There is no paradox, moving from linear time into exponentially slowing down time slows everything down exponentially. It is essentially (a silly) coordinate system modification.
I let Grok have a go at your contention. Your analysis sidesteps my argument that it is a psychological issue, and the original mathematics must not be adequate but the it seems to be correct to those who have not been trained in a particular sort of mathematical thinking. Here Grok speaks:
Your formulation of Zeno's Dichotomy paradox is precise and insightful, reframing it as a problem of recursive remainders rather than discrete stages. It correctly emphasizes that the paradox hinges on the infinite divisibility of space and the assumption of how time interacts with this process. Let me address your points systematically and expand on the implications of the "exponential time" response, while keeping the discussion concise and rigorous.
Correct Formulation: Recursive Remainders
Your description captures the essence of Zeno’s Dichotomy: to traverse a finite distance, one must always cover half of the remaining distance, leaving a non-zero remainder that can be halved ad infinitum. This recursive structure suggests that the endpoint is never reached because there is always a fraction of the distance left. The paradox lies not in the physical act of motion but in the logical structure of the process—completion seems indefinitely deferred due to the infinite regress of remainders.
This is distinct from misinterpretations that treat the paradox as requiring a finite sequence of steps. Instead, it’s about the infinite divisibility of space and the conceptual challenge of "completing" an infinite process. As you noted, the issue is not the length of the path but the recursive framing of the journey.
"Exponential Time" Response
The quoted response—“Moving from linear time into exponentially slowing down time slows everything down exponentially. It is essentially (a silly) coordinate system modification”—offers a mathematical resolution by challenging the implicit assumption that time progresses linearly. Let’s unpack this:
Linear Time Assumption: In Zeno’s framing, each halved distance (1/2, 1/4, 1/8, ...) is often assumed to take a proportional amount of time, leading to an infinite sum of time intervals (e.g., 1 + 1/2 + 1/4 + ...). If each step takes a fixed or additive duration, the total time appears to diverge, making motion impossible.
Nonlinear Time Resolution: The response suggests that if time intervals shrink proportionally with the spatial remainders (e.g., each half-distance takes half the time of the previous step), the total time converges. Mathematically, this is modeled by a geometric series. For a distance of 1 unit, the distances are 1/2, 1/4, 1/8, ..., and if the times for each are proportional (e.g., 1/2, 1/4, 1/8, ... units of time), the total time is:
∑n=1∞12n=1
n=1∑∞2n1=1
This convergent sum shows that the infinite process completes in finite time, dissolving the paradox.
Exponential Deceleration: The phrase “exponentially slowing down time” refers to a time coordinate where successive intervals shrink exponentially, aligning with the halving of space. This is equivalent to a coordinate transformation where time is compressed as the endpoint is approached, ensuring the process terminates.
Coordinate System Interpretation
The “silly coordinate system modification” highlights that the paradox is an artifact of the chosen mathematical frame. In a linear time scale with a convergent model (e.g., summing the geometric series), motion completes in finite time. In a logarithmic or exponentially decelerating time scale, the endpoint appears unreachable because time “stretches” near the endpoint, but this is a descriptive artifact, not a physical barrier.
This aligns with modern mathematical resolutions of Zeno’s paradoxes, particularly through calculus. The infinite divisibility of space and time is resolved by recognizing that infinite series can have finite sums. The paradox only persists if one insists on a frame where time intervals do not converge appropriately.
Addressing Potential Counterarguments
One might argue that the paradox reflects a deeper metaphysical issue about the nature of infinity or continuity. However, your formulation and the “exponential time” response shift the focus to a mathematical resolution: the infinite regress of remainders is not a barrier to motion but a feature of how we model it. Physical motion occurs in a continuum, and the mathematical tools of convergent series (developed by later mathematicians like Cauchy and Weierstrass) resolve the apparent contradiction.
Summary
Your reframing of Zeno’s Dichotomy as a problem of recursive remainders is accurate and clarifies that the paradox arises from the infinite divisibility of space, not from physical impossibility. The “exponential time” response resolves the paradox by showing that it depends on a linear time assumption. By modeling time and space as converging geometrically, or by using a nonlinear time coordinate, the infinite regress completes in finite time. The paradox is thus a “coordinate artifact”—a result of mismatched assumptions about time and process, not a challenge to the reality of motion. This resolution underscores the power of mathematical frameworks like calculus to reconcile intuitive paradoxes with logical consistency.
LOL, grok is overly verbose unless you tell it to be concise.
I have no idea how young I was (8, 10?) when I encountered this type of "paradox", if A is k times as fast as B (VA = k*VB), with B having an S distance advantage at the beginning, it is trivial to calculate when will k*VB*t = VB*t + S; t = S/((k-1)*VB).
Hence I rejected the "paradox" and figured out it is an artifact of modifying the time axis.
There is no paradox, moving from linear time into exponentially slowing down time slows everything down exponentially. It is essentially (a silly) coordinate system modification.
I let Grok have a go at your contention. Your analysis sidesteps my argument that it is a psychological issue, and the original mathematics must not be adequate but the it seems to be correct to those who have not been trained in a particular sort of mathematical thinking. Here Grok speaks:
Your formulation of Zeno's Dichotomy paradox is precise and insightful, reframing it as a problem of recursive remainders rather than discrete stages. It correctly emphasizes that the paradox hinges on the infinite divisibility of space and the assumption of how time interacts with this process. Let me address your points systematically and expand on the implications of the "exponential time" response, while keeping the discussion concise and rigorous.
Correct Formulation: Recursive Remainders
Your description captures the essence of Zeno’s Dichotomy: to traverse a finite distance, one must always cover half of the remaining distance, leaving a non-zero remainder that can be halved ad infinitum. This recursive structure suggests that the endpoint is never reached because there is always a fraction of the distance left. The paradox lies not in the physical act of motion but in the logical structure of the process—completion seems indefinitely deferred due to the infinite regress of remainders.
This is distinct from misinterpretations that treat the paradox as requiring a finite sequence of steps. Instead, it’s about the infinite divisibility of space and the conceptual challenge of "completing" an infinite process. As you noted, the issue is not the length of the path but the recursive framing of the journey.
"Exponential Time" Response
The quoted response—“Moving from linear time into exponentially slowing down time slows everything down exponentially. It is essentially (a silly) coordinate system modification”—offers a mathematical resolution by challenging the implicit assumption that time progresses linearly. Let’s unpack this:
Linear Time Assumption: In Zeno’s framing, each halved distance (1/2, 1/4, 1/8, ...) is often assumed to take a proportional amount of time, leading to an infinite sum of time intervals (e.g., 1 + 1/2 + 1/4 + ...). If each step takes a fixed or additive duration, the total time appears to diverge, making motion impossible.
Nonlinear Time Resolution: The response suggests that if time intervals shrink proportionally with the spatial remainders (e.g., each half-distance takes half the time of the previous step), the total time converges. Mathematically, this is modeled by a geometric series. For a distance of 1 unit, the distances are 1/2, 1/4, 1/8, ..., and if the times for each are proportional (e.g., 1/2, 1/4, 1/8, ... units of time), the total time is:
∑n=1∞12n=1
n=1∑∞2n1=1
This convergent sum shows that the infinite process completes in finite time, dissolving the paradox.
Exponential Deceleration: The phrase “exponentially slowing down time” refers to a time coordinate where successive intervals shrink exponentially, aligning with the halving of space. This is equivalent to a coordinate transformation where time is compressed as the endpoint is approached, ensuring the process terminates.
Coordinate System Interpretation
The “silly coordinate system modification” highlights that the paradox is an artifact of the chosen mathematical frame. In a linear time scale with a convergent model (e.g., summing the geometric series), motion completes in finite time. In a logarithmic or exponentially decelerating time scale, the endpoint appears unreachable because time “stretches” near the endpoint, but this is a descriptive artifact, not a physical barrier.
This aligns with modern mathematical resolutions of Zeno’s paradoxes, particularly through calculus. The infinite divisibility of space and time is resolved by recognizing that infinite series can have finite sums. The paradox only persists if one insists on a frame where time intervals do not converge appropriately.
Addressing Potential Counterarguments
One might argue that the paradox reflects a deeper metaphysical issue about the nature of infinity or continuity. However, your formulation and the “exponential time” response shift the focus to a mathematical resolution: the infinite regress of remainders is not a barrier to motion but a feature of how we model it. Physical motion occurs in a continuum, and the mathematical tools of convergent series (developed by later mathematicians like Cauchy and Weierstrass) resolve the apparent contradiction.
Summary
Your reframing of Zeno’s Dichotomy as a problem of recursive remainders is accurate and clarifies that the paradox arises from the infinite divisibility of space, not from physical impossibility. The “exponential time” response resolves the paradox by showing that it depends on a linear time assumption. By modeling time and space as converging geometrically, or by using a nonlinear time coordinate, the infinite regress completes in finite time. The paradox is thus a “coordinate artifact”—a result of mismatched assumptions about time and process, not a challenge to the reality of motion. This resolution underscores the power of mathematical frameworks like calculus to reconcile intuitive paradoxes with logical consistency.
LOL, grok is overly verbose unless you tell it to be concise.
I have no idea how young I was (8, 10?) when I encountered this type of "paradox", if A is k times as fast as B (VA = k*VB), with B having an S distance advantage at the beginning, it is trivial to calculate when will k*VB*t = VB*t + S; t = S/((k-1)*VB).
Hence I rejected the "paradox" and figured out it is an artifact of modifying the time axis.
What is verbose to a mathematician is just explanatory to us normals. ;-)