One of my favorite quotes about mathematics is from John von Neumann:
“In mathematics you don’t understand things. You just get used to them.”
On one level, this runs against everything I believe as a teacher. Mathematics should not be an intimidating collection of inscrutable methods! It should be a timidating collection of scrutable methods! We should accept nothing on authority. Everything in mathematics is there to be understood.
And I do believe that.
But I also know that mathematics is full of startles and shocks. I know that even the simplest objects can bury deep secrets.
Take the number line.
Boring, right? Nothing could be more prosaic.
Well, that’s only because you’ve gotten used to it. To my mind, the number line merits only one possible reaction:
It’s not the integers that spook me, although those are plenty weird.
To wit: although we distinguish numbers like “4986” vs. “4987,” not all human languages are so fussy. After all, does the difference between these numbers really matter? Are you good enough at counting to reliably tell one from another? The fact that we assign these distinct identities is a bit of a mind-popper.
No, what keeps me up at night is the stuff between the integers: the numbers a 12-year-old student of mine recently dubbed “the disintegers.”
When we draw a line connecting 0 to 1, what exactly are we drawing?
You might think we’re drawing a line of fractions. After all, fractions are like bedbugs: tiny and everywhere, living in every nook and crevice you can imagine.
And yet there are lots of numbers that aren’t fractions, perhaps the most famous being the square root of 2.
The square root of 2 can’t be written as a fraction (by which I mean, one whole number divided by another). Sure, you can get close – say, 665,857 / 470,832, which is the square root of roughly 2.0000000000045. But you can’t get the square root of 2 precisely.
Okay, so the disintegers aren’t all ratios of whole numbers. But that’s okay. Maybe, like the square root of 2 (which is the solution to the equation x^2 = 2), they’re just the solutions of equations. That’s not so bad, right?
We call such numbers – the solutions to simple polynomial equations – algebraic.
Unfortunately, some numbers escape even this widened circle.
There’s no nice algebraic equation that gives you pi or e. These numbers are called “transcendental.” For them, the best you can do is to describe a process by which a computer could, in theory, compute the number.
(For example, if you want to compute e, you can take a big number n, find 1 + 1/n, and then raise this sum to the nth power. It won’t give you e precisely, but the bigger the n you choose, the closer you’ll get to e.)
At this point, we’re forced to widen our conception of “number” again. Numbers, we shall say, are “things you can compute.” Maybe they’re not fractions. Maybe they’re not even the solutions to equations. But they are things that you could tell a computer to calculate.
Unfortunately… that’s not quite right.
There are non-computable numbers.
In fact, almost everything on the number line is non-computable.
What does this mean?
If you pick a random number between 0 and 1, you will arrive at a number that literally cannot be communicated or expressed. Sure, you can list the decimal digits, but if you ever stop, even after a trillion years, then you will have failed to specify your number – and there is no process by which you can tell a computer to carry on the job.
These are numbers that cannot be touched by the human mind.
And this is most numbers.
Well, you know what they say: In mathematics, you don’t understand things. You just get used to them.
FURTHER READING:
Fawn Nguyen taught a badass lesson where she posed kids the philosophical and mathematical challenge of finding the square root of 7 on a number line.
Simon Gregg calls noncomputable numbers “the dark matter of the number world” and points out that biology has its own version of the “most of what’s out there is beyond our imagination” problem.
Also worth reading is Mr. Sock Monkey’s comment below, which speaks for us all:
monkey still not sure of definition of word number or if number have existence independent of human mind. when monkey read definition of number by mr russell it make simple brain of monkey spin.
I think it frightened zeno of Elea to ! Thanks for another good post.
Great post!!! Thanks for giving me something to ponder today!!!!
Nice post. Thanks.
On the other hand, isn’t it true that non-computable numbers are essentially meaningless to human beings? I mean, they exist for completeness of the real numbers, but otherwise, if they’re non-computable, they can’t be the limit of a definable sequence, or in any other way serve a purpose in a proof, can they? In other words, we may not be able to get our heads around them, but would we ever need to? We can just leave them in the wild, untouched by mathematical domestication.
I disagree. There is so much in math and science we just don’t know. Maybe if we could wrap our heads aroubd bon-computable numbers they’dbecome useful in answering our questions and broadening our scientific knowledge.
…or maybe not. But we really don’t know.
What does it mean to “wrap our mind around non-computable numbers”? What, exactly, is there to know? They share exactly the same properties as the reals you and I know. We know how many of them there are. We know… I don’t know, what else is even there to know? What do these numbers have that’s so special? If anything, it should be the computables that are special, much fewer in number, weird because they have a finite description.
The point is that we don’t know. At one point in history and all the years before that, people didn’t know matter is made of atoms. People had trouble wrapping their minds around itty bitty tiny particles. Even Rutherford’s foil experiment didn’t make complete sense. How could there be any space within something that seems solid. But eventually we came to understand it as electrons that “orbit” a nucleus. We just don’t know…yet. Until someone researches and shows us.
(I am replying here because, for some reason, I can’t seem to reply to your other comment.)
It’s worth mentioning that nothing about non-computable numbers “doesn’t make sense”. Yes, they are counter intuitive, but we understand them pretty well. About as well as we understand the computables, at least.
There is no big mystery here. These numbers are not obscure unknown entities. They are just entities with the not particularly remarkable characteristic that we can’t give them a finite name or description, is all.
Meaning isn’t necessarily defined by function, so I would disagree. Neither does perceived “meaning” necessarily trump factual existence. That the world is round once mattered not even a little bit to human beings, who went about their days and “meaningful” tasks just fine assuming it was flat. But it was still round.
The universe depends on values of objective physical reality, such as spheres in space being round and acting upon each other accordingly. We could argue that this “matters” much more to the universe than whether you or I don’t realize it one millennia but do the next.
It isn’t so much domestication as description. There are only as many ways to describe a number, whether in language, as the solution to an equation or as the result of a computation, as there are integers. That’s because descriptions, equations and programs are strings of characters. There are a lot more numbers than the integers, so most numbers can’t be described.
On the other hand, we need them. Without all those in-between numbers there wouldn’t be any distance between zero and one. It takes a lot of numbers to put distance between zero and one. There just aren’t enough of them that can be described. Still, if we threw out every number that can be described, there would still be enough left over to put some distance between zero and one. There are just that many that can’t be described.
Non-computable ?
Well pick one then.
You can’t. If you could give any one of them a finite description, then it would be computable.
The reason we know they exist within the real number line, mind you, is because there simply are too many real numbers and too few possible algorithms. Since, by definition, any computable number has a finite description, the set of computable numbers is , in some sense, a subset of all possible finite strings in some given (finite alphabet). In less technical terms, we can assign every computable number to a finite description. And the thing with these finite descriptions is that we can make an infinite list of all of them, and any given description will be *somewhere* on that list.
But we can’t do that with the reals (read: Cantor’s diagonal proof) and, as such, there must be numbers we are missing. Numbers which are not in our list of “numbers with a finite description”. Those are, roughly speaking, the non computable numbers, and there are way more of those than computable ones.
Yes, but then what is the difference between i.e. pi number and such non-computable numbers?
I mean we can compute pi to an arbitrary precision (but never exactly), but still we cannot give exact value of it. Therfore we can caluclate pi because of some fortunate relations between the geometrical/aritmetical/physical entities.
Because as I see it the only difference between pi and is the fact that the pi is expressed for example as a ratio between circle length and its diameter. Same with e number (but with different relation). Therefore how can we be sure that there aren’t such formulas for all of the non-computable nubmers? Can we prove that enough such entities do not exist so we can find relations between them and therefore for this non-computable numbers, such as for pi or e?
Please pardon me if the answer is in the post above – i didn’t understood it then.
Here is the difference: if a number is computable, then, roughly speaking, there is a procedure to get us as many digits of the number as we want; as good an approximation as we want.
The point where the non computable numbers differ is that there simply is no such description for them, or arbitrary accuracy descriptions of them.
As I said, we only know they exist, not because we have constructed them (like the proof for existance of irrationals, though the same train of thought applied here can certainly be used to prove their existance), but because we simply looked at the reals and at the ways we describe numbers and though “there are too many real numbers comparatively to descriptions. Thus, there must be undescribable numbers.”
This seems weird, because, in theory, we should be able to describe an infinitude of numbers. And we can: the set of computables is infinite. But, as they say, some infinities are bigger than others, and the infinitude of descriptions we can make is nothing compared to the sheer mind-boggling infinitude of real numbers.
There are a LOT of real numbers, by the way.
The set of programs/algorithms/descriptions is countable. The set of computable numbers is defined as the numbers that can be described through a program or algorithm. Therefore the set of computable numbers is countable.
The set of real numbers is noncountable. Therefore there must be a noncountable set of real numbers that are not computable.
Specker Sequences are an example for definable but non-computable numbers.
monkey still not sure of definition of word number or if number have existence independent of human mind. when monkey read definition of number by mr russell it make simple brain of monkey spin.
Oooh, Real Analysis. The subject that separates the mathematicians from the engineers.
I think there were a few mind benders in that course. There are more real numbers than rational numbers, yet between any two real numbers there is a rational number.
As far as the non-computable numbers go, I think of them as objects waiting to be called on. If I need a number, even if I don’t know how to express it, I know that it is there (and feels nice) But as soon as I call on it, it must be computable!
I am a fan of Dedekind cuts, personally.
http://Www.webmail.Nsbbpo.in
I like to explode a few brain cells by asking my students how many numbers there are between 0 and 1 and we do this the same way. Eventually we begin talking about infinity. But when I ask them how many numbers are between -1 and 1 (or on the entire number line) and it is also infinite, you see them start to think. You have to love the kids who say “two times infinity”.
Funny how when you define God as something that doesn’t have a finite description (and therefore categorically imperceptible), you clearly are talking nonsense. But when you talk about mathematical objects that don’t have a finite description, yet still somehow exist, you’re making sense.
Platonism is weird like that.
doesn’t really work for me.
I was disappointed by what I thought was a promising title.
All you really mean to say is that there aren’t neat and tidy ways to write most numbers in two or more forms.
Every number can be written as it is, and sure the number of numbers drifts off into one of the many possible infinities, and yes the number of digits in most numbers itself also drifts off into its own infinity.
And sure, this makes it rather awkward to write down such numbers in an alternative form that is more convenient than the particular infinity any particular number is presenting you with with its particular string of digits.
So?
really all you are trying to say is there are fewer numbers that we can write and use neatly and conveniently in various spare forms than there are long and unwieldy numbers that we can only write with some meaningless level of precision in an extraordinarily long form.
Big deal.
The more interesting point is just that our world lies in a particular scale within the universe. This means that things far larger and far smaller than the general scale that we operate in are just not that meaningful (generally) or useful (readily).
And as far as your question about whether two large numbers separated by a single whole number are meaningfully different? well, you have contradicted the main point of your whole post. Context matters. So sure any too large numbers, one single number apart can be meaningfully different, or meaningfully the same, depending on,
That’s right, scale.
The same point applies to your observation that we can’t write down the square root of two in a tidy abbreviated form. Seriously is 1.4 so different from the square root of 2? or 1.41421356 so different than either?
Well,
it depends.
Get your questions straight.
I never thought I would be saying this but- this is one of those “math things” that I can talk about with my high school students (I currently teach only geometry) and most of them actually stop to THINK about it. In this age of teach/test/repeat all they care about is how to do a specific type of problem with a specific formula and heaven forbid if the next problem isn’t exactly the same. I have the math degrees but my students don’t. I want them to see the beauty, magic, and mystery of math, and if I can get them to think and question now, maybe, just maybe, there may be a few math majors coming back in 7 years to say hi.
Your posts are fantastic! Thanks!
I once had a VCR [that’s video cassette recorder, for you young folk who have never seen one] that, when recording the position an a tape, went ….-3, -2, -1, -0, +0, +1. +2, +3,….. Think about it – yes, there is a +0 AND a -0. There should have been a -0 BC and a +0 AD – that;s why the first century only had 99 years, and we have all these arguments over whether the century starts on 2000 or 2001.
Oh, your post makes me happy….I am in the mystery of medicine where the information changes…..and now all sorts of people are taking probiotics who would not have dreamed of doing so 20 years ago….
I am not sure I understand the discussion of the various classes of numbers in the post and the comments. I don’t think Von Neumann said what he said jokingly. We don’t really understand numbers, we just get used to talking about them. We can say, it is convenient to introduce a number which is a solution to the equation x^2 = 2, but try to explain what this really means to a kid. And once the kid accepts your idea of introducing numbers which are solutions of equations, she could say “So we can then introduce a number that is a solution to x^2 = -2 as well, can’t we?”. Now you are stuck with trying to explain the difference between “real” and “imaginary” numbers, and while you are doing that, you start thinking “OMG, it seems, all numbers are imaginary. They don’t really exist, do they?”
But we can step back one step: how many kids do really understand fractions? What are fractions?
And then one more step back: do kids really understand what 1, 2, 3, really mean? What are these things?
I think, kids (and most people) just get used to counting things without understanding what numbers are.
Even more provocative is the question: do mathematicians understand what numbers are? I claim, most mathematicians don’t think about what numbers are, and certainly don’t think about if they exist or not.
I don’t think numbers exist. Even 1, 2 or 3 are just in our imagination, and we introduced them out of convenience, so that we can describe our world.
Similarly, we introduced square root of two as the name of the length of the diagonal of a square with side length 1.
Similarly, we introduce real numbers since it’s convenient to talk about limits and continuity.
What do you get if you add 1 cup of milk to 1 cup of popcorn? So what exactly is 1?
By the way, did you think about adding the latex plugin to your site? https://wplatex.wordpress.com/
Love the ‘bad’ art and the idea of digging deeper into the impact of science and mathematics to how we really see things.
Hahahahaha… OMG that cracked me up! I actually love the number lines. xD
Your doodling is amazing. 🙂
uhh realmente mis neuronas están alborotadas, estoy buscando una forma de iniciar sistemas numéricos, cero que me dió una idea, mechas gracias
… then you have to start thinking about the numbers that don’t even get invited to the number line, like i. Poor i.
“To wit: although we distinguish numbers like “4986” vs. “4987,” not all human languages are so fussy. After all, does the difference between these numbers really matter? Are you good enough at counting to reliably tell one from another? ”
This made me think of that joke you probably know:
A dog goes into the post office to send a telegram. The man behind the counter asks him,
“What do you want it to say?”
“WOOF WOOF WOOF WOOF WOOF WOOF WOOF WOOF WOOF”
The telegram man says, “That’s just nine words. Did you know you can have ten words for the same price? I can add an extra ‘WOOF’ on if you like?”
The dog looks at him, confused, and says, “But that would mean something completely different.”
Ha! Hadn’t heard it. Like it.
But pi IS a rational number, it is the circumference of a circle divided by its diameter, ergo a ratio. Just because you can’t write the ratio with integers or in a finite amount of time is immaterial.
Also, it bugs me that we are going on “faith” that these pi algorithms are giving us the correct number. Has anyone really measured a circle precisely enough to verify?