There are many good treatments of this supposed loophole. I happen to like this one:<p><a href="https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is-deterministic-sorry-norton/" rel="nofollow">https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is...</a><p>It points out many flaws in Norton's reasoning, some fatal to his argument, some not. Putting it as simply as I can, Norton seems to claim that "Newton's Laws" are <i>non-deterministic</i>. That's not quite right. Rather, they are <i>non-complete</i>. I.e. they are <i>incomplete</i>. They're incomplete insofar as Newton's First Law (<i>"An object at rest remains at rest, and an object in motion remains in motion at constant speed and in a straight line unless acted on by an unbalanced force"</i>) establishes first-order and second-order derivatives (momentum and acceleration) as state variables but places no constraints on higher-order derivatives. However, higher-order derivatives <i>are</i> (as many as are needed) among a system's state variables. In many (but far from all), higher-order derivatives are zero and human experience with them is rare, so they're easy to overlook. Norton's (unphysical) Dome is a specific example of a general class of systems where higher-order derivatives are <i>not</i> zero. Given that, the two branches of Norton's equation of motion (for the <i>stable</i> and <i>unstable</i> trajectories) cannot both describe the same system (or the same particle) with the same set of state variables. <i>That's</i> the sleight-of-hand.<p>Again, all credit to Gareth Davies for working this out. I am absolutely not trying to pass off his work for my own. Just reporting it and trying to summarize it.
Related ongoing thread:<p><i>The Dome (2005)</i> - <a href="https://news.ycombinator.com/item?id=42583688">https://news.ycombinator.com/item?id=42583688</a> - Jan 2025 (22 comments)