# Debunking Zeno's Paradox with Galileo's Speed

Zeno's Paradox can be expressed by considering turning a page in a book half way the distance from being fully turned. Then, turning it half the remaining distance. Then, half the remaining distance, ad infinitude. Another premise is added that there are an infinite number of points between each halved distance. Finally, the question is posed: How can the page ever fully turn, since it would take eternity to pass an infinite amount of points?

Thanks to Galileo, this paradox can be proven not to be a paradox, but a confusion in physical properties of motion.

The page turns past the first half distance, half way across the book. Let us say it has moved with a constant speed.

Now, we use the same speed as it crosses the second half.

Now as we continually do this, what is occurring with time?

Time is being halved with a constant speed, because the distance is being halved. Since, the remainder of time to fully turn the page is being continually being halved, there will never be enough time to cross the full distance.

Zeno conveniently cuts the very last moment in two, halving time and distance, making it seem impossible to pass the very last distance. Ironically, Zeno uses the common sense premise that the page does pass the first half of prior distances to base his argument. But, why would it not pass another equal half successfully, as his argument is based upon?

It would. Even if the nonsense scenario that the last smallest half point is nothing, it would take no time to pass it, because it is not a distance at that point. It is not a paradox in terms of the basic formula that Galileo pointed out: speed = distance over time.

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chroot
Staff Emeritus
Gold Member
Zeno conveniently cuts the very last moment in two
There is no "last moment," since it's an infinite series.

All you need to disprove Zeno's paradox is a basic education in calculus. The sum of an infinite series need not be infinite -- a fact that would have startled Zeno.

- Warren

Hurkyl
Staff Emeritus
Gold Member
I posit that you first need a clear, precise statement of Zeno's paradox before it's worth considering at all.

I have refrained from commenting in these “Zeno” threads as I find them rather free of thought. Everyone seems to assume Zeno was stupid.
The sum of an infinite series need not be infinite -- a fact that would have startled Zeno.
I doubt it would have startled Zeno at all. Methods of summing infinite series had been developed before Zeno was born. I am of the opinion that he was pointing out a fact that everyone studiously avoids confronting. Now that is no more than an opinion but it is how I feel about the situation..
I posit that you first need a clear, precise statement of Zeno's paradox before it's worth considering at all.
And I agree one hundred percent. One must ask one’s self, “exactly what is Zeno’s complaint?” In my opinion he was pointing out the fact that no evidence can exist for continuity of reality as your knowledge of reality must be expressible in a finite amount of information. Ergo, it is an assumption that reality is continuous and not a “fact”. And the fact that it is an assumption yields some important consequences; an issue also avoided by everyone.

Have fun -- Dick.

fact that no evidence can exist for continuity of reality
I take it that all your perception are not continuous (especially those of movement). I would not put simplified model of convenience such as coordinate system above perception and have it asserted as "real".

Zeno's paradox seems to revovle around a map vs territory error. math is a language used to express or symbolize the world, but the world is not perfectly symbolized by math. Zeno's infinite divisions are fine in math, but do not apply to reality which is not infinitely divisible. there turns out to be a smallest step which cannot be halfed in quantum physics. there are only full jumps from state1-state2, no half jumps.

There is no "last moment," since it's an infinite series.

All you need to disprove Zeno's paradox is a basic education in calculus. The sum of an infinite series need not be infinite -- a fact that would have startled Zeno.

- Warren
Ok, there is no last moment, but since the series is infinite, there could be no sum, because you would be infinitely adding. Any sum deduced, could only be deduced in light of a finite series.

All words are symbols which represent some organization of matter and/or it's motion. Infinite and finite are words that represent properties of the material world.

If I say infinitely large, then I'm implying all matter in existence, which is finite. The assertion that infinite represents something beyond all matter in existence is wrong, because matter cannot be created. Therefore, no word could correlate with anything beyond all matter in existence, without creating matter.

If I say infinitely small, I imply finite, because zero cannot be represented. Zero has no inertia. An object with no inertia cannot be interacted with. An object that cannot be interacted with cannot be sensed, and therefore represented as a thought in a humans mind. Zero is represented in a human's mind, so zero has mass.

I think the illusion that infinity has a correlation with the time and matter is caused by circular reasoning.

chroot
Staff Emeritus