What is infinity?
Materialism 101, Pt. 3
Matthew Connally is a recent graduate of our World Journalism Institute mid-career course, but I first met him in 1992 when he was editor in chief of The Daily Texan, the student newspaper at The University of Texas at Austin. From there he moved on to earn a master’s degree at Trinity Evangelical Divinity School and become a pastor in Princeton and a campus chaplain at Princeton University. From 2012 to 2016 he was a teacher and principal in Nanjing, China, and since 2017 has been a pastor at a Houston-area Chinese church.
So let’s review: a Christian on a highly secularized campus newspaper, an evangelical at theologically liberal Princeton and in neo-Maoist China, and (as this essay shows) a critic of Darwinism. Matt is used to being in a minority, and by taking on Darwinism he’s cementing his position as a smart person who doesn’t believe what the smart set still believes—even though discoveries in recent decades about the complexity of cells, the fine-tuning of the universe, and the information coding in and around us have kicked the legs off materialism’s dining room tables.
This is the third in an occasional series of essays for our Saturday Series. In the last installment, he asked: Who is the Author of scientific truth? In this one, he follows up by exploring a concept that is both foundational to mathematics and devastating for materialism: the infinite.—Marvin Olasky
If we did not have infinity, we would not have calculus. And if we did not have calculus, we would not have smart phones or satellites or slushy machines or any modern technology at all. Suffice it to say that infinity is just as necessary a tool for the engineer as a wrench or a crane. Granted, the latter are tangible—you can hold a wrench in your hands or bang your head on a crane. But what about infinity?
It is intangible … immaterial. It is pure meaning.
This is not a philosophical statement, nor is it a presupposition. It is an undeniable fact. Infinity can never have any physical representation. For example, you can have three apples or 3 million apples, but you can never have infinite apples. Going the other direction, you can put 3.1 apples in your pie or, with a good laboratory scale, you can put 3.14 apples in your pie, but you can never put π apples in your pie because you cannot cut an apple with infinite precision.
Furthermore, there is no context for declaring that infinity is somehow less “real” than the iron in wrenches and cranes. For one thing, the meaning of the word real is just as abstract as the meaning of the word infinity, so before we ask whether the meaning of infinity is “real” we need to ask whether the meaning of real is “real”! And, more to the point, both are indispensable for modern technology. Just as engineers don’t create iron but rather discover it and then use it creatively to build wrenches and cranes, they don’t create mathematics but rather discover it and then use it creatively. Yet one set of tools is tangible while the other is intangible.
We are face to face with a nonphysical phenomenon.
So, how could the brain interact with or “know” about—much less create—something that cannot be seen, heard, felt, tasted, or smelled? This is not a philosophical question. Nor is the answer to it: The organ inside our skulls cannot do such a thing. Even if Darwinists wanted to believe that it could, they cannot even begin to articulate a theory—much less test a theory— as to how it could be possible. It’s like asking whether your colon can comprehend organic chemistry or whether your fingers can count to 10 or whether your smartphone can understand English or whether a dictionary knows what the word dictionary means. At the end of the day, the mystery is not what we do not know, but what we do know. It’s not that we do not know how the brain could possibly perceive infinity; it’s that we know, of course, it cannot.
It’s not that we do not know how the brain could possibly perceive infinity; it’s that we know, of course, it cannot.
Does that imply that the minds using our brains are just as immaterial as is infinity itself?
It really is that simple. This should be a slam dunk not only for spirituality, but for the knowledge of a divine Mathematician. In fact, the German mathematician who discovered set theory, Georg Cantor (1845-1918), saw a deep connection between mathematics and his Christian faith in God, whom he associated with the Absolute Infinite. He believed that God had entrusted him with an understanding of infinities so he could reveal them to the world. The evidence is so overwhelming that it almost seems unfair to the materialist. What is a good Darwinist to do? How in the world do they suppress this truth?
First, they often change the subject from talking about God and spirituality to talking about Plato and philosophy. After all, if you’re going to deny spirituality, which religion’s truth claims are you denying? It’s much easier to find a generic representation. That’s what Plato (428-348 BC) provided—a philosophy about a perfect immaterial world. So instead of saying, “I don’t believe in spirituality,” the materialist may say, “I’m not a Platonist … at least not an overt one.”
Next, they’ll turn on a nuclear-powered fog machine and start churning out all kinds of relentlessly esoteric ideas about formalism, intuitionism, and Hegelian idealism. They may quote guys like Immanuel Kant (whose name sounds familiar) and Henri Poincaré (no, never heard of him) and offer extrapolations of interpretations of speculations until they reach pathological levels of abstraction. Pretty soon you’re staring out the window, trying to remember whether you fed the dog and, oh dear, when was the last time you checked the air pressure in your tires!
Let’s just stick to the facts. Infinity is as indispensable for the mathematician as oxygen. “It appears to be a universal feature of the mathematics normally believed to underlie the workings of our physical universe that it has a fundamental dependence on the infinite,” writes Sir Roger Penrose, professor of mathematics at Oxford University.[i] It is impossible to argue that infinities don’t exist. It would be on par with pretending that you don’t believe that the sky is blue or that the sun will rise tomorrow. Cantor understood well why eternity haunted the materialists:
The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.[ii]
Alas, Darwinists simply cannot accept such unsophisticated conclusions. When push comes to shove, they’ll say, “Just shut up and calculate!” Some questions simply should not be asked.
But Cantor continued to ask and seek and knock with the zeal of a true pioneer. “In mathematics the art of asking questions is more valuable than solving problems,” he wrote in his 1867 doctoral thesis. He insisted that mathematicians were explorers of an objective reality, not just the describers of the physical world. And through his own explorations he discovered several types of infinities—countable infinities (such as 1, 2, 3, etc., or 5, 10, 15, etc.), uncountable infinities (they’re so dense that you never even make it from 1 to 2, much less to 3 or 4 or 5; uncountable means unlistable), geometric infinities, and infinities made of transfinite numbers. Since then, mathematicians have discovered many other infinities whose forms truly beggar the imagination.[iii]
Yet there were a couple of infinite sets for which, no matter how he tried, Cantor was unable to categorize their size. In 1878 he put forward what is called the Continuum Hypothesis, a proposition about the sizes of these infinite sets. Much to his dismay he never proved it, and his opponents said this failure justified all their skepticism. Their mockery drove Cantor into deep bouts of depression. He died in a sanatorium in 1918.
What can we really prove?
Then, in 1931 a young German mathematician named Kurt Gödel arrived on the scene. (He would become one of Einstein’s best friends.) He wrote two brief proofs that shook the foundations of both philosophy and mathematics. He showed that we can never actually decide whether any mathematical axiom is true or not.
Now that might almost sound like a self-contradictory statement: He mathematically proved that we cannot mathematically prove a statement to be true? Not quite: He mathematically proved that we cannot decide whether a statement is true. Our logic can be exhaustively complete or it can be provably true, but it cannot be both at the same time. There will always be statements which we know to be true even though we cannot prove them to be true. Our logic will always be incomplete. Communications engineer Perry Marshall explains it this way: “Anything you can draw a circle around cannot explain itself without referring to something outside the circle–something you have to assume but cannot prove.”[iv] In effect, when we know that mathematical axioms are true, we know it by faith.
The question “Where did mathematics come from?” is unanswerable. At least by us.
Now it’s not blind faith in arbitrary statements, but rather reasoned faith in self-evident facts. As Gödel famously put it, “I don’t believe in empirical science. I only believe in a priori truth.”[v] That doesn’t mean that empirical science is not useful. Rather, it means that empirical science ultimately rests upon objective, self-evident truths that we cannot wrap our minds around—that we will never, ever be able to wrap our minds around. For, at its foundation, mathematics is objective, infinitely complex information. As Galileo Galilei had intuitively realized 300 years earlier:
There are such profound secrets and such lofty conceptions that the night labors and the researches of hundreds and yet hundreds of the keenest minds, in investigations extending over thousands of years would not penetrate them, and the delight of the searching and finding endures forever.[vi]
Thus, although we can know axioms to be useful and consistent for all practical purposes, we do not get to be the authors of their absolute truthfulness. Today Gödel’s Incompleteness Theorems are just as mesmerizing as ever. They demonstrate that the question “Where did mathematics come from?” is unanswerable. At least by us.
What can we really know?
Okay, back to the Continuum Hypothesis. Cantor’s opponents had said his failure to prove the hypothesis revealed the lack of a rational foundation for all of his work on infinities and set theory. But Gödel showed the foundations of all of mathematics were likewise incomplete. He said Cantor’s theories were actually just as robust and meaningful as any other consistent branch of mathematics and that those who still believed a proof of the Continuum Hypothesis was necessary were in denial of the facts. To reject set theory because we cannot prove one axiom was not rational, for our descriptions of rationality and of nature will always be lacking.
Only someone who (like the Intuitionist) denies that the concepts and axioms of classical set theory have any meaning could be satisfied with such a solution, not someone who believes them to describe some well-determined reality. For in reality Cantor’s conjecture must be either true or false, and its undecidability from the axioms as known today can only mean that these axioms do not contain a complete description of reality.[vii]
Gödel’s proofs sent shockwaves across the world. Atheist philosopher Bertrand Russell had recently finished his massive Principia Mathematica, an ambitious attempt to replace all of religious belief and philosophical belief with mathematical logic. Russell spent several hundred pages laying the foundation for the proof that 1+1=2 because he wanted mankind to own rationality and be the author of axiomatic truth.
Gödel watched this herculean effort, figured out why it was futile, and in about 20 (very complex) pages proved that regardless of whether the statement 1+1=2 is true (and of course it is), we aren’t the ones who decide that it is true—no matter how badly we wish for it, no matter how hard we try. (The title of his paper was “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”.) We can know such things by reasoned faith, but we do not get to be the authors of them.
For that matter, we cannot be the authors of any such truth, only the believers, teachers, and preachers of it. Our math can be reliable enough to run a global economy or to put men on the moon, but we still do not control it any more than we control the tides. As Daniel Andrés Díaz-Pachón, research assistant professor at the University of Miami’s division of biostatistics, put it:
In the end, the most formal exercise in knowledge is an act of faith. The mathematician is forced to believe, absent all mathematical support, that what he is doing has any meaning whatsoever. The logician is forced to believe, absent all logical support, that what he is doing has any meaning whatsoever … Faith is the most fundamental of the mathematical tools.[viii]
So just as any mathematician or scientist must have faith that infinities are objective realities, Gödel showed that the same was true for all of mathematics: It is objective, not subjective. And again, that foundation is also undeniably immaterial.
So who authored it all?
Gödel described his own faith as “baptized Lutheran (but not member of any religious congregation)”.[ix] Although he seldom joined public worship, his wife, Adele, said, “Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning.”[x] He took some interest in apologetical arguments about God’s existence, but never published any. Perhaps he realized it was a moot point.
A moot point
Although this makes for a great story, we never really needed Gödel to prove what he proved. Our knowledge of the infinite reveals more than enough.
Still, what became of Cantor’s Continuum Hypothesis? In 1938 Gödel proved that it could not be disproven by standard methods. Then in 1962 another mathematician, Paul Cohen, proved that it could not be proven either! Gödel told Cohen his proof was “really a delight to read … Reading your proof had a similarly pleasant effect on me as seeing a really good play.”[xi]
Why insist on a world of make-believe? Remember that Darwinism is dependent upon the presuppositions of materialism.
Where does that leave us? The big takeaway is that everyone agrees we are exploring objective truth. And the immaterial nature of that truth confronts the scientific establishment with a catastrophic mystery—what Einstein called “the eternal mystery of the universe”—that they have to bury under philosophical fog. As Brown University mathematics professor Philip J. Davis and University of New Mexico mathematics professor Reuben Hersh explained in 1981, Darwinists have no other choice but to try to have it both ways:
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.[xii]
Why pretend? Why insist on a world of make-believe? Remember that Darwinism is dependent upon the presuppositions of materialism. There simply cannot be any immaterial/nonphysical phenomena in the universe, for that would open wide the door to spirituality, which is unacceptable. Furthermore, the rationality of mathematics begs for an Author. That’s also unacceptable. Now we cannot prove the existence of a divine Mathematician any more than we can prove the absolute veracity of the statement “1+1=2”. Any such attempt to do so would be irrational. Nevertheless, we can know by faith that both are true. As Díaz-Pachón put it:
In order for faith and reason to have a foundation, not merely from an epistemological viewpoint but also from an ontological one, there must be something that sustains it —a First Sustainer undergirding them all. There is no logic without a Logos. Faith’s only task is to accept that such a Logos does exist. The opposite is despair, meaninglessness.[xiii]
The materialist will argue that if we courageously embrace such despair, then we can give life meaning. As Russell put it, “Only within the scaffolding of these truths, only on the firm foundation of unyielding despair, can the soul’s habitation henceforth be safely built.”[xiv] As poetic as that might sound, the fact remains that the word meaningless is only coherent within the much broader context of objective meaning, and the word despair is only coherent within the much broader context of objective hope.
Thankfully, the reverse is not true for either meaning or hope, for evil is entirely parasitical. (This isn’t philosophy; it’s just semantics! But semantics can be just as breathtaking for the writer as it is for the mathematician.) And to the extent that we know there is both meaning and meaninglessness in life, we also know that we cannot be the authors of both.
[i] Roger Penrose, The Road to Reality (New York: Alfred A. Knopf, 2005), 357.
[ii] Georg Cantor, Gesammelte Abhandlungen [Collected Essays], eds. A. Fraenkel and E. Zermelo (Berlin: Springer-Verlag, 1932), 374. As quoted in Infinity and the Mind by Rudy Rucker.
[v] Kurt Gödel, Collected Works: Volume III: Publications 1938-1974, edited by S. Feferman et al (Oxford: Oxford University Press, 1995).
[vi] Galileo Galilei, 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.
[vii] Kurt Gödel, Collected Works: Volume II: Publications 1938-1974, edited by S. Feferman et al (Oxford: Oxford University Press, 2001), 181.
[ix] Hao Wang, Reflections on Kurt Gödel, (Mass: MIT Press, Cambridge, MA, 1987.
[x] Hao Wang, A Logical Journey: From Gödel to Philosophy. A Bradford Book, 1997. Print. p.316.
[xii] Philip J. Davis and Reuben Hersh, The Mathematical Experience (Boston: Birkhäuser, 1981), 362.
[xiv] Bertrand Russell, Mysticism and Logic and Other Essays (Heritage Books, Kindle Edition, 2019), Kindle Locations 897-899.
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