In the past months, I have pointed out a few times that real numbers have no reason to exist in physics, if only because our measurement instruments don’t have an infinite precision and range. In general, this point, which I believe should make any physicist nervous, does not get much traction. Maybe most people simply believe that it does not matter, but I believe on the contrary that one can show how it has a significant impact on physics.
A Mathematician’s objection to Real Numbers
Why Real Numbers are a Joke
According to the status quo, the continuum is properly modelled by the `real numbers’.[...]
But here is a very important point: we are not obliged, in modern mathematics, to actually have a rule or algorithm that specifies the sequence In other words, `arbitrary’ sequences are allowed, as long as they have the Cauchy convergence property. This removes the obligation to specify concretely the objects which you are talking about. Sequences generated by algorithms can be specified by those algorithms, but what possibly could it mean to discuss a `sequence’ which is not generated by such a finite rule? Such an object would contain an `infinite amount’ of information, and there are no concrete examples of such things in the known universe. This is metaphysics masquerading as mathematics.
I reached this post through one of MarkCC’s posts, where he criticizes Prof. Wildberger’s arguments. Both points of view are worth a read (although both are a bit long).
In general, MarkCC is a no-nonsense guy, and I tend to side with him. But for this once, I disagree with his criticism. As I understand it, Prof. Wildberger main point is that real numbers are defined in a way that is rather fuzzy. If it does not allow us to prove uniqueness, maybe something is flawed in the logic leading to how we build them? Again, it all boils down to the problem of defining specific, non-trivial real numbers. There are too many of them, so the set of mathematical symbols is not sufficient: it only allows us to make a countable number of mathematical propositions, and the set of real numbers is provably not countable.
And real numbers are only the beginning…
We spent a day with the family at the 2007 edition of Fête de la science (a sort of national science day). We went to the Valrose campus in Nice, where many experiments and shows presented science in a way that was accessible to children and distracting to adults.
Among other things, there were:
- Remains of various hominids on display, showing the progressive evolution
- Robotics experiments
- Mechanics experiments, like gyroscopes, balances
- Optics experiments, with lasers going through various materials
- A replica of Sputnik
- Chemistry experiments, which attracted a number of people, notably with liquid nitrogen ice-cream (yum!)
- Experimental psychology
- Marine biology
- Genetics and genetic engineering
I had a long discussion with members of the zetetics observatory. I don’t know how to define zetetics precisely, but let’s say that it’s a form of scientific skepticism, in the good sense of the term. A lot of what they do is debunk pseudo-science or low-quality science.
So naturally I started asking questions about UFOs and how one could address, in a scientific way, something which is by construction difficult to catch and relies entirely on witness reports (with all the associated sociological effects). This was a very interesting discussion, and he pointed me to a book, available on-line (but in French) which apparently demolishes the work of the GEIPAN. I did not find the studies of the GEIPAN too convincing, so I’m glad to hear that there is a more scientific and systematic verification of what they did, and apparently, it is not pretty (I did not read the book yet, it’s only hear-say at that point).
Unfortunately, zetetics will also tend to dismiss witness reports, for a simple reason. Between various explanations, they will always prefer an explanation that matches known laws. It turns out that this algorithm tends to select the option: “witness (or someone along the reporting line) is lying or at least distorting the observation”. This option is always valid, it obeys a known law. But I think this introduces a kind of methodological bias. I don’t know how to eliminate that bias. Do you?
Update: I started reading the book in question, and I got a very bad overall feeling about it. It is exactly what I talked about: the primary argument is casting doubt about the validity of the testimonies. This has some value, of course, but pointing that the work of someone studying a phenomenon is sloppy is easier than figuring out a non-sloppy way to do it.
How to Teach Special Relativity is a famous article by John Bell where he advocates that the way we teach relativity does not give good results. He describes an experiment now known as Bell’s spaceship paradox (even if Bell did not invent it):
In Bell’s version of the thought experiment, two spaceships, which are initially at rest in some common inertial reference frame are connected by a taut string. At time zero in the common inertial frame, both spaceships start to accelerate, with a constant proper acceleration g as measured by an on-board accelerometer. Question: does the string break – i.e. does the distance between the two spaceships increase?
Considered a difficult problem
The correct answer is that the string does break, even if the spaceships appear to always be at the same distance from one another as seen from an observer who did not accelerate with the spaceships. Yet, according to Wikipedia:
Bell reported that he encountered much skepticism from “a distinguished experimentalist” when he presented the paradox. To attempt to resolve the dispute, an informal and non-systematic canvas was made of the CERN theory division. According to Bell, a “clear consensus” of the CERN theory division arrived at the answer that the string would not break.
In other words, this problem was considered hard by a majority of serious physicists at the time Bell raised the question in 1976. I would venture to say that this remains the case today, except that this particular paradox is probably well known now. But the teaching of special relativity has not changed. This is how we explain the paradox today. I have a lot of admiration for John Baez in general, his “blog” is even in the sidebar of this one. But with all due respect, the explanation of the paradox posted on his web site is utterly complicated (I know he gives credit to someone else for it, but by hosting it on his web site, I would say that he condones it).
It should be easy
This particular formulation of the paradox was not known to me until someone recently asked me if the string would break. Using my little technique, it took me less than one minute to have the correct answer, without looking it up, obviously, but also without any computation or complicated diagram. Here is the mental diagram I used (click to see it in high resolution):
On this diagram, time is represented horizontally, and the two space ships are represented by the green and red curves, which are identical but separated by a distance along the vertical “spatial” axis. The distance at rest is represented by the blue arrow. The distance as measured between the two ships after they started moving is measured by the green and red arrows. The distance as measured “from the ground” is along the vertical axis, and remains constant.
Remember the only trick is that a “cosine” contraction on this diagram corresponds to a dilatation in relativity and conversely. On the diagram, the red and green arrows are obviously shorter than the blue arrow. The contraction factor is the cosine of the angle between these arrows and the vertical (space) axis, which is the same as the angle between the red or green curve and the horizontal axis. Therefore, relativity predicts that the distance between the two ships, as seen from the ship, will increase. Specifically, it increases by a factor usually denoted “gamma” (but which I prefer writing as the cosine of an angle myself), which can also be seen as a hyperbolic cosine, and which plays in Minkowski geometry the exact same role as the cosine in the Euclidean diagram above. You can find the precise mathematical relationship here.
Consequently, the string will break.
Accelerated solids in relativity
Another interesting observation one can make from the diagram is that you cannot draw a straight line that is perpendicular to both curves. What is “space” for one ship is not just “space” for the other. You need to draw a curved line between the two rockets if you want to always be perpendicular to the local “time” direction. In other words, the “time” direction for the string is not constant along the way, so all parts are not moving at the same speed. Someone sitting anywhere on the string will see other parts of the string move relative to him. That’s another way to explain why the string will break.
You can easily verify that this problem exists for any kind of accelerated solid. All parts of an accelerated solid in special relativity do not move at the same speed.
My own puzzle
Here is the interesting other thing that I realized within this short moment of reflection: there is a way for the two ships to accelerate “identically” (for a suitable definition of identically which remains to be given) so that the string will not break. Can you find it?
C’est tout de même dur à avaler.
Comment voulez vous qu’on enseigne cela à des élèves de Terminale?
Jean-Claude Carrière, in Entretiens sur la multitude du monde with Thibault Damour
My favorite Ig Nobel prize this year is in Linguistics:
LINGUISTICS: Juan Manuel Toro, Josep B. Trobalon and Núria Sebastián-Gallés, of Universitat de Barcelona, for showing that rats sometimes cannot tell the difference between a person speaking Japanese backwards and a person speaking Dutch backwards.
A close second is in aviation:
AVIATION: Patricia V. Agostino, Santiago A. Plano and Diego A. Golombek of Universidad Nacional de Quilmes, Argentina, for their discovery that Viagra aids jetlag recovery in hamsters.
Le pélican est, avec le kangourou, le seul marsupial volant à avoir une poche ventrale sous le bec.
Having just shared a concern about how science is being taught, it may be useful to explain what I think would be a better way to teach it.
Here is one of the many ways you can use the layman’s intuition to explain quantum mechanics, for instance the famous double slit experiment:
Imagine a surface of water, like a swimming pool, where some apparatus creates waves. In the middle, you build a wall with two relatively small vertical holes that you can open or close at will. When a hole is open, the wave goes through, otherwise, it does not. If the hole is small enough, it will behave as a “point emitter” for the wave, i.e. the waves will appear to be circles centered on that point.
When the two holes are open, the waves they emit are not totally independent from one another. They are correlated, since they ultimately come from the same source. Consequently, they will interfere in a predictable way. There will be spots where the wave amplitude will appear to be twice as high, spots were the water amplitude will be almost flat. On the other hand, as soon as you close one of the holes, or as soon as you significantly disturb one of the waves, the interference pattern disappears.
You can think of the amplitude of the wave as a “probability of presence” of water molecules: if the wave is high, then it means that there is a higher probability that water molecules will be here, and ultimately, you find more water here. Conversely, if the wave is low, it means that there are less water molecules at that point, so the probability of finding water molecules there is lower.
To predict the interference pattern, i.e. to add the two heights of water, you cannot simply consider the average height of the wave, i.e. the average probability of presence over time. Instead, you need to consider whether the two waves are correlated or not. In our example, they are correlated. The displacement caused by one wave is not independent of the displacement caused by the other. For this reason, there are locations where the two waves always cancel, and other locations where they always add up.
In the case of a photon for example, this probability of presence is not caused by matter like for water waves. But the same general idea applies. The photons arrange themselves according to a probability wave. The most surprising thing is that the wave exists even when there is a single photon, and that even a single photon does not necessarily follow a straight line, but will be found according to this probability of presence. The straight line is a special case of probability distribution, not the most general case.
I may be wrong, but I believe that by explaining something like the above, you have captured the key elements of the experiment without using a single word of math, without asking the person to give up his/her common sense or intuition at any time, and more importantly, remaining pretty faithful to what the math/physics actually says. In other words, you have actually explained, not asked the other guy to have faith.
Ce qui se conçoit bien s’énonce clairement, et les mots pour le dire viennent aisément.
The first article is about the following preconception: Science fights second-handed ideas (they translated as “second-handed ideas” what I translated as “preconceptions”, the original French being “idées reçues”). It is an interesting discussion of how the ideal model of Descartes to isolate “what is certain” is seldom followed and, instead, normal sociology applies where dominant ideas must be fought in science to introduce new ideas.
Highly recommended reading. If you speak French, that is…
Scale relativity corner
I’ve been invited to participate in a new blog on scale relativity. I find the opportunity extremely interesting, but I strongly cautioned the original author about what appears to be blatant copyright infringement of Nottale’s work. The author is trying to address that point now, but since he asked Laurent Nottale for advice, he prefers to leave the site in the state it was in when the e-mail was sent. In any case, I find that Nottale’s text may be appropriate for a book, but not for on-line reading (even if some chapters can be downloaded freely).
A New Kind of Science – Disappointed
I’ve been somewhat disappointed by A New Kind of Science. OK, the main point is as fascinating as ever, that we can study mathematics and possibly physics using computer simulations and relatively simple programs. But 1200 pages on this topic? Give me a break! The overall style is extremely verbose. And while Wolfram advocates that his emphatic style adds clarity, I personally find it annoying to read “I’m a genius” every other page. It’s the first science book in a long time that I ended up speed reading at about 2 seconds per page. I’ll probably return to this book later, but for the moment, I’m sort of fed up with it.
Thibault Damour explaining science to the layman
I am also currently reading “Entretiens sur la Multitude du Monde” by Thibault Damour and Jean-Claude Carrière. It’s a dialogue between a physicist and a “layman” (a cineast and author), which I find very interesting as a case study of difficult communication. Since I know what Damour refers to, I’m somewhat puzzled at what appears to be the understanding Carrière gets after the discussion. Often, I find myself thinking: “I wouldn’t have explained it that way”.
Carrière expresses my unease very well at one point. After Damour starts making references to rather complex topics (the book addresses general relatitvity, quantum mechanics, etc), Carrière observes that at that point, he has no choice but to “believe”, instead of actually understanding. And he points out that when this happens, science becomes indistinguishable from religion. The term “layman” which scientists often use to characterize non-scientists is not random.
I am especially puzzled by the fact that Damour simply asked Carrière to “give up” about general relativity or even the notion of curved space, when it is actually so easy to explain what it is about by using Earth as an example. You know the “layman” won’t understand what a stress-energy tensor is, fine. So explain general relativity without it!
“Special relativity for Dummies” back on-line
On a related topic, a recent thread entitled “Relativity without tears” on sci.physics.research practically brought tears to my eyes. How is it possible, 100 years after Einstein, that we still have folks who can do math but totally fail to grasp special relativity? This and a recent comment on this blog suggesting that Einstein was wrong prompted me to dig up in dusty corners of my hard drive an old page I had written 8 years ago in an attempt to make special relativity intuitive to the layman.
I know that this explanation usually works with any kid aged 10 or more. I wish this is how special relativity was taught, instead of the constant blabbering about “inertial reference frames” and “forget all your intuitions”. Crap!
La Cité des Sciences – On-line lectures
Another link mostly for French-speaking readers: the Cité des Sciences has a series of on-line videos about a large number of topics in science. They are generally extremely interesting. I came there after hearing about a debate between Thibault Damour and Lee Smolin. I recently listened to a talk on how to interpret quantum mechanics, which I found interesting, but which also left me the same bitter taste of “the layman will never understand anything about quantum mechanics if we keep presenting it this way”.
The fact that this guy worked so much on the interaction between human nervous systems and machines is a reason to hope that this is almost real. A lot of progress seems to have been made since last time I wrote about this topic.
An interview of A.C. Clarke reminded me of a topic I wanted to write about for a long time: why do we need to gain the ability to go into deep space?
50 years in space
During the first 50 years of space exploration, we have sent many satellites and, more importantly, developed a whole economy around space flight. Recently, private companies have entered the fray. Sir Clarke mentions the Google Lunar X-Prize foundation as one of our hopes to get back to the moon.
The current state of our space technology is largely due to many historical accidents, including World War II and the following competition between the Soviets and the Americans for the best long-range ballistic missiles. In the interview, Sir Clarke recalls Bainbridge’s observation that we were not necessarily due for space travel yet:
As William Sims Bainbridge pointed out in his 1976 book, The Spaceflight Revolution: A Sociological Study, space travel is a technological mutation that should not really have arrived until the 21st century. But thanks to the ambition and genius of von Braun and Sergei Korolev, and their influence upon individuals as disparate as Kennedy and Khrushchev, the Moon—like the South Pole—was reached half a century ahead of time.
Despite what may have been an early start, or maybe because of it, we only explored the immediate neighborhood of Earth. The initial rate of progress (and, in retrospect, pretty wild risk taking) led many science-fiction writers to confidently predict the colonization of the solar system by 2100. The chances of this actually happening now seem a bit more remote than in the 1970s. While we went to the Moon and back, we did not establish any permanent base there.
Scarcity of resources? Not in the solar system
The first step beyond that is to do for the solar system what happened for Earth orbit, which is developing some kind of successful economic model of space. One requirement for this to happen is to lower the cost of space launches, and our best bet for that so far is some kind of space elevator.
Such a technology would also lower the cost of satellite launches, but to me, that’s not the main point, and it is also not all-good thing knowing how much space junk there already is. I do not entirely share Stephan Scherer’s optimism:
50 years after Sputnik, Space below the geostationary orbit has become quite crowded. Fortunately, it is still wide enough
The main point of solar system exploration, as far as humanity is concerned, is really to find useful stuff there, like hydrogen, water, and who knows what else would become useful. Resources that are scarce or, at least, limited on Earth may be available in vast quantities out there, that much is certain. What is not certain is that we could ever lower the cost of exploiting this bounty to a point where it would make sense at all.
But even the solar system is only the beginning…
Deep space vs. Local space
Given our track record in the past 50 years, it may seem premature to talk or think about deep space exploration. First of all, let me explain that by “deep space”, I am referring to space beyond the solar system, warp drives and this kind of far-fetched stuff.
Why do we need that? Well, it’s simply a matter of not taking chances with the survival of our species. On a cosmic scale, there are just many events that could wipe out the entire human race. And, unlike a few, I do not consider this a good thing.
Many of these so-called extinction-level events (ELE) could impact the whole planet. The most well-known type of ELE is an asteroid impact, something so widely known that it even received the Hollywood treatment.
The problem is that it’s not just the whole planet. Some large cosmic events could easily impact the whole solar system. Many cataclysmic scenarios have been imagined, like a gamma ray burst a little bit too close, and in some cases the Earth’s magnetosphere might be insufficient to protect us, while the rest of the solar system would become even more hostile than it currently is.
Are we alone?
Another reason that is always behind everybody’s mind when we talk about space travel is: are we alone in the universe? In that respect, I find Sir Clarke’s comment in the interview highly illogical:
I have always believed in life elsewhere in the universe (though I don’t agree that some are visiting us secretively in flying saucers).
To me, it is quite illogical to believe in something while at the same time dismissing the only “evidence” there is about it, irrespective of how weak that evidence is. There is simply no better reason to believe in intelligent life out there than witness reports that seem hard to explain otherwise.
Sir Clarke’s position is about as logical as believing that one can win the Lottery, but that any testimony of alleged winners must be a big fraud.
Standards of skepticism are too low
Don’t get me wrong: it is possible, even likely, that the majority of this evidence is crap, and one cannot even rule out that all of it is fabricated. But in reality, most “debunking” sites are not more convincing than the believer’s sites, and don’t hold themselves to any stricter standard of analysis. It’s belief against belief, generally with a strong dose of the other guy is stupid ad-hominem rhetoric.
The real reason most scientists don’t believe in UFOs of extra-solar origin is that we have no science that would allow it. Einstein proved that there is an absolute speed limit in the universe, the speed of light, and we don’t know how to go over that speed limit. As far as we know, it’s impossible for any material object to even reach the speed of light. If Einstein is right, then there can be no extraterrestrials from other solar systems in our backyard, it’s really that simple. Interstellar travel based on what we know will never allow a human voyage to a remote star. And for that same reason, it also precludes the visit of biologically similar organisms.
How to cross the light barrier?
What if Einstein was wrong? It is not a stupid question. A few serious scientists have given it a try. It takes some humility to admit that there are still a few things we don’t know about the universe. Lord Kelvin’s famous quote is a mistake not to be repeated again:
There is nothing new to be discovered in physics now, All that remains is more and more precise measurement.
This does not mean we can accept anything. Einstein cannot be too wrong. His theory has been verified over and over again. So the solution is not to be searched by trying to add more fuel to rockets so that they break the light barrier. That simply won’t work. Clearly, we need some kind of breakthrough, something more subtle than brute force.
Maybe a new understanding of the structure of space-time would help, but based on my personal experience, even that may not be sufficient. As far as I can tell, my own “demolition” of space-time only made Einstein’s limit more solid, instead of weakening it. One does not need space-time if all the properties we attribute to this “background” can instead be shown to be properties of electromagnetic interactions. No space-time, great! Properties of electromagnetic interactions, not so great: the speed of light limit comes back with a vengeance. How can you ever hope to go faster than light if time, space, and the relations between them are actually defined by light itself? (For the most curious minds, it’s spelled out in section 3.2 of the article).
On the other hand, looking for a breakthrough does not mean that current space research is useless, on the contrary. Even if today’s automobiles rely on a propulsion system that would have seemed very “improbable” to the average middle-age horsecar driver, a lot of the technology developed back then remained useful, including: the wheel, the seat, roads, maps, and so on. It is very likely that even if we invent some new and fancy way to cross interstellar distances, we will still need space suits or protection against radiation.
Can it be done at all?
Based on whatever little evidence we have, I would give it a pretty good chances, precisely because in-depth analysis of the UFO phenomenon cannot really explain some cases other than using some hypothetical intelligence driving some apparently mechanical object with a behavior that, as Sir Clarke would say, is so advanced as to be indistinguishable from magic.
Let’s now assume that there is a good chance we can do it. The question in that case is: how? And I regret to say that my own hobby research did not bring me any closer to answering that question.
Someone recently shared with me an interesting quote from Immanuel Kant:
Out of the crooked timber of humanity, no straight thing was ever made
(German original: Aus so krummem Holze, als woraus der Mensch gemacht ist, kann nichts ganz Gerades gezimmert werden)
It looks like robots are not the solution to this particular problem:
A particular variant of Murphy’s law states that a demonstration’s probability of failure is directly proportional to the size of the audience. Since such a probability quickly exceeds one (unless you use super-advanced mathematics, that is), failure almost never fails:
I wonder if the presence of a camcorder measurably improves the rate of failure…
This cruel treatment to robots reminds me of one of the greatest robotic quotes ever:
The first ten million years were the worst, and the second ten million years, they were the worst too. The third ten million I didn’t enjoy at all. After that I went into a bit of a decline.