Math Paradoxes for English
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Math Paradoxes for English
I'm currently working on a research project in my English class. I decided to do mine on infinite as it relates to math and some of the paradoxes involved. So far, I've decided to do parts on Hilbert's Hotel and a few of Zeno's Paradoxes.
The problem is that I'm having trouble thinking up ways to describe a mathematical concept in terms nontechnical enough for an English class. Does anyone have any ideas of how to explain these concepts or other paradoxes to write about?
The problem is that I'm having trouble thinking up ways to describe a mathematical concept in terms nontechnical enough for an English class. Does anyone have any ideas of how to explain these concepts or other paradoxes to write about?
 skeptical scientist
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Re: Math Paradoxes for English
I don't see why either are too technical  simply explain what is going on in plain English, using complete sentences and avoiding any mathematical notation. You can say things like, "Suppose the hotel is full, and a guest arrives. How might the hotel manager squeeze him in? Well, even though the hotel is full, he can still make more room  simply ask the guest in the first room to move to the second, the second to the third, and so forth. Once all the guests have moved, the first room is free and can accommodate the new guest."
There's nothing in either of the things you mentioned that precludes explaining it to an English teacher without using mathematical symbols of any kind, in such a way that even the most mathaverse could understand and appreciate.
There's nothing in either of the things you mentioned that precludes explaining it to an English teacher without using mathematical symbols of any kind, in such a way that even the most mathaverse could understand and appreciate.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
Re: Math Paradoxes for English
talk about a super task and zenos paradoxes.
a super task is a task with an infinite amount of tasks and can be completed in a finite amount of time.
zenos paradoxes talks about movement and how it is a super task, an argument that says movement is impossible.
zeno http://en.wikipedia.org/wiki/Zeno%27s_paradoxes
super task http://en.wikipedia.org/wiki/Supertasks
a super task is a task with an infinite amount of tasks and can be completed in a finite amount of time.
zenos paradoxes talks about movement and how it is a super task, an argument that says movement is impossible.
zeno http://en.wikipedia.org/wiki/Zeno%27s_paradoxes
super task http://en.wikipedia.org/wiki/Supertasks
Re: Math Paradoxes for English
Given that the current thread as it stands has limited shelf life, might I suggest turning it into a general discussion area for explaning mathematical ideas to nonmathematicians?
This is a placeholder until I think of something more creative to put here.

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Re: Math Paradoxes for English
I was thinking of trying to explain bijectivity for comparing infinite sets, but I haven't been able to think of a simple way of explaining besides becoming extremely longwinded.
Re: Math Paradoxes for English
Really? Won't "can be put into onetoone correspondence" suffice? You can give the examples of {natural numbers, integers, rational numbers} and {numbers in (0,1), numbers in (0,x), real numbers} since you can find explicit formulae for those bijections.
This is a placeholder until I think of something more creative to put here.
Re: Math Paradoxes for English
I'd suggest explaining that when we count a finite set, what we're really doing is forming a bijection between it and the set {1,2,3,...,n}. Then using bijections to compare infinite sets becomes a natural, intuitive extension rather than just some random abstract definition.
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Re: Math Paradoxes for English
Yeah, I think I would use the word "match" to explain a bijection. You take the two sets and you "match up each element of one with a single element of the other, and vice versa." I dunno, seems to avoid mathematical jargon, besides "set" and "element."
Re: Math Paradoxes for English
It may be a bit into the "too technical" realm, but Gabriel's Horn seems like it would fit the bill.
Re: Math Paradoxes for English
The problem there is that it's sufficiently counterintuitive and quicklyintroduced calculus without analysis is sufficiently patchy that people will go away skeptical. The dragon fractal would be a nice one to showcase for an analytical "paradox"  it's an example of a continuous curve that covers every point in a square, and it doesn't need much maths at all. (Handwaving is the best approach for continuity, but in this particular case it's possible to handwave extremely convincingly and pretty rigorously  see Strange Curves, Counting Rabbits and Other Mathematical Explorations by Keith Ball for a great exposition.)
Alternatively, if you're feeling adventurous, you could try talking about Turing machines and specifically the Halting Problem  it requires no mathematical background to understand, but it's a pretty deep result when you consider (for example) a program that halts when it finds a counterexample to the Goldbach conjecture. Plus if you have time, you can go on to talk about the busy beaver numbers, which are really fun. (They're uncomputable and they grow faster than any computable function, since they can be used to give an upper bound to the maximum number of steps a Turing machine can take and still terminate.) Godel's theorem is also awesome, though probably too advanced for a presentation.
Alternatively, if you're feeling adventurous, you could try talking about Turing machines and specifically the Halting Problem  it requires no mathematical background to understand, but it's a pretty deep result when you consider (for example) a program that halts when it finds a counterexample to the Goldbach conjecture. Plus if you have time, you can go on to talk about the busy beaver numbers, which are really fun. (They're uncomputable and they grow faster than any computable function, since they can be used to give an upper bound to the maximum number of steps a Turing machine can take and still terminate.) Godel's theorem is also awesome, though probably too advanced for a presentation.
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 BeetlesBane
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Re: Math Paradoxes for English
As far as bijection is concerned, I'd use the model of a shepherd who can't count having a pile (or pocket full) of stones to tally his sheep. The model itself can then serve as a basis for deeper concepts, such as onetoone correspondence.
 Torn Apart By Dingos
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Re: Math Paradoxes for English
A bijection is just changing the names of the elements. For example, a bijection between N and Z changes the names of 1,2,3,4,5,6,... to "0","1","1","2","2",... . That's the reason why a bijection should preserve the size of a set: changing the names of the elements shouldn't change its size!Simply Curious wrote:I was thinking of trying to explain bijectivity for comparing infinite sets, but I haven't been able to think of a simple way of explaining besides becoming extremely longwinded.
Re: Math Paradoxes for English
The ball and vase problem is a really good one.
http://en.wikipedia.org/wiki/Balls_and_vase_problem
About infinite bijections: you should avoid using the word size initially. Stick with cardinality. People intuitively think there are more whole numbers than even numbers (because for finite values proper subsets always have a smaller size). If you say 'size' you're going to go up against that intuition.
Explain cardinality, show how hard it is to say if infinite sets are bigger than each other, then mention that cardinality extends 'size' to infinite sets.
http://en.wikipedia.org/wiki/Balls_and_vase_problem
About infinite bijections: you should avoid using the word size initially. Stick with cardinality. People intuitively think there are more whole numbers than even numbers (because for finite values proper subsets always have a smaller size). If you say 'size' you're going to go up against that intuition.
Explain cardinality, show how hard it is to say if infinite sets are bigger than each other, then mention that cardinality extends 'size' to infinite sets.
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 Cleverbeans
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Re: Math Paradoxes for English
BeetlesBane wrote:As far as bijection is concerned, I'd use the model of a shepherd who can't count having a pile (or pocket full) of stones to tally his sheep. The model itself can then serve as a basis for deeper concepts, such as onetoone correspondence.
I prefer the variant where they use notches on a post, because we know that ancient sheep herders did exactly that, and we still DO tally using a similar system where we draw 4 lines then a 5th crossing them to count. In many respects I think the bijection is really the fundamental building block of all counting and can be represented as such in countless ways  pun intended.
"Labor is prior to, and independent of, capital. Capital is only the fruit of labor, and could never have existed if labor had not first existed. Labor is the superior of capital, and deserves much the higher consideration."  Abraham Lincoln
Re: Math Paradoxes for English
I dislike any analogy which specifically mentions counting, be it in the form of stones in a pocket or notches in a post, because that only really works intuitively as far as countable sets, whereas the second set of examples that I gave explicitly covers uncountable sets.
This is a placeholder until I think of something more creative to put here.
Re: Math Paradoxes for English
True, bijections are considerably more general, but when you're proving extremely counterintuitive results to a nonmathematical audience, if you don't justify your definition then their response will be not "Hey, neat!" but "Well, yeah, I guess if you take that definition of size...". Making explicit the fact that "counting" is itself an example of "creating a onetoone correspondence" would seem to position bijections intuitively as not just a definition that works with finite sets, but the most obvious definition to choose that works with finite sets. That could make a fairly major difference. So mathematically it's slightly less sound, but pedagogically I think it's much more sound.
The other thing I think you need to tackle is that forming a surjection [imath]A\rightarrow B[/imath] can't imply that [imath]A[/imath] is "bigger" than [imath]B[/imath] in any meaningful sense, only that [imath]A[/imath] is at least as "big" as [imath]B[/imath]. It sounds obvious, but one of the most common responses when introducing something like a bijection from [imath]\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}[/imath] is "But you can map a tiny part of [imath]\mathbb{R}\times\mathbb{R}[/imath] onto the whole of [imath]\mathbb{R}[/imath]!"
The other thing I think you need to tackle is that forming a surjection [imath]A\rightarrow B[/imath] can't imply that [imath]A[/imath] is "bigger" than [imath]B[/imath] in any meaningful sense, only that [imath]A[/imath] is at least as "big" as [imath]B[/imath]. It sounds obvious, but one of the most common responses when introducing something like a bijection from [imath]\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}[/imath] is "But you can map a tiny part of [imath]\mathbb{R}\times\mathbb{R}[/imath] onto the whole of [imath]\mathbb{R}[/imath]!"
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Re: Math Paradoxes for English
Using counting as an example of a bijection is fine, but not as a model of bijections.
This is a placeholder until I think of something more creative to put here.
Re: Math Paradoxes for English
Fafnir43 wrote:The other thing I think you need to tackle is that forming a surjection [imath]A\rightarrow B[/imath] can't imply that [imath]A[/imath] is "bigger" than [imath]B[/imath] in any meaningful sense, only that [imath]A[/imath] is at least as "big" as [imath]B[/imath]. It sounds obvious, but one of the most common responses when introducing something like a bijection from [imath]\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}[/imath] is "But you can map a tiny part of [imath]\mathbb{R}\times\mathbb{R}[/imath] onto the whole of [imath]\mathbb{R}[/imath]!"
Please, consider us poor subsilver users! That paragraph was horrible.
All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.
Re: Math Paradoxes for English
You can work out what it says, though: a bijection from R^{2} to R.
This is a placeholder until I think of something more creative to put here.
Re: Math Paradoxes for English
I like to use the following example:
Consider an infinitely long table, with infinitely many place settings. We could never count up all the knives and all the forks, but we know that there are the same number of forks as there are knives because we can match them up.
Consider an infinitely long table, with infinitely many place settings. We could never count up all the knives and all the forks, but we know that there are the same number of forks as there are knives because we can match them up.
 skeptical scientist
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Re: Math Paradoxes for English
Token wrote:Fafnir43 wrote:The other thing I think you need to tackle is that forming a surjection [imath]A\rightarrow B[/imath] can't imply that [imath]A[/imath] is "bigger" than [imath]B[/imath] in any meaningful sense, only that [imath]A[/imath] is at least as "big" as [imath]B[/imath]. It sounds obvious, but one of the most common responses when introducing something like a bijection from [imath]\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}[/imath] is "But you can map a tiny part of [imath]\mathbb{R}\times\mathbb{R}[/imath] onto the whole of [imath]\mathbb{R}[/imath]!"
Please, consider us poor subsilver users! That paragraph was horrible.
Token, I think you spend enough time in the math forum that it's worth making the switch. Trust me, you get used to it.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson

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Re: Math Paradoxes for English
Thank you for the ideas and explanations everyone.
You have helped me a lot, and I hope to become more active in xkcd's fora as time goes on.
You have helped me a lot, and I hope to become more active in xkcd's fora as time goes on.
Re: Math Paradoxes for English
Token wrote:Fafnir43 wrote:The other thing I think you need to tackle is that forming a surjection [imath]A\rightarrow B[/imath] can't imply that [imath]A[/imath] is "bigger" than [imath]B[/imath] in any meaningful sense, only that [imath]A[/imath] is at least as "big" as [imath]B[/imath]. It sounds obvious, but one of the most common responses when introducing something like a bijection from [imath]\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}[/imath] is "But you can map a tiny part of [imath]\mathbb{R}\times\mathbb{R}[/imath] onto the whole of [imath]\mathbb{R}[/imath]!"
Please, consider us poor subsilver users! That paragraph was horrible.
Wait, hold on. *Spends a little while working out what the hell subsilver is and why it's a problem*
Subsilver is the only theme in which the imath tags work?! I think if I fully express my feelings on that matter I will be banned in short order for massive obscenity, so suffice it to say that this renders giving the board LaTeX support in the first place close to meaningless and is a highly undesirable state of affairs. I will moderate my typesetting in future  thanks for letting me know.
hnooch wrote:I like to use the following example:
Consider an infinitely long table, with infinitely many place settings. We could never count up all the knives and all the forks, but we know that there are the same number of forks as there are knives because we can match them up.
It's a nice example in some ways, but I think if I were giving the talk I'd prefer to emphasise that we have to leave our traditional notions of "number" at the door a bit. The problem is that the example suddenly becomes a lot less intuitive if we add one person eating spaghetti (so he has a fork but not a knife), but there's still a bijection between N and N U {0} and hence there are still the same number of knives and forks.
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Re: Math Paradoxes for English
I agree with your point about the person eating spaghetti.
No, prosilver is. In subsilver, they don't work.Subsilver is the only theme in which the imath tags work?!
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 jestingrabbit
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Re: Math Paradoxes for English
Fafnir43 wrote:Wait, hold on. *Spends a little while working out what the hell subsilver is and why it's a problem*
Subsilver is the only theme in which the imath tags work?! I think if I fully express my feelings on that matter I will be banned in short order for massive obscenity, so suffice it to say that this renders giving the board LaTeX support in the first place close to meaningless and is a highly undesirable state of affairs. I will moderate my typesetting in future  thanks for letting me know.
At this point LaTeX support is a couple of weeks old, so we're still trying to work out what's going on ourselves.
However, I see it as very significant that subsilver (where support *doesn't* work) is deprecated, so at some point in the future, probably some number of months in the future, subsilver won't be a choice people can make, because it won't work with some software update that the boards need.
So the people holding onto subsilver are inevitably going to have to change at some time. This seemed like a good time to change to me, but some people obviously believe that their kneejerk aesthetic judgements are in some way objective and not at all merely a consequence of conditioning, thereby rendering their current dislike of prosilver unalterable and absolute, and not at all something that can be overcome with conditioning. This position seems unscientific to me.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
Re: Math Paradoxes for English
Oopsie, brain fart. I didn't realise LaTeX support was only a couple of weeks old! That changes things from utter stupidity on an incredible scale into a temporary inconvenience until things are fully set up  sorry about that...
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Re: Math Paradoxes for English
jestingrabbit wrote:So the people holding onto subsilver are inevitably going to have to change at some time. This seemed like a good time to change to me, but some people obviously believe that their kneejerk aesthetic judgements are in some way objective and not at all merely a consequence of conditioning, thereby rendering their current dislike of prosilver unalterable and absolute, and not at all something that can be overcome with conditioning. This position seems unscientific to me.
Your logic fails. I'm still on subsilver, and I don't believe that my aesthetic judgements are objective. I just want to stick with my preferred choice as long as I can. I'm fully capable of reading TeX without having to have it translated, so I see no pressing need to switch.
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 jestingrabbit
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Re: Math Paradoxes for English
Token wrote:jestingrabbit wrote:So the people holding onto subsilver are inevitably going to have to change at some time. This seemed like a good time to change to me, but some people obviously believe that their kneejerk aesthetic judgements are in some way objective and not at all merely a consequence of conditioning, thereby rendering their current dislike of prosilver unalterable and absolute, and not at all something that can be overcome with conditioning. This position seems unscientific to me.
Your logic fails. I'm still on subsilver, and I don't believe that my aesthetic judgements are objective. I just want to stick with my preferred choice as long as I can. I'm fully capable of reading LaTeX code without having to have it translated, so I see no pressing need to switch.
So... okay... you know you have to change sometime, you could change now, changing now would have benefits and a temporary cost, a cost you will have to bear anyway at some time, but you're choosing not to change. Yet it is my logic that has flaws. I admit that my analysis has a hole as I had not forseen your particular stance.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
Re: Math Paradoxes for English
jestingrabbit wrote:So... okay... you know you have to change sometime, you could change now, changing now would have benefits and a temporary cost, a cost you will have to bear anyway at some time, but you're choosing not to change. Yet it is my logic that has flaws. I admit that my analysis has a hole as I had not forseen your particular stance.
Which part of my previous post did you not read/understand? I don't think the benefits are all that great, and I don't think the cost is temporary. It seems to me like you are the one who doesn't realise this isn't objective.
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 BeetlesBane
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Re: Math Paradoxes for English
Token wrote:jestingrabbit wrote:..., but some people obviously believe that...
Your logic fails. I'm still on subsilver, and I don't believe that my aesthetic judgements are objective. I just want to stick with my preferred choice as long as I can. I'm fully capable of reading TeX without having to have it translated, so I see no pressing need to switch.
jestingrabbit''s logic doesn't fail. I've redacted his message to detail the key wording "some people believe." While you, or I, may question the claim of obviousness, Token's choice to remain with subsilver while contending he does meet the rest of the descriptor is logically included by the adjective "some".
Re: Math Paradoxes for English
My taking issue with the word "obviously" was the point. In the absence of any evidence that people use subsilver for that reason, the only way it would be obvious would be if it was the only possible reason. Which it isn't.
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 Cleverbeans
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Re: Math Paradoxes for English
Perhaps we could split this thread so the "I'm on about something silverish" segment can frolic to their hearts delight without distracting the original topic.
"Labor is prior to, and independent of, capital. Capital is only the fruit of labor, and could never have existed if labor had not first existed. Labor is the superior of capital, and deserves much the higher consideration."  Abraham Lincoln
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