealing with an objective reality whose properties he is attempting to determine. But then, when challenged to give a philosophical account of this reality, he finds it easiest to pretend that he does not believe in it after all.[iv]
Twenty years later Hersh found the duality persisted but said, “It just doesn’t work, to be a Platonist in mathematics and a materialist empiricist in physical science.”[v] Together with fellow mathematician Philip Davis, they wrote, “The activity of mathematical research forces a recognition of the objectivity of mathematical truth.”[vi] They explored objections to this position and then concluded:
Mathematics is an objective reality that is neither subjective nor physical. It is an ideal (i.e., nonphysical) reality that is objective (external to the consciousness of anyone person). In fact, the example of mathematics is the strongest, most convincing proof of the existence of such an ideal reality.[vii]
As far as physicists can tell, the physical laws and constants of nature are uniform across the universe. Many have tried and failed to find exceptions.
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.[viii]
Non-materialistic 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 discover patterns. Consider, for example, the following chart.
Now we used computers to author many of these sentences that we could not otherwise have authored. But that is no different from using a telescope to see things that we could not otherwise see or using a crane to lift things that we could not otherwise lift. The point is that objective information is everywhere. Just as outer space conveys intelligent sentences such as E=MC2 and F=MA, so also your average rock will translate into volumes of rational understanding about geology, chemistry, and subatomic physics.
And of course, this also goes without saying for living things—again, so long as we listen rather than dictate. For many years, biologists thought that much of our DNA was meaningless junk because that’s how it appeared at first glance—a lot of random patterns. But they continued to listen carefully and realized that the way a single strand of DNA is folded matters tremendously. They discovered that just as the dents on a DVD carry several layers of meaning—a binary layer, a cinematic layer, a linguistic layer (i.e. a movie’s dialogue), etc.—so also the nucleic acid bumps on a DNA molecule carry astonishingly deep layers of meaning, revealing patterns astronomically more complex than they first thought. And so they press on with the quest to understand and then translate them so that students can have fun reading about it all in their textbooks.
The point is that all this information that we discover in nature is just as objective and useful as a block of iron ore. Engineers don’t create the metal in the tools that we use; instead, they discover the metal ore and then use it creatively. Likewise, they don’t create the math we use to design things; instead, we discover the math and then use it creatively.
In other words, the information that they use is objective. It is not something that only exists in our heads. But unlike our physical tools—cranes and tractors, etc.—numbers and equations have no tangible traits at all. As with any and all information, they are simply immaterial words and sentences, pure meaning. Hammers and tractors and protractors have lots of physical qualities; circles and triangles and integrals do not. We can feel a hammer as we use it to build, but we cannot feel a trigonometric function as we use it to create the plans for a building. We can measure the force of a tractor as we use it to dig the foundation of a house, but we cannot measure any physical qualities in the trigonometric equations that we use it to build that same house.
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 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.[ix]
The only reason to deny the existence of such information is due to an aversion to spirituality. Yet rational, creative sentences saturate us. “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.”
These Patterns Are an All-or-Nothing Phenomenon
These mathematical patterns that we discover in nature form an all-or-nothing phenomenon. (Now calling patterns mathematical is redundant, like calling rainbows colorful. So I will just continue to refer to patterns.) Just as you can’t have only one species of fish, or one word in a language, or one baseball player on a team, so also you can’t have just one branch of mathematics.
In fact, this is a mathematical fact, proved by German mathematician Georg Cantor in 1878. He used a simple proof to show that, for example, a 2-cm line segment has just as many points on it as are inside a cube the size of our solar system. “I see it, but I don’t believe it!” he wrote to a fellow mathematician, Richard Dedekind.
To understand the implications of Cantor’s discovery, 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 possibly have just one side? Consider not just the paper object but the idea that it represents.
What happened? 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 of each other then you’ll get a 3-D block of paper. However, if you “stacked” a thousand mathematical planes together, then you would still only have a 2-dimensional plane. Similarly, if you “stacked” a thousand 2-D ribbons on top of each other (though you would have to do it before twisting them in 3-D space) you still would not have any sides or surfaces. But this can be easy to lose track of, and so the tricks a good mathematician can do with such things would make any Las Vegas magician drool.
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 you can take all the information in a 3-D cube and translate it digitally into 1-D information—into a linear series of 1’s and 0’s. We have already recognized that mathematical truth is an objective phenomenon—that it is there to be comprehended (even on Sundays) like a book on a shelf—and now we are confronted with the fact that this “book” is a singular phenomenon.
If, like Cantor himself, you have trouble believing it, try this out: imagine for a moment 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. Arithmetic will always be in the context of geometry, etc. Although each of the three dimensions is perfectly unique and coherent in and of itself, they only exist—even in our minds—as a singular reality. We might compare the three dimensions to how white is composed of red light, green, light, and blue light.
We’re just observing the facts here. Yet how do we know all these things?
How Do We Know that Math is “Real”?
How do we know that the math that we used to put men on the moon was just as “real” as the rocket ships that took them there?
In answering this question we need to distinguish between presuppositions and self-evident truths. Presuppositions are beliefs that we do not feel are necessary to prove. For example, a thousand years ago most people had a good reason to assume that the earth was the center of the universe. After all, it certainly appeared that way if they watched the sky. Who would have guessed that they were standing on a giant spinning ball? Surely most people didn’t even know that they had a presupposition about the matter, so it never would have occurred to them to look for an alternative. And if someone tried to explain the heliocentric system to them, then their arguments wouldn’t have just sounded wrong, but also goofy.
Another example is a major topic of this blog: the scientific establishment presupposes that our minds are matter-in-motion and that science only works materialistically. They insist these presuppositions are necessary not only because they assume it would be impossible to study how something nonphysical could influence something physical, but also because they assume that any conclusions to the contrary would contradict The Standard Model of the physical universe. As I show in the article titled “When Do Words Occur?” all of these presuppositions are flat out wrong.
Presuppositions such as geocentrism and materialism can be believed, doubted, proven or disproven. By contrast, self-evident truths are things that we are not even capable of doubting.
We can pretend to doubt self-evident truths, but we cannot truly doubt them. We could pretend to doubt that the sun will rise tomorrow or that other people have minds like our own or that we are not a dream in the mind of an iguana, but we cannot truly doubt such things. After all, if we doubted absolutely everything then the word doubt would, literally, lose all meaning. (So a person who truly doubted such things would need serious medical attention to see if they are clinically insane.) It would be like saying that there is only a left and no right or only an up and no down. The word doubt derives its meaning from those self-evident truths that we do not doubt. “I know that I’m breathing and I know that the sun will rise tomorrow, but I doubt she’ll go out with me.”
Mathematics is a self-evident truth. To doubt its existence would be no different from doubting that language exists, for math and language are one-and-the-same. (See the article titled “What Are Words?”) You would have to use language to express doubt about the existence of language.
So back to the question, how do we know that math is real even though it is immaterial? We know it by faith. Granted, it’s a rock-solid faith, dependable enough to provide the infrastructure for the global economy. But it is faith nonetheless. You cannot see, hear, taste, touch, or feel numbers. You can only believe they are coherent. It’s the same kind of faith required to believe that money has value. (Economists usually use the word confidence rather than faith in discussing the basis for any economy.) That is to say, for example, that you cannot see or feel or taste the value of a dollar bill. Instead, you can simply believe what is written on it (provided you speak English).
To use another example, it’s the same kind of faith required to believe that George Washington was the first president of the United States. After all, we cannot actually see him being president. Instead, we have to simply believe what is written in the historical record. (If that sounds silly, consider how millions of people doubt the massively detailed historical record of the Holocaust.) We can also consider the circumstantial evidence, for the process that installed our current president links directly to the process that installed Washington.
In like manner, we can believe not only that mathematics is real but also that it saturates every speck of the universe. Indeed, 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 eventually discover and decipher rational explanations and patterns.
[i] Martin Rees, Just Six Numbers: The Deep Forces That Shape the Universe (New York: Basic Books, 2000), 1.
[ii] Richard Feynman, The Character of Physical Law, Modern Library; Modern Library ed. 1994, chap. 2.
[iii] Paul Dirac, “The Evolution of the Physicist’s Picture of Nature”. (May 1963). Scientific American. Retrieved 4 April 2013.
[iv] Reuben Hersh, “Some Proposals for Reviving the Philosophy of Mathematics,” in Advances in Mathematics, Vol. 31, 1979. (pp. 31-50).
[v] Reuben Hersh, What is Mathematics, Really? (New York: Oxford University Press, 1997), 236.
[vi] Philip J. Davis and Reuben Hersh, The Mathematical Experience (Boston: Birkhäuser, 1981), 362.
[vii] Philip J. Davis and Reuben Hersh, The Mathematical Experience (Boston: Birkhäuser, 1981), 409.
[viii] Richard A. Muller, Now: The Physics of Time. (New York: W. W. Norton & Company, 2016). Kindle Locations 4653-4657.
[ix] Martin Rees, Just Six Numbers: The Deep Forces That Shape the Universe (New York: Basic Books, 2000), 178-179.