Your most counterintuitive yet simple math problem

[Loren Booda]

Mine is the Monty Hall paradox. For an introduction, please see http://en.wikipedia.org/wiki/Monty_Hall_problem"

 

[Georgepowell]

How about the probability that two people share a birthday in a group of 23 people?

It is surprisingly just over 50%. 23 is the smallest value for which the probability is over 50%.

Perhaps even more surprisingly, if there are only 50 people in a room, the probability is 96.5% that two people share a birthday.

Most people would guess a much lower value.

 

[Loren Booda]

That one is great too, George. It always amazes me. (Great to try out on a third grade class!) I suppose that it could be phrased in any number of contexts.

 

[HallsofIvy]

How about the Banach-Tarski property: that a sphere in three dimensions can be partitioned into a finite number of subset and, by rigid motions reasssembled into two spheres of the same size as the first sphere.

 

[Georgepowell]

How about the Banach-Tarski property: that a sphere in three dimensions can be partitioned into a finite number of subset and, by rigid motions reasssembled into two spheres of the same size as the first sphere.

Do you understand this one? I don't... Not one bit. A friend told me about it and I immediately told him he had been misinformed. But he send me a link later and I was baffled... It is obviously illogical and it is a genuine paradox!

What is the explanation... It can't be done in real life, so what is the error?

edit: I have just been reading a book about maths, and there is a section about how our mathematical system of logic is fundamentally flawed. Something to do with sets of sets and so on. Is this to do with it?

 

[adriank]

No, the incompleteness of mathematics has nothing really to do with it. It has everything to do with infinite sets and the way you assign a "volume" to them.

The basic idea that the Banach-Tarski "paradox" is based on is that, say in 3D Euclidean space, you can assign a volume (a measure) to certain subsets of the space. However, any reasonable definition of a measure must have some sets that are unmeasurable; they really don't have a well-defined volume. The idea is that the ball can be split up into a few subsets that are unmeasurable; you can then transform these unmeasurable sets (by rotations and translations) such that they don't overlap, and then take their union to obtain another measurable set, which is in fact two balls of the same size as the original! The result depends crucially on the fact that you're in 3D space; it doesn't happen in 2D space. It also depends on the fact that unmeasurable sets are involved; the measure of a (countable) union of disjoint measurable sets is the sum of the measures of the original sets.

This isn't possible in real life, of course, as in reality balls are composed of finitely many atoms.

You may or may not gain some more insight by looking at Wikipedia.

 

[Fredrik]

edit: I have just been reading a book about maths, and there is a section about how our mathematical system of logic is fundamentally flawed. Something to do with sets of sets and so on. Is this to do with it?

It's not true that "our mathematical system of logic is fundamentally flawed". You mention sets, so you're probably thinking of Russell's paradox. If we're allowed to form the set X=\{Y|Y\notin Y\} (the set of all sets that aren't members of themselves), then we get a paradox. (Is X a member of X?). Such paradoxes can be avoided by putting restrictions on what sets we're allowed to form. See e.g. the ZFC axioms.

The Banach-Tarski paradox is just a result of the fact that it isn't possible to define a "size" of an arbitrary subset of the real numbers in a meaningful way.

 

[robert Ihnot]

Monty Hall paradox can be studied with cut up pieces of individual paper, but a very convincing way I use is to consider the three cases for door A, which is arbitrary and covers all cases:

We have: Behind A, Behind B, Behind C. Only in 1/3 of the cases does it pay not to switch. In the other two cases, since one of the wrong doors has been eliminated, by switching we have the right door!

It helps to draw a very simple grid indicative of that, and the answer is obvious. But, attempting to work it over in your mind, doesn't work so well as a simple diagram.

 

[adriank]

It's not true that "our mathematical system of logic is fundamentally flawed". You mention sets, so you're probably thinking of Russell's paradox. If we're allowed to form the set X=\{Y|Y\notin Y\} (the set of all sets that aren't members of themselves), then we get a paradox. (Is X a member of X?). Such paradoxes can be avoided by putting restrictions on what sets we're allowed to form. See e.g. the ZFC axioms.

When I read his statement I was thinking more of Gödel's incompleteness theorems.

 

[NoMoreExams]

For me it was understanding that there are more reals in (0,1) than all rationals

 

[Dragonfall]

The fact that there is a real to real function that is continuous, not constant on any interval, and has uncountably many zeroes.

 

[Georgepowell]

It's not true that "our mathematical system of logic is fundamentally flawed". You mention sets, so you're probably thinking of Russell's paradox. If we're allowed to form the set X=\{Y|Y\notin Y\} (the set of all sets that aren't members of themselves), then we get a paradox. (Is X a member of X?). Such paradoxes can be avoided by putting restrictions on what sets we're allowed to form. See e.g. the ZFC axioms.

Yeah that's what I was thinking of. So until that paradox was discovered, where there no restrictions on what sets were allowed to form?

 

[Hurkyl]

Yeah that's what I was thinking of. So until that paradox was discovered, where there no restrictions on what sets were allowed to form?

Yesish. Keep in mind that set theory had only recently been made explicit, and that there are obviously other restrictions (e.g. sets have to be sets). But in its initial form, set comprehension was indeed unrestricted.

 

[HallsofIvy]

Do you understand this one? I don't... Not one bit. A friend told me about it and I immediately told him he had been misinformed. But he send me a link later and I was baffled... It is obviously illogical and it is a genuine paradox!

What is the explanation...

As Adriank said, it involves using non-measurable sets

It can't be done in real life, so what is the error?

There is no error. Perhaps your "real life" is too restricted!

edit: I have just been reading a book about maths, and there is a section about how our mathematical system of logic is fundamentally flawed. Something to do with sets of sets and so on. Is this to do with it?

The fact that "naive set theory" is not rigorous only means that we need a more sophisticated concept of "sets"- and that has already been developed.

If you were thinking of Goedel's incompleteness theorem, that only says that we we will never have a "finished" mathematical system- there will always be more to do. I don't consider that a flaw!

 

[Borek]

If you were thinking of Goedel's incompleteness theorem, that only says that we we will never have a "finished" mathematical system- there will always be more to do. I don't consider that a flaw!

As I have stated on numerous occasions I am mathematically challenged, so I can be wrong, but from what I understand it says that even if we will do everything, there will be still statements that we will be not able to say if they are true or false.

Could be that implies that there is still something to do outside of the system in which we can't decide...

 

[Dragonfall]

Ooo, let's not forget that most real numbers are uncomputable.

 

[Tac-Tics]

I have just been reading a book about maths, and there is a section about how our mathematical system of logic is fundamentally flawed. Something to do with sets of sets and so on. Is this to do with it?

Here's something for your noggin. I don't know your familiarity with set theory, but I hope you can enjoy it.

By http://en.wikipedia.org/wiki/Cantor%27s_theorem" , the number of subsets you can make from a given set will always be strictly greater than the number of elements in the set itself. That is, if S is a set and P(S) is the powerset of S, S < P(S).

Take A to be the set of all sets, defined by A = {S | S is s set}. By Cantor's Theorem, A < P(A). However, P(A), the powerset of A, is also the set of all sets. So it's cardinality is both equal to itself and inequal to itself. A paradox.

The resolution to this paradox is something similar to what your comment above hints at. It's not a fundamental flaw, inasmuch as it's not really a flaw. Most mathematicians don't need to care about it at all, and boring old set theory will almost never get you into trouble as long as you're not looking for it.

 

[D H]

That an object with finite area can have an infinite perimeter.

For me it was understanding that there are more reals in (0,1) than all rationals

That's a good one, too. 2^{\aleph_0}&gt;\aleph_0 is a mathematical equivalent to relativity theory in the sense that both are crackpot magnets. Now that Fermat's last theorem has been put to bed, the majority of "interesting" unsolicited proofs that professors of mathematics receive from the lay community are attempts to disprove Cantor's result.

 

[NoMoreExams]

That an object with finite area can have an infinite perimeter.

Are you talking about fractals/space filling curves? What you said made me think of something else, objects with infinite area but finite volume (Gabriel's horn is the one I was thinking of).

 

[D H]

Sorry for not being clear. An unbounded planar object can clearly have finite area. Since it is unbounded, it obviously has an infinite perimeter. So yes, I was talking about fractals -- e.g., the coastline of Britain.

 

[Fredrik]

I think I read somewhere that there's a continuous bijection from \mathbb R into \mathbb R^2. (I hope someone will let me know if this is wrong). The existence of a bijection isn't surprising to me. Intuitively, that just means that the two sets have "the same number" of members. The existence of a continuous bijection is very counterintuitive however. That would be a curve that goes through every point in a plane without ever intersecting itself.

 

[NoMoreExams]

Yes, |\mathbb{R}| = |\mathbb{R}^{n}| I believe

 

[adriank]

Well, there is a continuous surjection from [0, 1] to [0, 1]², but no continuous bijection. The existence of such a bijection would mean that [0, 1] and [0, 1]² are homeomorphic, which isn't true.

This would be my pick for the counterintuitive math thing. :)

 

[Georgepowell]

For me it was understanding that there are more reals in (0,1) than all rationals

There are more reals between any two different rationals, than all rationals overall.

This is perhaps a slightly more interesting concept. Although it is essentially the same.

Besides, this is an unusual definition of "more" anyway. It is not the same sort of "more" that we usually use, all that we really know is that there isn't a way of counting (or pairing with the naturals) all the irrationals.

 

[adriank]

Yes, you have.

 

[Georgepowell]

Sorry for not being clear. An unbounded planar object can clearly have finite area. Since it is unbounded, it obviously has an infinite perimeter. So yes, I was talking about fractals -- e.g., the coastline of Britain.

That's a controversial example

 

[D H]

That's a controversial example

In what sense? Mandelbrot's paper "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" (Science, 156(3775), pp. 636 - 638) is one of the seminal papers on fractals.

 

[Georgepowell]

In what sense? Mandelbrot's paper "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" (Science, 156(3775), pp. 636 - 638) is one of the seminal papers on fractals.

I have not read that paper, but I presume that he only talked about the length of the coast of Britain as a concept, or an analogy. And did not suggest that it was actually definatly infinite. The most accurate way you could measure it is to measure around each atom... In which case it might not be infinite.

By some definition it might be still be infinite, but by other definitions it is not. Anyway, I said "controversial", and not "false".

 

[Count Iblis]

"[URL
The Löwenheim-Skolem Theorem[/URL]

Skolem's Paradox

LST has bite because we believe that there are uncountably many real numbers (more than 0). Indeed, let's insist that we know it; Cantor proved it in 1873, and we don't want to open the question again. What is remarkable about LST is the assertion that even if the intended interpretation of S is a system of arithmetic about the real numbers, and even if the system is consistent and has a model that makes its theorems true, its theorems (under a different interpretation) will be true for a domain too small to contain all the real numbers. Systems about uncountable infinities can be given a model whose domain is only countable. Systems about the reals can be interpreted as if they were about some set of objects no more numerous than the natural numbers. It is as if a syntactical version of "One-Thousand and One Arabian Nights" could be interpreted as "One Night in Centerville".

 

[HallsofIvy]

If you were thinking of Goedel's incompleteness theorem, that only says that we we will never have a "finished" mathematical system- there will always be more to do. I don't consider that a flaw!

As I have stated on numerous occasions I am mathematically challenged, so I can be wrong, but from what I understand it says that even if we will do everything, there will be still statements that we will be not able to say if they are true or false.

Could be that implies that there is still something to do outside of the system in which we can't decide...

Yes, and that is exactly what I said in what you quoted: we will never have a "finished" mathematical system.

 

[Loren Booda]

Do you all think that fundamental, visualizable math relations have mostly been considered?

 

[Hurkyl]

The existence of [a continuous] bijection would mean that [0, 1] and [0, 1]² are homeomorphic, which isn't true.

Is that obvious? I know that there are continuous bijections which aren't homeomorphisms (e.g. |X| --> X for any space X that is not given the discrete topology)...

 

[dvs]

Is that obvious? I know that there are continuous bijections which aren't homeomorphisms (e.g. |X| --> X for any space X that is not given the discrete topology)...

A continuous bijection from a compact space onto a Hausdorff space is automatically a homeomorphism.

 

[enigmahunter]

The "http://en.wikipedia.org/wiki/Continuum_hypothesis" " is my choice.

"There is no set whose size is strictly between that of the integers and that of the real numbers."

It is an interesting problem and considered as a fact, even though it cannot be proved or disproved in ZFC.

 

[Count Iblis]

Suppose you simulate a world in which mathematicians and physicists live on a huge computer. Then everything in that world will be discrete and countable. Nevertheless the virtual mathematicians and physicists will likely still invent uncountable sets, real numbers, Axiom of Choice, etc. and pretend that it applies to their world.

 

[csprof2000]

"Suppose you simulate a world in which mathematicians and physicists live on a huge computer. Then everything in that world will be discrete and countable. Nevertheless the virtual mathematicians and physicists will likely still invent uncountable sets, real numbers, Axiom of Choice, etc. and pretend that it applies to their world. "

And then when people said that the real world was really only 2-D and made of discrete bits and pieces, the mathematicians and physicists would say something like "Perhaps your experience is limited" or something. Then the mathematicians would continue proving there are numbers that don't have any value you can name and physicists would keep letting all functions equal the first term in their Taylor expansions.

Seriously, though... Set theory in general (particularly that which applies to infinite sets) seems dangerously close to being more mysticism than logic. I don't like it and, thankfully, computer scientists don't have to.

 

[Hurkyl]

Seriously, though... Set theory in general (particularly that which applies to infinite sets) seems dangerously close to being more mysticism than logic.

Then you need to learn more about logic.

I don't like it and, thankfully, computer scientists don't have to.

You're joking, right? How exactly do you plan to study the theory of computation if you can't talk about languages (or worse, classes of languages)? How do you plan to study generic programming if you don't allow type variables? How do you plan on discussing the semantics of types like java.math.BigInteger? How do you plan on doing asymptotic analysis without calculus?

 

[csprof2000]

We have the luxury of using potential, rather than actual, infinities in computer science. It's called constructive mathematics. Go read a book before you pretend to know things.

I don't think anybody has a problem admitting that there is no largest integer. Likewise, who could pretend to be able to list all the reals between 0 and 1? Still, there's no reason to linger on such things as a theory of infinities... unless that's what you like.

 

[Hurkyl]

We have the luxury of using potential, rather than actual, infinities in computer science. It's called constructive mathematics. Go read a book before you pretend to know things.

Before your retort, did it even cross your mind that maybe, just maybe, I have some clue what I'm talking about? Whether you call them "potential" or "actual", you're still doing set theory with infinite sets.

And really, I disagree with the judgement you're using potential infinities. e.g. if sets are expressed either through enumerating its elements via a Turing machine, or a calculation of its membership relation via a Turing machine... the Turing machine serves as a complete description of the set.

But really, at this point, we run into the problem that "actually infinite" and "potentially infinite" aren't (to my knowledge) well-defined notions, so there isn't really any substance to such a debate.

 

[Vanadium 50]

For me it was understanding that there are more reals in (0,1) than all rationals

Related to that - nearly all numbers are normal, yet there are very few - perhaps zero - concrete examples of one.

 

[csprof2000]

Hmmm...

If there were zero concrete examples of a normal number, would you still say they exist?

The ancients didn't believe in proof by contradiction. If somebody knows why people started believing in it, I'd love to know. Was it Aristotle? That seems too early. But I really don't have any idea.

I think that any mathematical object that can be shown to exist but which also has no concrete example is the most counterintuitive thing in mathematics. Most of the rest... some probability aside... is usually straightforward, if with some hindsight.

 

[Hurkyl]

The ancients didn't believe in proof by contradiction.

I would love to see a reference for that claim.

If somebody knows why people started believing in it, I'd love to know. Was it Aristotle?

http://planetmath.org/encyclopedia/ReductioAdAbsurdum.html cites Aristotle citing Euclid's use of RAA, so at least as far back as that.

If there were zero concrete examples of a normal number, would you still say they exist?

Logically, one would say a normal number exists if and only if one could prove the statement

\exists x \in \mathbb{R} : x \text{\ is normal}

(and it is a theorem of real analysis in classical logic)

 

[daniel_i_l]

I'm not sure what one this is called but consider the following trivial question:
You are given a family with two children, one of them is a girl, what are the chances that the other one is a girl? The answer is of course 1/3.
Now consider a similar problem:
You are given a family with two children, one of them is a girl named Florida (a very rare name), what are the chances that the other one is a girl? The answer in this case approaches 1/2 as the popularity of the name Florida approaches 0. (to see why this is write down all the possibilities)
It amazes me that knowing the name of one girl changes the chances that the other one will also be a girl.

 

[CRGreathouse]

So Hurkyl cited PlanetMath's citation of Aristotle citing Euclid's use of RAA?

 

[Tac-Tics]

"Suppose you simulate a world in which mathematicians and physicists live on a huge computer. Then everything in that world will be discrete and countable.

This reminds me of a quote by Einstein:

"God does not care about our mathematical difficulties. He integrates empirically."

Probably not fully applicable to the conversation, but I always thought it was a neat idea when considering what would otherwise be infinitely time-consuming integration.

Nevertheless the virtual mathematicians and physicists will likely still invent uncountable sets, real numbers, Axiom of Choice, etc. and pretend that it applies to their world. "

This seems like a realness fallacy. I don't know the proper name for it, but claiming that some things are more "real" than others because they are better understood or more classical. For example, some people think that "infinity" isn't real and zero is, because distances in the real world can be measured to be zero, but not infinity. But the fallacy is in that both are just an abstraction. If you have a loop, for example, and follow it in a circle, measuring how long you travel before you find the end, there IS no value to satisfy how long you've gone, so infinity, in some sense, is completely legitimate.

Mathematicians have created the notions of the real numbers because they are more useful in many instances than rationals. If we stick to rationals, then the number of allowed angles we can measure is crippled, as a triangle with a typical rational angle leads to an irrationally long hypotenuse. And the loss of the least upper bound property is death of many useful phenomenon. For example, you can have a continuous rational function which assumes both positive and negative values, but which has no zeroes.

The axiom of choice is just plain handy, and it doesn't matter if it reflects reality, because it makes life easier.

Then the mathematicians would continue proving there are numbers that don't have any value you can name and physicists would keep letting all functions equal the first term in their Taylor expansions.

Mathematicians solve problems faster than the real world can provide. Physicists need to know the answers faster than they can prove them. It works out for both teams.

Seriously, though... Set theory in general (particularly that which applies to infinite sets) seems dangerously close to being more mysticism than logic. I don't like it and, thankfully, computer scientists don't have to.

Perhaps if you haven't, you should read Godel Escher Bach. If you do, you should re-read it. If you assume only fundamentals of logic and peano's axioms of integers, you get a system which every bit as set theory does. Even if your formal system doesn't allow for functions, sets, or cardinality literally, you can easily write a computer program that can translate sentences back and forth between an axiomatic set theory and any other logical system, like Hofstadter's Typographical Number Theory (TNT). Once you have a way to translate sentences like that, you're in trouble, because you have problem your system to have the same logical power as the other. In other words, the infinite mysteries of the integers ARE the mysticism of set theory.

 

[Hurkyl]

If there were zero concrete examples of a normal number, would you still say they exist?

It just struck me; there's an interesting variation on this. In the arithmetic of complex numbers^*, there are two solutions to the polynomial equation x^2 + 1 = 0... however, neither one can actually be constructed!

(The reason is symmetry; any condition expressed using the elementary notions of arithmetic that is satisfied by a is also satisfied by the complex conjugate of a)

Of course, particular models of the complex numbers could provide an external construction for such roots. (e.g. building the complex numbers out of R², or as a quotient of R[x]) But such constructions cannot be carried out in a purely arithmetic fashion, and require properties specific to the ambient theory and construction of the specific model.


*: Of course, the theory could be formulated in other ways that don't have this property; e.g. one could provide an explicit constant symbol that is axiomatically defined to be a solution to x^2 + 1 = 0, in which case it is trivial to construct a solution

 

[Doodle Bob]

Seriously, though... Set theory in general (particularly that which applies to infinite sets) seems dangerously close to being more mysticism than logic. I don't like it and, thankfully, computer scientists don't have to.

Seriously, though, brick-layers also don't generally have to know and understand set theory, but that's not a particularly high mark of honor for the field of masonry.

Why you feel the need to denigrate a field just because you don't understand it very well, is beyond me.

 

[Vid]

My friend just sent me a proof that given any function from R->R if you tell me every value except at x_o, there is a strategy to guess the value with probability 1.

 

[adriank]

That sounds interesting. Do you mind posting it?

 

[daniel_i_l]

My friend just sent me a proof that given any function from R->R if you tell me every value except at x_o, there is a strategy to guess the value with probability 1.

Even if the function isn't continuous?