In my experience with children, one of the easiest-to-grasp concepts of infinity is provided by the transfinite ordinals, since it can be viewed as a continuation of the usual counting manner of children, but proceeding into the transfinite:
1,2,3,⋯,ω,ω+1,ω+2,⋯,ω+ω=ω⋅2,ω⋅2+1,⋯,ω⋅3,⋯,ω2,ω2+1,⋯,ω2+ω,⋯⋯
Presumably this person has no experience with 6 year olds? This explanation is horrendous haha
No it isn't. If you ask a child what comes after infinity, "Infinity + 1" is pretty much the default answer. Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two". The answer of "Infinity TIMES Infinity" is also popular for kids to say when they know a number bigger than their friend (who just proclaimed infinity is the largest number).
I thought that most of us learn at an early age, as a result of this kind of exchange, that "infinity" is not "the biggest number" or even a number at all, as far as the ordinary notion of "number" goes.
No math instruction I had ever discussed infinity with any rigor until calculus -- and even then, it was only infinity as a limit. Infinity as a concept was brushed off in the same way that the square root of negative one was brushed off until we were actually taught about it.
On the one hand I get why that is - the calculus notion of infinity is the one that tends to be useful in applied math - on the other hand it's a shame because the set theoretic notion of infinity has more to offer to someone trying to ponder the nature of the infinite.
Or put another way, "what's ∞ + 1" basically invites the non-answer "that's not a well-formed question" whereas "what's ω + 1" gives you a whole intellectual thread to pull on.
I've always been disappointed that number theory, set theory, etc aren't introduced in middle school or high school.
It makes sense, since those are a lot less useful than the subjects that are taught, but something like number theory is incredibly approachable to a middle school student. And it can show students that math can be a lot less about memorization and a lot more about creative thinking w.r.t. proofs.
I would argue that "that's not a well-formed question" is a correct answer, not a non-answer.
...and that the intellectual thread you are pulling on is a (more) artificial notion, constructed by set theorists for the sake of set theorists, not for the sake of counting or measuring in any real sense.
I’d disagree personally. The idea that you can add things to an infinite set, or multiply an infinite set are actually useful concepts. If you imagine the universe is infinite and as such has infinite stars in it (ω) you could discuss how some infinite universes have twice as much star density as our infinite universe (ω2). Or imagine taking a copy of our universe and adding a single star 10 light years from earth (ω+1). In a very real sense the second universe would have twice as much stuff in it as the first universe, even if you can countably map the two universes to each other.
Or maybe put another way, taking the idea that infinity is just infinity makes a lot of sense when you’re primarily considering non-infinite numbers. When you’re primarily considering the concept of infinity and what you can do with it mathematically though, using systems that let you describe infinity with more nuance makes a lot of sense.
It came up a bit in some physics classes, when you can mathematically make something go to positive or negative infinity being able to remove it from the simplified calculation of something is very handy.
You're not alone! Georg Cantor was deeply concerned about the theological implications of his work on transfinite numbers, to the point that he wrote letters to Pope Leo XIII to explain why the new infinities were consistent with a God of an even higher order of infinity.
> Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two".
Or is it "Two Times Infinity"? (Hint: It isn't, because "Two Times Infinity" = "Infinity", while "Infinity Times Two" = "Infinity + Infinity". Not sure every kid knows that.)
I think you have that backwards. “Two times infinity” is “infinity, two times” or “infinity, twice,” which maps to Infinity + Infinity. “Infinity times two” is 2 + 2 + 2 + 2 + 2… forever.
Is addition defined _by_ set theory, or is set theory one way of defining addition? If it's the later, then there could be other ways of defining addition that don't have the same results for infinity (because our math system doesn't really "work" for infinity, or 0, depending on the circumstances).
I am in no way a mathematician. My question about the definition of addition as it relates to set theory is just that; a question.
It's the latter; I'm also not a mathematician, just a guy who worked through Halmos's "Naive Set Theory" in intense detail...
But your question actually hints at my most profound takeaway from that whole book. I think what you're saying is right, AND that foundations-of-mathematics folks spent a long intense period searching for different set theory axioms that did NOT lead to transfinite numbers. But anything anyone could come up with that included "the axiom of infinity" led to transfinites leaking in.
Which begs the question of how to think about these things. Are they "real"? Are they an oddball side effect that we shouldn't take seriously?
I think you've arrowed right to the philosophical heart of all of this.
Does everything become a paradox given enough time and/or thought?
I think we often end up at the end of logical thought processes back at the original question - how can we observe and describe a system that we are inherently a part of?
There are many ways of definiting everything. Most of them are equivalent in the ways that matter, which is why math "works" so well as the language of science. Some of them are different in critical ways, which opens up vistas of new objects and concepts.
I majored in math and my biggest problem with this is that you don't get to "do" anything infinitely many times in the math that I'm used to. In discrete contexts where infinity is used, you instead can "do" something an unbounded but finite number of times. In a continuous setting you are allowed to pick an arbitrarily large (finite) number.
In that context the first quantity that you refer to above is nonsensical because you can't "increment infinitely many times".
Secondly, I'm not sure your construction is correct, since your Infinity+1 set cannot be a singleton (it must contain all the numbers less than Infinity).
Sorry, of course you're right on "Secondly". The right construction is ω, ω∪{ω}, ω∪{ω}∪{ω∪{ω}}...
For the first point, I went through the book long enough ago that I can't rebuild the proof here, but iirc the more rigorous idea is that you can construct a bijection between 1+ω and ω given the recipe I had above for how to represent numbers as sets, but you can't do it for ω+1, which is bijective with ω∪{ω}. The axiom of infinity declares that ω itself is a set, opening the door for transfinite numbers.
I had it explained at a very early age as "three lots of three", and to imagine it like three boxes of three ice-creams. Treating the multiplication symbol as one would to indicate quantity in a list, thus calculating how many ice-creams there are.
The technique used by the Oregon public school system in the 80s went something like "Hand the child a 10x10 grid of numbers, then tell them, absent of any other context, that they must be memorized." I like your way better.
Ontario's 1990s curriculum was pretty awesome. The idea of dimension and sets were both introduced simultaneously and joined, using multiplication. Started in the 2nd grade and they just kept elaborating. Number lines and groups of items. (Tied it into geometry, too. Square numbers came up by at least 4th grade.) What is 3 x 3 but moving 3 units, 3 times in one dimension? Now, memorize these tables up to 12 x 12, you won't always have a calculator at hand.
I do though. I still blow minds when I put my iPhone calculator in scientific mode. Math education is important for many reasons. But teaching it as a practical survival skill using no tools does a disservice to the student. Either it is useful as a problem solving exercise or it is a practical skill that should take advantage of tools. "Just memorize this stuff" isn't useful because it backfires into hating learning. Nothing about math makes it ideal for memorization and none of my math teachers spent any time on study skills.
Nah. As somebody who ends up doing a ton of mental math, I think it's valuable. Yes, they should also learn how to use tools. But developing a feel for numbers is valuable, and I think that is much harder to do if one always relies on a calculator. (And yes, of course, this should be learned in a way that doesn't involve the kids hating it. But that's possible.)
Phrases like "you won't always have a calculator at hand" only serve to erode trust in the educator. It's simply not compelling, and for all practical intents and purposes is untrue. Even on backpacking trips I have a cell phone, even if it is off. If you believe mental math is useful then say that and explain the benefits. Students can smell a lie.
That sounds like an excellent thing to somebody who actually said "you won't always have a calculator at hand". Maybe you should find someone like that.
I'm not sure that HN readers-- a community that will disproportionately skew towards folks educated &/or employed in STEM fields-- are indicative of other people's contact with 6-year old kids.
Number blocks is a great show, my 5yo watches and, being entertained by it, absorbs more than I could easily get him to sit still for. Then he asks me questions about what he watched and is more engaged with my answers as a result.
Making a subject "fun" is alright, but making it entertaining (IME) makes for more productive engagement.
Yes. Simple multiplication and even division and fractions are part of the national curriculum at ages 5 to 6 in the UK. Which is about the age when I remember learning them decades ago too. I think we learned how to add and multiply fractions too.
By age 6 to 7 they're expected to understand that addition and multiplication are commutative, while subtraction and division are not.
They dont usually know formal arithmetic multiplication but they well understand the concepts of repeated addition and subtraction. Most places in the world do start teaching multiplication at age 6/7.
Yes, I learned long division fairly well around that time. I was fortunate (/ disruptive) enough to be sent to a "Montessori school". Long division was definitely pushing it when I was about 5 or 6, but, honestly, given steadier instruction in math starting earlier, I suspect I could have been entirely solid on long division by that time and moving on to algebra. And, I think this is true for a reasonable proportion of children.
My experience, ultimately, was much less ... 'high-quality', let's say. When I left the Montessori school (by 3rd grade), I learned practically no math from then until after high school. First, in normal 'elementary' school (US), multiplication was still being covered in 6th grade. Then, suddenly (from my perspective), letters were being brought into the picture in 7th or 8th grade. So, in my arc, math started to not make sense, at all.
From my perspective, we had spent multiple years on multiplication and long division, which I already understood very well by the end of 2nd grade ... so, there was the period where I basically didn't learn anything, where it seemed like we'd reached the end of math or something. Or, perhaps, like there were some sort of subtleties remaining in multiplication and division. It just gave me a chance to be bored with all of it, boredom correlates heavily with mistakes with kids with attention issues (IMO), this fed into some sort of doubts about my understanding of everything etc., and then, suddenly, there was new material again starting in 7th grade. Material that was 'mechanical', and that didn't seem to have explanations I could understand.
Ultimately, I struggled along with that garbage through high school, then, after, took a course where we actually did PROOFS. Basic number theory stuff - modular arithmetic, etc. Bam, suddenly, the subject started to make sense.
Typing this out actually makes me slightly angry. I'm not sure I previously connected it all together - why I had so much trouble with math for some years ... how this 'arc' was pretty much perfectly engineered to make math a problem, for me. In any case, schooling through high school can be a really low quality experience at times - for some students, subjects, etc. The math curricula, methods of teaching, and progression I was exposed to, worked together, in some sense, to make the subject a problem for me. To do almost the opposite of what was intended - to pretty well impede learning. There's no one factor in that story I can point to and say 'here, fix this' ... no one involved in the story was actively attempting to do anything other than what they thought was best or what they were required to do, but, the net result was honestly worse - I now believe (and believed some years ago, even without quite this analysis) - than if I'd just been given some selection of math material to pick from and been allowed some sort of semi-self directed coursework.
Even better, though, if I'd simply had that course with proofs / basic number theory in, say, 8th grade ... guh, would have avoided so much pain, I'm pretty sure...
I explained basically this to my 4 year old nephew recently. He wanted to count to infinity. I asked him what is the biggest problem with counting to infinity? It's too slow. I said ok let's take bigger steps. We counted by 2's then 10's then hundreds and millions and then zillions and other ridiculous superlative numbers. It doesn't really matter because everything is still too slow. So then we said ok lets make up a number ω that is half way there, One ω, Two ω, done. He's happy. Then I told him to add one more and sent him back to play fetch with the dog.
I taught my kid that the way to think of infinity is that it's like hugs, there's always one more, unlike candy, which is limited and can be counted, infinity cannot be counted.
Hmm, that could potentially cause confusion later. There are 'countable' and 'uncountable' forms of infinity / infinite sets.
A countably infinite set could be 'counted' (i.e., you could sit around labeling elements using the 'natural' or 'counting' numbers) in the sense that we might count candy. The issue for a human being is that you'd run out of time but not elements to count, at least, proceeding in the sense one might count the candy - a piece at a time. Of course, you can, instead, simply provide a 'bijection' (between the natural numbers and the set you wish to prove is countably infinite), and in a sense, you are done.
The subject of infinity and infinite sets can be kind of subtle, and for years the best mathematicians made many mistakes and had many difficulties handling these concepts in ways that didn't cause potentially serious problems (absurdities, paradoxes, etc.). I think that with the development of things like Zermelo-Fraenkel set theory, Gödel's incompleteness theorems, etc., things became a lot clearer. It's a lot easier, with all of the groundwork laid by people who worked on these, to get a good sense of what is possible and what isn't - what gets you into trouble and what doesn't. But, boy, did it twist the minds of the people trying to work it out at the time. In part, this is because it was less clear, without development in these areas, what math even is and what its limits are ... what its relationship to the structure of the universe, say, even is (something along those lines, in my opinion / experience).
> Hmm, that could potentially cause confusion later [...]
(Q: Do you have kids?)
Our experience is that pretty much everything parents tell young children could potentially cause confusion later.
In no particular order: Father Christmas aka Santa Claus, The Tooth Fairy, Where Babies Come From... it's a long list, our eldest is 13 and we're not done yet.
(sorry for responding after so many days - didn't see reply before)
Ha! Certainly a fair and good point.
I would propose that there is a spectrum when it comes to the 'damage', as a term that comes to mind right now, (likely to be) caused by various kinds potentially confusing information.
Given differences in the way different people understand, well, pretty much anything, I'd propose that it might best be thought of as some set of statistical distributions. Using this kind of framework*, we might be able to reasonably improve thinking about what these distributions might look like, how we might tailor the information we provide and how much work we put into trying to avoid introducing possibilities for confusion, etc. Further, I suggest 'set' as we might benefit from 'parameterizing' (thinking about distinct distributions) in terms of traits - autism, ADHD, anxiety, etc.
In my mind, and based on my experiences, I would (in part, thinking terms of the model I'm proposing here) be much more wary of asserting potentially incorrect information in the realm of math and some of the more 'abstract' subjects that people tend to have more trouble in the first place. A concept like 'Santa Claus' isn't something that a child may need to be able to use as a basis for building serious skills on, say. Of course, 'Santa Claus' can be helpful for building imagination, ability with storytelling, developing narratives, etc. ... but the fundamental information regarding some specific entity 'Santa Claus', is not really problematic, in terms of the perspective I'm trying put forward here. On the other hand, statements that are 'too strong' (or 'too weak' possibly) or using terms in ways that aren't standard in mathematical discourse ... these sorts of things can make it feel like the ground is really slipping away as you try to learn other bits about a subject that, again, for many people is ... nebulous ... it's not (so) visual, tactile, ... it's very strange in many ways, early on.
That's the best I can do, right now, in response, I think.
You raise a good point, for sure. And I'm sure there are entire books, there are papers out there in the literature, etc. Personally, I can HIGHLY recommend books like Polya's "How to Solve It" ... as a starting point regarding 'math pedagogy'. That book is a gem, IMO, and gives some real insight into how to think and problem solving in general. And, it's a good gateway to many more resources and research into these areas.
As with everything human and 'complex', there's really no 'optimum' or chance of finding any such thing, I think. Avoiding the worst impacts ... essentially, in terms of opportunities and establishing bases etc., that's doing pretty well - raising children / 'new humans' is hard.
* Which is a way I've been trained to think, sorry if it's not a great model for you - kind of best I can think of off the top of my head and with limited time this moment
> this person has experience teaching children mathematics
Just as a FYI, there are plenty of countries in Europe where many 6 year-olds are still in kindergarten not at school, as a result they most likely have not have properly started learning numbers or reading and writing.
unless the kindergarten is playfully toying with numbers already, usually with no obligation but as an enrichment for those kids who love such activities.
Also my first thought. I assume he's writing this to other people who know what transfinite ordinals are (I don't understand the explanation) and would frame it differently with an actual kid. Even in context it's a hilarious quote though, I think it's possible this was on purpose
I think the big assumption that kids can't get "complicated" ideas is faulty.
Sure, they lack rigor, and often will just get the sketch of the idea.
And it's a lot more work to think about how to put things in the terms that a kid will understand given their knowledge so far.
But this idea? "Infinity plus one?!@" --- this is a conversation elementary school kids have on their own. Pulling it a little closer to a sane footing in ordinal analysis is not hard. Half of six year olds can handle it.
On the other hand, there's not a lot of obvious utility to teaching a six year old this particular concept early. On the gripping hand, there is a cost to keeping kids in a bubble where you don't talk about any big ideas (of whatever sort-- mathematical, philosophical, historical, linguistic) at all, or excessively dilute them to the point where they're meaningless.
Richard Feynman would be making disapproving noises.
Explain everything like you're talking to a fifth grader. If you can't, you don't understand your problem fully.
He spend much of his professorship agonizing about how to fit all of physics into a freshman lecture. When he couldn't, he knew we needed to think more about that area.
Transfinite ordinals also known as hyperreals should really be taught in school as they make many parts of math easier: algebraic definition of derivatives (including algebraic derivative of step functions without dirac 'density') and yes: natural addition and multiplication.
> Transfinite ordinals also known as hyperreals should really be taught in school as they make many parts of math easier: algebraic definition of derivatives
Q: What proportion of children study maths long enough to understand derivatives?
Having mechanical formulae for solving closed form equations involving the notation for derivatives… does not mean that derivatives have been understood, in my experience of tutoring not-especially-mathematically-inclined folks.
Do you think 90% of attendees of Gymnasium (which I don’t think is the majority) understand derivatives? My friend’s wife who attended Gymnasium and got reasonably good grades most certainly did not, but she is my only example of a non-mathematician Gymnasium graduate, so I’m quite willing to be convinced she is an outlier.
They are not explaining to a 6 years old, they explains to somebody who will in their turn explain it to a 6 years old, which is a different task and has to be optimized in a different way.
In the comments of the answer the author says they have a 4 and a 9 year old:
"Bill, despite your emphatic comments, I know for a fact that counting into the ordinals is something that children can easily learn. I have two young children (ages 4 and 9), who are happy to discuss ℵα
for small ordinals α---although my daughter's pronunciation sounds more like Olive0, Olive1---and my son can count up to small countably infinite ordinals. The pattern below ωω is not difficult to grasp. Below ω2, it is rather like counting to 100, since the numbers have the form ω⋅n+k, essentially two digits"
Ordinals are hard to grasp for people that know the standard school curriculum, know about countability and uncountable sets, cardinality, and the basic properties and arithmetic of cardinality.
I don't know why would it be hard for people that haven't been familiarized with a similar but different concept?
He has children (not sure about age right now) and discusses mathematics often with them. His tweets have had many interesting examples.
I do not think he means he would use symbols to explain to children, but that the notion of counting natural numbers that children have easily generalises to counting transfinite numbers.
