In Search of Wilde Bohr
Aphorisms and paradoxes through the metallic looking glass
Tops on the Paradox hit list
My favorite quote of all time comes courtesy of Niels Bohr…
The opposite of a correct statement is a false statement. The opposite of a profound truth may well be another profound truth.
I love it because…
- It sounds profound, and not just because it contains the word profound.
- When I read it/think it/speak it I feel like I understand it and don’t understand it at the same time. It’s the equivalent of stepping out on a verbal tightrope without a linguistic net. It’s paradoxical.
- Niels Bohr is one of the greatest scientists of all time, period. Unless you ask a physicist, however, most people on the street would not include him in a list of “famous scientists” — but they would definitely nominate Einstein. And to think that Niels wiped the floor with Albert over the quantum nature of reality!
Perhaps I love the quote because it is a perfect scientific aphorism — whatever that is.
This Paradox is False
A wonderful thing about the words and sentences that we use to communicate facts and ideas to each other is that they can also be used to create paradoxical statements if the word-order, or phrase and sentence structure is just the right, magical blend of proper grammar and illogic.
Sometimes the paradoxical nature of the statement is evident as a sort-of troubled feeling you get in your gut when reading it. Your mind can’t hold onto the seeming contradiction between the words in the sentence itself, and what the sentence is saying. These words make up the atomic structure of the paradox, but they combine into a statement that can’t possibly be true — or false — that is the paradox. It’s like driving over a small hill at just the right speed to cause the normal force pushing up on your butt to zero out, giving you a brief instant of virtual weightlessness, a lose-your-lunch sensation — even when you know you’re carrying plenty of weight around this planet.
This limbo-state of truth-falseness often happens when the statement is self-referential. A classic example:
This sentence is false.
If the sentence is true, then This sentence is false is a true statement. But then the sentence is false. If the sentence is false, then This sentence is false is false and therefore the sentence must be true.
Got that?
Then there are statements that seem paradoxical, but don’t quite have the same gut-punch — they are more clever than they are troubling. Bohr’s quote certainly has that flavor. Maybe it’s not a paradox, then, or perhaps it is at the low end of a paradox spectrum.
On the other hand, the message of Bohr’s two sentences should also apply to the sentences as well. Does this mean that the opposite of Bohr’s combined sentences is also profound? Or is it false? Now I’m not sure whether the set of Bohr-sentences is a paradox or not, or just too clever for its own good.
Is there a scale then, a continuum of paradoxical statements from clever to lose-your-lunch?
If so, there must be a position on the paradox-spectrum for ones that have a darker undertone:
Death never happens but once, yet we feel it every moment of our lives. — Jean de La Bruyere
It’s clear then that verbal paradoxes can be anywhere on the spectrum from “hey that’s a great quote, I am going to memorize it so I can write a blog post that begins with it” to “holy bleep, Batman. I better leave my will on the dining room table before heading to work today.”
Some may find this discussion of paradoxes a bit frivolous — just simplistic word play. Is a paradox more serious — more worthy of our attention — more memorable — if it is darker? And what makes a paradox a paradox anyway?
Burrowing to Inconsistency
To consider this question, take a page out of the old physics playbook and go all reductionist on it, burrowing deeper into the individual bits of the paradoxes presented so far.
For Bohr, opposite statements of a truth can be both false and profound. For Bruyere, death is with us all of the time, and also not with us until the end. It appears that a paradox is necessarily made up of two mutually-exclusive conceptual threads existing simultaneously when it’s clear that this is impossible.
In his classic The Ways of Paradox, W.V.O Quine found nothing frivolous about the study of paradoxes. He even classified three types: veridical paradoxes, falsidical paradoxes, and antinomies. According to Quine, the first two types aren’t really paradoxical at all — we just need to improve our thinking…which often means our mathematical logic.
Veridical paradoxes occur when what seems to be a nonsense conclusion arises from a set of valid premises, yet the conclusion is true. The Raven Paradox , in which a green apple can be considered a fact that supports the inductive argument that all ravens are black, is often given as a veridical example.
Zeno’s paradoxes of motion, which appear to definitely prove that motion is impossible, are considered falsidical paradoxes. These occur when a set of what appear to be valid premises result in false, or illogical results. In these paradoxes, the mathematics of infinite sums must be used to remove the offending paradox.
Then there are antinomies — truly ghoulish statements whose truth or falseness can’t be determined. According to Quine, they
…bring on the crises in thought. An antinomy produces a self-contradiction by accepted ways of reasoning. It establishes that some tacit and trusted pattern of reasoning must be made explicit and henceforward be avoided or revised.
This sentence is false is such an antinomy. Does Bohr’s quote constitute an antinomy? I would hate to think that Bohr — a giant among physicists — would unwittingly unleash something into the world that would bring on a crisis in thought — something that must be avoided or revised.
(Ironically, this is exactly what Bohr did with his seminal work on the the foundations of quantum theory, and especially his Copenhagen interpretation of quantum reality.)
Counting on Paradoxes in Mathematics
Paradoxes occur with some frequency in mathematical logic. However, perfectly logical mathematical statements that lead to absurd results are typically not characterized as paradoxical. Instead, they are described as contradictions, or, in the specialized language of logicians, inconsistent statements. (Try to think of a paradoxical mathematical statement that isn’t a contradiction — for example 0 = 1)
Mathematical inconsistencies are often nasty, unwanted, theorem-killers, and their presence is usually a sign of things gone awry — or perhaps something crucial that is missing in the theory.
Actually, inconsistency and the related problem of incompleteness is built into all of the mathematics we do. After all, in 1931 Austrian logician Kurt Gödel showed that mathematics cannot be both consistent and complete in his two Incompleteness Theorems — results that shook the world of mathematical logic, and from which we have never recovered.
Gödel’s Theorems contain some of the densest instances of mathematical logic ever put to paper. Full of self-referential mathematical statements that are antinomies at their heart, bursting with contradiction and paradox, they are almost impossible to decipher unless one is brilliantly adept at symbolic logic. It is ironic, then, that Gödel marvels at our ability to understand each other given our language. It is really his language — the language of abstruse logical symbols and their manipulation — that may be the reason we don’t understand him.
We need to focus on our everyday language to get to a better understanding of paradox, contradiction, and inconsistency. And what does all this have to do with aphorisms?
It’s Only Words, And That is All
Leaving Gödelian logic behind then, consider basic words. The same set of words can tell different stories depending on how they are arranged into sentences. (Note: Let’s assume that these arrangements follow the basic subject-verb-object rules of English so the sentences are not total gibberish.) Some of these different sentences may be understandable — they make sense in our language. For example, the bus hit the car can be rearranged without losing meaning. I am more interested in those that are still appropriate English sentences, but there is something not quite right about them. They don’t really make sense, but are close to it. For example,
The cowboy rode the horse
with nouns interchanged becomes
The horse rode the cowboy.
The latter is a legitimate English sentence, although it definitely sounds wrong. However it is possible to imagine it being said for emphasis after a particularly spirited horse had the better of an over-confident cowpoke.
This type of word switch — one that yields a sentence that sounds wrong, but perhaps isn’t, is occasionally found at the heart of a verbal paradox.
Switching words and ending up with perfectly valid sentences, but other times leading to nonsensical sentences that are still usable as sentences fascinates me. Thinking like a physicist, words are the atoms of sentences, and if the sentence doesn’t quite make sense, it’s because there is something inherently inconsistent about where these word-atoms are situated with respect to all of the other word-atoms in the sentence.
Quick! Freeze!
If this discussion of word-atoms has you glassy-eyed, consider metal atoms that are quick-frozen into a glassy, amorphous state. Without the super-rapid cooling used to produce them, they would be relaxing at home in a nice, regular, crystalline matrix. These metallic glasses are tangible inconsistencies because their measurable properties are so radically different from both metals and glasses. However, the atoms in these glasses are, on average, not too far away from where they would be if the material had been allowed to form as a crystal. It’s just that there is a more random distribution of these atoms as compared to a crystal lattice.
For me there is this interesting connection between words in a sentence that we recognize as a sentence, but does not really make sense, and the atoms in a metallic glass.
I’m not sure that any of this discussion moves us closer to an understanding of paradoxes and language, but metallic glass — the name itself sounds paradoxical, or at least oxymoronic — is one of the coolest materials on earth, with all sorts of useful properties.
But the best thing about metallic glass is this: it is a little piece of paradox that you can hold in your hand.
This talk of paradox vs. inconsistency suggests that verbal and mathematical paradoxes (and, by extension, scientific) are different in kind. Which brings me back to aphorisms — maybe that is the umbrella that will encompass these two forms of paradox.
Aphorismsic Conclusions
The OED presents two definitions of aphorism…
1. A pithy observation which contains a general truth.
2. A concise statement of a scientific principle, typically by a classical author.
Even though Umberto Eco claims in On Literature, his 2002 book of essays, that “there is nothing more difficult to define,” many authors have tried just that, attempting to classify the slippery aphorism through the ages. Perhaps none have done this with more fun and completeness than James Geary in his Geary’s Guide to the Worlds’ Great Aphorists. In addition to the OED meanings, Geary adds a whole taxonomy of aphorism-types, including the paradox, the observation, the moral, and several others.
I am glad to see that paradox is one of the aphorism templates listed by Geary. Are all paradoxes aphorisms, then? Clearly that can’t be true — or can it? Let’s take a look at a classic paradoxical statement that appears deliciously antinomy-like because of its self-referential structure:
This sentence no verb
Note however, that this is quite different from this sentence is false. In the current sentence, truth or falsehood is really not the issue. Here the paradox is that the sentence is not a sentence because it does not contain a verb. It is incomplete, as our Godelian logicians would describe things. However the “sentence” seems like a real sentence because it provides information in a way that sounds very much like a declarative sentence. The fact that it describes itself as a sentence without actually being a sentence makes for one very clever “sentence,” because we understand it as if it is a sentence.
So we have a paradox, but unless your definition of a pithy statement which contains a general truth is different from mine, it sure ain’t no aphorism. Maybe a paradoxical statement without self-reference to the actual statement is the way to a valid aphorism.
Whatever an aphorism is or isn’t, my image of the King of Aphorisms is Oscar Wilde, who skewers any and all quite paradoxically:
“An ethical sympathy in an artist is an unpardonable mannerism of style”,
a haughty comment that has its inverse aphorism in Groucho Marx’s classic
“I don’t care to belong to any club that will have me as a member.”
Both of these have a nice whiff of paradox to them. Note that the Wilde and Marx quotes are not really self-referential because they don’t refer to themselves as sentences in the way that This sentence no verb does. But Marx’s quote is self-referential in the sense that he refers to himself in it. In other words, care must be taken in using self-referential to describe a property of paradoxes that either are, or aren’t aphorisms.
Bohr was a master at generating profound-sounding quotes that, because of his stature among 20th-century physicists, survived and multiplied on pages in notebooks, blackboards and whiteboards, and, now, dedicated quotation websites.
Case in point:
Now this is a good one. Definitely pithy. It’s got that self-reference thing working. Paradoxical — the atom is looking, but physicists look at atoms. It even uses the word atom as a word-atom in a sentence. I like that. But there’s something not quite right about it. There’s something about it that just doesn’t make sense. Somehow the words are in the right places in the sentence. There is subject-verb-object structure. It resonates…but it is the resonance of panes in a metallic glass …inconsistent and slightly discordant.
Not great Niels, but good.
Now what about the Bohr quote at the beginning of this story? After all of this musing about aphorisms, paradoxes, and self-reference, I am now convinced that it is an excellent aphorism. It has the right sequence of word-atoms that make the sentences consistent. It is pithy. And it does contain a general truth that is meta-general because it refers to the nature of true and false statements. In other words it has quite a bit of self-reference in it. But at what level? Is the reference to Bohr’s sentences, or to something larger? Because Bohr’s passage is about true statements and their opposite, I assume that Bohr’s sentences themselves are true in a profound way, which means their opposite is also profound…
If you, dear reader, are feeling a bit vertiginous as I am, it is best to end this discussion now. But we can’t escape the fact that we have followed Bohr into the rabbit hole of self-reference and infinite regress. Of course, he didn’t stay there for long. He rocketed out to live again, to be quotable again, to lead the world in the acceptance of quantum reality. To be one of the greatest scientific figures of all time.
And I now really understand why I love Bohr’s quote at the beginning of this story. I may be stuck in that rabbit hole, but at least I have his quote to turn over in my head over and over, to hold up to the light and see that its reflection on the wall is different every time I look, as if it is a polished piece of metallic glass.
It is profoundly true…
It is nothing less than The Greatest Aphorism of All Time.
Based on an earlier post at www.fractalog.com: December, 2007.
