Let me open this blog post by quoting a Zen koan from Mumonkan, along with the comments and poem that Zen master Mumon added to it:
Koan:A monk asked Zen master Nansen, “Is there any teaching no master has ever taught before?”
Nansen replied, “Yes, there is.”
“What is it?” asked the monk.
Nansen answered, “It is not mind, it is not Buddha, it is not things.”
Mumon’s comments about the koan:Being asked a question, old Nansen gave away his treasure words. He must have been greatly upset.
Mumon’s poem about the koan:Nansen was too kind and lost his treasure,
Clearly words have no power.
Even though the mountain becomes the sea,
Words cannot open another’s mind.
Now that I’ve put you in the mood, I’d like to tell you the story of my theorem about “indescribable numbers”. I can’t believe I didn’t write about this thing years ago. This theorem was one of the most exciting episodes in my love affair with math, and it’s a tale worth telling.
The year was 2005. I was about 19 years old, with only a general and vague idea of what I wanted to do with my life. I enrolled to study Electrical Engineering at the Technion, a leading university in Israel. Why did I choose Electrical Engineering? Because (a) it sounded science-y and (b) it was said that there was a lot of money in it.
Contrasting my present self with my 2005 one, the most striking thing is how empty my mind was from the various strong opinions and views that inhabit it today. Not quite the emptiness of mind that a Zen master would aspire to, but rather an emptiness in which the two promises above, of scienciness and money, were all I needed to decide that EE was indeed for me, at least for a little while.
One morning I was walking to class, or perhaps I should say climbing; The Technion, built upon Mount Carmel, is famous for its brutally steep slopes. I lived in the Canada dorms, which were located at the lowermost point in the campus, and I had to walk up all the way to the campus’ vertical center. I’m not the athletic type, so this daily ordeal left me sweaty and tired every time I got to class.
Just as I was passing the faculty for Civil Engineering on my right, the first beads of sweat already formed on my forehead, a funny thought came to me.
That thought was: Could there be numbers that cannot be described?
I should explain this idea in detail.
Imagine that you’re solving a homework problem in physics. Say the problem introduces you to a helium-filled balloon as it is making its way towards the ceiling, and you are tasked with finding its
Y coordinate at
3 seconds, air resistance neglected. After some work you come up with an answer. If the homework in question is given in junior high or high school, the resulting number describing the height is likely to be a simple one. Perhaps
7 meters, or
11 meters; thanks are due to the invisible compassion of high school physics teachers who have a hard enough time as it is teaching physics to teenagers who are much more interested in sex. The teachers, knowing they have only a precious modicum of the students’ attention at their disposal, compose their problems in such a way that the final number would be so simple, usually an integer and in wild cases a simple fraction, that when the student finally hits upon it, he can put his mind at ease; after all, if after these complex and error-prone calculations, fraught with quadratic equations, you got a result of
7 meters, that’s a good enough reassurance as any that an error was not made and that the answer you came upon is the correct one.
As high school pupils become university students and are gradually weaned from the soft and cozy world of integers, the numerical solutions gradually become more complex. You could be hard at work on a more advanced variant of the above physics problem in Physics 101, air resistance emphatically not neglected, only to arrive at a result of
3*Sin(57) meters, which the last few pages of the book reassure you is the correct answer.
You stare at that number,
2.5160117038362720(...) in all its decimal glory, and you imagine the fictional God of that fictional universe who placed each and every one of the sextillions of helium atoms in that balloon in just the right position, so that when
3 seconds, the height of the center mass of the balloon would be at precisely
3*Sin(57) meters, to the last of its infinite digits. And you realize, of course, that this invisibly compassionate God is nothing more than a braver version of the invisibly compassionate high school teacher, who was so careful not to frighten you with an unholy number such as
So you imagine what a real God would do, and what the real height of a real helium balloon would be. Since we’re dealing with pure math, where two numbers would be absolutely different even if they differ from one another only in the billionth digit, it is obvious that physical experiments are not going to help us here. We turn to thought experiments instead.
We imagine a real God, creating a real balloon with sextillions of real helium atoms, each of which emphatically apathetic to our desire for beautiful numbers as our solution. And we imagine that balloon at
3 seconds. We think, what would be its height? Definitely not
3*Sin(57) meters. There is no chance in hell that the solution would be as pretty as that. In fact, intuition tells us that the number would be as ugly a number as we’ve ever seen. Next to that number,
3*Sin(57) or even
3*Sin(57)^74*Arctan(0.42^3.5) would look as pathetically simple as
7. No, that number would be so ugly, that… That there won’t even be a way to describe it. And now we’re finally reaching the original thought with which I opened this discussion: Indescribable numbers.
Let me humbly take upon myself the job of being that real God of that apathetic universe; and let me hereby declare, with the full power of my omnipotence, that the height of the balloon in meters is, as suspected, not
3*Sin(57), but rather the following number:
Yes, be impressed by the glory of these digits that I have created in my infinite wisdom; look at them and wonder what pattern might lie behind them, knowing full well that I am too terrible a God to have planted there any pattern you might be able to understand with your limited human mind; stare imploringly into the siren call of the final ellipsis, for you know that no matter how often you expand it, I will always smile when giving you more and more digits, because you and I will both know that the final wisdom of The Number will always be mine and never yours.
Please excuse me, it’s not that often that I get to be an omnipotent being and I’d like to have fun with it as long as it lasts. I’ll retire now and join you in the pathetic human race.
So the question is: Are there really such numbers as
3.749(...), whose pattern is not only unknown to us, but could never be known, will forever be beyond the grasp of us mere mortals? Could there be a number that simply cannot be described by any means known to us? A number which is definitely not an integer, obviously not rational, desperately not algebraic; we know that it is at some innocuous-looking spot on the line of real numbers, somewhere between honest old
3.749 and honest old
3.750, but we know that it is a different creature entirely than these two model citizens of the Reals. Our number carries secrets infinitely more profound, and therefore we know that unlike the above duo, we’ll never be able to pinpoint the exact spot in which this number resides. Like a telephone number that is unlisted in the Yellow Pages; if God didn’t give you the number as he spoke from behind a burning bush, you’ll never find it using the scientific method.
Those were my thoughts as I was wiping sweat off of my forehead and passing the faculty for Civil Engineering. Being 19, my mathematical mind not yet fully developed, I thought: What an interesting philosophical conundrum. That’s pretty cool. I doubt it could ever be solved though; yet another unanswered, vague philosophical question in the long history of unanswered, vague philosophical questions. I got to class, I believe it was Calculus, and concentrated on that instead.
Fast forward a few months.
I was into my second semester by now, which would prove to be my final semester. I had a little bit more training in math by now, given to me not only in the courses I’ve taken, but also from random math-related articles I would read on the web.
It was morning again.
I was taking my usual morning climb from Canada dorms to class, passing Civil Engineering on my right.
That is when the answer came to me. As Zen master Mumon would say: At that moment, I became enlightened.
I found an answer to the question of indescribable numbers, and I found a mathematical proof to the answer. I managed to take this philosophical question, so vague and soft, and not only define it mathematically, but find an actual goddamned water-tight proof to validate my answer.
I was amazed. I was shaken. I could hardly believe this was really happening.
The answer to the question is this: Yes, there are such things as indescribable numbers. In fact, the vast majority of real numbers are indescribable, and only a tiny fraction of real numbers are of the familiar describable variety.
I didn’t really know what to do with this. Is my proof even right? Did I just make a first-rate scientific discovery? Perhaps I’ve rediscovered a known truth, perhaps a theorem my Calculus 3 teacher will teach to us next semester as my classmates will yawn and doodle in their notebooks? Perhaps I am simply deranged?
Then I got an idea who to turn to: My Calculus 2 teacher. He was a friendly old man. He was clumsy, had a huge unkempt beard, and his great love and fascination for math showed through in our Calculus 2 lectures. He was your classic math professor. I would later learn that he was originally from Australia, which helped explain how clumsily energetic he was.
I found his love for math contagious, even when it was hard for me to keep up with the technical aspects, which was often; my fellow Electrical Engineering students, more enchanted by the promise of great money in EE rather than a love for science, were less cooperative with him. It was sad to see someone so passionate trying to inspire those who simply did not want to be inspired. (No disrespect intended to these students; I love money too.)
After the class ended, and all the students were putting their notebooks back in their backpacks and leaving, I went to the professor’s desk. He was packing as well. I told him I had a mathematical thought that I didn’t really know what to do with. He was intrigued. He was too busy at that time to continue the conversation, but he told me to email him, and via email we set a time for an appointment.
I arrived at his office. It was small and cramped and incredibly messy. There were boxes of papers everywhere. Fortunately, there was a whiteboard.
I was excited. I’ve never been to a professor’s office before. I’ve never had a serious conversation about math before with someone who could be actually considered an expert in it. I tunneled my energies of excitement to making sure that I’m explaining the original question and my proof in a clear, calm, but relentlessly watertight way.
I will recreate the explanation of the proof for you now, leaving out the more technical parts.
The proof is not complex; any BSc of Mathematics would understand it quite easily.
The only way us humans have of describing numbers is using language. Sometimes that could be a non-math-specific language like English. For example, “seven” is a description of
7; “the sum of ten and three” is
13; and “the ratio of a circle’s circumference to its diameter” is a concise expression of
Quite often though, we prefer to describe numbers using mathematical language. “
7" is a simple number described by that language. Our previous acquaintance "
3*Sin(57)" is a more complicated one, and "
3*Sin(57)^74*Arctan(0.42^3.5)" is an even more impressive specimen. If I’ve given the impression that sines and cosines are the most exotic symbols that this language has to offer, feel free to feast your eyes here.
This mathematical language is more powerful for describing numbers than the English language, but it’s still a language; every such description of a number is just a finite string of symbols, taken from a finite pool of available symbols: “
3”, then “
*”, then “
S”, then “
i”, then “
n”, then “
(”, then “
5”, then “
7”, and finally “
)”. Descriptions can get much longer than that, as long as they’re finite. But there is an infinite number of descriptions, since we can combine symbols in infinite many ways to make descriptions as long as we want.
Now this is where things start to get interesting. I’ll try not to get too technical. You can read more about the technical points here.
In math, we have several different kinds of infinity. The smallest kind is “countable infinity” also known as Aleph zero. It is the infinity of natural numbers, the infinity of
The infinity just a step bigger than it is Aleph one, the infinity of real numbers: The infinity of an impossibly dense line of numbers, between each two, no matter how close, resides yet another infinite spectrum of numbers, itself bigger than the previously mentioned infinity of natural numbers.
It is well proven that Aleph one, which is the infinity of the real numbers, is undeniably bigger than the infinity of the natural numbers. What this means, is that you can never “cover” the real numbers with the natural numbers. In technical terms, you can’t have a surjective mapping from the set of naturals to the set of reals. In more intuitive terms, if you try to pair up each natural number to a real number, you will run out of natural numbers way before you’ll run out of real numbers. (We can never really imagine the moment where we’ve run out of natural numbers, since there are an infinity of them… But bear in mind that the set of real numbers is “even more infinite”, and that’s the closest I can give you to an intuitive description.)
Let’s get back to describing numbers. We’ve said that our descriptions of numbers using mathematical language are nothing but finite strings of a finite language. There’s infinitely many of them, but that infinity can be easily shown to be Aleph zero, the smallest infinity. Just consider that any such description, like “
3*Sin(57)”, could be saved as a text file on a computer, and every file on a computer is just ones and zeroes.
However, the real numbers have the bigger infinity of Aleph one. You can feel the proof forming, can’t you?
When we take a description like “
3*Sin(57)”, we know it has an obvious counterpart in the set of real numbers: The number it’s describing, the very real number close to
2.51. We can in fact pair every such description to the number it describes. “
7" could be paired with the number
7, and the venerable “
3*Sin(57)^74*Arctan(0.42^3.5)" will also be paired with the number it describes. We’re in effect pairing up each member of the set of descriptions to a member in the set of real numbers.
But, remember what we said before, that if you try to pair up an infinite set of size Aleph zero with an infinite set of size Aleph one, you will never be able to cover the entire Aleph one set. There will always be unfortunate members in that bigger set that would be left without a counterpart in the smaller set.
Those unfortunate members are indescribable numbers.
Why? Because consider what they are: They are real numbers, for which we have just proven it is impossible to find a description that will match them. We have proven that no description will ever describe them.
We have proven that indescribable numbers exist. Q.E.D., and please someone hand me a cigarette.
I explained all this to the professor. He asked a bunch of questions, and I managed to answer all of them without having a “Oh shit, you just discovered a critical flaw in my proof, my entire proof is wrong now fuck me” moment. At the end, when he was quite convinced that my proof was correct, he said something along the lines of “that’s pretty cool.” He said he was unfamiliar with this area of mathematics, but that he thought my proof was correct and really interesting.
I felt so proud. I managed to find a proof that impressed not only me, but a genuine crazy math professor! I’m smart!!!
The professor arranged an appointment for me with a different professor who did specialize in logic, and was familiar with my theorem. He said that my theorem and proof have been well known to logicians since the 1940s. He was still impressed that I was able to prove them despite being a first-year Electrical Engineering student.
I was a bit saddened that my theorem was old stuff for logicians, and not a new discovery. There goes my Nobel Prize…
But I was still so happy just to have found the proof. The experience of having taken a vague philosophical question, and using the precise machinery of math to state it rigorously and then actually prove it, was amazing for me. It was like discovering a net with which I managed to catch a beautiful butterfly. Imagining those indescribable numbers out there, the mathematical equivalent of dark matter, occupying most of the space in the set of real numbers despite being completely invisible and unattainable… I was in love with math.
That episode was one of the reasons that I quit Electrical Engineering and spent the next 2 years of my life studying mathematics. But that’s another story :)