This Sentence is False

Rev. Josh Mason Pawelek

“Deep in the human unconscious is a pervasive need for a logical universe that makes sense,” said the science fiction writer Frank Herbert.[1] This is likely not an earth-shattering revelation to any of you. Herbert is not alone in making this observation. A close look at the history of both science and religion reveals at their cores a common, profound human longing to make sense of life, of the world, of the universe, of all existence. I detect this longing at the heart of those words we said earlier from Nicaraguan priest Ernesto Cardinal—his proclamation of a harmonious universe, a unity behind apparent multiplicity.[2]  I detect this longing at the heart of our fourth Unitarian Universalist principle, the free and responsible search for truth and meaning. I detect this longing at the heart of Religious Humanism which has been a central identity for so many Unitarians and Universalists over the past century. For me, this longing—this pervasive need, as Herbert calls it—is at the heart of what makes us human.

Scientists John Casti and Werner DePauli, in their biography of the twentieth century European logician, Kurt Gödel, write, “Humans have always hungered for a certain knowledge, the kind that transcends millennia.”[3] They, too, are referring to the human longing for a logical universe that makes sense. Gödel’s incompleteness theorem says something about this. But before I offer some muddled Sunday morning musings about this, I want to remind you whose idea it was that I preach on Gödel. For the eighth year in a row, Fred Sawyer purchased a sermon at last year’s goods and services auction. He asked me to preach on the significance of Gödel’s theorem for us. This theorem goes far beyond anything Fred has suggested before in terms of complexity. I certainly appreciate and enjoy the challenge, but I confess the math is utterly beyond me. (I take some comfort knowing it’s beyond most mathematicians.) Hopefully I will convey it well. And as always, I will be offering more sermons at this year’s goods and services auction, Saturday evening, Febraury 11th. Tickets on sale now. Please come, please bid!

Kurt Gödel (1925)

What is the path to the knowledge that would enable us to make logical sense of the universe? And how can we be sure such knowledge is true? Casti and DePauli write, “we most assuredly can’t find that kind of knowledge in the natural sciences where theories even as fundamental as Newton’s laws of mechanics can be overthrown by relativity theory, which itself may be cast in doubt by observations yet to come. Thus it is always to mathematics, especially the realm of pure numbers that we turn for the kind of certainty that we can really count on, if you’ll pardon the poor pun. In this domain, the truth-generating mechanism we employ is the process of logical deduction bequeathed to us by Aristotle.”[4]

Aristotelian logic begins with a set of assumptions or axioms we take to be true without proof. From those axioms we infer certain rules; with those rules we deduce further truths. For example, axiom: all German Shepherds are dogs. Axiom: fluffy is a German Shepherd. Rule: If all German Shepherds are dogs, and if Fluffy is a German Shepherd, then Fluffy is also a dog. Sounds straightforward, but there’s a problem. (I love it when there’s a problem.) When one digs down deep into the rules of any mathematical system (arithmetic, geometry, calculus, set theory) one is likely to find contradictions—paradoxes—which suggest that maybe the axioms we first accepted as true aren’t entirely true. Paradoxes defy the system’s rules. They are statements that are both true and false. Somehow, Fluffy is both a dog and not a dog. It shouldn’t be possible. A flaw lurks somewhere in the foundation of our knowledge. Such paradoxes are the mathematical equivalents of the statement, “this sentence is false,” which is known as the Epimenides or Liar’s Paradox. Let your mind ponder this for a few moments. This sentence is false.

If it’s false, then it’s actually true … which means by its own definition it’s false … but wait! Isn’t that what it says? This sentence is false? So it’s true … which means it’s false. And so on. It’s a paradox. It can’t be resolved using the system’s rules. Another example is the Barber Paradox. The village barber shaves all those who do not shave themselves. If that’s true, then who shaves the barber? If the barber shaves himself, then he doesn’t shave himself, because he shaves all those who do not shave themselves. But if he doesn’t shave himself, then he shaves himself, because he shaves all those who do not shave themselves.[5] It cannot be resolved using the system’s rules.

A (hopefully) fun mathematical example comes from the twentieth century British logician and philosopher, Bertrand Russell: “The set of all sets that are not members of themselves.” Consider this question: Is the set of all sets that are not members of themselves a member of itself? The philosopher and novelist Rebecca Goldstein writes, “if the set of all sets that aren’t members of themselves is a member of itself, then it’s not a member of itself, since it contains only sets that aren’t members of themselves. And, if it’s not a member of itself, then it is a member of itself, since it contains all the sets that aren’t members of themselves. So it’s a member of itself if and only if it’s not a member of itself.” To which she reacts with two sharp words: “Not good.” Why not good? “Paradoxes,” she says, “have often been found lurking about in the deepest places of thought. Their presence is often a signal (like the canary dying?) that we have managed, sometimes unwittingly, to stumble on a deep and problematic place, a fissure in the foundations.”[6] Why not good? Because they don’t make sense, and we humans long for a logical universe that makes sense.

Throughout the late 19th and early 20th centuries mathematicians and philosophers tried to create mathematical systems completely free from paradox. The holy grail of such efforts was known as a formal system. I won’t get into the details of formal systems, because I’m not sure I can explain them and keep you awake at the same time. Suffice it to say, those seeking this holy grail believed paradoxes existed in mathematical systems because the numbers and words that made up those systems had certain intrinsic meanings. Paradoxes, they argued, arose from those meanings. If you could drain all meaning from the system you could get rid of paradoxes. Formal systems attempt to do just that. To each of the meaningful numbers, words, axioms and theorems in, say arithmetic, they assign a meaningless symbol. Get rid of meaning, get rid of paradox.[7]

Related to this, remember in logical deduction we start out with axioms we accept as true without proof.  If we can’t prove them, then we must admit we’ve arrived at them by some other means: intuition. They are intuitively true. Yet, Goldstein reminds us, intuitions “are a tricky business…. An intuition is supposed to be something that we just know, in and of itself, not on the basis of knowing something else…. But not all … intuitions are genuine… and how is one to tell when one is in possession of the genuine article? Murky motivations … not only abound but also tend to hide themselves…. You might think that in mathematics … murky motives for beliefs are at a minimum. Still, even in mathematics we can get suckered. Accidental features can insinuate themselves into our most pristine mathematical reasoning, presenting us with propositions that seem intuitively obvious when they are not obvious at all—maybe not even true at all.”[8] Intuitions, said the formalists, also lead to paradox. So, a formal mathematical system—which drains all the meaning out of the numbers and words—also, in theory, removes intuition. Without intuition, without meaning, presumably those pesky paradoxes disappear. A formal system would finally give us that logical universe that makes sense, that knowledge transcending millennia, that hidden r half of Lir’s plan for creation,[9] that unity behind apparent multiplicity.

The faith that such a formal system could be established was widespread in early twentieth-century Europe. It seemed as if a logical, sensible universe was within reach. On September 30, 1930, at a symposium in Konigsberg, Germany Kurt Gödel—at 25 years old—announced his incompleteness theorem. From what I’ve read, nobody was paying attention. It was the last day of the symposium; people were tired and ready to leave. Eventually his theorem was published, became widely accepted, and effectively ended the search for math’s holy grail.

Gödel’s theorem says this: “For every consistent formalization of arithmetic, there exist arithmetic truths that are not provable within that formal system.”[10] Casti and DePauli write, “What Gödel discovered is that even though there exist true relationships among pure numbers, the methods of deductive logic are just too weak for us to be able to prove all such facts. In other words, truth is simply bigger than proof.”[11] In every system there are certain truths—we can intuit them—but we cannot prove they are true using the system’s rules. Therefore our mathematical systems are inherently incomplete. Our knowledge—in terms of what we can prove—will forever be incomplete. The mathematical holy grail does not exist. “Deep in the human unconscious is a pervasive need for a logical universe that makes sense,” writes Frank Herbert, “but,” he continues, “the real universe is always one step beyond logic.”[12]

What Gödel did is fascinating, innovative, thrilling, a testament to his genius, and even funny. When the theorem was first published

Gödel and Einstein were close friends at Princeton

many called it a trick. They called him a conjurer, a magician. But today the incompleteness theorem is regarded as the most important discovery in mathematics since Aristotle. Gödel presented a formal system modeled, I believe, after the system—known as a type system—established by Bertrand Russell and Alfred North Whitehead in their three volume work Principia Mathematica. He re-coded that system by assigning a special number to each meaningless symbol in the formal system. That’s the part I don’t understand. These are astronomically huge numbers which came to be known as Gödel numbers. Using these numbers he then created a statement similar to “This sentence is false.” He created “This sentence is not provable.” And then he proved it. Hear this: He proved the sentence is not provable. There’s no paradox here. We don’t get caught up in an endless stream of provable, not provable, provable, not provable. He proved it’s not provable. The formal system worked. No paradox. But watch: he proved the sentence is not provable, which means it’s true. Within this system there is a statement that is unprovable, but also true. There are truths we cannot prove. It turns out in any mathematical system (as long as it is consistent) there are unprovable truths. All mathematical systems are incomplete. There are truths that reside beyond proof.

What significance might this hold for us? Part of me that wants to throw up my hands and scream, “I have no idea!” Another part of me needs to remind us Gödel’s theorem is not religion; it’s not theology, spirituality or ethics. It’s cold, hard math and any attempt to draw a spiritual conclusion from it is risky. Gödel once wrote to his mother that “sooner or later my proof will be made useful for religion, since that is doubtless also justified in a certain sense.”[13] While I can find no indication of what he meant by that, he did at one point attempt to prove the existence of God and the afterlife. (I’m not impressed with his theology, which is quite distinct from the incompleteness theorem.) I also wouldn’t be surprised if some more traditional religious thinkers might be tempted to find proof for God in the incompleteness theorem. That thing we can’t prove but we know is true beyond the limits of our mathematical systems, beyond the limits of human knowing? It might look like God to some. But I don’t think the numbers are saying that.

What I take from my brief study of Gödel is this: First, if our mathematical systems and all systems derived from them are incomplete, then we ought to be skeptical of any religious, ideological, political or social claim to completeness. Human motives are often murky. In response to any world-view we ought to remain open to the possibility of truths residing beyond its claims. We ought to accept and embrace the mystery at the edges and perhaps at the heart of any world-view. We ought to align ourselves with the old liberal religious axiom, “revelation is not sealed.” As we sang, “Creative love, our thanks we give that this our world is incomplete.”[14]

Second, Gödel’s theorem does not signal the end of reason and logic. Rather, it was a triumph of reason and logic. It was a triumph of the human mind and a testament to the value and necessity of reason and logic in all areas of our lives including our spiritual lives.

Finally, the incompleteness theorem also confirms that reason and logic, while essential, are not the only path to truth. There are truths they cannot prove. How do we access these truths? It seems to me we do so through intuition, through poetry, art, dance, exertion, prayer, meditation, silence. We access unprovable truths not only through the mind, but through the body, the heart, the spirit. All these ways of searching for truth are necessary if we are to come to the knowledge we long for, if we are to meet that pervasive need. We’ll never fully know a logical universe, but if we learn to trust our intuitions and search for truth in all these ways, maybe—just maybe—we’ll come to know a universe that makes sense nevertheless. Perhaps that is the ultimate paradox, a universe that makes sense, yet its deepest truths lie beyond reason and logic.

Amen and blessed be.

[1] Herbert, Frank, Dune (New York: Berkley Books, 1965) p. 373.

[2]Cardinal, Ernesto, “The Music of the Spheres,” Singing the Living Tradition (Boston: Beacon Press and the UUA, 1993) #532.

[3]Casti, John L. and DePauli, Werner, Gödel: A Life of Logic (Cambridge, MA: Perseus Publishing, 2000) p. 3.

[4]Casti and DePauli, Gödel, pp. 3-4.

[5]Casti and DePauli, Gödel, p. 24.

[6]Goldstein, Rebecca, Incompleteness: The Proof and Paradox of Kurt Gödel (New York: W.W. Norton & Co., 2005) p. 91.

[7] Math meets philosophy here. What makes a symbol meaningful? What makes a symbol meaningless? I’m not sure.

[8] Goldstein, Incompleteness, pp. 122-123.

[9] This is a reference to the Composer Henry Cowell’s  “Voice of Lir.” Lir of the half tongue was the father of the gods, and of the universe.  When he gave the orders for creation, the gods who executed his commands understood but half of what he said, owing to his having only half a tongue; with the result that for everything that has been created there is an unexpressed and concealed counterpart, which is the other half of Lir’s plan of creation. See:

[10]Casti and DePauli, Gödel, p. 50.

[11]Casti and DePauli, Gödel, pp. 4-5.

[12] Herbert, Dune, p. 373.

[13]Goldstein, Incompleteness, p. 192.

[14] Hyde, William DeWitt, “Creative Love, Our Thanks We Give,” Singing the Living Tradition (Boston: Beacon Press and the UUA, 1993) #289.