Why have so many physicists shrugged off the paradoxes of quantum mechanics?

The only reason it appears to be paradoxical is because you're mandating someone move from a real coordinate (a, b, c) to another real coordinate (a', b', c') on the interval [x, y] while also passing through the set of all real points between them, without first defining a notion of distance of time. That's not possible for the same reason you can't ask someone to count all reals on an interval, because continuity implies uncountability. Between any pair of real numbers is another real number, and it takes an equal amount of effort (and time) to count any given number.

To a first glance, this seems like a paradox because we can clearly move from (a, b, c) to (a', b', c), yet we shouldn't be capable of any movement whatsoever. Calculus solves this problem by formalizing Zeno's demand as a geometric series with a notion of distance. The requirement is that you move from one position to another position while passing through every halfway position between them. Equip the vector space ℝ^3 with the Euclidean metric so you have a metric space (defined distance). Then we have the sequence of steps

    (a, b, c) -> |(a, b, c) - (a', b', c')|/2 -> ... -> (a', b', c')

    (1/2)^1 + (1/2)^2 + (1/2)^3 + ...

More concretely: an infinite expansion such as 0.99999999... is equal to 1. Each half step will take only half as long to traverse as the half step preceding it to it once you've defined Euclidean distance on a continuous space. The first step to formalizing sequences and series like this is by constructing the real numbers as a continuous set and distinguishing between different types of infinities. Then you can define limits, and from there you're essentially done.

Note that at no point am I talking about what happens when you reach 1, or (a', b', c'), or anywhere else. I'm just explaining how you reach it in finite time. If you can get arbitrarily close to a point, you can get to the point itself.

I guess I should be more technical and say that real analysis solves this problem, because what's really doing the heavy lifting here is the topology induced by defining a metric on ℝ^3 in combination with the notion of limits.

Your geometric series is not a summation of the steps or positions, but rather the time required to complete each step. Therefore your example is characterized by an identical geometric series to the model I used in my previous comment.

More generally, Zeno’s paradox can be succinctly resolved by citing the monotone convergence theorem. Every bounded, monotonically decreasing function converges. The time required to complete the infinite series of half steps converges, because (again, with the definition of a metric) the time required to complete each individual step decreases commensurate with the change in distance.

I am not sure what you mean here. You can calculate the sum of the time series, but you can't calculate Achilles' final position, which is the question at issue. The question remains: if it's possible in general to traverse an infinite sequence of steps in space, why is it not possible to traverse the one that I specified? "Solving" Zeno's paradox by admitting the possibility of traversing an infinite series of points in space or time seems to give rise to paradoxes just as deep as the originals.

> The time required to complete the infinite series of half steps converges, because (again, with the definition of a metric) the time required to complete each individual step decreases commensurate with the change in distance.

Yes, that was Aristotle's observation and a key part of his proposed solution to the paradox. The problem is that this explains why it's possible to sum the series, not why it's possible to traverse it. You seem to be taking the position that any series that cannot be summed cannot be traversed. But why should that be so?

The observer effect is a real thing, and even has interesting effects in quantum mechanics, e.g. the quantum zeno effect https://en.wikipedia.org/wiki/Quantum_Zeno_effect

Yet the observer effect is not the reason why we can't know an object's position and velocity at the same time. There are two ways we can see that this supposed explanation is a red herring:

- We don't need to interact with (e.g. 'bounce a photon off') a quantum system in order to observe it ( https://en.wikipedia.org/wiki/Interaction-free_measurement ). I particularly like the "quantum bomb detector" ( https://en.wikipedia.org/wiki/Elitzur%E2%80%93Vaidman_bomb_t... ), which can tells us whether a photon detector connected to a bomb is working or not, without hitting it with a photon and hence triggering the bomb (50% of the time, at least).

- The actual reason, the uncertainty principle ( https://en.wikipedia.org/wiki/Uncertainty_principle ), doesn't require any notion of observation at all, let alone interaction. It's a simple property of waves (the relationship between duration and frequency). 3blue1brown did a nice video on this https://www.youtube.com/watch?v=MBnnXbOM5S4