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As you can see in this link, the following argument originated from Georg Simmel was put forward against Nietzsche:

Even if there were exceedingly few things in a finite space in an infinite time, they would not have to repeat in the same configurations. Suppose there were three wheels of equal size, rotating on the same axis, one point marked on the circumference of each wheel, and these three points lined up in one straight line. If the second wheel rotated twice as fast as the first, and if the speed of the third wheel was 1/π of the speed of the first, the initial line-up would never recur.

I'm not here to ask about the philosophy, but rather the validity of Simmel's argument. Can't they really recur back to their original state even if it takes so much time?

My take: Even though one of the wheel's speed is irrational, it's still a constant, so I think we can pass his assumption of having an irrational number as a wheel's speed. Having that, shouldn't we be able to reach a point where they will end up back to their original state like many least common multiple problems do (although as I have found while trying to solve it, it's not as those problems)?

Acnologia
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    It can get arbitrarily close if you give it enough time of spinning, but never quite exactly the same position. – mathreadler Dec 10 '17 at 14:05

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If you just look at the first wheel going 1 rotation per unit time, and the other wheel going $1/\pi$ rotations, they will never again line up (I do not know why that example has 3 wheels).

Let $r_1(t)$ and $r_2(t)$ be the radians spun by the first and other wheel over time $t\geq 0$. Then:

  • $r_1(t) = 2\pi t$

  • $r_2(t) = \frac{2\pi t}{\pi} = 2 t$

An integer number of rotations means an integer multiple of $2\pi$ radians. So to line up again at some time $T>0$, we need positive integers $n,k$ so that

\begin{align} \underbrace{r_1(T)}_{2\pi T}&= 2 \pi k\\ \underbrace{r_2(T)}_{2T}&= 2\pi n \end{align}

So we need $2 \pi T = 2 \pi k, 2 T = 2\pi n$, so $T = k = \pi n$, that is: $$ \pi = \frac{k}{n} $$ This is impossible since $\pi$ is irrational.

Michael
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  • So, do we conclude that the argument is valid since we have seen that an irrational speed is contradictory, or is it invalid since we can't have an irrational speed for the wheel? – Acnologia Dec 11 '17 at 02:42
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    @Acnologia This then becomes not a question about mathematics, but rather about the nature of our world - whether we can achieve irrational speeds. – Wojowu Dec 11 '17 at 09:50
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Since this is Math SE, I’ll address the mathematical content of his question only. There are good questions to be asked about if this set up is physically possible or if it amounts to a serious philosophical challenge to Nietzsche, but those are not within the preview of this website (for what it’s worth, my answer to both are “no”).

First, let’s start with a simpler version:

Suppose three wheels are set up in the fashion described, one moving at a rate that we’ll normalize to $1$, one moving twice as fast, and one moving five times as fast. Will they ever line up?

WLOG, lets suppose the circumference of the wheel is a natural number, $n$. Let’s mark the wheel at unit distances around with $0,1,2,\ldots n-1$. The dots start at position $0$. The three dots will line up at their original positions eventually. This happens at time $t$ that satisfies $t=2t=5t\pmod{n}$. This has infinitely many solutions, and in particular has a solution whenever $t$ is a multiple of $n$. There may be others as well, depending on the prime factors of $n$.

Another way to think about this is to consider three lines, marked at every multiple of $n$. Take a point starting at $0$ and moving at the aforementioned rates. Now we wish to know if there will be a time where all three points reach a marked multiple of $n$ at the same time.

The advantage of this formulation is that it more easily accommodates transcendental velocities. This formulation of the problem (with the speeds that the OP specifies) will never have a solution. To see this, note that you would need there to be a $t$ and $k_1,k_2,k_3\in\mathbb Z$ such that $nk_1=t,nk_2=2t,nk_3=\frac{1}{\pi}t$ are all simultaneously satisfied. Combining the first and third gives us $nk_1=\pi nk_2$. Since $n,k_1,k_2$ are integers and $\pi$ is irrational, there is no solution.

So yes, the underlying mathematics of this objection is correct, independent of the physics or philosophical content.

Stella Biderman
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  • Sorry, I'm not that too knowledgeable in some math notations, but how do you satisfy t=2t=5t (mod n)? – Acnologia Dec 10 '17 at 15:15
  • @Acnologia $a=b\pmod{n}$ means that $a$ and $b$ are the same mod $n$, aka that $n|(b-a)$ aka that $a$ and $b$ leave the same remainder when divided by $n$. – Stella Biderman Dec 10 '17 at 15:17
  • Also, is the problem's designations of wheels the same as how you designated your subscripts in your third paragraph? – Acnologia Dec 10 '17 at 15:18
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    @Acnologia It is now. I had a typo that I fixed. – Stella Biderman Dec 10 '17 at 15:20
  • Where does the number "5" come from? – Michael Dec 10 '17 at 15:21
  • @Michael i made it up to demonstrate a simpler version that does have solutions. – Stella Biderman Dec 10 '17 at 15:23
  • @StellaBiderman So, do we conclude that the argument is valid since we have seen that an irrational speed is contradictory, or is it invalid since we can't have an irrational speed for the wheel? – Acnologia Dec 11 '17 at 02:42
  • @Acnologia That’s the philosophical/physics question that I said was off-topic. The answer to that question is very long. – Stella Biderman Dec 11 '17 at 02:44
  • @StellaBiderman No it's not. It's just like how you prove things in math (as far as I know). When you prove or disprove a relation or argument, or something like that, you should have a conclusion as a result of the proving process. So with the proof you've given us, what can you say about the argument, math-wise? – Acnologia Dec 11 '17 at 02:55
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    @Acnologia I have proven that if the set up described is built and time is a linear progression the way people usually naively assume, then it is true that the wheels will never line up. You cannot draw conclusions about Nietzsche’s eternal return without a lot of non-mathematics though. – Stella Biderman Dec 11 '17 at 02:57
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    @StellaBiderman Yes I know that. I just want to know if the argument is valid math-wise. Thanks! :) – Acnologia Dec 11 '17 at 02:59
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    @Acnologia Sorry! I thought I had already communicated that and so assumed that you were talking about the broader context of the argument. – Stella Biderman Dec 11 '17 at 02:59
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Atleast in the example given above George Simmel is wrong. Can u really give a wheel the speed of 1/pi? Since pi is an irrational eventually you'd have to truncate the (1/pi) number upto certain decimal point This will hence make the speed rationally representable. And because of that the wheels eventually will coincide. The main question is ,does there exist a physical process that can impart a non rational speed to an object.If answers is yes, then (only for this example) Simmel would be correct

Yashashavi Ym
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    Although pi is irrational, it is a constant, so even though it's hard for us to imagine, I think we can make it pass as an assumption that the wheel could actually have a speed of 1/pi. – Acnologia Dec 10 '17 at 14:48
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    You could equally ask "can you really give a wheel the speed of 2"? – Michael Dec 10 '17 at 14:51
  • @Acnologia The problem with speed exacltly 1/pi is entropy. You would need infinite entropy to give something the speed of 1/pi. Why? How many digits do you need to exactly represent pi? Entropy is essentially the amount of info needed to accurately represent the state of a system,Since digits of pi show complete randomness and are infinite in size. You'd thus need infinite entropy – Yashashavi Ym Dec 10 '17 at 14:57
  • @Michael Unlike the speed of "2" .Speed of 1/pi has a fundamental problem.As mentioned above – Yashashavi Ym Dec 10 '17 at 14:59
  • This has nothing to do with mathematics. – YoTengoUnLCD Dec 10 '17 at 15:04
  • @Michael Can u answer that question ? The truth is the Heisenberg uncertainity priciple is laughing at us both right now.U can never have exactly 2 , 3 , 4... velocity – Yashashavi Ym Dec 10 '17 at 15:05
  • @YashashaviYm, I think it is implied in the problem that we'd have to take these assumptions as they are. I mean, this is a hypothetical problem. We really wouldn't be able to get a speed to infinite accuracy as what's needed for an irrational number in real life, but we can think of doing so in theory, unless someone proves that by doing so would introduce contradiction to the problem. – Acnologia Dec 10 '17 at 15:06
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    @Acnologia Sure. – Yashashavi Ym Dec 10 '17 at 16:55