Image: Mathematician and philosopher René Descartes, in a thoughtful mood.
Modern science has been built, for some time now, on an edifice of mathematics, with some remarkably useful results.
For example. When I’m suffering from a toothache, I can punch a certain number into my iPhone, and be instantly connected with my dentist’s answering machine.
In a sense, numbers are magical in that way. Assuming you know your dentist’s number. It’s important to know stuff.
But even the mathematicians don’t know everything.
In 1900, the eminent mathematician David Hilbert announced a list of 23 key problems to guide the next century of mathematical research. His 23 problems reflected an ambitious vision… a vision that all mathematical statements could be proved true or false. Or so he thought.
From a recent article by Joseph Howlett in Quanta magazine:
In the 1930s, Kurt Gödel demonstrated that this is impossible: In any mathematical system, there are statements that can be neither proved nor disproved. A few years later, Alan Turing and others built on his work, showing that mathematics is riddled with “undecidable” statements — problems that cannot be solved by any computer algorithm…
…Some mathematics can simply never be known.
If you believe the mathematicians and the journalists who write about mathematics, Hilbert’s dream was dead. Murdered by Kurt Gödel.
The Quanta article dealt mainly with Hilbert’s 10th Key Problem, which we might have assumed was dead, along with the other 22 problems. Although I didn’t understand most of the article, I was struck by a sentence near the end:
This is just one of many questions, according to Andrew Granville of the University of Montreal, that “reflect the philosophical side of what in the world is true.”
Obviously, David Hilbert had a vision of knowing things that cannot be known. But he didn’t know that they couldn’t be known. (Until Kurt Gödel let him know.)
My question for today. Can we know what we don’t know?
By that, I mean, “can we know which are the exact things we don’t know”? And can we know “which are the exact things we don’t know, that would be a waste of time trying to know, anyway?”
We’re all familiar with the simple equation, “1 + 1 = 2”. Mathematicians long ago agreed that this was a “true” statement. If this statement were not “true”, then, in my opinion, the whole edifice of mathematics falls apart.
But several years ago, writing here in the Daily Post, I offered compelling proof that, in certain instances, “1 + 1 = 1”.
The ambiguous truth of this equation, “1 + 1 = 1”, came to me one rainy day, as I sat by the window watching the droplets of rain run down the glass. I noticed that, occasionally, one of the droplets would bump into another droplet, and the two droplets would instantly combine into a single droplet.
“1 droplet + 1 droplet = 1 droplet”
I have since demonstrated scientifically that this equation applies in other instances. For example, when I pour a partial cup of leftover (cold) coffee into a partial cup of freshly brewed coffee, I end up with a single cup of (lukewarm, but still palatable) coffee.
If, in fact, the equation “1 + 1 = 2” is not always true, then what can we really know about the world?
The great French mathematician and philosopher René Descartes once wrote, “I think, therefore I am.” A simple, but nevertheless profound, statement.
(Actually, he wrote, “Cogito, ergo sum” which means essentially the same thing, although it sounds a bit more elegant.)
He was thinking when he wrote this, and he also existed when he wrote it. But were his thoughts proof that he existed? Or did he only think that he existed?
I’m just asking, because I don’t know. And I may never know.
It could be one of those things that a person can’t know.
Just as we will never know if René Descartes liked wearing women’s clothes, or it that was simply the style in those days.