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I have been tasked with performing a "spoof" lecture for the new math undergraduates at my school. It is supposed to be a bit intimidating and confusing, but not too bad, and still humourous (at least for those that understand the stuff). I also want everything I say to be true, so no jokes about all functions being linear or anything.

I have already done this before, but that time I found out about it just a few days in advance, so I ended up telling the students a little bit about functions and sets, and proceeded to give some definitions and theorems from homological algebra. It was kind of made up on the spot, but it was confusing and not very consistent. This time it is supposed to be in late August, so I can plan a lot more. I already have a few ideas, but I am interested in knowing what the community thinks. Maybe you have done something like this before, and can share your experiences. Maybe you are yourself an undergraduate, and know what would catch you off guard. Regardless, I am open for any ideas. Keep in mind that it is not only about the content, but about the presentation too; if the level makes a big jump at the right time, then all the better.

Thanks for any input, and I will share my ideas later.

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    $\begingroup$ There is, for example, the famous "mathematical ways to catch a lion" lecture $\endgroup$
    – lulu
    Jul 8, 2017 at 15:10
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    $\begingroup$ @lulu I believe math.ucdenver.edu/~wcherowi/mathmajor/archive/catchlion.pdf is the original article, although I don't know the full story behind it. $\endgroup$
    – Chappers
    Jul 8, 2017 at 15:14
  • $\begingroup$ @Chappers Thanks! I just grabbed the first article which listed a lot of lion-catching methods. I'm sure I didn't get the original. $\endgroup$
    – lulu
    Jul 8, 2017 at 15:23
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    $\begingroup$ I don't know why you would want to intimidate or confuse students. Math is already hard enough, and math anxiety exists in many students - it doesn't seem wise to add fuel to the standard insecurities (by making undergrads feel that it is expected that they follow a proof in homological algebra). I understand how this could be funny if done tactfully, it could also be traumatizing (and deter students) if done carelessly. It is worth being cautious. This concern aside, maybe you can deliver a lecture out of the book "mathematics made difficult." $\endgroup$
    – Elle Najt
    Jul 8, 2017 at 15:28
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    $\begingroup$ I sometimes think of the Curry-Howard correspondence between excluded middle and call-with-current-continuation as "somebody with the power to rewind time can convince you they know the answer to every true/false mathematical question". The setup is roughly that they can either answer false without a proof, or answer true with a proof - and then the adversary tries to catch them in a contradiction. The algorithm is roughly: at first, they answer false to every question, and it turns out any proof of a contradiction will prove one of the things they said was false - so then they... $\endgroup$ Jul 8, 2017 at 17:18

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Proof that $2=4$. Take an equation with an infinite tower of exponents: $$2=x^{x^{x^{x^{....}}}}.$$ Then $2=x^2$ so $x=2^{1/2}.$ And another such equation: $$4=y^{y^{y^{y^{....}}}}.$$ Then $4=y^4$ so $y=4^{1/4}$ so $y=4^{1/4}=2^{1/2}=x.$ Therefore $$2=x^{x^{x^{x^{....}}}}=y^{y^{y^{y^{....}}}}=4.$$

My point is not the joke but rather to illustrate the importance of not making unwarranted assumptions: The paradox occurs because the assumption that there can exist such $x,y$ with the required properties is false. Also this illustrates a proof by contradiction: That at least one of the two main equations (the second one, actually) has no solution.

Likely, many students will think that the paradox occurs because the infinite tower is an "invalid move", but that is not the reason.

On a different topic, you might ask for a show of hands for the Q of whether $0.\overline 9$ is $1$ or less than $1.$ My experience is that most high-school graduates were not taught properly about the logical foundations of $\mathbb R$ and are confused about this Q. It is hard to learn about limits and calculus on $\mathbb R$ without knowing what $\mathbb R$ is. You might consider asking someone who says it's less than $1$ for their reaction to $1=3\times (1/3)=3\times 0.\overline 3=0.\overline 9.$.... And let the students know that (1) this Q does have an answer, and (2) the answer is more complicated than they might expect.

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  • $\begingroup$ Good one. This seems elementary enough for them to actually understand (instead of the stuff just going over their heads). $\endgroup$ Jul 8, 2017 at 19:57
  • $\begingroup$ The equation in x was a Q on an Ontario high-school math contest (multiple-choice Q's) when I was in high school in the Pleistocene Age. At that time I wondered what would happen if you changed the 2 to something else,like 4......BTW One teacher wrote that he would tell his students to differentiate the real function $\ln \ln \sin x$ and most of them would of course use the chain rule. The explanation of why the rule seems to work for a non-existent function requires a non-trivial excursion into complex calculus, $\endgroup$ Jul 9, 2017 at 0:33
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This is from A Path To Modern Mathematics by W.W. Sawyer. We want to solve $f(x)=1+\int_0^xf(t)dt$. Re-arrange and abbreviate as $1=f-\int f=(I-\int)(f)$. Therefore $$f=\left(I-\int \right)^{-1}(1)=(I+\int +\int \int + \int \int \int +...)(1).$$ We have $\int 1=x$, and $\int \int 1=\int x=x^2/2$ and so on. Therefore $f(x)=1+x/1!+x^2/2!+x^3/3!+...=e^x.$

He wrote that the response by many students to this is laughter. (As if it were just a lucky trick.)

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