A new challenge to physical reality

Question

So recently I was thinking about Zeno's paradox (of infinite sum of 1/2^n in motion). Although I love calculus, I still don't get how it could possibly solve the paradox in Physical world, because although in theory we can bypass it by using derivatives, our legs don't know derivatives. Nor does our rest of the body-then how can we essentially travel a distance when our legs would need to constantly travel 1/2 first? Is it that even in the physical world, we can never actually reach a specific point but near or around it?

Answer 1

Short Answer

Mathematics describes the universe, and Xeno's ignorance of calculus does not determine how the universe works. It would be disconcerting if an infinite series of diminishing lengths could not add up to a finite length, but mathematically they do, and that is consistent with empirical observation. Paradoxes are word problems involving logical contradiction, not bugs in the physical universe. This is an important metaphysical claim, so I will repeat:

Math is constructed language which describes our physical experience, it does not directly determine it. Any claims to the contrary are anti-scientific.

Long Answer

Physicalism and Motion

First and foremost, as a physicalist, calculus at best describes motion with language, it does not determine it. It's an old-fashioned idea that the laws of physics govern the universe; they merely describe it, which is why laws have so often been revised. Science is full of cases where laws, such as the Laws of Motion, have been revised in the face of empirical evidence. Newton-to-Einstein is the obvious and insurmountable example.

Furthermore, to say that our legs "know" is poetry, not literal fact. Our legs are part of an indivisible spacetime and the explanation of motion might be enriched by calculus, but is ultimately and metaphysically rooted in notions of time and space. Specifically, motion is a function of distance traveled during time elapsed: v(x,t), and this was rather apparent long before calculus as a system was constructed by mathematicians. If you want an explanation of why legs move, a standard and scientific explanation is rooted in physics, chemistry, and biology in the hard sciences. Calculus just provides a rigorous mathematical tool set to address instantaneous rather than claims involving arithmetic means.

Paradoxes Are Contradictions in Language, Not Physical Phenomena

What Xeno did was cook up a paradox, an apparent contradiction OF LANGUAGE that presumes there are an actual infinity of motions required to get between two points. No one has ever had a physical problem moving from point A to point B, generally speaking. Now, one has two philosophical choices. One can accept the rather platonic notion that actual infinities exist as physically real mathematical objects (whatever that means), or one can like the intuitionists reject that claim and seat actual infinities as a useful construct or concept of the mind. As someone with strong empirical ideas, I and many choose to do the latter. Actual infinities are abstractions, members of the map, not concrete object, members of the terrain. These are admittedly metaphysical presumptions, but they are consistent with a position midway between scientific realism and instrumentalism, like van Fraassen's constructive empiricism.

So, to say that the "legs need to know calculus" or something similar is figurative language and at best might qualify as metaphysical speculation, the sort the logical positivists hated. Legs don't know. Minds know. The connection between the mind and body is a contentious issue in philosophy, so to presume that somehow the body needs to have access to the mind is at best a controversial statement, and likely in this question an unexamined presumption. I would simply assert that legs obey the law of physics, and calculus obeys the theorems of algebra, and to conflate the two issues is unproductive. The idea that somehow problems in calculus affect the motion they describe puts the cart before the horse.

What Calculus Provides

The epsilon-delta definition and the limit are merely a logical tool to address the claims of potential and actual infinities and infinitesimals. By breaking up motion into an infinite series of segments that approach the infinitesimals, it is possible to show that intuitively contradictory claims are actually logically consistent. For instance, one can add an infinite number of things to a finite sum. Take a look at this answer (MathSE) which addresses that fact. If a paradox rests on the argument that an infinite amount of segments must itself be infinite, than clearly calculus shows otherwise! Intuitively, it's not even outrageous a claim, because as segments become infinitely small, then they approach zero and are very similar to adding zero. So, an sum of infinite addends which themselves become infinitesimal is finite, and rigorously can be shown so.

The Psychology of the Paradox

Ultimately, what happens psychologically in many paradoxes might be considered a deepity (YT). The trivial reading is that an infinite sum is finite is true, and mathematically speaking unimpressive by today's standards. The math which models physical motion and physical motion clearly show it is possible to get from A to B in spacetime. Ho-hum. On another reading, just imagine if it weren't true! Imagine if Xeno's ignorance of calculus determined motion in space (it doesn't and never did), and we weren't able to get from point A to B?!? That would be deep and disconcerting. The result is a psychological experience of wonder and awe that stems from nothing other than mathematical and logical ignorance. That is a deepity, and the apparent paradox in logic that has no bearing on how the universe empirically behaves, and ultimately like many paradoxes can be resolved by additional context and information.

Answer 2

According to calculus: The sum of (½)^n from n=0 until infinity is 2. It is the sum of the geometric series with q =1/2, a standard example for a convergent series.

You can easily check by running in the „physical world“ that you cover the unit lenght in two units of time if you have a constant speed of ½ length/time.

The error of the paradoxon lies in the viewpoint of the argument: It restricts itself by considering at each step never more than just half of the remaining distance.

A second, hidden prejudice assumes that an infinite sum cannot converge towards a finite value.

Answer 3

The paradox assumes that we can do as small movements as we want. Our muscles are not able to produce movements smaller than some multiple of the Planck constant. Assuming this the paradox vanishes.

Answer 4

While the other answers here have discussed how calculus addresses the issues arising from infinities and infinitesimals here, there is another issue here regarding the physical soundness of Zeno's argument. In particular, Zeno claims that, in order for an object to go from one point to another, it must reach half way before it can reach the final point. There is a problem with this claim in the way that it assumes our mathematical descriptions are making ontological statements about the physical world, namely through its requirement that the object always has a well-defined position. This assumption is problematic because quantum mechanics tells us that objects cannot have well-defined exact positions, due to the fact that position eigenstates are not normalisable. Thus, there is no need to cite calculus to solve Zeno's paradox in the physical world because there is no paradox in the physical world—physical objects will do whatever it is that they do without needing to know any math.