Are real numbers real?

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.

In mathematics, Skolem’s paradox for instance can be used to invalidate some far-fetched claims. But I was not considering that real numbers in mathematics were otherwise a problem.

A Mathematician’s objection to Real Numbers

Well, it looks like at least one mathematician is also making similar points, but for mathematics instead of physics. Prof. Wildberger writes:

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.

Contrarian view

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…


Fête de la science 2007

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

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

And the winner is…

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.

Meanwhile, a more serious prize was awarded for really useful physics.

Le pélican est, avec le kangourou, le seul marsupial volant à avoir une poche ventrale sous le bec.

Quantum mechanics for dummies

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.

News of news

La Recherche

La Recherche just issued a catalog of preconceptions in science. It is especially interesting in light of the recent controversy surrounding books by Lee Smolin and Peter Woit.

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”.