I am about to get really wonky, and for me that can be a pretty dangerous thing. So settle in and put that red pen away. I’m going to talk about logic, proofs, and the virtue of curiosity. (Disclaimer: I didn’t know any of this would be on the test, I swear!)

If I have three apples and give you two, can you prove to me that I will only have one apple left? By counting, we can represent mathematically that *x* = 3 - 2 and solve for *x*; 3 - 2 = 1; *x* = 1. But without empirically measuring the initial condition against the assumption—that I had three apples—and without measuring the result, you cannot *prove* that I have one apple, and therefore, if your theorem of the apples is based on the math, you can’t convince me it’s true.

The initial conditions governing any system are supremely important in dealing with the validity of that system. Austrian mathematician Kurt Gödel came up with what’s now a foundational work of logic called the “incompleteness theorems.” I am no expert in formal logic, so to my high school physics teacher, and those unfortunate enough to have taught me math in college, I apologize for this “in a nutshell” attempt to dumb it down to something even I could understand.

Essentially, Gödel said that any logical proof involving mathematics cannot be both consistent—provable within itself—and complete, meaning it can prove its own consistency. Therefore, without empirically measuring that I began with three apples, ended with one, and you could count two more apples than you began with, in a closed system in which there were only you and me and a finite number of apples, you cannot prove the truth of my initial statement. The proof is external to the theory.

The human mind is capable of abstractions beyond the bounds of representational mathematics. Yet the human mind is somewhat limited in its ability to perform recursive functions in representable sets of numbers, i.e. “computing” (that’s what we have computers for, thought Alan Turing). We excel in thinking abstractly, but not so much in formal, consistent mathematical proof.

In other words, brilliant people like Einstein (and others) can come up with thought experiments to support theories like General Relativity and Special Relativity. We can all conduct the thought experiments. The hard part is in the proof. Though empirical measurements required to prove some aspects of these theories and their axioms have been known for many years, it was decades before we had the engineering ability to conduct them. We are still proving them today.

In contrast, there remain plenty of unproven mathematical theorems. If you want to decorate the bathroom walls of the MIT Mathematics Department with the proofs, you can become the real-life Will Hunting. Humans are great at imagining, but not so good at proving. That’s what makes proof so magical and engaging.

It’s also what makes curiosity the most coveted human character virtue, outside of charity and mercy. Genuinely curious people don’t have enemies; they only have people who don’t understand them, and whom they do not understand. Incurious people count everyone they don’t know as a potential enemy.

The problems emerge when reality meets the incurious mind. In just about every area of society, and our planet-wide human civilization that’s quickly fusing into a global cultural community, reality can bite back very hard when groups of people approach life with axioms derived from systems that began with bad assumptions about initial conditions.

For example, Jonah Goldberg recently threw a Jackson Pollock palette at a blank computer screen to try and explain the current state of Republican politics, centering on the meme of Leeroy Jenkins, charging into battle in defiance of a well-planned computer game attack, and getting every player killed. The politics of Romanticism argues for fighting over formal argument and proof. Romanticism, he explained, is the belief that any construct of civilization is artificial, and therefore inauthentic to the “real” self, which if left to its own devices, yields a noble savage. The counterpoint to that theory is the well-known Bible verse Jeremiah 17:9, “The heart is deceitful above all things and beyond cure. Who can understand it?”

There are no noble savages, only tribal savages. In a two-tribe closed system, this is easily shown by which side has the open mind, and which side seeks to dominate. When the incurious meet the curious, eventually one becomes the enemy. Want proof? Attempt to visit the Sentinelese living in the Indian Ocean. They are not curious about outsiders; they will kill you on sight. History reinforces this picture. Many Native Americans were curious about European visitors and colonists. They even helped the colonists survive in the “New World.” But when their curiosity was returned with imperial designs on every part of their lives, many turned to fight. The treatment of the original inhabitants of North America is a story of tribal savagery, not the nobility of man.

Revolutions are replete with butchery of the ruling class being deposed. The French revolutionaries penned the high-minded “Declaration of the Rights of Man and of the Citizen,” before Robespierre and his Jacobins decided that the mob must be given license to slaughter every single member of the old regime in the First and Second Estates (the nobility and the clergy), wiping away history for a glorious proletarian future. For all the “inviolable” rights granted, savagery was the main driver of most revolution, even among the “enlightened.”

In American politics, especially in the GOP, the mass thought infection—a failure of imagination, really—that many far-left Democrats are in fact incurious and closed-minded about what conservatives really believe or want, has led to the intellectual death of conservative curiosity as a virtue. The new way is as the Jacobins had it—legislate them away, and enforce it with state power, backed by mob power. Rebellion for the sake of doing away with the old in favor of a new order is the expression of this new incuriousness. In reality, it leads to nothing but savagery.

The latest case in point, as I write this, is happening in Brazil. Jair Bolsonaro lost his election, but now a mob of his supporters, a la Trump, stormed congress, the presidential offices, and the supreme court in Brasilia, in support of Bolsonaro. Incuriosity has the unfortunate property of being easily transmissible, being an abstract idea that easily infects other humans.

This idea of abstract incuriousness also works as a poison in religion. Extremists and dogmatic fundamentalists of all stripes ironically share their devotion to it. You can’t get a Sunni *jihadist* and an Indian *Hindutva* to agree on anything, except the fact that they don’t care about each other’s philosophy or ideals, only that they themselves are right beyond argument. Therefore they engage each other only in the language of power, and savagery. Thousands of years of religious wars prove this human failing beyond a doubt.

There is no inherent nobility in owning the truth, even if it is externally provable in an experiential sense. Mahatma Gandhi famously said, “I like your Christ, I do not like your Christians. Your Christians are so unlike your Christ.” Christ, met with curiosity, needs little formal proof. He is experiential, historical, documented thoroughly, and in words and actions, counter to human nature. Individually and as a group, Christians, on the other hand, struggle to overcome Gödel.

Christ said “you shall know the truth, and the truth shall set you free.” But it’s never been the truth of the Gospel itself that transforms a person. Sanctification is, Biblically, and in Christian philosophy, a continuous effort of human will and applied discipleship. Christ’s “Great Commission” is to make disciples, not to mass produce truth at wholesale prices. Truth, when gained, is often experientially self-evident, but the proof of it is not contained within itself, as Gōdel theorized.

There is no formal proof of Christianity that can survive consistency tests without a supernatural faith. Any faith ending in human war or slaughter is not a proof, but an exception, or at worst, a failure of the system. Incurious minds judge all systems by their exceptions and failures—this leads to “straw man” arguments, which unfortunately make compelling conspiracy fodder. (Ask any Jew.)

Religion is only one step adjacent to economics. How we treat each other is expressed in how we treat our property, and the property of others. In a fully communist, atheist, Marxist society, there is no personal property, which means there is no individuality. Each is trusted to become his or her authentic self based on the needs of the cooperative. Of course, the state determines those needs, and the state is made up of people. People with the privilege of determining everyone’s needs. Unless the “noble savage” assumption holds true (it doesn’t), as Goldberg noted, there’s no end to this but tyranny. Until we re-engineer the human mind, “real communism” can never be tried. But plenty want to keep trying.

Of course, Marxism is an easy, extreme case. Socialism, to the degree that capitalism unrestrained leads to some perverse incentives and absurd results, has a more nuanced place in our abstract battles with Herr Gödel. What portion of scarce resources should be in the state’s grasp versus the public market? Building and maintaining roads? Snow removal? Electromagnetic spectrum? Airspace? Can you imagine if all airspace was privately owned? Every day would be a potential Southwest Airlines meltdown.

Now the harder questions: medical care? Pharmaceuticals? Here’s a thought experiment: If there were a single cure for every form of cancer known to medical science, but that cure cost $1 million per patient to administer, who would decide how much money the U.S. government would spend on it? Some would argue that the market should decide, so zero. Others would go the route that COVID-19 vaccines took—100%. Our government has leaned very socialist lately, despite four years of so-called “conservatives” in the Trump administration. “America First” handed out at least $2.3 trillion to Americans in cold hard cash. We took in a lot less than that in foreign (China) tariffs, and the lion’s share of it went to farm payments.

In reality, the battle between conservatives and liberals these days is not fiscal. It’s over initial conditions. Is America racist or not? If America *was* racist (undoubtedly it was) in its history, does that make today’s America responsible for undoing history by turning racism upside down against the formerly privileged class? Is there such as a thing as “anti-racist” without being a racist? Should speech be free, or just anti-hate, in the same vein? Law is not immune to Gödel either.

Our political, social and economic battle is not about spending money versus not spending money, it’s about where the money goes. And lately neither side is open to curiosity about what the other side believes is the right answer. They’re more interested in conducting drawn-out political investigations on each other.

Finally, there’s a battle over the nature of intelligence itself. Or at least artificial intelligence. Today’s AI, or attempts at Artificial General Intelligence (AGI, the Holy Grail of AI research), is based on big data sets, internally consistent logic, and (theoretically) infinite training evolution, to produce a “real” mind that can pass the Turing test, that Turing called “The Imitation Game.” Arguably, ChatGPT can produce human-quality prose and fool many humans into believing it was written by a person, not a bot. But even the best prose can be filled with empirically disprovable, dangerous nonsense.

Computers are not equipped with the ability to discern abstract thought from disconnected facts. Not yet at least. Computers have not been programmed with the virtue of curiosity. AI does not have the drive to explore and test whether an answer it is constructing is true, or even provable, because that requires external means.

Humans have the ability to engage curiosity. This is the best road to reach reality, and to build systems, axioms, theorems, philosophies, policies, and processes to support real things, which don’t necessarily have completely consistent, internally logical arguments for every conceivable case. There will always be outliers, even outrages. There will always be rebellions and rebels against the orthodoxy and the common way. But curiosity will help to understand the reasoning—or the irrationality—of those cases, and at least properly categorize them within the taxonomy of our real world experience.

In other words, nut jobs should be labeled as such, and any axioms flowing from their incurious and degenerated minds should head to the appropriate dustbin along with their benighted conclusions. Failure to do so will only make the splat hurt more when gravity carries the lofty ideals into the rock hard surface of reality.

Reality is a harsh teacher. Politics, government, and society only works when the people are invested in politics working, and politicians recognize the value of external proof of their own axioms and theorems. The danger is that our system is headed for failure, when of the above are being ignored in favor of raw power.

You get an A in geeking out.

edited Jan 9, 2023"Essentially, Gödel said that any logical proof involving mathematics cannot be both consistent—provable within itself—and complete, meaning it can prove its own consistency."

Another way to think about it is this way: logical systems both have a language for expressing statements ("2 + 2 = 4") and rules for how those statements may be constructed so that the "truth" of any given statement can be evaluated to be "true" or "false". What Gödel showed was that language for expressing these statements is ALWAYS more "powerful" than the evaluation portions of the logical system, because in ANY logical system, you can say things (using the language) that your system cannot evaluate to determine if what you are saying is true or false.

This first(?) emerged when Georg Cantor was building a mathematics around the rules of set theory and encountered two sizes of infinity: the "smallest" infinity that is the size of of all countable numbers (integers) and a larger infinity, which is the size of all real numbers (continuous, infinitely divisible numbers, including irrational numbers). He was trying to prove if there were any OTHER infinities of a size between these two ("aleph-zero" for the smaller infinity, and "aleph-one" for the larger infinity). There's a fun pop-science book called "The Mystery of the Aleph"[1] that goes over Cantor's discovery and him pretty much driving himself crazy trying to figure out a proof about the relationship between these two sets of numbers.

In the absence of a proof one way of the other, mathematicians added a new "rule" to evaluate this special case (Zermelo–Fraenkel set theory), and Gödel showed that adding this rule enabled the expression of new mathematical statements that came along with the new rule that were similarly undecidable. Doug Hofstadter explores this in his books "Gödel, Escher, Bach" (GEB) and "I Am A Strange Loop", hypothesizing that this may be an emergent property of self-referential systems. (I'm still working my way through GEB, so don't take that as a complete or authoritative interpretation of Hofstadter's work.)

We run into a similar issue in computing (another logical system subject to Gödel's theorem) with something called the Halting Problem, which basically states that it's impossible to build a computer that can inspect at ALL programs that might it may be able to run and determine if it contains an infinite loop (hence the "halt"). You can build more powerful computers with capabilities that allow them to determine if all of prior generations' programs will halt or not, but in the process, the new architecture will allow NEW programs to be written that were not possible on the older architecture, and among these new programs will be NEW programs that the new computer cannot determine whether they halt or not. This is the fundamental reason antivirus and malware-related work will always be a running arms battle into the future.

Anyhoo - apologies for geeking out over this small point in Steve's post. This is a fun corner of math and computer science, and I love sharing how odd and interesting it is.

[1] https://www.goodreads.com/book/show/5786.The_Mystery_of_the_Aleph