Monday, 11 February 2008

Zeno's Bisection Paradox - A New Angle


Apropos of my last posting - in my latest lectures I have found a much better tool to convey the idea of time bisection so crucial to CTF. Up until recently I have applied the analogy of half life (possibly now invalidated by Mr MacIntyre) or Zeno's bisection paradox. This is called Thomson's Lamp in honour of James Thomson, Professor of Philosophy at the Massachusetts Institute of Technology who first suggested it back in 1954. It goes like this:

Consider a lamp, with a switch. Hit the switch once, it turns it on. Hit it again, it turns it off. Let us imagine there is a being with supernatural powers who likes to play with this lamp as follows. First, he turns it on. At the end of one minute, he turns it off. At the end of half a minute, he turns it on again. At the end of a quarter of a minute, he turns it off. In one eighth of a minute, he turns it on again. And so on, hitting the switch each time after waiting exactly one-half the time he waited before hitting it the last time.


QUESTION: At the end of two minutes, is the lamp on, or off?

The answer is rather startling. If we represent the successive states of the lamp by the series of increasingly short periods in which it is on and off, we obtain something that has no last member: 60, 30, 15, 7.5, 3.75, 1.875 .... The series, in other words, is infinite. So at the end of two minutes, the lamp has been switched on (and off) an infinite number of times. Now in my opinion there is nothing mathematically incoherent in the description of the experiment (unlike, possibly my analogy of half-life discussed in the last post). The sum of all the series of periods is not infinite: it approaches without quite reaching, 120 seconds.


This, I argue, is exactly what happens to the dying person as they approach death. Let us assume that I am a skydiver and in 120 milliseconds I will hit the ground and be killed. At 120 milliseconds before I do so the glutamate flood in the brain slows down my perception of time by increasing my metaboloic rate. Each subjective millisecond takes twice as long to pass in my perception as the one before. Exactly the same situation as Thomson's Lamp applies. I will never hit the ground because I will always be a fraction away from it - just like the lamp's switch I am trapped in an eternity of subdivisions. As such I never die.


Max Payne of the Scientific & Medical Network suggested at a recent presentation I did for the SMN that this bisection of time would come to an end when no more space and time was available for a further subdivision. He cited Cantor's Infinity Argument in support of this position. I am unaware of this and it is my intention to check this out when I have time (subjective or objective).


Of course I am hopeful that somebody out there can clarify this argument once and for all.


If you are interested in the source of my Thomson Lamp material it can be found in a wonderful book called Travels In Four Dimensions - The Enigma of Space & Time by Robin Le Poidevin, Professor of Metaphysics at Leeds University (ISBN 0-19-875255-5)


17 comments:

Hurlyburly said...

I consider things of this nature to be the inevitable cracks in the framework of our reality. The little details that cannot be ironed out to keep us from questioning the nature of things.

I think this is a perfect analogy that demonstrates the problem of time and it's duality with reality.

Carenza Waters said...

I think this is entirely plausible. Its comparable to falling into a black hole. From outside we go 'shoooop', gone!

But to the person going in, it goes on forever. The singularity is infintely distant so takes forever to reach, even though the event horizon may be only a few millimeters across (externally).

Anyone who has an old fashioned electrical meter can see this scaling effect in action on the 'Resistance' range. You see '0' at the extreme right of the scale and a few finite values in the middle.

But the left end of the scale gets more and more crowded as the numbers increase. Inifite resistance is on the extreme left end of the scale.

Karl L Le Marcs said...
This comment has been removed by the author.
Anthony Peake said...

Carenza,

I agree. The 'Event Horizon' image is another classic example of bisection.

Interesting point though - is there a 'distance' that cannot be bisected? For example I have head much of the term 'The Planck length' which I understand is the smallest possible size thout our science and maths can conceive of. It has a length of approximately 1.6 × 10−35 (to the minus 35) metres, 6.3 × 10−34 inches, or about 10−20 times the diameter of a proton. This is where quantum gravity will take effect (as per the recent discussions on gravity).

What we then have is Planck Time.

Planck time is the time it would take a photon travelling at the speed of light to across a distance equal to the Planck length. This is the ‘quantum of time’, the smallest measurement of time that has any meaning, and is equal to 10-43 seconds. No smaller division of time has any meaning. With in the framework of the laws of physics as we understand them today, we can say only that the universe came into existence when it already had an age of 10-43 seconds.

As such will there be a point whereby the distance between my skydiver and the ground cannot be bisected anymore because of the dual effect of both Planck Time (10 to the minus 43 seconds) and Planck Length (10 to the minus 35 metres)?

My brain hurts - Or am I being a bit of a plank myself?!!!

Karl L Le Marcs said...

Hey Tony, is that you adopting my Quantum Gravity theory already !!
Blimey !
You must have grasped the concept faster than the look on your face suggested.
*impressed*

Anthony Peake said...

Karl,

After you applying NLP to your Zennor cards on Saturday I had the vague feeling that you could actually tell what I was thinking!!

But then again maybe not (chuckle).

For those of you unaware of what I am talking about (and only a bloke called Richard was witness to this) Karl has a frightening skill of applying elements of Neurolinguistic Programming to predicting what card an individual will/has selected. He has clearly learned a lot from his close friend Derren Brown. What I could not quite make my mind up was whether the NLP bit was a misdirection and it was simply a clever card trick or whether it was really applying NLP. If it was I, for one, was very impressed.

Hurlyburly said...

Yeah but neither one of you two genius's could place that quote from your favourite-theory-related film though could you!

"Oh yeah..... don't drive on the railroad track!"

Anthony Peake said...

cause there might be a train a comin'!!!

You have me interested. What was the movie?

Karl L Le Marcs said...

Tony,
I can assure you that the NLP stuff I showed you was pure Psychology and NLP and not in anyway any misdirection or card-trickery.
Indeed when I next see you, I'll try to do all five cards in a row and not just one at a time - this IS possible but is extremely difficult.
I'm just composing an email to you which may help explain further.

Hurlyburly said...

It's the same things your whole life. "Clean up your room.", "Stand up straight.", "Pick up your feet.", "Take it like a man.", "Be nice to your sister.", "Don't mix beer and wine, ever.". Oh yeah, "Don't drive on the railroad track."


I'll give you a clue... there have been two posts very close together with the title of this movie this month.

Anthony Peake said...

Of course ... "Groundhog Day".

Now I understand the significance of your cryptic text.

Very good, mucho impresso.

Karl L Le Marcs said...

Ah! Now I understand Martin, I just thought you'd been drinking again.
*hic!*

Hurlyburly said...

Karl your schtick called...

It's batteries need changing!


xxx

Karl L Le Marcs said...

My dear fellow, have you just kissed me? Thrice !!
*swoons*

Hurlyburly said...

Damn straight....


Oh my, what an outstanding play on words!

Karl L Le Marcs said...

Trumpet
Blow
Own
*tee hee*

Karl L Le Marcs said...

Tony.
Thompson's Lamp IS fundamentally capable of being disproved mathematically without resorting to Cantor's infinity paradox.
I will try and explain simply (but knowing me that may be quite difficult).
Immanuael Kant's First Antinomy is a good example but basically the difference lies within the infinity divisibility of material objects against the altogether different aspect of the invinite divisibility of time.
For example, what Thompson's Lamp does NOT take into effect is the amount of time taken to actually switch the switch. Admittedly this is a relatively small amount of time, but given the infinite division of time you will soon reach a moment when the remaining divisible time is less than the functional time of the action.
As there is a definite limited block of time to work within, the switching would have to get progressively faster and faster the closer it drew to the two min mark, in a seemingly tangential fashion. Always closer but never reaching, achieving greater than light speed, double light speed, quad, etc. Given the parameters of the problem, it sounds like a fallacious problem on the face. By the 'end' of two min, the light would appear, for all intents and purposes, continuous.
Like 'motion' and 'time'.
*BLIMEY!!*
On an aside it was a delight to meet Max Payne at the SMN last year and, Tony aside obviously, he is one of the most fascinating men I've met in years.
Although he is wrong on Cantor, but I'm not even going to attempt to document how here, expect a book sometime in the future (pun very much intended).
*needs a lie down*