Rabbit and Turtle Race: A Mathematical Analysis of Infinity and Probability

[dom_quixote]

TL;DR Summary

turtle, not tortoise

A Mathematical Fable: Rabbit versus Turtle Race

a) Rabbit and turtle combined a race along a road of infinite length.

b) As time progresses to infinity, the distance between the rabbit and the turtle tends to infinity.

c) As time progresses, both animals approach infinity.

d)The closer time approaches infinity, the closer the rabbit is to the end of the race.

e)However, the turtle's distance from the start of the test tends to infinity.

Does this reasoning have logic or is it pure rubbish?

It is certain that by biology, that is, the lifespan of the turtle is much longer than the lifespan of the rabbit. So it is likely that the turtle will overtake the rabbit, even if it never reaches infinity.

 

[topsquark]

A Mathematical Fable: Rabbit versus Turtle Race

a) Rabbit and turtle combined a race along a road of infinite length.

b) As time progresses to infinity, the distance between the rabbit and the turtle tends to infinity.

c) As time progresses, both animals approach infinity.

d)The closer time approaches infinity, the closer the rabbit is to the end of the race.

e)However, the turtle's distance from the start of the test tends to infinity.

Does this reasoning have logic or is it pure rubbish?

It is certain that by biology, that is, the lifespan of the turtle is much longer than the lifespan of the rabbit. So it is likely that the tortoise will overtake the rabbit, even if it never reaches infinity.

How can the race end if they are on a road of infinite length? Does the race end when they are both dead?

Who is further up the road when they both die depends on what their relative speeds are. Without that we can come to no conclusion.

-Dan

 

[dom_quixote]

Thanks, Dan!

This Fable, actually a thought experiment was intuited to try to unravel the problems of different infinities, if they really exist.

 

[fresh_42]

Thanks, Dan!

This Fable, actually a thought experiment was intuited to try to unravel the problems of different infinities, if they really exist.

The problem with your comparison is that you treated infinity like a number. This does not work at all.

 

[topsquark]

Thanks, Dan!

This Fable, actually a thought experiment was intuited to try to unravel the problems of different infinities, if they really exist.

Different infinites do exist. The cardinality of the natural numbers ( $\aleph _0$) and the cardinality of the real numbers ($\aleph _1$) for example. The infinites you are referring to would be the cardinality of the reals since you are talking about times and distances.

The only way that I am aware of that you can use to compare the size of infinite sets is to see if they can be related by a bijection, ie. a 1 to 1 matching of the set elements. Cantor's diagonal example is probably the best known, though it is not the only one.

-Dan

 

[fresh_42]

The last chapter in
https://www.physicsforums.com/insig...er-systems-that-we-have/#Continuum-Hypothesis
is about infinities.

Your wording "approaches infinity" makes not much sense. Imagine $\pi$ "written out". Which digit would you call "approaches infinity"?

 

[phinds]

d)The closer time approaches infinity, the closer the rabbit is to the end of the race.

No, the distance from the rabbit to the end never changes. It is always infinite.

Does this reasoning have logic or is it pure rubbish?

As fresh42 pointed out, you seem to think infinity can be treated as a number, which is incorrect and makes the whole thing problematic.

 

[jack action]

At the start of the race, both the rabbit and the tortoise are exactly in the middle of the infinite-length road.
As the distance between them increases, they still both are - each of them - in the middle of the infinite-length road. Because there are no ends.

You could imagine the race happening on a road going around a planet. At the beginning of the race, both have the same distance of road ahead or behind each of them (since the road brings them back to their starting point). Once they start racing, even though there is a distance increasing between them, they still have the same distance ahead or behind them, whether you look at it from the point of view of the rabbit or the tortoise. To correlate with your problem, just imagine the diameter of the planet is infinite.

 

[PeroK]

A turtle could live up to 80 years (or 150 years for a sea turtle) and a rabbit lives up to 10 years. A turtle can move about 3 km/h on land and the rabbit more like 30 km/h.

If both creatures could be induced to race along a road for their entire natural lives, the result might be quite close. That said, it depends how many hours in the day each animal could sustain its optimum speed.

I don't see any problem with measuring the distance traveled along a road of infinite extent.

 

[dom_quixote]

I am fervently grateful for the contribution of the participants of this thread. I hope the discussion goes on as it has been very helpful to me.

I would like to touch on a point, apparently "off topic", but inspired by the race problem.

For years, I've been trying to discern what the main difference between physics and mathematics is (if that difference is real).

I realize that in mathematics, "infinity" is linked to numerical intervals.

On the other hand, physics directly or indirectly depends on the time variable. This is a fact for dynamics.

However, even in statics time is hidden, because in buildings the acceleration of gravity is taken into account, and in electrostatics it takes time to charge a body electrically.

 

[PeroK]

Physics fundamentally is concerned with natural phenomena, such as gravity, electromagnetism and atomic structures. Mathematics studies abstract mathematical structures, such as numbers, groups, vector spaces etc.

 

[mcastillo356]

Hi PF, in my personal opinion, Zeno paradox fails at posing the concept of infinity. An annoyed pupil stood up and left the place. Paradox solved

 

[topsquark]

Hi PF, in my personal opinion, Zeno paradox fails at posing the concept of infinity. An annoyed pupil stood up and left the place. Paradox solved

Actually, Zeno failed to utilize the concept of infinitesimals properly. He felt that it should take an infinite amount of time to cover an infinite number of infinitesimals. However he clearly didn't believe his own arguments since he could get up cross the distance to get a snack while he was thinking about it!

-Dan

 

[Svein]

The original "Zeno's paradox" is an example of logic short-circuit. If the Tortoise starts 100m in front of Achilles and Achilles runs 10 times as fast as the tortoise then after 15 seconds (assuming Achilles runs at 10m/s) Achilles passes the 150m mark and the Tortoise passes the 115m mark. Therefore Achilles has overtaken the Tortoise in less than 15s. Now if Zeno wants to know exactly when Achilles passes the Tortoise, then the infinite series may be relevant in some sense.

 

[dom_quixote]

Infinite numeric intervals are permissible in mathematics. But infinite temporal intervals, eternity itself, is a matter of theology

 

[PeroK]

Infinite numeric intervals are permissible in mathematics. But infinite temporal intervals, eternity itself, is a matter of theology

Mathematics, thankfully, is not constrained by theology. Let $$t \in (-\infty, \infty)$$

 

[Svein]

Now if Zeno wants to know exactly when Achilles passes the Tortoise

Assuming 100m start difference, 10m/s for Achilles and 1m/s for the Tortoise, Achilles will pass the Tortoise at t=t₁ where t₁ is given by $10\cdot t_{1}=100+1\cdot t_{1}$. Solving this equation gives $t_{1}=\frac{100}{9}$ (with apologies to all physicists for leaving out the units). This can of course be solved for different velocities and head start distance. Note that if they run at the same speed, the denominator is zero, and Achilles will never catch the Tortoise...

 

[Drakkith]

This Fable, actually a thought experiment was intuited to try to unravel the problems of different infinities, if they really exist.

They exist just as much as the rules regarding touchdowns in American Football exist. That is, both are essentially useful definitions and/or logical statements within a larger set of rules on some subject.