How to explain infinity to kids • Hacker News Board

Re: How to explain infinity to kids

#131

Post by enedil » Fri, Aug 03, 2018, 9:52 PM UTC

jgeralnik wrote: post #109

jpulgarin wrote: post #78

Earlier quoted context omitted.

If you're referring to real numbers, then virtually no real numbers have a name. If all real numbers had a name then you could order them alphabetically and put them in one-to-one correspondence with the natural numbers, which we know is impossible: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

Not quite, you can't order all natural numbers alphabetically either but that doesn't mean you can't count them. 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.

Yes, you can. Order is just a relation, i.e. a set that contains pairs of elements. So for example, let X be the relation <. X is a subset of NxN, the cartesian product of N. We say that n<m if the pair (n,m) is in X. Moreover, we say that (n,n+1) is in X, and whenever (n,m) in X and (m,k) in X, then (n,k) in X. This constitues order on naturals.

Re: How to explain infinity to kids

#132

Post by enedil » Fri, Aug 03, 2018, 9:56 PM UTC

TuringTest wrote: post #92

Izmaki wrote: post #28

I don't understand this explanation, and I'm an adult with an interest in mathematics and a degree in software development (although not mathematical per se, they usually go hand-in-hand). I like how "w" is the smallest number you cannot count to - if I understand it right it's almost 0 but not quite - but I don't understand the bonus questions and answers. If this is the best "explain it like I'm 5" explanation of i…

I believe that approach comes naturally to us software developers, as we tend to think about mathematics in procedural terms (which is how programs run in the execution environment) rather than in string rewriting (which is how mathematical proofs are made). I also consider infinity as a process that never finds its ending condition, my mental model is the "infinite loop". "ω" is not almost 0, that would be epsilon "…

Notice that mathematicians don't take real numbers for granted too - we need to assume their existence first. Looking at this, complex numbers stop being that arbitrary, just another extension of reals.

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).

Re: How to explain infinity to kids

#133

Post by cokernel » Fri, Aug 03, 2018, 10:17 PM UTC

empath75 wrote: post #48

gowld wrote: post #42

Earlier quoted context omitted.

Get a sheet of graph paper. Imagine that it goes on forever in 2 of the 4 directions. Name the intersection points in order -- left to right, top to bottom. 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?

I guess I’m mixing up ordinals and cardinals but it seems odd that if you order the points differently ((0,0), (1,0), (0,1), (2,0), (1,1), (0,2), etc...)you never get to omega and cover all the same points.

Yup, you never get to \omega that way. Cardinality is a coarser notion than "ordinality". So your example shows that Card(\omega * \omega) = Card(\omega) = \aleph_0, even though \omega * \omega and \omega have different order types.

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).

Re: How to explain infinity to kids

#134

Post by enturn » Fri, Aug 03, 2018, 10:26 PM UTC

I like to think about the length of the outside of a circle. It's only when you use a line with a beginning and end in the shape of a circle that you can measure it. The idea can be extended to 3D with a spiral staircase that has no beginning and end. The steps of the staircase cannot be measured until you give it limits.

Re: How to explain infinity to kids

#135

Post by Sniffnoy » Fri, Aug 03, 2018, 10:27 PM UTC

romwell wrote: post #97

Sniffnoy wrote: post #60

Earlier quoted context omitted.

> The other kinds of infinity - cardinals, for example - are encountered early on, and there are things you can do with them. 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 c…

Sure, oridnals are interesting, but in the context of explaining infinity to kids transfinite induction is probably not the easiest thing to throw into the fun basket. Even first-year undergrads often have to learn regular induction! >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, oridnals are interesting, but in the context of explaining infinity to kids transfinite induction is probably not the easiest thing to throw into the fun basket. Even first-year undergrads often have to learn regular induction!

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.

Re: How to explain infinity to kids

#136

Post by asdffdsa321 » Sat, Aug 04, 2018, 8:45 PM UTC

I thought infinity was most always described in terms of limits. I have trouble thinking about it as a number

Re: How to explain infinity to kids

#137

Post by axilmar » Tue, Aug 07, 2018, 1:56 PM UTC

enedil wrote: post #128

axilmar wrote: post #107

What an awful way to explain infinity to kids and adults. How about something simpler? like 'infinity is when something never ends'?

Your explanation is worse, since it doesn't allow to distinguish different infinities.

It doesn't have to explain different infinities. When we talk about infinity, we need to realize the concept of 'never ending', and not anything else.

Re: How to explain infinity to kids

#138

Post by JeanMarcS » Fri, Aug 10, 2018, 7:03 PM UTC

likeclockwork wrote: post #39

JeanMarcS wrote: post #36

Teaching mathematics is lying to kids all along, at least in France : - 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...

> - No, you can’t do 2-3. If you have 2 apples, you can’t give 3 Well, how soon is too soon to teach children about debt?

Apparently the 18th century :-D !

https://betterexplained.com/articles/a-visual-intuitive-guid...

Re: How to explain infinity to kids

#139

Post by jgeralnik » Fri, Aug 17, 2018, 5:52 AM UTC

enedil wrote: post #131

jgeralnik wrote: post #109

Earlier quoted context omitted.

Not quite, you can't order all natural numbers alphabetically either but that doesn't mean you can't count them. 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.

Yes, you can. Order is just a relation, i.e. a set that contains pairs of elements. So for example, let X be the relation <. X is a subset of NxN, the cartesian product of N. We say that n<m if the pair (n,m) is in X. Moreover, we say that (n,n+1) is in X, and whenever (n,m) in X and (m,k) in X, then (n,k) in X. This constitues order on naturals.

True, but the GP claimed that if we can order alphabetically we can create a one to one correspondence with natural numbers.

Obviously we can order reals (using <), I interpreted the original post as referring not to your definition but to "order and count"