7 ms·

A Child Thinking About Infinity (2001) [pdf]

102 points by Phithagoras 10y ago

thrownear 10y ago
Some guy brags about his son..
- estrabd 10y ago
  Maybe, but it's interesting because I had quite a few discussions with my kids about infinity. It came up when playing the "how much do I love you" game. I introduced infinity, and from there I was able to teach them a little about it. They were very young and I am still impressed with how easily they grasped some of the concepts.
  - thrownear 10y ago
    Young kids easily grasping things is interesting. It is also commonplace...
    - estrabd 10y ago
      I agree. I am not claiming my kids are geniuses. They are smarter than me I feel, but that's not saying much. :)
  - Phithagoras 10y ago
    Do you think it's possible that they accepted your ideas of infinity in a dogmatic sort of way, similar to how many young children accept god? I remember first having discussions with my father about infinity as young as 6, but I didn't really start exploring infinity for myself until my mid teens. It would be interesting to see how your kids thoughts change as they mature.
    - estrabd 10y ago
      I do think they accepted it in a dogmatic way. Incidentally, I am raising our children Catholic so it's an idea that fits pretty well with the idea of God, the Holy Trinity, and all that. Three in One, yet one is the Father and the other is the Son? Yes, dear. Oh, ok. <grin>
- castis 10y ago
  It seems that you may not be the target market for this information if that's all you have to contribute.
- Snackchez 10y ago
  You're being downvoted, but you're right. This article was somewhat silly.
- coherentpony 10y ago
  The guy is David Tall: https://scholar.google.com/citations?user=IV1rmyEAAAAJ&hl=en&oi=ao https://scholar.google.com/citations?user=IV1rmyEAAAAJ&hl=en...
- dang 10y ago
  Please don't post snarky dismissals to HN. They're corrosive of the higher-quality discussion we're hoping for here, especially because they generate replies that end up taking the thread in a less interesting direction.
TuringTest 10y ago
Interesting read. Though Socratic method much? The writer uses leading questions[1] as a primary expository method, directing the kid towards conclusions that "should" be made within the proposed system, rather than merely exposing a small set of rules and letting the child freely explore their consequences. All the talk about the different infinities felt that way. This leading style may be good for teaching the history of mathematics, i.e. introducing concepts that have been relevant to solve problems in the past. But for teaching math skills I prefer methods where a problem is stated, and the student is left to build their own abstractions about it; the IOUs and thermometer examples were more in this line. This competence will prove more useful when solving new problems than merely following the trodden path of already invented mathematics. [1] https://en.wikipedia.org/wiki/Leading_question https://en.wikipedia.org/wiki/Leading_question
- jmilloy 10y ago
  I think it's noteworthy that the article you point to pertains to common law and testimony, rather than to philosophy, pedagogy, or education. When a student freely explores a concept, that clearly demonstrates a higher ability or skill in mathematics. But "leading questions" are scaffold to show the student how to explore and build abstractions, and most will have to learn how to do that from example.
  - TuringTest 10y ago
    The Wikipedia article may be primarily written about legal cases, but leading questions appear in many other contexts. In psychology and interviews they're often advised against, as they induce a precise answer and thus make it hard to learn about the subject of study, instead reinforcing the interviewer preconceptions; in user interface design they can produce the wrong conclusions about how easy an interface is to use. Leading questions have their use in education, but as the other commentator below points out, the child might be giving the answer that the questioner wants to hear, rather than showing a true understanding of the topic. Having a stated problem to solve may prevent that, as the child can instead answer the question in a way that solves the problem.
Wile_E_Quixote 10y ago
I know this isn't the point of the article, but did anyone else LOL at the bottom of page 3 when the difference of 10^100 - (ninety nine 9's) was given as 1? The author seems proud to attest "There really are ninety nine 9s!" The problem though is that this doesn't equal 1. You would need to subtract a number with one hundred 9's in a row to get an answer of 1.
dpflan 10y ago
I would like to see fMRI studies on how human experience the infinite or perhaps near infinite. There is an instinctual appeal of vastness (near infinite, perceptually infinite?) - mountains are climbed, sunsets into far away horizons over seemingly infinite oceans, skyscrapers are built and their height and subsequent view are valued,... - These are times when humans first experience vastness which may trigger pathways in the brain related to comprehending / thinking about the infinite. I would like to see is the parts of the brain associated with self-awareness are active during these experiences.
- pbhjpbhj 10y ago
  Mathematical concepts like nothing, straight lines, circles, are also abstractions that have no complete reference in nature - is it just infinity that you consider needs self-awareness? I'm not quite sure what you're hinting at?
  - dpflan 10y ago
    I was just musing, no concrete connections at the moment. I don't know if infinity needs self-awareness, but I was thinking that it could prompt one to call into question his/her self relative to the infinite that is being experienced, and in that perhaps adjust one's concept of self through comparison. Or even just thinking about how one can conceptualize such a thing based upon all life experiences where almost everything encountered is finite.
    - hodwik 10y ago
      In my experience, children (and most people) just conceptualize infinity as a very large set, just as they perceive a tire a "circle", and think a square drawn on paper is made up of "lines". It requires years of thinking about these mathematical entities before people get a first glimpse of what they really mean, beyond a simplistic qualitative experience. To most people, children included, infinity is conceptually no different from the amount of sand on a beach. They just allow that it is simply not quantifiable. That turns off the brain. The more interesting thing to test, I would argue, would be the human mind conceptualizing a very, very large but ultimately finite set of something. That requires that we make a distinction between the discreteness of something and our ability to quantify it. That is a much more difficult concept to comprehend, and it must be understood before we can even begin to think about teaching a child about mathematical entities like infinity, circles, or lines.
CookieCats 10y ago
I vaguely recall having a "night terror" centered around infinity when I was very young. I think I was trying to truly grasp how tiny and insignificant I am and how immensely huge everything else is and it just freaked me out.
- Tervis 10y ago
  I think I know what you mean. When I was a kid, I got a feeling of complete emptiness lasting maybe couple of seconds when I was thinking about what if there wouldn't be anything.
- geierdmtr 10y ago
  I still fall into a rabbit hole every-time i imagine infinity. it started when i was 5 or so. brain goes into a loop and imagines the infinite. for me it was and is connected to space. you have to hit a wall at some point, but then there is something behind the wall.. there can never be something that isn't there. the limited donut universe theory framed the space imagination. but the donut has to be surrounded by something. here goes the infinite loop again.
illumin8 10y ago
Just a few days ago I had an interesting conversation with my 4 year old son that went something like this: "Daddy, what's the biggest number?" "There is no such thing as the biggest number; numbers keep getting bigger and bigger forever. This is a concept called infinity." "Which number is the biggest?" And so on; he's only 4, so I don't expect him to grasp the concept of infinity, but I was impressed that he's already asking about it. My son is in his first year of Montessori education, and has learned to count into two digit numbers already, and understands zero, but doesn't yet understand negative numbers. Montessori is fairly interesting because they teach math in a different way, by learning powers of 10 and using a bead system similar to an abacus, so Montessori educated kids can actually do simple arithmetic with large numbers (in the thousands) by around age 5.
- branchless 10y ago
  There's a great anecdote about this. Son: the biggest number is 1081 Dad: what about 1082 Son: well, I was close
  - overcast 10y ago
    Classic :)
  - vmilner 10y ago
    Tony Benn (UK politician) relates his grandson doing this in his diaries, but I've heard it related many times and suspect it doesn't come from that single source.
    - branchless 10y ago
      I expect as well as being borrowed many times it's also happened many times.
- knicholes 10y ago
  I'm not sure if this is normal or expected, but my two-year-old can count to 20. I'm not trying to brag-- maybe just seeing if maybe I should start bragging? ;)
paulvs 10y ago
On a side note: In the photo given of 10^100 - 999...(99 9s) = 1, there should be 100 9s, not 99 9s, for the result to be 1.
cLeEOGPw 10y ago
Sounds like a child was picking up on OPs intonation when he asked the questions. That's how children answer. They pick on body language about what the correct answer is and basically tell what the one who asks wants them to tell. This was made especially easy by the binary format of questions. Yes/no answers are very easy to guess based on the way you ask. And people tend to ask extra suggestively when talking with children.
oaktowner 10y ago
I'm not sure my son was even 6 yet (he may have been five) when he learned about subtraction. We started by talking about things: to illustrate 4 minus 2, you start with four marbles and remove 2 of them, leaving 2. For a couple of weeks he would ask me to subtract numbers, then he moved on to asking me to ask him numbers (and writing down simply arithmetic problems to solve). Then, one day, out of the blue, he asked with a look of incredulity in his eye: "Daddy. What is one minus TWO???" He clearly realized that he had come up with a question which had no representation in the marble-based construct. I was so proud. I sat down with him and drew a number line...and then showed him how we could extend the number line to the other side of zero. He does know about infinity now (he's 7) -- and his most recent revelation that there is no such thing as an infinite number of any physical object. Even germs. Even atoms -- no matter how tiny, an infinite number of them can't exist in the universe (which he certainly pictures as finite).
- sethammons 10y ago
  I had a similar experience when my oldest was still young. The conversation was a little flipped. We live in a cold climate and she had started taking note of the thermometer. I had asked her "what is 5 - 5?" > "zero!"; "what is that minus 5?" > "you can't do that!" > "What happens if it is 0 degrees out and it gets 5 degrees colder?" > * eyes open wide * "there are all the same numbers below zero!?" Some time later, when she was learning algebra, I told her "0.999... == 1, can you tell me why?". I was expecting an algebra lesson to show her how that was true. She surprised me when, a second or two later, she looked up and said, "1/9 is 0.111... so 9/9 is 0.999... and 9/9 is 1". I was very proud.
OJFord 10y ago
Amusing typesetting error: > We also discussed powers of ten, so that he knew that > 102 was 100, 103 is 1000, and 10100 is a 1 followed by a > hundred noughts.
mswen 10y ago
So this resonates with me. In general I don't have a particularly detailed memory of my early childhood. However, I do recall one day when I was 4 years old sitting at the end of the driveway waiting for my dad to get home from work. I had picked up 2 digit counting from my listening to my mom work with my older sister. As I am sitting there by myself counting and then I got to 99. At that moment something partly clicked and I realized that 3 digits let me continue counting and I could just carry on the pattern. I remember being pretty excited at this and counted for awhile longer ... maybe another 20 or 30 digits and then in a flash I realized that the same thing works when I run out of 3 digits and I intuited the fact that there is really no end to counting. And, even though I didn't have a name for it I remember being kind of awestruck at something that could just go on and on without end. It makes me smile today at 54 to be reminded of that rush of discovery.
jeena 10y ago
I was born in communist Poland and back then everybody was catholic there, and so was I. So everyone had to go to the church, so I did too. I listened what the preast had to say about the heaven and the hell, but particularly that he said one would end up there for eternety. That sunday lying in my bed I couldn't sleep. I was thinking about this infinity thing. I thought to my self, going to hell is bad because you get tortured and stuff, but going to heaven for eternity isn't really much better either, because it never ends! This was so frightening to me I couldn't sleep. I was ok with going to hell for some really long amount of time, say 10k years, as long as I knew it would end some day. But being in heaven for ever? I still get a bad feeling when I remember my thoughts from back then, that would be unbearable, you couldn't even commit suicide, nothing would change anything, you'd be trapped forever with no hope of escape. That was when I was seven or something, when I was eleven we moved to Germany where not everybody was catholic, half of the people were protestants. Then later when I was 27, so 10 years ago, I moved to Sweden where most people are atheists, and with time I also liberated myself from this scary religion. Now I feel really happy about the fact that there most probably is no god and thus no inifinite amount of time in heaven either.
- irrational 10y ago
  By the conservation of matter, matter cannot be created or destroyed. Thus in a certain sense you were not created but have always existed to infinity in the past (as far as we know the singularity which came before the big bang had no beginning). So is it a strange thing to think of yourself as continuing to exist in some form to infinity in the future?
- tn13 10y ago
  For the same reason I have found the notion of Christian God to be very scary. A father who does not die and keeps judging his children and is always powerful.

102 points by Phithagoras 10y ago

thrownear 10y ago

Some guy brags about his son..

estrabd 10y ago
Maybe, but it's interesting because I had quite a few discussions with my kids about infinity. It came up when playing the "how much do I love you" game. I introduced infinity, and from there I was able to teach them a little about it. They were very young and I am still impressed with how easily they grasped some of the concepts.
- thrownear 10y ago
  Young kids easily grasping things is interesting. It is also commonplace...
  - estrabd 10y ago
    I agree. I am not claiming my kids are geniuses. They are smarter than me I feel, but that's not saying much. :)
- Phithagoras 10y ago
  Do you think it's possible that they accepted your ideas of infinity in a dogmatic sort of way, similar to how many young children accept god? I remember first having discussions with my father about infinity as young as 6, but I didn't really start exploring infinity for myself until my mid teens. It would be interesting to see how your kids thoughts change as they mature.
  - estrabd 10y ago
    I do think they accepted it in a dogmatic way. Incidentally, I am raising our children Catholic so it's an idea that fits pretty well with the idea of God, the Holy Trinity, and all that. Three in One, yet one is the Father and the other is the Son? Yes, dear. Oh, ok. <grin>
castis 10y ago
It seems that you may not be the target market for this information if that's all you have to contribute.
Snackchez 10y ago
You're being downvoted, but you're right. This article was somewhat silly.
coherentpony 10y ago
The guy is David Tall: https://scholar.google.com/citations?user=IV1rmyEAAAAJ&hl=en&oi=ao https://scholar.google.com/citations?user=IV1rmyEAAAAJ&hl=en...
dang 10y ago
Please don't post snarky dismissals to HN. They're corrosive of the higher-quality discussion we're hoping for here, especially because they generate replies that end up taking the thread in a less interesting direction.

TuringTest 10y ago

Interesting read. Though Socratic method much? The writer uses leading questions[1] as a primary expository method, directing the kid towards conclusions that "should" be made within the proposed system, rather than merely exposing a small set of rules and letting the child freely explore their consequences. All the talk about the different infinities felt that way. This leading style may be good for teaching the history of mathematics, i.e. introducing concepts that have been relevant to solve problems in the past. But for teaching math skills I prefer methods where a problem is stated, and the student is left to build their own abstractions about it; the IOUs and thermometer examples were more in this line. This competence will prove more useful when solving new problems than merely following the trodden path of already invented mathematics. [1] https://en.wikipedia.org/wiki/Leading_question https://en.wikipedia.org/wiki/Leading_question

jmilloy 10y ago
I think it's noteworthy that the article you point to pertains to common law and testimony, rather than to philosophy, pedagogy, or education. When a student freely explores a concept, that clearly demonstrates a higher ability or skill in mathematics. But "leading questions" are scaffold to show the student how to explore and build abstractions, and most will have to learn how to do that from example.
- TuringTest 10y ago
  The Wikipedia article may be primarily written about legal cases, but leading questions appear in many other contexts. In psychology and interviews they're often advised against, as they induce a precise answer and thus make it hard to learn about the subject of study, instead reinforcing the interviewer preconceptions; in user interface design they can produce the wrong conclusions about how easy an interface is to use. Leading questions have their use in education, but as the other commentator below points out, the child might be giving the answer that the questioner wants to hear, rather than showing a true understanding of the topic. Having a stated problem to solve may prevent that, as the child can instead answer the question in a way that solves the problem.

Wile_E_Quixote 10y ago

I know this isn't the point of the article, but did anyone else LOL at the bottom of page 3 when the difference of 10^100 - (ninety nine 9's) was given as 1? The author seems proud to attest "There really are ninety nine 9s!" The problem though is that this doesn't equal 1. You would need to subtract a number with one hundred 9's in a row to get an answer of 1.

dpflan 10y ago

I would like to see fMRI studies on how human experience the infinite or perhaps near infinite. There is an instinctual appeal of vastness (near infinite, perceptually infinite?) - mountains are climbed, sunsets into far away horizons over seemingly infinite oceans, skyscrapers are built and their height and subsequent view are valued,... - These are times when humans first experience vastness which may trigger pathways in the brain related to comprehending / thinking about the infinite. I would like to see is the parts of the brain associated with self-awareness are active during these experiences.

pbhjpbhj 10y ago
Mathematical concepts like nothing, straight lines, circles, are also abstractions that have no complete reference in nature - is it just infinity that you consider needs self-awareness? I'm not quite sure what you're hinting at?
- dpflan 10y ago
  I was just musing, no concrete connections at the moment. I don't know if infinity needs self-awareness, but I was thinking that it could prompt one to call into question his/her self relative to the infinite that is being experienced, and in that perhaps adjust one's concept of self through comparison. Or even just thinking about how one can conceptualize such a thing based upon all life experiences where almost everything encountered is finite.
  - hodwik 10y ago
    In my experience, children (and most people) just conceptualize infinity as a very large set, just as they perceive a tire a "circle", and think a square drawn on paper is made up of "lines". It requires years of thinking about these mathematical entities before people get a first glimpse of what they really mean, beyond a simplistic qualitative experience. To most people, children included, infinity is conceptually no different from the amount of sand on a beach. They just allow that it is simply not quantifiable. That turns off the brain. The more interesting thing to test, I would argue, would be the human mind conceptualizing a very, very large but ultimately finite set of something. That requires that we make a distinction between the discreteness of something and our ability to quantify it. That is a much more difficult concept to comprehend, and it must be understood before we can even begin to think about teaching a child about mathematical entities like infinity, circles, or lines.

CookieCats 10y ago

I vaguely recall having a "night terror" centered around infinity when I was very young. I think I was trying to truly grasp how tiny and insignificant I am and how immensely huge everything else is and it just freaked me out.

Tervis 10y ago
I think I know what you mean. When I was a kid, I got a feeling of complete emptiness lasting maybe couple of seconds when I was thinking about what if there wouldn't be anything.
geierdmtr 10y ago
I still fall into a rabbit hole every-time i imagine infinity. it started when i was 5 or so. brain goes into a loop and imagines the infinite. for me it was and is connected to space. you have to hit a wall at some point, but then there is something behind the wall.. there can never be something that isn't there. the limited donut universe theory framed the space imagination. but the donut has to be surrounded by something. here goes the infinite loop again.

illumin8 10y ago

Just a few days ago I had an interesting conversation with my 4 year old son that went something like this: "Daddy, what's the biggest number?" "There is no such thing as the biggest number; numbers keep getting bigger and bigger forever. This is a concept called infinity." "Which number is the biggest?" And so on; he's only 4, so I don't expect him to grasp the concept of infinity, but I was impressed that he's already asking about it. My son is in his first year of Montessori education, and has learned to count into two digit numbers already, and understands zero, but doesn't yet understand negative numbers. Montessori is fairly interesting because they teach math in a different way, by learning powers of 10 and using a bead system similar to an abacus, so Montessori educated kids can actually do simple arithmetic with large numbers (in the thousands) by around age 5.

branchless 10y ago
There's a great anecdote about this. Son: the biggest number is 1081 Dad: what about 1082 Son: well, I was close
- overcast 10y ago
  Classic :)
- vmilner 10y ago
  Tony Benn (UK politician) relates his grandson doing this in his diaries, but I've heard it related many times and suspect it doesn't come from that single source.
  - branchless 10y ago
    I expect as well as being borrowed many times it's also happened many times.
knicholes 10y ago
I'm not sure if this is normal or expected, but my two-year-old can count to 20. I'm not trying to brag-- maybe just seeing if maybe I should start bragging? ;)

paulvs 10y ago

On a side note: In the photo given of 10^100 - 999...(99 9s) = 1, there should be 100 9s, not 99 9s, for the result to be 1.

cLeEOGPw 10y ago

Sounds like a child was picking up on OPs intonation when he asked the questions. That's how children answer. They pick on body language about what the correct answer is and basically tell what the one who asks wants them to tell. This was made especially easy by the binary format of questions. Yes/no answers are very easy to guess based on the way you ask. And people tend to ask extra suggestively when talking with children.

oaktowner 10y ago

I'm not sure my son was even 6 yet (he may have been five) when he learned about subtraction. We started by talking about things: to illustrate 4 minus 2, you start with four marbles and remove 2 of them, leaving 2. For a couple of weeks he would ask me to subtract numbers, then he moved on to asking me to ask him numbers (and writing down simply arithmetic problems to solve). Then, one day, out of the blue, he asked with a look of incredulity in his eye: "Daddy. What is one minus TWO???" He clearly realized that he had come up with a question which had no representation in the marble-based construct. I was so proud. I sat down with him and drew a number line...and then showed him how we could extend the number line to the other side of zero. He does know about infinity now (he's 7) -- and his most recent revelation that there is no such thing as an infinite number of any physical object. Even germs. Even atoms -- no matter how tiny, an infinite number of them can't exist in the universe (which he certainly pictures as finite).

sethammons 10y ago
I had a similar experience when my oldest was still young. The conversation was a little flipped. We live in a cold climate and she had started taking note of the thermometer. I had asked her "what is 5 - 5?" > "zero!"; "what is that minus 5?" > "you can't do that!" > "What happens if it is 0 degrees out and it gets 5 degrees colder?" > * eyes open wide * "there are all the same numbers below zero!?" Some time later, when she was learning algebra, I told her "0.999... == 1, can you tell me why?". I was expecting an algebra lesson to show her how that was true. She surprised me when, a second or two later, she looked up and said, "1/9 is 0.111... so 9/9 is 0.999... and 9/9 is 1". I was very proud.

OJFord 10y ago

Amusing typesetting error: > We also discussed powers of ten, so that he knew that > 102 was 100, 103 is 1000, and 10100 is a 1 followed by a > hundred noughts.

mswen 10y ago

So this resonates with me. In general I don't have a particularly detailed memory of my early childhood. However, I do recall one day when I was 4 years old sitting at the end of the driveway waiting for my dad to get home from work. I had picked up 2 digit counting from my listening to my mom work with my older sister. As I am sitting there by myself counting and then I got to 99. At that moment something partly clicked and I realized that 3 digits let me continue counting and I could just carry on the pattern. I remember being pretty excited at this and counted for awhile longer ... maybe another 20 or 30 digits and then in a flash I realized that the same thing works when I run out of 3 digits and I intuited the fact that there is really no end to counting. And, even though I didn't have a name for it I remember being kind of awestruck at something that could just go on and on without end. It makes me smile today at 54 to be reminded of that rush of discovery.

jeena 10y ago

I was born in communist Poland and back then everybody was catholic there, and so was I. So everyone had to go to the church, so I did too. I listened what the preast had to say about the heaven and the hell, but particularly that he said one would end up there for eternety. That sunday lying in my bed I couldn't sleep. I was thinking about this infinity thing. I thought to my self, going to hell is bad because you get tortured and stuff, but going to heaven for eternity isn't really much better either, because it never ends! This was so frightening to me I couldn't sleep. I was ok with going to hell for some really long amount of time, say 10k years, as long as I knew it would end some day. But being in heaven for ever? I still get a bad feeling when I remember my thoughts from back then, that would be unbearable, you couldn't even commit suicide, nothing would change anything, you'd be trapped forever with no hope of escape. That was when I was seven or something, when I was eleven we moved to Germany where not everybody was catholic, half of the people were protestants. Then later when I was 27, so 10 years ago, I moved to Sweden where most people are atheists, and with time I also liberated myself from this scary religion. Now I feel really happy about the fact that there most probably is no god and thus no inifinite amount of time in heaven either.

irrational 10y ago
By the conservation of matter, matter cannot be created or destroyed. Thus in a certain sense you were not created but have always existed to infinity in the past (as far as we know the singularity which came before the big bang had no beginning). So is it a strange thing to think of yourself as continuing to exist in some form to infinity in the future?
tn13 10y ago
For the same reason I have found the notion of Christian God to be very scary. A father who does not die and keeps judging his children and is always powerful.