32

Whether one can or can not double the brilliant in real life has nothing to do with the Banach-Tarski paradox. Various mathematical objects are models of various aspects of our universe. $\mathbb R^3$ is typically taken as a model of three-dimensional space. But, it is only a model. Some things that are true in the model are false in space and some that are ...


27

I think that actually trying to get students at this age to contemplate infinities in a rigorous way is probably ill advised. I do think that exploring counting from both an "ordinal" and a "cardinal" point of view is probably a good idea. Example for a 5 year old: Something you could do is have 20 stuffed animals, only 18 of them wearing hats. You can ...


22

I think that the following story is quite illuminating for introduction. Naturally, one can/should adapt/change it to better fit the audience, I just wanted to sketch the general idea. Suppose you came here by bus. Which bus was that? You say it was the 42, which is the line that goes from the main station to the university. However, was it really the 42? ...


17

This explanation from http://www.kuro5hin.org/story/2003/5/23/134430/275 might be useful: One important difference between S and a real, physical sphere is that S is infinitely divisible. Mathematically speaking, S contains an infinite number of points. This is not true of a physical sphere, as there are a finite number of atoms in any given physical ...


14

Matter is no bounded set of points with a non-empty interior The sets in the Banach–Tarski theorem have to fulfill some requirements: They need to be bounded. They have to have a non-empty interior. They need to be, well, sets of points. In particular their elements are identical, i.e., do not contain any other information than position. No matter how you ...


13

I would think a good first example is the rational numbers. (Note the "quotient" terminology here, too.) In particular, the rationals can be written as the set of integer pairs $(a,b)$ with $b\neq0$, modulo the following equivalence relation: $(a,b) \equiv (c,d)$ iff $ad = bc$. Note that the rule described here is commonly referred to as "cross-...


12

You can easily make them draw $\aleph$s; however the rest is much more demanding. There is a nice analysis from a researcher group taking a constructivist perspective. They distinguish potential infinity from actual infinity. The first one is covered by the idea of "counting on forever" the second one needs infinitely many things to exist in your thinking at ...


12

I think the level of the student is very important to this question. If the student has never had an abstract math course (like my students), then the lack of a definition of "number" is a great way to introduce the idea of abstract algebra. They are very happy to initially believe a definition like A number is anything that you can add and multiply, such ...


12

Not every property is preserved by limits. Here is a more basic situation in which the same reasoning is used: For each natural number $n$, there are only finitely many natural numbers in the interval $[0,n]$. However, letting $n \to \infty$, there are infinitely many natural numbers in the interval $[0,\infty)$. There is clearly no contradiction here, ...


12

Not a complete answer (could there be one?), but too long for a comment. why is it [Set Theory] not being taught at the very outset of math education? It has been tried, most likely still is in certain places at different degrees. For some history and background, lookup the New Math of the '60s, possible keywords Belgium, Willy Servais, Georges Papy. ...


11

I think, the most important point is to emphasize that the proof relies on the axiom of choice. If this connection is made, you "only" have to convince the students that the use of the axiom of choice is not constructive. I would ask them the following question: Take an arbitrary subset $A\subset \mathbb{Q}$. Please give me a concrete constructive way ...


10

A quintessential example of a cultural clash over math education in US schools, I would say, is the entire Math Wars, an ongoing struggle over standards and pedagogy in K-12 schools. It is a struggle of curriculum reformers trying to bring research-based innovations of mathematics thinking, learning, and teaching over the last three decades into schools ...


9

Consider a Sumerian person, living around 2500BC, who owns a flock. She hires shepherds to take the flock to pasture. Not being a scribe, she does not know Sumerian numbers- instead, for each sheep that passes her, she places a stone in an urn. When the shepherd returns, for each sheep that comes back, she removes a stone from the urn. If the last sheep ...


9

Failing argument Where one body is, there cannot be another. (Physical axiom) If you wanted to decompose and recompose the brilliant, you'd have to move some parts of it through some others, which is physically not possible. I don't know the proof, but I'm told, that there are decompositions and recompositions which avoid intersecting word lines. Valid ...


9

The problem is not one of reconciling analysis and set theory at all; it's one of understanding analysis properly (and how things go sideways when dealing with the infinite). One of the first and primary questions of analysis is, "when can limits and operators be interchanged?" To see why this is not trivial, consider the following exampe, which is a ...


8

I like this question very much. But I think the best approach is via a plethora of examples meant to demonstrate the variety of uses of equivalence classes. I doubt there is a singular example that can open every student's mind to the concept of equivalence classes. That said, here are some examples I have used effectively: 1) I have taught a "transition to ...


8

Whenever "the new math" comes up, I must point out that there was no one "new math" curriculum and much of that post WWII exploration into math curriculum and pedagogy had different aims and philosophies (Davis, 2003). (This doesn't directly pertain to your question, but the actual use of new instructional methods and curricular content during this period ...


8

My very incomplete understanding: I know from Henry Pollak (first-hand, but mentioned briefly here and in an article below) that one of the driving forces behind SMSG (with whom New Math is often associated) was Ed Begle. It should be noted that Begle's influence on mathematics education moving forward was also nontrivial: He served as advisor (along with ...


8

They are isomorphic. While $A\times B \neq B \times A$ for any arbitrary distinct sets $A$ and $B$, by defining the map $\phi:A\times B \to B \times A$ by $\phi(a,b)=(b,a)$, we can show that algebraically (or set theoretically) that they behave the same. Just a quick check of the set theoretic parts: $\phi(a,b) = \phi(c,d)$ iff $a=c$ and $b=d$. Hence $\phi$...


8

I think your specific situation with the exam question is similar to a pre-Calc class that spends a good amount of time on solving trig equations, but then asks on the exam for the student to write down the equation to describe a given scenario. These are just different skills. In a similar Discrete Math course I learned the hard way (as you have now) that ...


7

Perhaps this historical example fits the bill: Khovanova, T., & Radul, A. (2012). Killer problems. American Mathematical Monthly, 119(10), 815-823. The piece was published earlier on the arXiv as Jewish Problems. Here is the abstract: This is a special collection of problems that were given to select applicants during oral entrance exams to the ...


7

Here is another attempt. Consider the function $f:\mathbb{N} \to \mathbb{R}$ defined by $f(n) = \textrm{ The number of elements in the set $\{n,n+1,n+2,...2n\}$}$. In other words $f(n) = n+1$. Clearly $$\lim_{n \to \infty} f(n) = \infty$$ However, the limit of the sets $\{n,n+1,n+2,...2n\}$ is the empty set, since no element would be a member of such a ...


7

I like the idea here, but I agree that it misleads students, and might have the opposite of intended effect. Why not hand out a paragraph to the students, and ask them to critique it. Say that the paragraph is a fake student response to an exam question. One sentence in the paragraph could be something like ``Since $A \cap B$, there must be an $x$ so that $...


6

It depends on the context of the course where Banach-Tarski is being presented. My answer relies on knowing a bit of measure theory. Let's call our intial set which we're going to split into some pieces $A$, and the reassembled version of the pieces $B$. If $A$ is a bounded and measurable set in $\mathbb{R}^3$ and $A= \bigcup_{k=1}^n A_k$ is a pairwise ...


6

1. One main point that has been made is that Banach-Tarski Paradox needs to use non-mesurable sets; I want to insist on this by comparing with another similar looking statements. Theorem (Wallace-Bolyai-Gerwien). Given any two polygons of equal area, it is possible to cut one of them into finitely many pieces and rearrange the pieces into the second polygon....


6

I like Chris C's answer; I will offer another point of view. Perhaps the difficulty lies in the example that has been chosen: a deck of cards isn't naturally a cartesian product for exactly the reason that there is no natural ordering of rank and suit. Instead, you might look at a (simplified) menu, and consider all of the meals that consist of a main ...


6

First, although you talk a bunch about cardinality, I don't see how that makes sense, so I'm going to assume you mean that you have them determine if the set corresponding to p is a subset of the set corresponding to q. (Otherwise, in your second example, you'd also have $x=2$ implies $x^2=9$, for instance.) In formal terms, your method requires ...


6

I would not recommend to teach this method since there are some downsides. Take $A(x) \iff x \text{ is divisible by } 2$ $B(x) \iff x \text{ is divisible by } 42$ Is $A(x) \implies B(x)$ or $B(x) \implies A(x)$? Since there are infinitely many $x\in\mathbb Z$ fullfiling $A(x)$ and $B(x)$ this cannot be answered unless your students already know about the ...


6

I think the issue here is really more one of problem solving skill or abstraction ability. You could see the same thing in various problems in computer science or in college algebra where some students can solve an apparently complex problem simply and others get caught up in the complexity. Or you see the same thing on problems that contain extraneous ...


5

(I'm giving this as another answer, since it is completely unrelated to my other one.) A very simple example of a natural equivalence relation is (for two straight lines on a plane) to be parallel. The equivalence classes may be thought of as directions on the plane. This may have an added benefit of showing beginners that in mathematics, properties (i.e.,...


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