Counterintuitive Facts

[mjc (Mark)]

Mathematics is a rich source of counter-intuitive facts.

There are some infinities that are "bigger" than others.

http://en.wikipedia.org/wiki/Infinit..._the_continuum

Take the natural numbers (counting numbers - whole numbers) - one can have an infinite number of them.

Now consider the real numbers (those numbers that can have arbitrary decimal precision). There are an infinite number of numbers between 0 and 1 and between 1 and 2 - in fact between any two decimal numbers.

These two infinities are of different scale.

Cantor had real resistance to this idea.

Mark C.

[mjc (Mark)]

Originally Posted by Robh

Hi Craig and all,

For counter-intuitive, you can't go past this one from the mathematicians.
It is possible to take a solid sphere S and divide it up into a finite number of pieces and then reassemble them (using only translations and rotations) into two identical copies of the original sphere. This is known as the Banach-Tarski Paradox.
In fact, ignoring the original sphere's centre, it can be divided into precisely 4 pieces S1/S2/S3/S4 that can be re-assembled into two spheres S1+S2 and S3+S4 of the same volume as the original.

A more impressive statement of the theorem says that it is possible to take a solid sphere the size of a pea and divide it into a finite number of pieces that can be reassembled into a sphere the size of the Sun.

Regards, Rob.

Rob

That's one that I've heard of - but that is seriously beyond my comprehension.

Mark C.

[Screwdriverone (Chris)]

Originally Posted by mjc

Or very intuitive - but counter expected logic-

A tourist in a village outside of Dublin asks a local, "What's the best way to get to Cork?"

Local: "Well that depends - are you on foot or do you have a car?"

Tourist: " I have a car"

Local: "Well, that's the best way!"

Mark C (living in suburbs of Dublin)

That one is a bit like the blonde walking along the banks of a river, she sees another blonde across the river and calls out: "HEY!, how do you get to the other side?" To which the other blonde says: " You ARE on the other side!".....

Cheers

Chris

[Robh (Rob)]

Originally Posted by mjc

Rob

That's one that I've heard of - but that is seriously beyond my comprehension.

Mark C.

Mark. I think the proof is one for mathematicians with specific knowledge of set theory.
But your previous post gives an analogy.
Consider the set of positive integers Z:
1,2,3,4,5,6,7,8,9,...
This is, of course an infinite set.

Now lets split it into even numbers:
2,4,6,8,10,12...
and odd numbers
1,3,5,7,9,11,...
Both infinite sets.

Take the even numbers and divide by 2:
2/2,4/2,6/2,8/2,10/2,12/2... = 1,2,3,4,5,6... which is the same as set Z
Take the odd numbers subtract 1 then divide by 2:
(1-1)/2,(3-1)/2,(5-1)/2,(7-1)/2,(9-1)/2,(11-1)/2... = 0,1,2,3,4,5... which, if the zero is ignored, is also the same as set Z.

Thus set Z is split into two sets each of the same size as Z.

Rob

[sjastro]

I suppose this thread can evolve from the counterintuitive to the incomprehensible, such as the Klein bottle which is a two dimensional object that can only exist in four dimensions.

Klein bottles however can be purchased.
http://www.kleinbottle.com/

I like the Acme Klein Bottle lifetime guarantee.

In addition, Acme's provides this exclusive LIFETIME GUARANTEE: We guarantee that you will live your entire lifetime, or double your money back.

Regards

Steven

[CraigS]

Thanks all. I appreciate everyone's contributions on this thread .. I think its terrific.

I suppose if I had to disclose my point in raising this thread, I would have to make use of the following two quotes (thanks to Bert and Mark):

Originally Posted by avandonk

Nearly all of science is counterintuitive. It has taken thousands of years by many very smart people to work this out.

Originally Posted by mjc

Mathematics is a rich source of counter-intuitive facts.

Science would have to be counterintuitive, because science is fundamentally built on rationality. Wiki defines rationality as:

In philosophy, rationality is originally the exercise of reason, the way humans come to conclusions when considering things most deliberately.
…
Humans are not rational by definition, but they can think and behave rationally or not, depending on whether they apply, explicitly or implicitly, the strategy of theoretical and practical rationality to the thoughts they accept, and to the actions they perform.

And mathematics I think, is an essential part of that strategy.

The counterintuitiveness arises when we compare our expectations, (which aren't always rational ... as the above quote asserts), with the outcomes of a rational process.

Thanks to everyone for the previous examples. I don't think any one of them could escape being put to the test using mathematics, as the litmus test for rationality and ultimately, factuality.

The embedding of mathematics within Science, is so often accused of divorcing science from reality and yet, all I can see is that its presence prevents us from deluding ourselves, and being solely dependent on frequently flawed human 'intuitiveness'.

I see this as being particularly important in astronomy, as I find it easy to get caught up in the aesthetics and the deceptiveness of the image ..

Cheers

[CraigS]

Originally Posted by Robh

Hi Craig and all,

For counter-intuitive, you can't go past this one from the mathematicians.
It is possible to take a solid sphere S and divide it up into a finite number of pieces and then reassemble them (using only translations and rotations) into two identical copies of the original sphere. This is known as the Banach-Tarski Paradox.
In fact, ignoring the original sphere's centre, it can be divided into precisely 4 pieces S1/S2/S3/S4 that can be re-assembled into two spheres S1+S2 and S3+S4 each of the same volume as the original.

A more impressive statement of the theorem says that it is possible to take a solid sphere the size of a pea and divide it into a finite number of pieces that can be reassembled into a sphere the size of the Sun.

Regards, Rob.

Rob;
I'm havin' trouble with this one, too.
There's some tricky wording around how to do this …

Originally Posted by Wiki

The reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are complicated: they are not usual solids but infinite scatterings of points.

So what exactly is an infinite scattering of points if:

Originally Posted by Wiki

... the original ball is decomposed into a finite number of point sets

I guess I need a refresher on my set theory definitions …

Cheers
PS: Clearly a pea and the Sun have different masses. So, there's some trickiness around what these 'points' actually are. Seems to me, this maybe one which exists in set theory, but may not necessarily have a one-to-one mapping with the physical world

[Robh (Rob)]

Originally Posted by CraigS

Rob;
I'm havin' trouble with this one, too.
There's some tricky wording around how to do this …


So what exactly is an infinite scattering of points if:

Craig,

The concept challenges the way we think about volume. When we calculate volume, we give it in terms of so many m^3, cm^3 or mm^3. But why stop there? We could give it as so many nanometres^3 or picometres^3. In fact we could give it as the number of atoms of hydrogen gas that make up that volume.
I think the idea is that the sphere is considered to be made up of an infinite number of lattice points rather than a finite number of individual atoms. Analogous to the concept that an interval consists of an infinite number of points rather than a finite chain of atoms.
In that sense, the sphere could not be physically divided into 4 pieces (ignoring the centre) that can be reassembled into two identical spheres as the original. Simply because the sphere cannot be physically divided down past the size of an atom. It is probably best to think of each divided piece as a cloud of an infinite number of points scattered in the lattice of the sphere.
From my understanding, the paradox says nothing about the shape of the pieces. They may appear as a random framework of an infinite number of points that are loosely connected. However, remember also that physically atomic nuclei are actually strung out in a lattice with spaces between them.
As a theoretical possibility, the concept of division of a solid sphere into a finite number of pieces and then reassembly into two copies of itself is still extraordinary.

Regards, Rob

[multiweb (Marc)]

Here's another one: drinking more alcohol dehydrates you... not fair.

[CraigS]

Yes thanks for that Rob;

I was thinking along similar lines, but I wasn't sure if I was getting the point.

If (theoretically), you keep dividing an atom into more and more smaller particles, you end up drifting towards all that string theory stuff, which we've discussed before. However, from a QM point of view, we can't get smaller than Planck size so, this would seem to be where theory departs from practice, eh (??) (Ie: in practice, there's an asymptotic limit ?)

If this is the case, then I think this paradox may fit in the theoretical box.

Cheers

[CraigS]

Originally Posted by multiweb

Here's another one: drinking more alcohol dehydrates you... not fair.

I reckon you should keep working on that one, Mark.

Here's the next step … drinking too much water can kill you …
Water intoxication .. looks like Andy Warhol, (& others), died from it.

Better stick with the alcohol … (safer).


Cheers

[multiweb (Marc)]

Originally Posted by CraigS

I reckon you should keep working on that one, Mark.

Here's the next step … drinking too much water can kill you …
Water intoxication .. looks like Andy Warhol, (& others), died from it.

Better stick with the alcohol … (safer).


Cheers

Wow! Scary... You gotta die of something. I'd rather go with a nice after taste of good Port than chlorine

[troypiggo (Troy)]

2 of my favs:

You can't slam a revolving door.
You can't grow beer in a beer-garden

Not very scientific, sorry.

[gman (Grant)]

Something simple
-40 Celcius = -40 farenheit

Somthing hard
Double slit experiment
www.youtube.com/watch?v=DfPeprQ7oGc

[CraigS]

Hi Folks;
Its been bugging me, so the best way (I figure) for me to get this off my chest, without annoying everyone forever after, is to resurrect this thread and record the fact.

So here comes the counterintuitve fact …

The laws of Physics govern what happens when a brick on a rubber band is pulled up an inclined plane of carborundum paper. No matter how many times you run the experiment every outcome is different. It is called a complex system.

(Full credits to Bert: Thanks Bert).

(There are many other examples, but I love the simplicity of this one).

So now for the non-fact (speculative side of this):
I can't help wondering about the possibility that life in the universe may have emerged following this same pattern, ie: it can still obey all of the known Laws of Physics, but may still not have a repeatable outcome. I emphasise that there is no evidence that this is possible, but there is also no evidence to the contrary. Inferences can be made, and these are entirely dependent on the frequency of occurence of the 'right conditions', suitable for the emergence of life.

So, I emphasise that the possible non-deterministic nature of exo-life emergence idea, is purely speculative (and somewhat tenuous) on my part.

I can't resist refraining from looking at the huge the diversity of exo-environments, staring us square in the face, and wonder about mainstream science's propensity for leaning towards classical deterministic physics, to explain it all. Whilst predictable, replicated patterns clearly exist in the universe, on other scales, they don't. We seem to be living amongst a mixture of systems defined by both deterministic and non-deterministic physics. Scientists look for constraint limits, when exploring seemingly complex phenomena, and if observations align with classical, deterministic physics, the unknowns are then unable to influence the explanations. (Ie: we don't know what we can't explain with deterministic physics).

Quantum mechanics has taken us part of the way into this world, and may apply at the smaller scales, but does this mean that non-deterministic physics can't explain phenomena at other scales, and in other areas ?

All of this exo-life idea will instantly change of course, once a single instance of exo-life is found.

I just love the frailty and precariousness, of the exo-life situation in science. Tremendous stuff !

Another comment (by Robh) the other day nudges me along even more:

But just how much more complicated are things going to get? If dark energy exists, its nature is going to be even more elusive than dark matter.

… And the pattern continues in other areas …

Incredible stuff !
I'm done ! Comments welcome.

Cheers

[avandonk]

Craig even life on Earth would not follow the same evolutionary path given the exact same starting conditions. Evolution is a random process only constrained by the laws of physics and chemistry.

Of the many life forms with different body morphologies that were produced in the pre Cambrian explosion only a few survive today. All mammals have essentially the same body plan. From whales to mice.

The same rules would apply to exolife and exoevolution.

The brick pulled by a rubber band on an inclined plane of carborundum paper is a very good model for fault behaviour driven by tectonic plate movement. Where the pull is tectonic plate movement. The rubber band is the energy in the rock and the sudden movement of the brick is the earthquake.

There is a new program on SBS on 0830 PM tues 5th of April all about fractals chaos etc and the associated mathematicians and mathematics. Did you see the very early fractal animals on David Attenborough's First Life?
I only heard about this on the Science Show yesterday.

The counter intuitive thing that comes out of chaotic systems and fractals is that self regenerating order can arise from total randomness. LIFE!

The underlying quantum mechanics just gives us a glimpse of the connectedness of all matter and energy at a multidimensional level we cannot perceive directly.

Bert

[Brian W (Brian)]

Hi Craig, you were doing great right up to the point where you said that all of your speculation changes once a single instance of exo life is found.

No it doesn't it just means one and only one form has been found.

Humanity has always made itself the centre. Centre of the universe... wrong, centre of the galaxy... wrong, centre of the solar system... wrong.

Admittedly deciding where a ship is going next from its wake is a pretty dumb idea but if history holds true carbon may well not be the unalterable necessity for life many think it is.

Not that there will not be carbon based life forms but there may not be. And not there are life forms using a non carbon structure but there may be.

It is after all one incredibly huge place we inhabit with unfathomable stores of energy and a really really really long life span where pretty much anything is possible.

Brian

[CraigS]

Hi Bert;
Thanks for your words. I really do appreciate, read and consider them very carefully, whenever I see them.

Originally Posted by avandonk

Craig even life on Earth would not follow the same evolutionary path given the exact same starting conditions.

Bert, I’m a little confused by this. I’m just trying to understand. Previously you said:

We can however by observation and inference conclude given the right conditions life will inevitably occur given the correct conditions. This is only due to the laws of the Universe.

So, just trying to understand, in my own words, given the same starting conditions, we can infer and conclude that life will inevitably occur, but I’m trying to understand what other paths might occur and would we ever recognise the end result as ‘life’ ?

This is tricky !

Originally Posted by avandonk

Evolution is a random process only constrained by the laws of physics and chemistry.

So, in my own words, physics and chemistry are determinate and predictable, so the process constraints are predictable. However, the ‘end’ result of an unpredictable evolutionary path is unknown (due to lack of data) ... other than it would result in life, which we may, or may not, recognise as life ?

Originally Posted by avandonk

Of the many life forms with different body morphologies that were produced in the pre Cambrian explosion only a few survive today. All mammals have essentially the same body plan. From whales to mice.
The same rules would apply to exolife and exoevolution.

So the same evolutionary process ‘rules’ would apply elsewhere, but the evolutionary paths might differ, because the starting conditions may be the same or different.

Originally Posted by avandonk

The brick pulled by a rubber band on an inclined plane of carborundum paper is a very good model for fault behaviour driven by tectonic plate movement. Where the pull is tectonic plate movement. The rubber band is the energy in the rock and the sudden movement of the brick is the earthquake.

So, this was a example of a Complex System ... and from the same rationale, I’m pondering the chances of whether the emergence of life may also follow complex system principles. Interesting !

Originally Posted by avandonk

There is a new program on SBS on 0830 PM tues 5th of April all about fractals chaos etc and the associated mathematicians and mathematics. Did you see the very early fractal animals on David Attenborough's First Life?
I only heard about this on the Science Show yesterday.

Haven’t seen Attenborough’s, but I did pour through Mandelbrot’s ‘Colours of Infinity’ (several other books on Chaos, Complexity and Small Worlds).

Originally Posted by avandonk

The counter intuitive thing that comes out of chaotic systems and fractals is that self regenerating order can arise from total randomness. LIFE!

I think this is the definition of ‘Emergence’ (??)
Simply amazing stuff !
I appreciate your responses.
This is absolutely fascinating !

Cheers & Rgds

[CraigS]

Originally Posted by Brian W

Hi Craig, you were doing great right up to the point where you said that all of your speculation changes once a single instance of exo life is found.

No it doesn't it just means one and only one form has been found.

I suppose so, Brian.
But sceptics would be known as outright 'Deniers' if they didn't flinch to some degree !

I don't think they could hold up against the sheer pent-up tide of humanity and emotion, should such a discovery be confirmed !

Originally Posted by Brian W

It is after all one incredibly huge place we inhabit with unfathomable stores of energy and a really really really long life span where pretty much anything is possible.

Perhaps your mind is less constrained than mine !

.. "just the facts, m'aam, just the facts ..."

Cheers

[Brian W (Brian)]

Sergeant Friday quoted on Iceinspace...gotta love it.

My reality may have been more exposed to things that just don't make sense in the rational world a little more than yours.

One example; there is a graveyard that I once had the right to wander in. It was, and is, a truly culturally historic area. There were many unique headstones which I photographed. There was however one headstone which I was unable to photograph. I tried 4 times and every time the pictures in front of the headstone were fine as were the ones following but the pictures of the headstone were a mass of lines.

I have my beliefs about why no picture turned out but this is not the time nor place for such a discussion.

For myself there is very little difference between saying this world view as supported by this (put in your favourite religious institution) is correct because it is impossible for it to be another way than those who say this is the way it is because of (put in your favourite scientific paradigm) and math or physics or geometry shows that it has to be this way.

I have simply experienced too many things that I cannot explain to believe in a finite set of either questions or answers.
Brian