**Information
theory: **Imagine you are an alien from a totally different
world - so different that even the building blocks of life are completely different
on your planet. Now say that you land on planet Earth. You find three huge sheets
of paper (i) one containing English text (ii) one listing a DNA sequence and
(iii) one listing a computer program. You have no idea which is which, but you
recognize you are looking at ordered bits of information - three very strange
and different languages. The question is, using statistical principles and principles
from information theory can you (the alien) detect any fundamental differences
between the three sheets of paper so that you can distinguish between a human
language, a machine language and a biological coding language? Or is it in fact
impossible to distinguish them in principle?

**Physics: **Einstein's
*principle of equivalence* says that you can't tell the difference between
being in a stationary elevator in a gravitational field and an accelerating
elevator in free space. However, if the stationary elevator is in a field
that has the magnitude of that of a black hole, your temperature will increase
due to Hawking radiation. If you are accelerating in free space (at a value
equal to *g *of the black hole) your temperature will increase according
to the Davies-Unruh effect. However, the Hawking formula predicts a different
temperature to the Davies-Unruh formula. So, we can imagine a thought experiment
where we can measure this difference, and hence tell if the elevator is accelerating
or not. Does this violate Einstein's principle of equivalence? Explain.

**Photonics: **A
monochromatic beam of, say, red light appears red to the human eye. However,
if we chop the beam with a mechanical shutter, we get a square wave. Fourier's
theorem predicts that this beam will no longer be monochromatic as a square
wave contains an infinite number of frequencies. So why does chopped red light,
still appear red? What happens as the frequency of chopping increases?

Response
from Brad Ferguson, RPI, New York, 21 Jan 2002

**Mathematics: **The
famous Hungarian mathematician, Paul Erdos, did not rest until he found a
short and elegant proof to a problem. Whenever he achieved this he would call
it the "book proof." However, perhaps it is futile to search for short proofs
in all cases, as we know (eg. from the theory of chaos) that seemingly simple
things can generate great complexity. So can an information-theoretic approach
be used to prove that there are infinitely many more problems with complex
proofs than elegant proofs? Also I have a related question: if a proof is
very long, can we really believe it? Can we use, again, an information-theoretic
approach to produce a probabilistic confidence measure of the correctness
of a proof that declines as the number of bits of information in the proof
increases? Is there something special we can say when the number of bits of
information in the shortest proof exceeds the number of bits of information
in the shortest way of expressing the question?

**Quantum computation: **Using
classical numbers, pick an integer at random between 1 and *n, *where
*n* is large. The probability that this number is divisible by 3 is 1/3,
by 4 is 1/4, by 5 is 1/5 and so on. However, in the 1930s Erdos & Kac
pointed out a remarkable property: the probability of divisibility by 12,
say, is 1/12 which is 1/3 times 1/4. The remarkable thing about this is that
the multiplication is the Bayesian rule for combining *independent* probabilistic
events! So the question is if we replace this rule with Feynman's rules for
combining quantum probability amplitudes, can we work backwards and find what
number system we would need for probability of divisibility to obey quantum
rules? If we can find such a number system, would this be a more natural system
for inventing new quantum algorithms?

**Information theory: **If
we have a long sequence of, say, integers and we want to test them for randomness
we can measure the Shannon entropy. The more entropy you have, the more disordered
or random the sequence is. Alternatively, we could apply Chaitin's compressibility
test. If you can generate the sequence of integers with an algorithm that
takes up less space than sequence, itself, then the sequence had redundancy
- if you cannot compress the sequence then it contains maximum information.
However, a truly random sequence is incompressible! Therefore it seems that
maximal information corresponds to disorder! The question is what is the best
framework to adopt so that we can accommodate this view without getting confused?
Also a related question is this: the digits of pi are not random, in the Chaitin
view, because we can compress the sequence by writing an algorithm to generate
pi - however if I presented you with the digits of pi but with the first 100
digits deleted, could you compress the sequence? I can compress the sequence
because I know that all I have to do is write down an algorithm for pi and
remove the first hundred digits. But to you the sequence looks totally random
and it might take you forever to guess that the sequence was pi (in disguise),
so you will quickly give up trying to compress it and conclude it is incompressible.
So another question would be to ask if Chaitin's viewpoint is truly a useful
one, as it seems to depend on one's foreknowledge - in other words ignorance
of the data can affect your viewpoint.

**Optics: **When
you have a sandwich of three layers of differing refractive index, you can
eliminate reflections at the incident surface by tuning the refractive index
of the middle layer. However, how does an incident photon know what the refractive
index of the middle layer is, so it knows not to be reflected when it is incident
on the first layer?!

**Ethics and
Law:** Imagine you have been sentenced to jail for 50 years for murder.
But you are really innocent (someone set you up). You serve your sentence
and when the 50 years is up, you are so angry that you kill the person who
set you up. The question is: should you go to jail for a further 50 years
or have you already served your sentence in the last 50 years?

**Convention:**
Why do clocks go clockwise?

**English
language: **What is the origin of the English expression third world
country? It begs the question, what is a second world country and why do
we not hear that expression?

**Theology: **In
the Judaeo-Christian tradition, body & soul are holistically one. The
notion that the body is merely a vessel for the soul is actually a hang over
from Plato's dualistic philosophy and Gnosticism. The Biblical texts, in contrast,
support an inseparable unity between body & soul, which theologians call
a hypostasis. However, the Bible texts also support resurrection of the body
in which you go to heaven with both a body and a soul. However, as your present
body rots in the ground, you get a new or transfigured body in heaven. But
if your soul is inseparably connected to your present body in a hypostasis,
how can you get a new one? Is there a singularity or discontinuity at the
swap over point between old and new body? How can we elucidate the present
theological framework to resolve this paradox?

**Sociobiology: **Is
the phenomenon of falling in love with someone, a trick of biology to get
you to mate or is it deeper than that? Is it just a biochemical rush of hormones?
If deeper, why does the feeling wear off after a while? If it is indeed a
trick of biology, then what are the consequences (if any) of pairing and procreating
with someone you didn't fall in love with? Would that affect in anyway the
efficiency with which our genes are propagated? Would it affect our social
happiness, given that the initial feeling would have worn off anyhow?

**Physics: **Do
a thought experiment where all the vacuum fluctuations in the universe magically
disappear. What would happen to the universe? For example, black holes would
stop emitting Hawking radiation, presumably, and become truly black. But would
all atoms collapse into points? Would the whole universe collapse? If it is
too hard to imagine removing all vacuum fluctuations, try perhaps halving
their intensity, as a first step.

**Philosophy: **As
quantum mechanics runs counter to the rationality of the classical world,
does this mean that Nature or even God is ultimately irrational? Could it
be that our concept of what we think is rational is not the full story? Are
we the irrational instead?

**Urban
legend: **Is
it true that the Great Wall of China can be observed with the naked eye from
the moon? Although the wall is very long, the width is surely smaller than
the resolution of your vision from that distance?

**Philosophy:
**What is truth?

**Ethics and
Information Technology:** Would you feel offended if your partner engaged in telesex with a live
movie star across the internet? What are the ethics here? Would there be any
difference if the movie star wasnt really at the other end but it was a prerecorded
session copied out to thousands of other people? What if it was a virtual
session created by a machine?

**Law and Sociology:**
It could be argued that when we lock someone in jail for a crime, we cut them
off from society and inevitably interfere with any possibility of normal social
development of that person, and therefore perpetuate their condition. If the
person is a danger to society, is there an alternative?

**Astronomy:**
Why does the man in the moon still appear to be the same way up no matter
whether you are in Sydney, Australia, or New York, USA, for example?

**Physics:**
Does water really have a preferred direction when going down a plughole? We
hear all these theories about the plughole, but have rigorous experiments
been performed? I once tried observing the water in a basin in an aircraft
toilet. I repeated the experiment as I flew between hemispheres and crossed
the equator. I even repeated it many times coming back the other way. To my
disappointment the water went straight down the plughole without rotating
this happened everytime! Perhaps you need a private jet with a whole bath?

**Philosophy:**
If there were no evil in the world, would good cease to have meaning?

**Physics:**
Why does the sun & moon appear larger at the horizon than when they are
at their highest points in the sky? Can refraction really account for all
that difference? Has anyone proved it? Is the ratio of the biggest to smallest
moon sizes the same as the ratio for that of the sun? How can we account for
those observed ratios? Why are their colours more yellowy-orange, nearer the
horizon? At a certain time of the year, the moon remarkably appears fairly
high in the sky, but then sets below the horizon fairly rapidly why does it
appear to move more slowly all the other nights of the year?

**Philosophy:**
Given that the atoms in your body get replaced over each seven-year period
and that your mind both develops and forgets old data, how can you define
identity? Are you really the same person, you were yesterday?

**
Law:**
Many paradoxes in law arise because law is black & white and real life
is a continuum of grey. Law takes continuous variables and sets a threshold
or boundary. Either side of the boundary is 1 or 0, ie. right or wrong. Is
this for convenience because we have no better way or is there a deeper reason?
If it is a matter of convenience, can we someday use technology to evaluate
all the main variables and produce continuum based laws? Could we trust machines?
Would it be fairer than binary laws?

**
Convention:**
Why is the English alphabet in the order it is? Why is Z on the end and not
O for example? Whose idea was it to chose the order for the alphabet?

**
Physics and Metaphysics:**
If a billiard ball hits a stationary billiard ball, it comes to rest. Then
the stationary billiard ball begins to move off. Kinetic energy is said to
transfer from one ball to the other. However, *why* does the energy transfer?
And what is the mechanism? If you stand back from the physics, it appears,
at present, that the kinetic energy transfers from one ball to the other rather
like a "ghostly spirit." Other than just citing the law of conservation of
energy can we explain this mechanism more deeply?

**
Game theory:**
If a country has the capability to fire a nuclear missile at New York city,
then conventional game theory seems to tell us that New York should point
a nuclear missile at the other country. This creates a stalemate, so that
no actions are taken and we are all safe. However, straight game theory needs
to be expanded to include error analysis. The question is what if a nuclear
missile was fired at New York and was said to be by mistake? What is the best
strategy for New York now? From a game theory point of view, should New York
still strike back? Also there are two possibilities to consider: the declared
mistake could have been genuine or a bluff. Also another related question
is that if the missile is intercepted whilst in the air, will the resulting
explosion create more deaths and than if the bomb was allowed to hit the ground?

**Probability
theory: **If I say to you "give me any amount of money you like and
I will toss a fair coin, if it is heads you win and I will double your money,
but if it is tails you lose and I will give you only half your money back,"
do you play the game? The answer is yes because if the amount you bet is X
then you can win either 2X or X/2, giving the expected value of winnings as
1.25X. But now I change the game slightly and present you with two envelopes
and say one envelope contains 2X and the other contains X/2, pick an envelope.
Then when youve picked an envelope, I say do you want to keep that envelope
or swap it for the other one? Now it doesn't seem that swapping makes sense,
because you could have just as easily picked the other envelope in the first
place, but yet again it seems at first sight that the expected value of the
other envelope is 1.25X, just as in the previous coin tossing game. Now obviously
something has to be wrong with what I've said above, that has created this
dilemma. So the question for you is this: what is the fundamental difference
between the two games described above? Also consider two cases: (1) when you
don't bother looking inside the first envelope and (2) when you see the amount
in the first envelope. So another related question is: does the act of observing
the amount change the probabilities in question?

**
Physics:**
Here is an interesting experiment. Go in to a totally dark room and pull open
the seal of a self-adhesive envelope. Something amazing will happen. The glue
glows with a bright light. Experiment to see if the intensity increases the
faster you rip open the envelope. What happens? What is the physical explanation
for this transduction between mechanical energy and photons?

**
Psychology:**
The concept psychoanalysts call *ego lacuna *refers to a gap in our moral
thinking. An example would be the terrorist who helps a baby outside a supermarket,
by picking up a toy the baby has dropped, and then proceeding to plant a bomb
in the supermarket. The fact that the terrorist didn't make the connection
in his mind (that the bomb will be blowing up the same baby) is an example
of *ego lacuna. *We all suffer from this effect to perhaps a lesser degree.
So the obvious question is: where is the evolutionary survival value in all
this? Does this effect have a purpose? Also does it occur in other realms?
For example a person might be really good at solving a particular problem
-- now change the context of the problem: give that person a problem with
the same type of solution, but make it a new problem that looks different.
Very often we all fall into the trap of getting totally stuck with the new
problem. Does this inability to make that leap of judgement have the same
origin as *ego lacuna*? Do psychoanalysts have a word for *ego lacuna*
in these other contexts?

**
Biophysics:**
Stare into the distance and squint your eyes. You will notice faint tiny circles
floating around in your field of vision. To do this experiment it helps if
the background is uniform. A dull-grey cloudy sky works as a great background
to stare at. I used to think that these tiny circles were little dust particles
floating around on the fluid film of my eye, that come into focus as I squint.
However, I've rejected that hypothesis because the circles don't move with
Brownian motion. Their paths are very regular. So where do these circles come
from?

**
Cosmology:**
Look up at the stars at night. Obviously you infer the existance of stars
from the fact that the light you see comes from many tiny little points. If
the night sky was uniformly lit, you wouldn't think there were any stars there.
Now look at the sky in the microwave or gamma ray parts of the spectrum. What
do you see? You see the sky is virtually uniformly lit with microwaves and
gamma rays. So if visible light emanated from points (that we call stars),
where on earth did the microwaves and gamma rays come from? There are no satisfactory
theories about the gamma rays. Some think it is a vacuum fluctuation effect
- but where is the energy coming from? With the microwaves, it is believed
that it is the blackbody thermal background radiation of the universe, and
that any tiny ripples you see in it are from the time of the birth of the
universe. However, where are the microwave photons actually coming from so
as to appear uniform across the whole sky over time? If I could magically
transport to any part of the universe, would microwave & gamma rays always
appear uniform in the background? Is there a part of the universe where I
would get different behaviour?

**Botany:**
Is a fig a flower or a fruit? If it is a fruit, why is it that it has a hole
at the base? To me it looks like a flower turned in on itself. How many times
a year does a fig tree fruit? Is it true that there are male & female
figs? Is it true that it is the female ones that are edible and the male ones
that are dry & not very sweet? If so, why? Explain why figs are so mysterious.
Are there any other examples with "fruits" that have
holes in them?

A/Prof. Derek Abbott

EEE Dept

University of Adelaide

SA 5005, AUSTRALIA.
dabbott@eleceng.adelaide.edu.au

Ph: +618-8303-5748

Fx: +618-8303-4360