Achilles: “Well, the best way I know to explain it is to quote the words of another old Zen master, Kyōgen. Kyōgen said: ‘Zen is like a man hanging in a tree by his teeth over a precipice. His hands grasp no branch, his feet rest on no limb, and under the tree another person asks him: “Why did the Bodhidharma come to China from India?’ If the man in the tree does not answer, he fails; and if he does answer, he falls and loses his life. Now what shall he do?”
Tortoise: “That’s clear; he should give up Zen, and take up molecular biology.”
– Gödel, Escher, Bach: An Eternal Golden Braid (244 – 245)
In 1931, Kurt Gödel (an Austrian/American mathematician who had sweet glasses and looks kind of like Toby from The Office) published the groundbreaking paper “On Formally Undecidable Propositions of Principia Mathematica and Related Systems,” and effectively broke humanity’s understanding of mathematics with a proof of what has come to be known as ‘Gödel’s Incompleteness Theorem’. The proof which this paper presents is really complicated and counterintuitive (it relies on building paradoxes about mathematical statements using mathematical statements), and I know that my poor understanding of the proof wouldn’t do it justice, so I won’t attempt to explain the paper itself here. However, in his book Gödel, Escher, Bach: An Eternal Golden Braid, Douglas Hofstadter describes Gödel’s work in layman’s terms, and provides a concise definition of what Gödel’s Incompleteness Theorem actually proves. In describing the crux of Gödel’s theorem, Hofstadter writes: “All consistent axiomatic formulations of number theory include undecidable propositions” (17).
Now to me, that still sounds like utter gibberish, so let’s take Hofstadter’s summary and put it in even simpler terms: Gödel’s Incompleteness Theorem implies that there are mathematical statements which are true, but are not provable, where ‘provable’ just means that we can understand how a given statement was derived from the basic axioms of arithmetic. More concretely, this is equivalent to saying: “There are mathematical statements out there in the wild which are just as ‘true’ as 2 + 2 = 4 — but we cannot prove that these statements are true, given the basic axioms of arithmetic.”
And not only does this ‘flaw’ exist in our mathematics now, but this will always be the case, regardless of how good our great-great-great grandchildren (and/or Schwarzenegger robotic overlords) get to be at calculus or geometry. This ‘incompleteness’ is inherent in all sufficiently strong mathematical systems, and it always will be until the end of time.
Let’s take a second here — this is an absolute bonkers claim. It’s so counterintuitive and separated from how we unconsciously believe the world operates that it’s easy to skim over the implications this has. Our day-to-day life depends on math just working, completely devoid of mystery and ambiguity. We trust that when we balance our checkbooks and pay our bills and expect our sweet sweet tax return to come in the mail, the system we’re using to manipulate numbers and predict future events is complete (every true statement is provable) and consistent (every provable statement is true). Gödel’s theorem throws completeness out the window — it argues that it is impossible to have a consistent set of mathematical axioms which approximate number theory which is also complete (see Hofstadter’s summary again). We obviously don’t hit these mathematical oddities in daily life — if we did, simple everyday tasks would become much more difficult! But importantly, they do exist within the very same formal mathematical systems we depend on in almost every single moment of our lives. There is an inherent amount of mystery and contradiction and paradox embedded deeply within the incredibly useful (and ultimately essential) monotony of mathematics.
While I won’t dig deeply into the proof itself, I do want to give an overview of a particularly cool chunk of it which relies on a system called Gödel numbering. Hofstadter describes this portion of the proof as follows:
“The proof of Gödel’s Incompleteness Theorem hinges upon the writing of a self-referential mathematical statement, in the same way as the Epimenides paradox is a self-referential statement of language. But whereas it is very simple to talk about language in language, it is not at all easy to see how a statement about numbers can talk about itself” (17).
The Epimenides paradox is the classic self-referential statement: “Epimenides was a Cretan who made one immortal statement: ‘All Cretans are liars’” (17). See where the paradox comes in? Hofstadter also reformulates the paradox more bluntly: “This statement is false” (17).
So the key to Gödel’s Incompleteness Theorem, which again proves that there are mathematical statements which are entirely ‘true’ but also entirely un-derivable, depends on importing a similar self-referential ‘this statement is false’ declaration into number theory. The key to getting this self-referential paradox to show up is the nifty transformation provided by Gödel numbering. Hofstadter continues:
“In the Gödel Code, usually called ‘Gödel-numbering’, numbers are made to stand for symbols and sequences of symbols. That way, each statement of number theory, being a sequence of specialized symbols, acquires a Gödel number, something like a telephone number or a license plate, by which it can be referred to” (18).
So essentially, Gödel had the insight to map number-theoretical strings of characters (i.e. the string ‘2 + 2 = 4’) onto actual numbers (for example, maybe ‘2 + 2 = 4’ maps onto the number 23645293). Every mathematical string maps onto its own unique number, and ultimately, deriving mathematical statements is mapped directly onto performing actual arithmetic using Gödel numbers. Using this transformation, Gödel was able to create the following ‘true’ sentence in pure arithmetic, which is the straw that breaks the proverbial camel’s back (if math were a feisty humped ungulate that spits on people): “This statement of number theory does not have any proof in the system of Principia Mathematica” (18). Where Principia Mathematica is a formal system of mathematics which was published by Bertrand Russell and Alfred North Whitehead between 1910 and 1913.
Phew. Ok. That was a lot of information to dump out at once — I know this short bit of writing flies through a ton of material, and if you’re interested, I’ve linked a good introductory YouTube video (above) which provides some additional insight into how Gödel’s theorem works. Of course, if you’re really interested, Hofstadter’s book is fantastic. I highly recommend it. But you’re probably asking the same question I was while writing this — why in the world would this theorem be relevant for a Mockingbird post? What’s the payoff for reading a crappy description of some math paper from the 1930s, when many more important things happened in that decade, like the release of the Milkybar by Nestle? Or Clyde Tombaugh identifying Pluto, the single best almost-planet there ever was which, incidentally, is also the setting for the single best scene from The Magic School Bus of all time?
Working backwards, I think there’s something really profound about how Gödel numbering binds strings of characters to pure arithmetic. Again, Gödel used this transformation to map strings of characters which represent mathematical statements onto pure numbers, but if we loosen the mathematical rigor of Gödel’s original paper a bit, the same mapping would also apply to any string of characters. For example, if every character on my keyboard were mapped to a number, we could convert this entire essay into a single (very large) number which would represent the same information. What if we multiplied that number by two, and decoded it back into text? Would it say anything meaningful? Is there an arithmetic operation I could perform on my essay-number to convert it into, for example, the Gettysburg Address? The U.S. Constitution? The script of Nacho Libre?
Through Gödel numbering, a deep connection is shown to exist between between language and mathematics. There is a direct mapping between strings of text and numerical operations. In the creation account provided by Genesis, God “speaks” creation into existence, and creates order out of chaos through the development of physical laws which govern how the universe operates. Similar to our own language, the laws of physics can ultimately be boiled down to the manipulation and transformation of pure numbers, but rather than using the ‘Gödel Code’, Physics operates more directly through formulas and equations to ‘converse’ with God’s created order. There is a deep connection between our language and the mathematical dialect of the Creator, which undergirds everything in the universe, from the force with which my fingers strike this keyboard to the freaky black holes that sit somewhere out there in the void. It’s all the same “language,” in that each ‘dialect’ can be mapped into the other; and strangely, the language of the Creator was shown by Gödel to have a mapping onto the language we, as image-bearers, use to understand the Creator’s handiwork and each other. Gödel showed this specifically for strings in formal mathematical systems, but the language connection can certainly be generalized far beyond this as a thought experiment, even if the mathematical rigor is lost due to the flexibility of conversational grammar.
In what is arguably the strongest metal album to come out in 2014, Language by The Contortionist, the song “Language I: Intuition” focuses on the underlying communication which takes place between all living and nonliving things, or put in Christian vocabulary, the underlying meta-language which speaks through all of the Creator’s handiwork:
Begin hyper-communication
Restore our vision
Of natural progression
Rise in groves to reclaim the source
The center
We will be the salvation the Mother seeks
Traversing in all directions
Reaching
Expanding
Balance finds its place
Reaching for the Mother Sun
Rooted to intuition
You are the language
Ever flowing
Ever echoing
Drift with the ebb and flow
Drift with the ebb and flow
Ebb and flow
Intuition sets in
Branching out from your seed to seek
Contrived sense of inception
Intuition speak to me
The line “You are the language, ever flowing, ever echoing” obviously has close parallels with God’s spoken creative work, which continues to reverberate throughout the cosmos.
However, the same Creator who spoke all things into existence, and whose words continue to give life to all living things through the mathematical underpinnings of the laws of physics, is also the God who speaks to Job “out of the whirlwind” (Job 38:1 NRSV). It is the same Creator who has constructed the new reality about which Karl Barth writes:
“In the light of this coming world a David is a great man in spite of his adultery and bloody sword: blessed is the man unto whom the Lord imputeth not iniquity! Into this world the publicans and the harlots will go before your impeccably elegant and righteous folk of good society! In this world the true hero is the lost son, who is absolutely lost and feeding swine — and not his moral elder brother! The reality which lies behind Abraham and Moses, behind Christ and his apostles, is the world of the Father, in which morality is dispensed with because it is taken for granted.” (“The Strange New World Within the Bible”, pg. 40)
In his commentary on The Epistle to the Romans Barth also writes:
“‘Now, Spirit is the denial of direct immediacy. If Christ be very God, He must be unknown, for to be known directly is the characteristic mark of an idol’ (Kierkegaard). So new, so unheard of, so unexpected in this world is the power of God unto salvation, that it can appear among us, be received and understood by us, only as contradiction.” (pg. 38)
In some strange, contradictory, and (conveniently enough) paradoxical way, the incompleteness and mystery in the Creator’s meta-language, which undergirds creation, is a shadow of the fundamental Incompleteness Theorem presented to us by the Gospel and the Resurrection, which fully puts the mysterious nature of God on full display. The Gospel is the Epimenides Paradox, transplanted into the existential manner in which we think about life, existence, creation, morality — ultimately, everything. It should come as no surprise to us that the mystery of God, so easy to forget and concretize into Law, is always subverted by the reality of the Gospel, which is played out once again, albeit on a much smaller scale, in our tendency to concretize and treat mathematics as complete when it has been proven otherwise.
The Incompleteness Theorem proves that the Law is not God’s final word; rather, he reveals himself through the most profound paradox of all, abounding Grace, extended to all of us. We cannot, and never will be able to work out, how the grace of God is freely given to us in this way. And yet, the truth of the Gospel remains, confronting us with a much larger paradox and contradiction than even the one written about by Gödel in 1931. And this paradox, I believe, is much more important for us than Nestle’s release of the Milkybar, or the discovery of Pluto, or even me finally working out how the heck Gödel’s proof actually works, which would be nothing short of a miracle, and would only be attributable to the handiwork of the same Creator who made mathematics so inconceivably mysterious in the first place.
COMMENTS
3 responses to “Incomplete Math and the Paradox of Grace”
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The computer science approach to Incompleteness is considered more approachable, if that helps. But I love Gödel, Escher, Bach.
I was excited to see the theological parallels or insights from this connection, but I think you overreach with this idea that the incompleteness theorem proves something about the law or paradoxical grace. For me the idea that there are true things which can not be proven or disproven is more along the lines of Aquinas. Our faith needs to be reasonable, but can not be established by reason. When we want to communicate grace, it is by story, by example, not by argument.
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Yeah ive felt for years now that my faith in the gospel is a hope against hope. Like, there are plenty of things in my life and in Scripture itself that undeniably prove to me that I am doomed. But Im really counting on a gospel that contradicts all of that.
I’m a math teacher and this is wonderful.