Looking for more examples of absurd math constructs that go against our intuitive biases.

[MLFinley]

So my general premise is to make a persuasive video essay. The argument I want to make is that math is not just invented, but downstream of the axioms chosen . There is nothing inherently true about it besides its utility and internal consistency.

I am interested in this subject for two different reasons.

The first one is that I took a Modern Algebra class last semester and all of the abstractions creating these really absurd structures just tickled me every time. The entire class was just abstracting away everything you knew about addition and multiplication down to groups and rings and it created these wonderful scenarios where, when you're being a bit handwavy, you get absurd nonsensical results.

The other is that I am a trans woman and quite often (and especially in the current climate) people will try to make arguments about definitions from this idea that there is some absolute truth. Sometimes you will hear the argument that saying trans women are women is like saying that 2+2=5. That it is just made up on no basis. The problem with this is there are good solid reasons for saying trans women are women and it is not only sometimes possible but also practical to construct mathematical systems which go against our intuitive biases.

What I am looking for is examples of math producing seemingly absurd results that can be explained simply to a lowest common denominator.

The only two ideas I have right now are different base systems (binary, hex, etc) and modular arithmetic. In both of these areas, the definition of addition changes in such a way that 1+1 = 10 or 2+2=1 (mod 3) are sensible reasonable statements.

An overly complicated unusuable example that I think is applicable to the idea I am trying to get at would be the controversy surrounding the axiom of choice. I can't really wrap my mind around the AoC but the fact that there is not some inherently right choice as to whether to include or not include the axiom is the kind of thing I am looking for. Examples where there is not a strictly right or wrong way to do math.

Sorry for the long rambley post.

[alexmoon01]

Here's two cool ones.

1. There's a lot of cool counterexamples in topology that may fit your criteria. One of my favorites is the Topologist's sine curve, a space which is connected but not path connected. I will give a fair amount of explanation since I think it's fun, but the topic is very visual and can be explained to a layperson with very little exposition.

A space is "path connected" if every pair of points in the space has a path connecting them. A space is "connected" if there's no way to decompose it into disjoint open subsets. Informally, a subset S of R^2 is disconnected if it's possible to draw a curve disjoint from S that splits the plane in such a way that part of the space is on one half of the plane, and part of the space is on the other. The union of the black circles in this image is an example of a disconnected space. You can see that the red line splits the plane into two halves, with one circle on each side. Also, the red line never touches either circle. However, a single circle is connected, since any curve that splits the set into pieces would have to pass through the set itself. Intuitively, one might think that path connectedness and connectedness are equivalent, but the topologist's sine curve provides a beautiful counterexample.

The topologist's sine curve is the graph of sin(1/x) for x > 0 (the sine), unioned with the set of all points of the form (0, y) for -1 <= y <= 1 (the bar). You can see it here on Desmos. You'll notice as you zoom in to the blue line that Desmos can't really handle sin(1/x). The key trait of the topologist's sine curve is that there's no path connecting any point on the sine to any point on the bar. Yet, every point on the bar is infinitely close to the sine! It's impossible to split the space like we did with the two circles.

2. Another one that is more related to your modular arithmetic ideas is a toy puzzle I came across a few days ago. It also suits itself to the visual format, since you have to draw a lot of octopi. Say you have 45 octopi living in an octopus commune. 13 of them are blue, 15 of them are red, and 17 of them are yellow. When two differently colored octopi bump into each other, they both turn into the third color. For example, if a blue octopus were to bump into a red octopus, both of them would turn yellow. Does there exist a sequence of collisions that will make all of the octopi end up the same color? Spoiler under the "details" tab in case you want to solve the puzzle yourself.
   Surprisingly, the answer is no! You can see this by considering the problem mod 3. There are 1 (mod 3) blue octopi, 0 (mod 3) red octopi, and 2 (mod 3) yellow octopi. If you play around with the possible bumps, you will see that there always has to be one color that's 1 mod 3, one color that's 0 mod 3, and one color that's 2 mod 3. However, if all 45 octopi were the same color, every color would be 0 mod 3. Contradiction!

[Illumimax]

Examples from logic: -The axiom of determinacy contradicting the axiom of choice (but likely consistent with the axiom of dependent choice) -Martins axiom ++ implies Woodins axiom (*) making a case against the continuum hypothesis -The anti-foundation axiom and its suprising usefulness Also inaccessible cardinals are just completly wild. (Dont know enough about them yet though.) (And of course the classics like Gödel showing the incompletness of arithmetics or more than two valued logics.)

[Nikolaj-K]

A right list of shock-and-explain statements can be found at

https://en.wikipedia.org/wiki/List_of_paradoxes

I feel your examples (like 2+2=1 (mod 3 )) of "absurd mathematical statements" all just exist for the second where we don't recognize that the words and symbols are used differently. They are "absurd" in the same way "A beetle has 3 legs. And I drive a beetle. And my favorite song writer was a beetle." If nothing else, you can take that as an example, albeit not a mathematical one.

I can't really wrap my mind around the AoC but the fact that there is not some inherently right choice as to whether to include or not include the axiom is the kind of thing I am looking for.

But the inherently right choice is of course not to adopt it?

[kevinb9n]

The world is overflowing with lessons to the effect that we cannot trust our intuitions (or our senses) as a rock solid guide to what's true. And yet we can always use more. It's not necessarily that there isn't any absolute truth, but it is certainly much, much more difficult to perceive/access than we think it should be.

The people who believe gender is binary and immutable, which is scientifically absurd, might as well be flat earthers; they aren't interested in the lessons. I don't say any of this to discourage you from your goal, though! Godspeed.

[jkenderes]

I'm very interested in the "truth vs. usefulness" idea, and spend a lot of time talking about this with my students. The classic example here is geometry (I reference the H.J. Poincare quote: "our mind has adapted itself to the conditions of the external world. It has adopted the geometry most advantageous to the species or, in other words, the most convenient. Geometry is not true, it is advantageous"). If you're an architect who wants to use trigonometry, assuming the parallel postulate is pretty useful. However, if you're mapping out flight paths or shipping lanes on the surface of the earth, you'll want to reject the parallel postulate and use spherical geometry/trigonometry. This doesn't make the parallel postulate true or false, just useful in some contexts and not so much in others.

But before diving into non-Euclidean geometry I like to talk a bit about Boolean algebra where it is useful to define an arithmetic and algebra where some of our typical rules hold, like A+B = B+A but others are different, like A+A = A. It is not more or less true than the algebra we learn in HS, it's just useful in different contexts.