The World of Social / Scientific Imponderables 1:
Imponderables: The Solution to the Mysteries of Everyday Life – David Feldman
Life is filled with imponderables. The older you get, the more imponderables you discover. Collecting “imponderables” or interesting unanswered questions has never been one of my hobbies. Still, Imponderables aroused my interest after reading Waheed Ogunjobi’s micro book, “Why did the Chicken Cross the Road?”. I decided to research for quotable Imponderables to encourage others to think creatively and identify deep questions. The questions posed are to encourage interdisciplinary thinking for students & scholars. The most exciting scientific problems in the century following 2001 will require a multidisciplinary approach. The under-listed imponderables were collected and written by Prof. Derek Abbott, EEE Dept, University of Adelaide, SA 5005, Australia. He posted them on his website to encourage students to think deeply. Please read them and answer the questions if you can.
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?
Mathematics:
The famous Hungarian mathematician, Paul Erdos, did not rest until he found a short and elegant proof of a problem. Whenever he achieved this, he would call it “book proof.” However, perhaps it is futile to search for short proofs in all cases, as we know (e.g., 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? If the proof is very long, can we 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 randomly between 1 and n, where n is large. The probability that this number is divisible by 3 is 1/3, by four is 1/4, by five 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 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 backward and find what number system we would need for the 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 the sequence itself, then the sequence has redundancy. If you cannot compress the sequence, then it contains complete 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 I have to write down an algorithm for pi and remove the first hundred digits. But to you, the sequence looks 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 genuinely helpful, as it seems to depend on one’s foreknowledge – in other words, ignorance of the data can affect your viewpoint.
Human Psychology:
Pathologically hypersensitive people are sensitive to other people’s feelings about some things, which is understandable. However, notice that they can be very insensitive to other people on issues that do not happen to worry them specifically. Is there a name for this phenomenon and a hypothesis to explain it?
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, then proceeds 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 an 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 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. We often all fall into the trap of getting stuck with a new problem. Does this inability to make that leap of judgment have the exact origin of the ego lacuna? Do psychoanalysts have a word for ego lacuna in these other contexts?
Molecular biology:
Mitochondrial (mtDNA) is the part of the DNA transferred from the female parent to the child. The textbooks say that mitochondria do not divide, so what happens in the case of identical twins?
Information theory:
Imagine you are an alien from a different world – so different that even the building blocks of life are entirely different on your planet. Now say that you land on planet Earth. You find three vast sheets of paper:
- One containing English text
- One listing a DNA sequence
- One listing a computer program
You have no idea which, but you recognize you are looking at ordered bits of information – three bizarre 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:
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 genuinely 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.
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 a black hole, your temperature will increase due to Hawking radiation. If you accelerate 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 than the Davies-Unruh formula. So, we can imagine a thought experiment where we can measure this difference and tell if the elevator is accelerating. Does this violate Einstein’s principle of equivalence? Explain.
Physics: Does water 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 every time! Perhaps you need a private jet with a full bath?
Physics: Why does the sun & moon appear more prominent at the horizon than at their highest points in the sky? Can refraction account for all that difference? Has anyone proved it? Is the ratio of the biggest to smallest moon the same size as that of the sun? How can we account for those observed ratios? Why are their colors more yellow-orange, nearer the horizon? At a particular time of the year, the moon remarkably appears relatively 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?
Physics: Here is an exciting experiment. Go into a dark room and pull open the seal of a self-adhesive envelope. Something terrific 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?
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.” Can we explain this mechanism more deeply than just citing the law of energy conservation?
