Q&A: Determinism and Norton’s Dome
Determinism and Norton’s Dome
Question
You wrote (in the context of free will) that the laws of “ordinary” nature (non-quantum) are deterministic.
I recently read about “Norton’s dome”—a thought experiment that demonstrates a very simple Newtonian system that is not deterministic.
https://en.m.wikipedia.org/wiki/Norton%27s_dome
I’d be glad to hear your opinion.
Thanks!
Answer
I wasn’t familiar with it. Amazing. Worth thinking about. Of course, it’s always possible that the mathematical description of Newton’s laws is not the correct description, at least for such edge cases. In other words, it’s mathematically possible but not physically so.
My first thought was that at the top of the pyramid there is a sharp point (not differentiable). In nature such a thing cannot arise, so this is basically an impossible case.
Discussion on Answer
The question is whether a shape that is not twice differentiable and beyond exists in nature. My feeling is that it does not.
It is indeed difficult, because the laws are ostensibly deterministic. In fact, they too present this as a paradox. Something is happening here without a cause, and that goes against the principle of causality.
It would be nice if the Rabbi wrote a column about this sometime, because it really does sound amazing, except that for me the above entry is basically like Chinese Wikipedia.
1. Regarding shapes in nature—I think it depends on how much you zoom in. At the atomic level, it seems to me there isn’t even differentiability once (and maybe not even continuity).
2. Norton explains that there is no paradox at all if you formulate the first law this way: “If the sum of the forces is zero, the acceleration will be zero”: until the time the body begins to move (t<T), its acceleration is indeed zero (and its velocity is also zero). At the moment t=T its acceleration is still zero (because of the shape of the curve). For T<t there is acceleration, but the sum of the forces is no longer zero. There is no point in time at which the law is violated, so there is no paradox.
https://sites.pitt.edu/~jdnorton/Goodies/Dome/
3. The analysis does not show that something happens without a cause (physically), but rather that in this system, at any moment something like this can happen. In my opinion (and hope), perhaps this opens up a possibility for events whose cause is not physical, yet they do not violate the laws of physics. In other words: maybe the laws of nature are kind enough to allow additional factors to influence reality from time to time.
4. Note: not every convex curve can produce this phenomenon. If we take a spherical dome, for example, the time required for the mass to gain velocity is infinite, and therefore the motion actually cannot begin. Norton’s innovation is in creating a curve for which the time is finite.
1. At the atomic level one has to use quantum theory, and the whole calculation is irrelevant.
2. I didn’t understand. At the moment the motion begins there is still no force. That is apparently an expression of the non-differentiability there at the tip.
3. I didn’t understand the difference. The problem is precisely that something like this can happen.
4. That’s clear. In Wikipedia they pointed this out from the other side. If you look at a ball climbing upward and coming to rest at the top, that would take an infinite amount of time.
I just uploaded it now (Column 687): https://mikyab.net/wp-content/uploads/2025/01/%D7%98%D7%95%D7%A8687-%D7%A4%D7%A8%D7%93%D7%95%D7%A7%D7%A1-%D7%A0%D7%95%D7%A8%D7%98%D7%95%D7%9F.pdf
1. The curve is differentiable at the top point (once).
2. I read between the lines of your answer that intuitively it’s hard for you to accept a theory that allows non-determinism in certain edge cases.
Is that correct? And if so, why exactly? Maybe that’s actually a “good” feature of the theory?