thicko's paradox question

[thicko]

Out of my depth but vaguely interested. maths teacher told me that he could prove that it's impossible to finish a 100m race due to infinity! His point was that at 99m  (or any point before or after) the remaining distance left could be extrapolated to infinity using tenths/hundredths/thousandths etc of a metre, thus making it impossible to actually cross the finishing line.

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[Strange]

Sounds like he was talking about Zeno's Paradox(es): https://en.wikipedia.org/wiki/Zeno's_paradoxes#Paradoxes_of_motion

[thicko]

That may well be so, but have these paradox (es) been proven or otherwise. I'm just an inquisitive plumber so I'd hate to be wasting anyone's valuable time here btw. If you need to do something else, crack on - but if you can spare a finite amount of your time to explain this to me I'd appreciate it to the power of 9.9 recurring?

Thanks in advance  

[Ken Fabian]

I was never that impressed by Zeno's paradox - which seems to me to be more a misframing of a question than a true paradox. I mean, you wouldn't want Zeno as your running coach... "Now run to the place the person ahead of you is now - damn he's moved! To where he is NOW! No, NOW, NOW! Oh, you just went past him - I really didn't expect that!"

Apart from that, consider that the existence of an infinity of fractions of a finite distance does not effect the rate at which distance is traversed; it does not take any longer to traverse an infinity of them than traversing the finite total and is irrelevant to the time it takes.

 

 

Edited April 30, 2018 by Ken Fabian
fixed typo

[thicko]

Dear me, surely you can't all be saving the universe with scientific research at this very moment in time?

May I just post this with a finite amount of mirth and an infinite amount of sarcasm please?

Find the difference between the two equations below :

Mathematician has problem with heating system + seeks help via plumbing forum online = conscientious plumber replies eruditely, irrespective of the banality of said problem!

Thicko plumber has problem with mathematics + seeks help via mathematicians forum = elitest mathematician replies dismissively, due to the banality of said problem!

Answer  : one is far more worthy than the other! 

 

 

 

 

[Ken Fabian]

Thicko, not everyone here is a mathematician - certainly not me. I can prove pythagoras' theorem but not much beyond that. But I believe the response I gave, that an infinity of fractions of a finite distance does not effect the rate at which that finite distance is traversed, is a 'solution'. Or you can use a search engine - google is popular. I did and found this - http://www.iep.utm.edu/zeno-par/#H2

[Velocity_Boy]

1 hour ago, thicko said:

Out of my depth but vaguely interested. maths teacher told me that he could prove that it's impossible to finish a 100m race due to infinity! His point was that at 99m  (or any point before or after) the remaining distance left could be extrapolated to infinity using tenths/hundredths/thousandths etc of a metre, thus making it impossible to actually cross the finishing line.

Expand  

Oh....sounds like your math teacher ain't very original, mate.

He's just using a slightly tweaked version of the centuries old Zeno's Paradox that featured the rabbit and tortoise.

Which I've always thought of ad a big yawn.

Both zeno and you teacher's stories are pure thought experiments. That is, in reality they're of course totally wrong. Just like the rabbit who'd never catch the much slower turtle since he could only continually halve the distance he was behind him after giving him his huge head start. To me...thought experiments like this do little to improve science and are basically just forms of mental masturbation.

I dunno..maybe I'm being unduly harsh...being a former track guy who always hated losing? LOL

[thicko]

Ken Fabian, I was obv mid-text when you replied so I hereby exclude you from my pseudo-vitriole, and I thank you very much for your response. However as a thicko in your field of knowledge, can you explain when or why distances become finite or infinite?

In maths you can have .3 recurring (or any other number for that matter), aye? 

How can a millimetre not be defined in the same way, therefore making it infinite?

 

 

 

[Velocity_Boy]

1 hour ago, thicko said:

That may well be so, but have these paradox (es) been proven or otherwise. I'm just an inquisitive plumber so I'd hate to be wasting anyone's valuable time here btw. If you need to do something else, crack on - but if you can spare a finite amount of your time to explain this to me I'd appreciate it to the power of 9.9 recurring?

Thanks in advance  

You can disprove your teacher's thought puzzle...I refuse to call it a paradox since those are real....by simply going outside and marking off 100M and then running it. Plus...you get a bit of cardio in!

Just ad easily could ol Zeno be busted up by you choosing a much slower opponent to race against and then beating him after allowing him a head start. One that you're sure you can overcome, that is.

Cheers.

[thicko]

I watched Alan Wells cross the finishing line in Moscow so I'm damned sure that if tubby Scotch people can run 100m then I can. 

I could be missing a fundamental point of maths/physics here, and if so I apologise profusely  (maybes I was off school that day?) but if an integer can be divided into infinitesimal amounts in maths how can that not be so in reality?

 

[Strange]

6 hours ago, thicko said:

I could be missing a fundamental point of maths/physics here,

I don't think so. 

I have never understood why anyone takes Zeno's paradoxes seriously. They obviously don't apply to the real world.

However, if you do take them seriously, then you can prove they are wrong but it means using some rather advanced maths (calculus and the mathematics of limits and infinitesimals) which you probably would know unless you had done an A-level.

[thicko]

Thanks for the post, my friend, but I'm afraid I don't meet your stipulated criterion, as I have not done an A-level.  

I'm sorry!  

[swansont]

On April 30, 2018 at 6:04 PM, thicko said:

That may well be so, but have these paradox (es) been proven or otherwise. I'm just an inquisitive plumber so I'd hate to be wasting anyone's valuable time here btw. If you need to do something else, crack on - but if you can spare a finite amount of your time to explain this to me I'd appreciate it to the power of 9.9 recurring?

Thanks in advance  

The concept of infinitesimals wasn't known at the time. The paradox existed when the tools to analyze and solve it didn't exist.

On April 30, 2018 at 7:12 PM, Velocity_Boy said:

You can disprove your teacher's thought puzzle...I refuse to call it a paradox since those are real....by simply going outside and marking off 100M and then running it. Plus...you get a bit of cardio in!

Just ad easily could ol Zeno be busted up by you choosing a much slower opponent to race against and then beating him after allowing him a head start. One that you're sure you can overcome, that is.

Cheers.

That's why it's a paradox. The experiment cannot be explained with the available theory, such as it was.

[Strange]

9 hours ago, thicko said:

Thanks for the post, my friend, but I'm afraid I don't meet your stipulated criterion, as I have not done an A-level.  

I was just suggesting the reason why you don't have the maths background to follow the explanations. (I didn't learn any of these things at school either.)

[studiot]

On 01/05/2018 at 12:12 AM, thicko said:

Ken Fabian, I was obv mid-text when you replied so I hereby exclude you from my pseudo-vitriole, and I thank you very much for your response. However as a thicko in your field of knowledge, can you explain when or why distances become finite or infinite?

In maths you can have .3 recurring (or any other number for that matter), aye? 

How can a millimetre not be defined in the same way, therefore making it infinite?

 

 

 

 

There are two types of 'things' in our universe.

English calls them concrete nouns (such as brick)  and abstract nouns (such as happiness).

We do not know if any concrete things can or do go on forever (I have never heard of an infinite brick) - we do not even know if the whole universe is infinite or not.

 

But  all concrete things we know of (except perhaps the universe) of have a beginning and an end, which is how a brick is 9 inches long.

But we have found abstract things that have a beginning (or they would not exist at all), but have no end

The difference between 0.9 inches and 0.9 recurring is exactly this.

And 9, 0.9 0.3 are abstract things.

/Edit

In respect of Zeno's paradoxes,

The path and time are concrete things with start and finish lines so are finite.

The division is an abstract thing which has a start but may have no finish, although obviously you can stop dividing at any point if you wish

/Endedit

Does this help?

Edited May 2, 2018 by studiot

[Strange]

On 01/05/2018 at 1:12 AM, thicko said:

In maths you can have .3 recurring (or any other number for that matter), aye? 

How can a millimetre not be defined in the same way, therefore making it infinite?

Important to note the difference between having an infinite number of digits (in a particular representation) and an infinite value. 

So 1/3rd is a finite value even though it requires an infinite number of digits to represent it (in decimal; in base 3 it is just 0.1).

 

[Doozel]

The paradox lays in the way it's proposed. If you think in terms of speed instead it appears more comprehensible: you can divide the lenght of the track in infinite many segments but the sum of all of them is always the total lenght of the track. If you are travelling at a constant speed it will take you t=s/v to reach the end, independently on how many segments you traverse.

[swansont]

2 hours ago, Doozel said:

The paradox lays in the way it's proposed. If you think in terms of speed instead it appears more comprehensible: you can divide the lenght of the track in infinite many segments but the sum of all of them is always the total lenght of the track. If you are travelling at a constant speed it will take you t=s/v to reach the end, independently on how many segments you traverse.

But how would you propose it if you had no concept of infinite division, and the formulation of speed as s/t had not entered the thinking?

[Doozel]

2 hours ago, swansont said:

But how would you propose it if you had no concept of infinite division, and the formulation of speed as s/t had not entered the thinking?

Since Zeno formulated this paradox I think they understood the concept of adding more and more divisions and, without going into integrals, it should be a simple logic exercise to conclude that if you procedurally divide a segment in more pieces its overall lenght doesn't change. This is obviously without taking into account the infinitely small/large.

I can't tell for sure how to approach the speed problem since, as with my previous statement, I'm biased by what we know today. An idea is to think about the fact that it takes the same amount of time to go from A to B even if you divide it multiple times. That would go around the immediate concept of speed but I doubt it would be a convincing explanation.

Edited May 3, 2018 by Doozel

[studiot]

6 hours ago, Doozel said:

The paradox lays in the way it's proposed. If you think in terms of speed instead it appears more comprehensible: you can divide the lenght of the track in infinite many segments but the sum of all of them is always the total lenght of the track. If you are travelling at a constant speed it will take you t=s/v to reach the end, independently on how many segments you traverse.

 

Sorry, that won't wash as an explanation.

Yes this bit is correct

Quote

The paradox lays in the way it's proposed.

 

But

There are several Zeno paradoxes.

One of them is about an arrow.

An arrow will never reach its target because

Before it can cover the whole distance it must cover half that distance, leaving half the distance uncovered.

No matter what speed it travels it there will always be half the distance remaining, however small.

So it will never reach the target.

[thicko]

Genuine thanks to all who've sacrificed their time and/or effort to reply to my initial post. The response has been informative and sagacious, yet unpretentious, and quite frankly beyond all expectations - believe me, you's have all warmed my cockles!

Tbh tho, the only reason I posted this "paradox", which has indeed troubled my inquisitive mind since 1984, was coz I'd fallen out with my wife and duly dismissed to the "huffy room" with no Sky TV, no DVD player and no supper - BUT WITH WiFi!!

Thence I decided to try to rid my thicko brain of the conundrums she finds most perplexing,  my initial post was the first one of myriad others! 

Maybe I should just stick to assessing/diagnosing/ and solving plumbing and heating problems in future, eh? 

[Ken Fabian]

On 04/05/2018 at 7:01 AM, studiot said:

 

Sorry, that won't wash as an explanation.

Yes this bit is correct

 

But

There are several Zeno paradoxes.

One of them is about an arrow.

An arrow will never reach its target because

Before it can cover the whole distance it must cover half that distance, leaving half the distance uncovered.

No matter what speed it travels it there will always be half the distance remaining, however small.

So it will never reach the target.

I'm not sure I can agree with this. I was going to be flippant and say the arrow has to cover the last distance remaining, just by momentum. A bit more seriously ...

The sum of the infinity of portions of the total distance equals the total distance.

The time to traverse each portion is proportional to the distance of each portion; the sum of the times taken to traverse the infinity of portions of the total distance equals the time taken to traverse the total distance. It cannot take longer than the time to traverse the total distance to traverse that infinity of portions of the total distance.

Edited May 15, 2018 by Ken Fabian

[swansont]

9 hours ago, Ken Fabian said:

I'm not sure I can agree with this. 

Obviously, because Zeno was wrong, and the understanding of motion was incomplete.

9 hours ago, Ken Fabian said:

The sum of the infinity of portions of the total distance equals the total distance.

The time to traverse each portion is proportional to the distance of each portion; the sum of the times taken to traverse the infinity of portions of the total distance equals the time taken to traverse the total distance. It cannot take longer than the time to traverse the total distance to traverse that infinity of portions of the total distance.

Now try to formulate this without the concept of infinity.

[studiot]

12 hours ago, Ken Fabian said:

I'm not sure I can agree with this. I was going to be flippant and say the arrow has to cover the last distance remaining, just by momentum. A bit more seriously ...

The sum of the infinity of portions of the total distance equals the total distance.

The time to traverse each portion is proportional to the distance of each portion; the sum of the times taken to traverse the infinity of portions of the total distance equals the time taken to traverse the total distance. It cannot take longer than the time to traverse the total distance to traverse that infinity of portions of the total distance.

 

2 hours ago, swansont said:

Obviously, because Zeno was wrong, and the understanding of motion was incomplete.

Now try to formulate this without the concept of infinity.

 

Not sure if you  correctly understood my words or if I was not clear enough.

I was trying to avoid discussion of 'infinity' since every Zeno paradox mixes an abstract infinity with a concrete situation. (continuing the terminology I already introduced).

Even in the Zeno paradox (Ken, you do understand I don't find this 'paradox' without explanation?) the abstract infinity is introduced by the word 'never'.

The ancient Greeks acknowledged two kinds of infinity - They called these actual infinity and potential

https://en.wikipedia.org/wiki/Actual_infinity

So for instance if they had decimal numbers they could have made the distinction between a non terminating decimal of finite magnitude and the totality of all numbers.

The infinities introduced by Zeno are all non terminating processes, usually the indefinite division of a finite space or time.

Some of the (more complicated) paradoxes transfer the result of the indefinite division from one parameter to another to try to establish a fallacy.

For instance "The arrow will never reach its target or the runner finish the race because there is always some remaining distance to go which take a finite amount of time (however small) to transit so the motion will never terminate."

This is still wrong thinking (take him away ), but one has to be careful to observe where the slip occurs.