How to explain infinity to kids
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blog.plover.com
The thing is, this kind of infinity just doesn't come up that often when dealing with other objects in math. Even though, as Tom Lehrer sang, one can count up to infinity - or somewhere in that vicinity - and that's mathematics, the question then is - so what?
The other kinds of infinity - cardinals, for example - are encountered early on, and there are things you can do with them.
The first time I've seen the notion of infinity was in a Russian children's book. There, they made a bijection between all the (infinite number) of points in a small segment and a larger segment - and even with all the points of an (infinite) line! I didn't really get it then; while I could find nothing wrong with the argument, it certainly looked like bullshit that a short segment could have as many points as a long one.
But some things don't have to make perfect sense right away.
The next time my understanding of infinity really improved (ignoring the notation for "x growing without bound" of calculus) was in the first year of college, with Cantor's diagonal argument. And I think that's when the picture from the book I read in kindergarten made sense, at last.
The bijection in that picture would have been boring if one could always make it. But with the diagonal argument, one sees that's not the case. That's what makes these infinities interesting and fun, to me.
So, I might be biased in that, but I think that the cardinals are the most playful type of infinity. And really the kind you can explain to kids.
One night I've had a long tea on a rooftop of a Brooklyn apartment building with a friend who is an artist, and by the sunrise, she understood Cantor's diagonal argument - and enjoyed it.
It is quite regrettable that this is the kind of knowledge that's only generally shown to math majors in college. This is reason #712889 why we need to change the way we teach and talk about mathematics.
If the child is old enough to understand basic addition and multiplication, you can probably run through a short explanation of the different (elementary) number systems.
E.g.: on a pasture, there are black and white sheep. How can you know, without counting, whether there are more black sheep than white sheep, the other way around, or there's the same number?
Well, start taking them out in pairs, black and white sheep in each pair. If at some point you have a black sheep, but no white one to pair with, you know there at least as many black sheep as there are white ones. Same for the other way around. And if all sheep can come out in pairs like that, you conclude there must be the same number of them!
My wording is not the clearest here, but you can get the idea. The notion of "same size" for sets via putting things side-by-side is something kids can get before they learn numbers.
What's the name of the first point in the first row? 0
What's the name of the first point in the second row? \omega
How is that more complicated or less interesting than cardinals?
EDIT: Just wanted to add that an order isomorphism has two requirements:
(1) it needs to be a bijection (so order-isomorphic objects have the same cardinality); and
(2) it needs to preserve all inequalities (so a strict inequality among items in one object turns into a strict inequality in the same direction among the corresponding items in the other object).
The notion of the cardinality of sets comes up everywhere in mathematics, and often enough you end up showing that something holds up to a countable number of exceptions. These two kinds of infinities - cardinality of naturals and reals - are so pervasive, you can't get away from them.
But you can do a lot of math without ever having to deal with the ordinal numbers.
For that reason, cardinals are more interesting to me - not just as a concept in and of itself.
There's plenty you can do with ordinals too! Being able to interate a function transfinitely many times can be quite useful.
> So, I might be biased in that, but I think that the cardinals are the most playful type of infinity.
Definitely disagree. Once you know the basics, doing things with cardinals tends to be either boringly easy or impossibly hard. Ordinals, on the other hand, you can just play around with and actually get somewhere.
>There's plenty you can do with ordinals too
Any examples that you could introduce to a kid who just asked you "What is infinity?" - genuinely curious.
Sure, but do you really have to explain that?
When I imagine introducing kids to ordinals, like the OP talks about, I'm assuming it's basically taking the approach in, say, John Baez's blog posts on large countable ordinals[1].
[1] https://johncarlosbaez.wordpress.com/2016/06/29/large-counta... https://johncarlosbaez.wordpress.com/2016/07/04/large-counta... https://johncarlosbaez.wordpress.com/2016/07/07/large-counta...
> Any examples that you could introduce to a kid who just asked you "What is infinity?" - genuinely curious.
Hm, maybe not. Maybe some of the classic examples of weird things that can happen with transfinite-time processes... but explaining any of that might be hard. And also that might not really be the right time to introduce people to discontinuity.
Really like I said I was basically thinking of the approach above, without application. I think it stands on its own pretty well, it's fun, you can play around with it -- it's basically the "is too, times infinity+1!" game except formalized (so it kind of comes naturally out of something kids already try to do) -- and the questions have actual answers.
That question was unresolved for me until recently, when I revisited the question and thought that at some point you could start naming them a, aa, aaa, aaaa, etc instead of seventen hexadecillions five hundred and sixty octillions ... or something like that.
I would love to remember all the other questions I had.
the proof crucially relies upon each real number being represented as an infinitely long (perhaps ending in an infinite sequence of repeated digits for less interesting numbers) sequence of digits. i.e. the representation that Cantor uses for reals is infinite sequences -- effectively representing each real number infinitely long (in the countable sense) strings.
Cantor represents each real number as an infinite sequence of digits. apart from technical details`+` you can think of this as the base 10 or base 2 expansion of the number.
for simplicity, you need only consider counting the real numbers between 0 and 1. there are plenty enough of those. each such number can be addressed in base 2 as some infinite sequence of 1s and 0s after the binary point. Cantor's argument shows that if you try to enumerate all such numbers (i.e. count them) by their series expansion, then you can generate a new number that doesnt apppear in the enumerated list, which is a proof by contradiction, refuting the assumption that you could enumerate them all in the first place.
`+` :
technical details include that some numbers have non-unique representations as infinite sequences of digits.
For example, in base 10, 1 can be represented as 1.000... (where the 0s keep going) or 0.999... (where the 9s keep going).
if this irritates you, let z = 0.999...
then 10 * z = 9.999... then 10 * z - z = 9 then 9 * z = 9 so z = 1
one of the smaller kinds of infinities is "countable". we say a collection of countably infinite things is countable if, intuitively, we can count them (as the parent says, put them in 1:1 correspondence with the natural numbers 1, 2, 3, ....).
there are only a countably infinite number of rational numbers, since each rational number has the form p / q, where p and q are (perhaps negative) integers. if we made a large 2d grid of all integer grid points, we could regard each rational number p/q as a grid point (p, q). Then we can count the grid points by starting at the origin of the grid (0, 0) * and spiraling outwards. This will eventually count every grid point, so this gives us a way to count all the rationals.
as the parent post says, by Cantor's diagonalisation argument we can demonstrate that we can't count the real numbers, so there are a lot more reals than rationals. it's a strictly bigger kind of infinity.
even if we start inventing new notation for particular reals we care about (e.g. pi, e, pi^e, door, super(door, |^|^bat)man, ) -- whatever you like provided it is well-defined, we can only name at most countably many reals, leaving a remainder of uncountably many un-named reals. we can single out any particular real that can be well-defined, and mint a new name for it, but we can only do this for at most countably many such reals, while the bulk of the reals escape naming.
* the origin (0, 0) corresponds to 0 / 0 which isn't a rational number, and some rational numbers such as 4 have multiple representations as coordinates. For example we could write 4 as 4 / 1 or 8 / 2 or 40 / 10 or -16 / -4 ... so strictly speaking by demonstrating we can count all of the 2d integer grid points shows that there are at most a countably infinite number of rationals. but since each natural number is a rational, and there are countably many rationals, we know there's at least a countably infinite number of rationals.
A better way to count based on names would be to first count all single letter names alphabetically, then all two letter names alphabetically, etc.
Obviously we can order reals (using <), I interpreted the original post as referring not to your definition but to "order and count"
This also maniests itself when people talk about infinite universes. "If there are infinite universes, there has to be one in which X". Not necessarialy. Maybe only even universes exist and universes in which X is the case might all be odd universes.
In all these infinite universes, surely there's one where there's an ice cube in the middle of the sun, since that's one possible configuration of matter.
But no! There's no story that ends with an ice-cube in the middle of the sun. There's no sequence of events that could have resulted in that so it's not in the space of possible universes.
One scenario I would not rule out is a large icy comet moving at large speed straight for a sun. Is there no combination of size and speed for which that would result in a small ice cube in the centre of a sun?
Going back to the sequence of events argument, I think that implies that all universes agree on similar rules like ours (Physics), as we could only deduce this kind of argument with this assumption.
Maybe there is a universe where ice is scorching hot and all sun cores consist of it, and as matter cools it evaporates? Maybe there is even a universe where gravity and orbital mechanics work in a way to produce cube shaped stellar bodies?
If you teleport a block of ice to coordinates which happen to be smack dab inside a star, then maybe for an instant, like a single "tick" of the processing engine, there would be an ice cube in the middle of the sun.
Could there be? This assumes that
1) different universes have different physics
2) there is at least 1 self-consistent rules of physics that allows this. I suspect that this is a harder problem.
It's fair to say that if our universe has uniform physics, then even if it is infinite, there are no solar ice-cubes anywhere in it.
> 1) different universes have different physics
Yes this is the presumption. Remember infinity allows all possible skewing of our current physics and all possible permutations of it. Further reading: https://en.wikipedia.org/wiki/Multiverse
No, it does not. Compare:
> If there are infinite numbers, and all numbers have a name, then there must exist a number with a different kind of name to the usual ones.
> If there are infinite universes, and all universes have laws of physics, then there must be universes with different laws of physics.
It's the same basic fallacy. There might universes with different physics but it does not follow from infinity.
Ah, this is the philosophical distinction between every possible thing, and every conceivable thing.
The set of possible number names may be very large or infinite, but it is assembled from a small set of symbols like "one", "two" "seventeen" "million", "and", "times" and "three" "to the power of", etc.
A number called "door" is conceivable (like superman) but it is not a possible under the rules of maths (or physics in superman's case) because "door" is not in that set of symbols.
The set of the possible is a subset of the conceivable. Both sets might be infinite.
Certainly you are able to represent "door" in a base-64 numbering system? (or base-36, or even base-28)
And it's rules in which "door" is a "numbery name". And "Достоевский" is not.
You could still ask "If there are infinite base-64 numbers, and all base-64 numbers have a name, then must there exist a number called "Достоевский" ?" and the same argument against would apply.
>If there are infinite numbers, and all numbers have a name, then there must exist a number called door
If this is the best "explain it like I'm 5" explanation of infinity, I believe I can think of a few examples that give a better idea of it. Heck, even the concept of "never ending" seems simpler to me. "Never ending" + 1 is still never ending.
This is actually the definition that the Simple English wikipedia article [1] gives. The very first sentence in that article:
> Infinity is about things which never end.
"ω" is not almost 0, that would be epsilon "ε". Omega ω is a number higher than any other natural number, i.e. "to the right" of the infinite line of numbers.
Mathematicians "will" this number into existence, so to say, starting from a contradiction. It's the same that they do with irrational numbers ("imagine there's this number "i" that, when multiplied by itself, it gives you -1"). With infinity, it's like: "you know this process that never ends? Well, imagine that it finishes, and let's call the result "ω".
Once they have this new number defined, they do lots of mathematical operations with it, trying to find its properties. What they never remember again after that is that the number dit not appear as the result of following the initial process to completion; they had to assume that it existed independently from the process.
* BTW, this is also why they have different kinds of infinities. They are using different never-ending processes in their respective definitions, and using the same name for all of them.
Also, it's nitpicking, but i is imaginary, not irrational. Also, complex numbers were accepted by mathematicians before negative numbers (it's something that boggles minds of some people).
This is the opposite of what Feynman thought. He said, and I agree, something like "the hard part about teaching is making concepts simple without saying things that are false." The whole challenge of teaching is NOT lying, but instead saying simple things that are still true.
That said, I see the appeal of choosing omega.
https://chem.tufts.edu/answersinscience/relativityofwrong.ht...
What do you mean, a number I can't count to? If I'm 8 years old this is like throwing a null pointer exception in my brain.
This is only because they/you haven't tried.
Source: counted to a million once.
That stops being true long before you reach the finish line. Try timing yourself counting from 147,895 to 147,900 and see how long it takes.
Well put! I'm in the process of explaining my 3yo daughter different molecules (there are some cheap kits on Aliexpress) and cells and life on a micro scale and this description of moving from simple but inaccurate models to more complex and accurate is something that I've also noticed in my explanations.
What Physics Teachers Get Wrong About Tides! | Space Time | PBS Digital Studios: https://www.youtube.com/watch?v=pwChk4S99i4
IIRC, one of the examples from the book thing had to do with the explanation and illustration of how light gets refracted when passing through glass. The overall effect is supposedly explained by the fact that light slows down as it enters glass, and typically a wavefront illustration is used to depict this. He went on to explain how the illustration is wrong and what the correct illustration should be instead, but I don't remember the details. And now that we have meta-materials which can bend light the wrong way, the whole "slowing down causes refraction" idea may be wrong, too. In fact, I remember reading an early potential explanation of this new effect (which as I recall had to do with the notion that the magnetic aspects of light might be affected differently than how the electrical aspects of light are, or vice versa) and thinking to myself "Aha - now that makes perfect sense!" And if that explanation makes sense for meta-materials then it also probably makes sense for regular materials. But once again I don't remember the details.
BTW, I understand the need to often simplify things quite a bit when you're dealing with students, but if we have "simple but incorrect" vs. "simple but correct" then we should be working hard to eliminate the former as soon as possible. If his critique of the situation is valid then it kind of beggars belief that we are still teaching so many things the wrong way!
> Well put!
Shudder. Consider a thought experiment - a military briefing. A captain briefing generals. One must necessarily simplify. But imagine a briefing that is grossly incomplete, assortedly incorrect, very misleading, written without understanding and without mentioning and characterizing that lack, and pervasively incompetently bogus, and that captain later on the carpet before the generals, defending the briefing with "well, all briefing is a process of lying". Shudder. I consider the "teaching is lying" meme to be vile. I've heard it most often associated with pre-college chemistry. Chemistry education research describes pre-college chemistry education content as "incoherent", leaving both students and teachers deeply steeped in misconceptions. Oh well.
You mentioned microscale, so I write just to note the top "How to remember sizes" section of my slowwwly-loading wasn't-intended-to-be-public page http://www.clarifyscience.info/part/Atoms . It might help you provide a framework for your kid to think about small things. FWIW.
Couldn't agree more. And when I read this in the article I also thought of high school chemistry even before I read your comment. I was put off chemistry in high school precisely because of its incoherence.
The best thing to do, as always, is to be honest. Tell your students that what you are teaching them is a simplification; a model that is useful at this level. Many will be satisfied with this answer and not want to go to a deeper level. But some children, those with a greater need for coherence and for things to make deep sense, will be reassured to know it's only a model and either wait for the deeper model to be taught or want to start exploring the deeper model at that time. Which is fine.
I found the approach taken by the author of this article quite patronising and rather distasteful.
I also do not like his suggested definition. It comes across as smartarse-ish and I think most children would feel the same way.
teaching is a process of lying is kind of provocotaive, but teaching is a set of progressions, each progression necessarily leaves the edges blury while trying to make one aspect clear.
There's a massive difference between sharing/communicating intelligence and teaching. The former relies on a common understanding of terminology, technique, etc, the later involves one side attempting to understand all of that to be able to facilitate the former.
> infinity (date, timestamp) later than all other time stamps
I think you could tell a kid who wasn't quite ready for Aleph numbers that:
> infinity is a useful made-up number that's bigger than all other numbers
Which is useful when kids first hear about it, I guess.
I think my first "practical" introduction to infinity came with "Space is infinitely big", and both mjd's explanation and my own fails at that point.
I love the emphasis on this being a choice, and I wish this kind of thinking was taught more in math. So many math explanations act like these things are immutable facts of the universe, rather than human constructions. We lose some of the history and character of math when we teach it as law rather than invention.
Zero is also a useful made-up number, and it didn’t always exist as a concept. We understand 0 so deeply now, it’s not longer possible to imagine a world without that idea. Another fun one, 0^0 gets defined (chosen) to be 1 sometimes, because it makes other formulas work out, not because it’s the correct or only answer.
ω - 1 makes sense in the surreal numbers (https://en.wikipedia.org/wiki/Surreal_number). The surreal numbers are the "greatest" ordered field, in a sense. They contain all ordinal numbers, which in turn contain all cardinal numbers. The cardinality of a set is just the least ordinal it can be put into a one-to-one correspondence with (this is called the von Neumann cardinal assignment).
How is 'adding one more' not the same thing as counting?
Chapter 15: Paradoxes of probability theory in Jaynes's "Probability Theory: Logic of Science" is a great reading on the topic (you can find a pdf easily on google). It starts with a quote from Gauss:
"I protest against the use of infinite magnitude as something accomplished, which is never permissible in mathematics. Infinity is merely a figure of speech, the true meaning being a limit." -- C. F. Gauss
Anyway, there are plenty of theories in mathematics which use infinity implicitly, but one should perhaps be cautious.
Consider limits. One can say that a function converges to a certain value v when its input approaches infinity. This seems all subjective and mystic and non mathematically trained people would come up with all sorts of interpretation of that. But this just means that no matter how small a number ε you chose, then there exists a number x such that for any input greater than x, the function's output will be contained within v-ε and v+ε. That's it. It's as simple as that. There is no mystery to it, nor judgment. "Infinity" is just a name that is involved in such formal property.
- No, you can’t do 2-3. If you have 2 apples, you can’t give 3
- Well, in fact you can, it’s negative numbers. But if you have 5 apples you can’t split them equally between 2 persons.
- Well, in fact you can. They got 2.5 each. But you cannot solo 2 apples in 3 equally !!
- Well in fact you can. That’s 2/3 each.
- etc...
Well, how soon is too soon to teach children about debt?
https://betterexplained.com/articles/a-visual-intuitive-guid...
It communicates that you can't work with infinity like with a number, because it is a group of numbers.
It swallows all the operands as infinity does: 2 groups = still a group; half a group is still a group; a group plus one is still a group, etc.
Then you have groups of groups, or infinity powers which work the same way. A group of groups is clearly greater than just a group.
At least, I think this is what he meant. He never really expanded on it beyond the garden analogy.
I see a problem with saying that. One of my earliest troubles when dealing with infinity in algebra was understanding, that you cannot add or subtract real numbers from infinity to make it something else.
e.g. inifnity - infinity is not 0. Suddenly saying that infinity + 1 \neq inifinity would just make it more confusing.
The article is a nice mind exercise but IMO not really helpful in explaining infinity to a child.
Disclaimer: I'm an adult that doesn't understand infinity. If you "add one more" you're still counting.
The correct form should be "all the numbers reachable by counting in finite time".
E.g. you can count up to 100 in a minute. Up to a million in a month[1]. Up to a billion in quite a long, but finite time. You take all such numbers, and you say that a number named Omega comes just after all of them (just like million and one comes just after all numbers that are less than or equal to ine million).
Omega is your first infinte ordinal - or, simply, infinity.
And now you're counting in a new way.
- ok, take number n, it's smaller than omega, so what about n+1, n+2, etc. ?
You should be telling yourself - ok, take number n. It's smaller than omega. Now check any other number m. If this property holds for any given number, then omega is greater than any given number, and thus infinite.
The new type of counting also allows you to take a different step and say, "take a collection of things and add a new thing at the end" (a limit ordinal). It's how you get from the set of finite numbers to ω. There's nothing that's "one before" ω, but all the finite numbers are before it.
Repeat these steps as much as you want (making sure none of your sets are circularly defined), and you're counting with ordinals.
[1] https://www.babelio.com/livres/Ekeland-Le-Chat-au-pays-des-N...
Funny. If explaining infinity to a kid had been my problem, I'd have tried hard to avoid calling infinity a number.
I can't wait to use his answer to what infinity is to my kid when he gets old enough to ask such a question.
He has a pretty solid understanding of basic addition, subtraction and multiplication.
He was asking about infinity and although he knows it's very big he was still asking about infinity plus one. Maybe it's not too soon to explain the concept.
but that's much the same as "It's the smallest number you can't count to."
Imagine you are 2 feet from the wall, and with every step you move forward 50% of the remaining distance. How many steps does it take to the wall.
And the answer of course is that you never get to the wall, no matter how many steps you take
There are infinitely many steps on the interval [0,1]. But if you add 1 "step" at any point of the interval greater than 0, you've still "passed" it.
I don't remember whether I was ever taught that. But I do remember the line from Marilyn Manson's "Posthuman".
> God is a number you cannot count to.
In philosophy of sciences, it is called "idealization" and "concretization". Many philosophers from Poznan even name this philosophy of science as "Marxist philosophy of science", since Marx uses the technique of idealization and concretization in his work "Das Kapital".
that's how
If infinity + 1 exists, then that number would be infinity. It’d be more accurate to say that Infinity + 1 = NaN
Just ask them to do i++ and never stop. That is also infinity.
(or in simpler words, imagine you have a table than you put one apple to it, then one again, amd again and again and never stop. And yes, they would also soon understand, that the table needs to be infinitly big)
edit: of course it is about the sum of the process. they should imagine the pile of things or the number if you never stop adding.
I love ops explanation because it gives more natural picture: there are _other_ numbers beyond naturals. Then you tell kids there are also negative numbers (integers) and rational numbers and real and complex and so on. I believe that way it is much easier to grasp that very abstract (but fundamental) idea of 'number'. That it is not just 1,2,3...
Because otherwise, we get people thinking "Complex numbers do not exist, that is just a silly thing used by mathematicians that has nothing to do with the real world, it is useless."
The idea that real numbers are real, imaginary numbers are not real, and complex numbers that have a real number part and an imaginary number part are also not real is a natural consequence of unfortunate naming choices.
For me, back in school days (or was it university?) it was a revelation when I came across quaternions. It suddenly clicked. I finally understood that there was nothing special about complex numbers (despite their special names and weird look). It was just an extension and a very intuitive one!
I finally understood that real numbers are the same thing: tuples. They just happen to have exactly one element, hence we omit parens and everything else and just write that element (number)! Complex numbers have 2 elements (real and imaginary). Quaternions - four. And so on.
Yes, but if you go too far with thinking this way it becomes easy to confuse complexes, quaternions, octonions, etc with 2-,4-, and 8-dimensional vectors. The additive and multiplicative behavior of complexes, etc changes in increasingly pathological ways as you increment the dimension. This is not the case with vectors.
In many cases you can safely replace R^2 with C. But there are specific cases where you cannot, because treating the complexes as just a pair of real numbers doesn't work the same way as if it was just a vector. Differentiable functions come to mind because they behave differently in R^2 than C, and when you're working with rings (instead of fields and vector spaces) they are also different. In a lot of places the isomorphism between R^2 and C is actually a happy accident rather than a definition.
And I thought we are talking about kids? There are not many kids who would understand the professors way.
So isn't it first about making them understand the concept of infinity? That they can imagine it, before you bombard them with other abstract math concepts?
I love ops explanation because it separates infinity (as a number) and infinite process (counting).
Because ifinity is complicated.
>Just ask them to do i++ and never stop. That is also infinity.
What, specifically, is infinity here? You are being vague here. Is it the process of counting up and never stopping? (that's the calculus infinity) The number you get in the end? (that's the infinite ordinal, but that is, as you noted, complicated) The number of numbers? (that's the cardinality of the set of integers)
Note that you can't say that the infinity is all that. The infinities I defined here are all radically different notions!
>In simpler words, imagine you have a table than you put one apple to it, then one again .. and never stop. [..] they would also soon understand that the table needs to be infinitely big
That's another infinity here - we're talking about size of geometric object here.
For starters, what happens if each next apple is half the size of the previous one? (Your assumption is false then!).
With equally-sized apples, it's still close enough to the notion of cardinality.
The problem with this explanation is that it doesn't help understand the nature of infinity, so it's not an explanation really - but a good start.
Some questions to explore from here:
1. Do that process twice. Now you have two infinitely big tables, but each has an end. Put them together end to end. Did you just get a longer table? Are there more tables on this combined table than you had on just one table?
2. Perhaps after you're done putting apples on the table, you realize that one apple rolled under the table. The table is absolutely full, but you want to put another apple on it. What do you do?
3. You find a pile of apples, each having a label consisting of a rational number on them. You notice that no two apples have the same label. Will they fit on the table? Can you put them without removing any apples already them? (The answer here is, mind-bogglingly, yes!)
4. Same question, but now the labels are infinite strings. (The answer is no, by the way, but it's by no means ovbious![3])
Etc, etc, etc... Essentially, Hilbert's Grand Hotel[1][2] with apples. I highly recommend reading this to get a decent understanding of one of the infinities.
[1]https://medium.com/i-math/hilberts-infinite-hotel-paradox-ca...
[2]https://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert...
"Do that process twice" "Perhaps after you're done putting apples " "..."
the point is, that the process is never finished ... just like you can never have a biggest number...just like the table has no end.
Defined as infinity. Kids do get that.
What they usually don't get are abstract mathematic concepts.
Thinking back to that bug last week, where someone forgot that you can't just compare ever-incremented sequence numbers with 'x < y'... Maybe we don't want to encourage that idea :)