If and only if
Knowing for sure we’ll never know anything for sure
If you were in the Milwaukee area Saturday, here’s hoping you were outside. Weather is subjective, but it was objectively beautiful. Happy to report I was outside at a park, sitting at a picnic table, reading about Kurt Gödel, a metamathematician living in Vienna in the 1930s. Hashtag spring!
This was not the other Gödel tome that’s been on my bookshelf intimidating me since the ‘90s, Kurt Hofstadter’s highly regarded Gödel, Escher, Bach: The Eternal Golden Braid. That’s next, I promise. Maybe. We’ll see how this goes. Because I’m more than halfway through Rebecca Goldstein’s brilliant Incompleteness: The Proof and Paradox of Kurt Gödel, using the dust jacket’s back-flap for a bookmark instead of its front. So, yeah. Deep enough into the cavern to have learned a lot, but probably having missed more than I should along the way. Enough to blog about it, in any case. You’ve been warned.
Goldstein is all kinds of awesome, and my other next-book option is her The Mind-Body Problem, as that’s become a top-three subject matter for me in the past year (the other two are equally obtuse and pretentious). She’s a Carl Sagan-level translator and communicator of complex ideas, aiding and abetting my delusion that I totally understand everything about Gödel. I highly recommend you read her work if you have any interest in what follows here.
Very dopely — and in my overly simplified version of it — Gödel proved through logic that nothing can be proven true or untrue through logic.
Still with me? Cool, thanks, I promise this is totally worth exploring.
Gödel’s first incompleteness theorem says mathematical disciplines are great for proving mathematical disciplines make sense. Essentially a bunch of people agreed on the rules, the symbols, and the syntax for mathematics — the axiomatic framework upon which all formulas are based. In philosophy the same goes for logic — somewhere along the line we all agreed on symbols strung together to make words that have definitions we’ve collected in books so we know what each other are talking about.
Gödel is said to have once stated, “The more I think about language, the more it amazes me that people ever understand each other.” Not surprising given what he’s best known for putting into the universe. A regularly present member of the famed Vienna Circle of logicians, he kept to himself in meetings, a low-key Platonist in the midst of militant Sophists, popularly labeled “logical positivists.” This circle was one of many in Austria, a silver-lining confluence happening between the two big wars that ravaged Europe in the first half of the 20th Century. Philosophers, artists, writers, scientists, psychologists… They all hung out in coffee shops, which they quickly outgrew, moving into living rooms and eventually classrooms. And for the most part, they thought they were getting a lot done.
Gödel was lurking on the periphery, taking it all in, meticulously working in solitude on his own ideas. Goldstein draws him in stark contrast to another brilliant member of the Circle, Ludwig Wittgenstein. Good ol’ Lu was magnetic in his performative tortured-genius act. “Why must I be so smart?” he probably anguished loudly a lot through gritted teeth. “No one will ever understand me!” As a result, everyone wanted to understand him so as to make him stop yelling so much and maybe even smile every now and again (or maybe not, idk). Our guy Kurt, though? He’d go to conferences and drop truth-bombs (insofar as they blew up the very possibility of objective truth) that went unnoticed (and definitely unappreciated) because he was all substance and no flash.
And here we go. Because this is not a book report. This is a pep-talk.
Nobody wanted to hear Kurt. Here comes this guy questioning empirical truth, something the Circle folks had agreed on to this point. Logic is based on math, which would have us believe something definitely totally makes sense because math is how we solve problems. One plus one equals two. Of this there can be no debate. And Gödel would agree. To a point. For him, though, it would be more like (this is all me here, please correct me if I’m wrong), ‘one’ ‘plus’ ‘one’ ‘equals’ ‘two.’
‘One’ is a symbol representing a natural number. And ‘plus’ is a concept. ‘One’ could easily be replaced by the symbol for a variable, ‘x.’ So x plus x equals two might be true. But it will only be true “if and only if” x=1. I mean ‘x’ could also be ‘two.’ But if we’re representing something in the real world? One apple plus one apple equals two apples, definitely. But the symbols don’t necessarily mean apples. I can see two apples in front of me, yeah. But numbers are, like, a construct, man!
Because that language of numbers, the symbols, could always be true when we change those symbols to variables. Gödel made up his own language to double-check what we know, using only nine symbols/numbers/meanings. Goldstein calls them ‘Gödel Numbers’ (he would never). And with those Numbers, he was able to prove in only 20 pages that we can’t prove anything at all is true through logic. Which he proved through logic. I love it.
“If and only if” is the qualifier that rears its head all too often in logic. It’s part of the syntax of proofs, the rules upon which words carry weight and meaning, if we play by the agreed-upon rules. Something is absolutely this (if and only if it belongs to this accepted set, though)!
Go back to my boiled-down version of his first incompleteness theorem, proving through logic nothing can be proven true or untrue through logic. “True” and “proven” are very different in this world. A logical or mathematical “proof” can prove all the symbols and syntax work out in a tidy conclusion that makes sense. But it’s the sets of those problems where we get into trouble. Arithmetic — one plus one equals two — is essentially fine. But?
But ‘infinity.’ That’s where things break down — things like language, our puny brains, rules of mathematics, philosophical debate. According to the sets of numbers and variables we’ve developed and agreed upon, the rules and syntax to explain the various states of how those numbers act in relation to each other… It should work that way all the way to the end of all numbers. But numbers don’t, you know, end. So we can’t know for sure numbers don’t line up how we think they’ll always line up somewhere down the road. Hell, numbers might even turn into kittens at some point. There’s no way of knowing, and so we’ll never know, so we can’t prove they do through logic using words and symbols… Logically, if something is what we expect and predict thanks to patterns in all that’s observable, then it’s proven. Gödel points out that maybe those patterns fall apart in ways we can neither see nor predict. So while it’s logically proven, it’s never really true.
Goldstein often takes us back to the “liar paradox,” a fun little thing Epimenides came up with around 600 BCE, which has since been translated and boiled down to, “this very sentence is false.” How fun! If it’s true then it’s false, if it’s false then it’s true. Much as we explored with Fermi’s Paradox a couple months back, it’s definitely totally paradoxical. It just can’t be a sentence!
But it is. It’s right there. It’s words made up of letters, letters made up of pixels, pixels each a state of either one or zero… It’s just symbols. It can’t possibly be false and true at the same time (even on a quantum computer) but it is, and the fact that it isn’t is why it can be. Still, it’s not. At all. It’s just words, man.
We don’t have to worry about it, then. Guys in the Vienna Circle had a habit of committing suicide over such syntactical word-games when rules fell apart. The nature of reality is called into question, so off we go to escape reality, which might not even be real. Personally, I find it liberating. (Side-note, here’s a speech Philip K. Dick delivered in 1978 on authenticity and reality, a long-read I think you’ll love because you’re still reading this.)
And ‘liberating’ not in an anarchist way, something I outlined in the first of these posts last year. Words might be symbols, but it’s the best thing we’ve got to work with. We might not know for sure if something works out mathematically all the time for infinity. But it also might. Because one of the very cool and human aspects of Gödel’s first incompleteness theorem is, while nothing can be proven, it can’t be disproven, either.
It just is.
We can debate the meaning of symbols and syntax, the value to which we assign them, what we know on paper to make sense or fly in the face of how x is defined… That’s not reality. Reality is disorder and discord, which is far from ideal. But it’s what it is, and no argument will change that. There’s no debating about how we perceive reality or how our senses feed information to our brains. Because we do and they do so let’s deal with it.
Or at least let’s try to. Because that’s the best we’ve got. It can be a contradiction, and often it is. But reality is never paradoxical.
There’s comfort in knowing we’ll never know anything for certain. I’m writing this on Søren Kierkegaard’s 205th birthday according to Maria Popova, who writes and administers the Brainpickings site (which also guided me to Goldstein’s book). Today Maria went down a Kierkegaardian path, quoting him as having said, “Man … is the synthesis of psyche and body, but he is also a synthesis of the temporal and the eternal.” The mind-body problem doesn’t begin or end with Descartes, as Maria demonstrates through Søren here. There are any number of theories about what the mind is, especially as it relates to the body. And we don’t know what’ll happen when we leave this realm. Are we a synthesis of the temporal and the eternal? Or are we a mass of cells powered by electricity that comes from breathing air and eating donuts, and we’re too dumb to know for sure nothing happens when that electricity cuts out?
Well, that’s where eternity comes in, and it’s why we’ll never know for sure, what with our puny brains and also… You know what? Doesn’t matter. Because whatever we believe to be true, we’ll never prove it. I’ve been laying off social media of late, as it too often seems to consist primarily of arguments over syntax. Tweets are about other tweets, and I’m just out here looking for some substance.
And that’s my main takeaway from what I’ve read so far about Gödel. We argue in bars and in courtrooms, on social media and outside Starbucks. But it feels like more than ever, nobody’s gonna budge on where they stand. So why do we argue if we’re not gonna prove anything to be true in the first place?
Because it’s the best we’ve got, and the stakes are too high to give up. So it’s the best we can do. Let’s do our best to use the words and the symbols and the electricity in our brains at our disposal — the only things upon which we can all agree — in a positive, constructive way. Let’s stop gleefully pointing out when someone seemingly contradicts their own words, because that’s ultimately not what’s real. Words are ephemeral, and will only become more so (as I’ve stated previously, and as Radiolab jaw-droppingly documented). Let our words be persuasive, but only if our hearts be inarticulately, objectively good.
Cheesy way to end, so I won’t.
I started a new full-time job this past week. This after freelancing for close to two years. I’m back in an office, with smart people 40 hours, Monday through Friday now. And I’m incredibly happy already. Turns out working remotely from an apartment for me was a dead-end, in my career path and my global outlook. Too much of my world was in my own mind, trying to make sense of what was going on out there. It wasn’t real, though. Didn’t matter how much Camus I read, working in an office with people every day is how it’s supposed to be. For me it is, anyway. But I think it stands to reason most of reality happens outside our minds (unless all of it happens there). Get out and take it in. It’s finally spring after all.