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This sentence has 13 words, one comma and a fullstop at the end.
Go ahead, check it, I’ll wait.
All correct, I trust? Statements like that are called “self-referential", meaning they speak about themselves. Like this sentence you are reading right now. In fact, this is a self-referential sentence too.
They are a novelty for a while, but self-referential sentences that say something straightforward about themselves quickly lose their charm. I mean, isn’t this one dull? So try something that’s not quite so straightforward.
Like this: This sentence contains exactly thiree mistooks.
What’s the third mistake? How will you correct it?
Or try this: This sentence is false.
What do you make of that? If it’s true, it’s false. But if it’s false, it’s true. An innocent sentence, and it’s diabolical!
The ancient Greeks, those constant philosophers, grappled with this dilemma. It came to them via a man from Crete named Epimenedes, who once said: All Cretans are liars.
Not very different from things many of us say without thought. Except that this is really a variant of “This sentence is false." For remember, Epimenedes was himself a Cretan. If he’s telling the truth, his pronouncement is false, because at least one Cretan—Epimenedes himself—tells the truth. But if his pronouncement is false, he’s a liar, and voila, the pronouncement is true again.
In its very statement, Epimenedes’s pronouncement contradicts itself.
So they’re intriguing and maybe you’ll make up your own. But aside from that, what’s so special about self-referential sentences?
One answer to that involves Kurt Gödel, arguably the most influential mathematician of the 20th century. In 1931, Gödel published a paper called On Formally Undecidable Propositions of Principia Mathematica and Related Systems. What’s now called his “incompleteness theorems" challenged basic assumptions of logic and mathematics and greatly shaped modern scientific thought. Awarding Gödel an honorary degree in 1952, Harvard University called his 1931 work one of the most important modern advances in logic.
The fascinating thing about the incompleteness theorems is that it is in effect a mathematical translation of what Epimenedes said.
A little background here. In a science like astronomy, we observe celestial phenomena (like the Doppler shift of galaxies) and use them to build theories (the Doppler shift tells us the age of the universe, see my earlier column Night Shift). This is different from geometry, which is founded on certain fundamental axioms. Like: any two points can be joined by a straight line. Or like: all right angles are equal.
From these axioms, we derive logically, step by step, the whole edifice of geometry. The ancient Greek mathematician Euclid pioneered this “axiomatic method", using it to develop the complex and evolving science we know as geometry today. And this idea—that a few axioms underpin the logical structure of a system like geometry—is one that thinkers throughout history have found seductive, alluring. Here was scientific knowledge and method at its best, they thought: a spare base of axioms, married to elegant, inexorable logic. Naturally, they wondered if other branches of mathematics were similarly constructed. By the early 20th century, there was a general belief that each branch had a set of basic axioms that were enough to deduce all theorems in that branch. Just as the Greeks had done so ingeniously for geometry.
Unfortunately, Gödel came along. His 1931 paper showed this belief up as the pleasant pipe dream it was. There are limits to the axiomatic method, he said. For even as “simple" a mathematical system as arithmetic and numbers, it is impossible to find a sufficient set of fundamental axioms. But going further, Gödel also showed that axiomatic systems that use logic eventually cannot be logically consistent. Eventually the logic itself runs into logical contradictions.
By their very definition, such systems will have contradictions. Are you beginning to see a connection to Epimenedes?
Of course, I will need extensive mathematical training to even begin to understand Gödel’s reasoning. But the essence of his incompleteness theorems is this: in any axiomatic system, it is logically possible to construct a statement that says “This statement cannot be proved" (compare to “This sentence is false"). Such a statement can be proved if, and only if, its opposite can also be proved.
You see the paradoxes. “All Cretans are liars" is true only if at least one Cretan, Epimenedes himself, tells the truth. But that makes it false. By obeying “disobey this command", you disobey it. If “this statement cannot be proved" can be proved, it cannot be proved.
Gödel arrived at the only possible conclusion: the axioms of logical systems are forever incomplete. Mathematics is essentially incomplete.
The computer scientist Douglas Hofstadter explained: “Somehow the full power of human mathematical reasoning eludes capture in the cage of rigour."
Now Epimenedes could never have imagined his link through the centuries to Gödel. I know that last sentence is true.
But I also know this one says nothing at all.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences. Comments are welcome at dilip@livemint.com
Also Read | Dilip D’Souza’s previous columns
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