Muons and Relativity

Fifty years ago this year, David H. Frisch and James H. Smith published an ingenious confirmation of special relativity (see D. H. Frisch and J. H. Smith, “Measurement of the Relativistic Time Dilation Using µ-Mesons,” American Journal of Physics, 31 (5): 342–355, 1963).

Image from an underground muon detector pointed at the moon. Since the moon filters out a small fraction of cosmic rays, it leaves a visible shadow.

Frisch and Smith studied muons, which are unstable electron-like particles. Muons have a mass roughly 200 times that of the electron, but decay moderately rapidly, with a half-life of 2.2 microseconds. Muons are formed in the upper atmosphere, when high-energy cosmic rays collide with atomic nuclei in the air. This raises the question: why don’t they all decay before they reach the ground? Even travelling at 99.5% of the speed of light, the time taken to reach the ground is many half-lives.

Sources of radiation exposure for a typical person in the United States. Cosmic ray exposure includes muons.

Frisch and Smith counted muons at an altitude of 1910 metres on Mt. Washington, New Hampshire, as well as at (close to) sea level. An average of 563 muons per hour arrived at the top of the mountain, and an average of 408 at sea level. The muons took 6.4 microseconds to travel the extra distance, but very few decayed during that time. Frisch and Smith calculated that this reflected a time dilation factor of 8.8±0.8. In other words, from the muons’ point of view, the 1910-metre journey took about one-ninth the time: less than a microsecond. This agreed very nicely with the relativistic prediction of time dilation by 8.4±2 times, thanks to some very elegant experimental technique (the large theoretical uncertainty reflects the variation in muon speeds). Well done, guys!

Mt. Washington (photo: “wwoods”)

The Elements: a book review

The Elements by Theodore Gray

Some time ago, I purchased the photobook The Elements: A Visual Exploration of Every Known Atom in the Universe by Theodore Gray of periodictable.com. A paperback edition has now also become available. Gray’s book is informative, brilliantly designed, and beautifully illustrated, displaying a real passion for the elements. Here are two sample spreads (click to zoom):

Everyone interested in science should probably have this one on the coffee table. Owners of the book can find some online resources here.

The Elements by Theodore Gray: 5 stars

Fly or swim?

Uria lomvia, the Thick-billed Murre

A recent study of the wing-propelled diving bird Uria lomvia shows that it has the highest recorded flight cost among vertebrates. That is, this bird (which consumes 146 watts/kg while flying) is so well-adapted to using its wings for diving that its flying suffers. In other words, the Thick-billed Murre is half-way to becoming a penguin.

Spheniscus humboldti (the Humboldt Penguin) swimming (photo: “Joccay”)

The top ten new species for 2013

According to news reports, scientists at the International Institute for Species Exploration at Arizona State University have produced a list of the top 10 new species discovered during 2012:

1. Viola lilliputana, the tiny Lilliputian Violet

2. Chondrocladia lyra, the carnivorous Harp Sponge (see the video above from MBARI)

3. Cercopithecus lomamiensis, the Lesula Monkey (photo above from this paper)

4. Sibon noalamina, a snail-eating snake

5. Ochroconis anomala, a fungus found in the Lascaux caves

6. Paedophryne amauensis, a frog which is the world’s smallest vertebrate – the name of this frog was miss-spelled in news reports: the name comes from the village of Amau (photo above from this paper)

7. Eugenia petrikensis, an endangered shrub

8. Lucihormetica luckae, a cockroach that glows in the dark

9. Semachrysa jade, a lacewing discovered, in a triumph of citizen science, by Malaysian photographer Guek Hock Ping (who took the photograph above)

10. Juracimbrophlebia ginkgofolia, a fossil hangingfly

Hyperion’s chaotic tumble

The Saturnian moon Hyperion, seen above in an image from Cassini, is known for its chaotic, tumbling rotation. The video below gives a sense of what this cosmic potato looks like, using the Celestia software (running at 100,000 times real time). Thanks to YouTube user “Beth F” for the video.

Lorenz and Chaos

In March 1963, Edward Lorenz founded chaos theory with his paper “Deterministic Nonperiodic Flow” (Journal of the Atmospheric Sciences, 20, 130–141). In honour of the anniversary, Physics Today has a survey paper, “Chaos at fifty.”

Lorenz’s classic paper described the Lorenz Attractor. The image above (by “XaosBits”) shows two orbits of the attractor. Microscopically different starting points give totally different trajectories. The image below (by “Wikimol”) gives another view of the attractor.

Lorenz’s paper was followed by the discovery of the Rössler attractor in 1976, and then what can only be called an avalanche of further work. Prior to Lorenz, signs of chaos had been seen in the logistic map, but Lorenz showed that the world itself (weather, specifically) was chaotic. Chaos also shows up in orbital motion, in turbulence, and in mixing. The image below (by “Eclipse.sx”) shows the result of simulating a magnetic pendulum.

Lorenz went on to coin the term “butterfly effect.” This effect, more formally known as “sensitivity to initial conditions,” is one of the key marks of chaos. Another is “topological mixing,” which means that the system will evolve over time so that any given region of its phase space will eventually overlap with any other given region. The image below (public domain) shows a circular region (blue) progressively being mapped (purple, pink, red, orange, yellow, and further iterations not shown) to a set of points which leaves no blank areas in the phase space.

Lorenz died on 16 April, 2008, at the age of 90. Physics Today also published an obituary when Lorenz died.

Alternatives to mathematical Platonism (2)

In my previous two posts, I outlined the Platonist view of mathematics, and the empiricist alternative. There are also two other alternatives:

Formalism

The truths of mathematics appear to be different in nature from the truths of physics. Formalism accepts this, but suggests that the nature of mathematics is inherently cultural. Different branches of mathematics are essentially just games with symbols and arbitrary rules – games that don’t have any particular meaning. Mathematicians simply work within the chosen rules. However, apart from the problem of the “unreasonable effectiveness of mathematics,” the idea that these rules are chosen arbitrarily runs counter to the experience of most mathematicians. When the concept of “number” was extended to include the imaginary numbers, for example, consistency with the existing rules meant that there was very little choice about how imaginary numbers behaved. In the words of mathematician Jacques Hadamard: “We speak of invention: it would be more correct to speak of discovery… Although the truth is not yet known to us, it pre-exists and inescapably imposes on us the path we must follow under penalty of going astray” (from the introduction to The Psychology of Invention in the Mathematical Field).

Many officially formalist mathematicians are Platonists at heart. Jean Dieudonné once wrote with refreshing honesty: “On foundations we believe in the reality of mathematics, but of course when philosophers attack us with their paradoxes we rush to hide behind formalism and say: ‘Mathematics is just a combination of meaningless symbols,’ and then we bring out Chapters 1 and 2 on set theory. Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working on something real. This sensation is probably an illusion, but is very convenient.” (from “The Work of Nicholas Bourbaki,” American Mathematical Monthly, 77(2), Feb 1970, p. 134–145).

Logicism

Most mathematicians feel that the truths of mathematics are indeed in a different category from the truths of physics – that the truths of mathematics in a sense come first. Logicism is a way of rescuing this aspect of Platonism while avoiding the more mystical aspects. The basis for logicism is that logic also comes before physics – all sciences assume logical thought as a starting point. Logical truths “exist” in some sense, and logicists assume that there are no philosophical difficulties about this kind of existence. In other words, a mystical Platonic world is not needed to explain logic. Consequently, if we can provide a foundation for mathematics in terms of pure logic, we can retain all the benefits of Platonism without any of the problems.

In logicism, numbers are defined as being particular kinds of sets. Logicism began with Gottlob Frege, who published two volumes of his Die Grundgesetze der Arithmetik in 1893 and 1903. Sadly for Frege, his fellow mathematician and philosopher Bertrand Russell found a major flaw – now known as “Russell’s paradox” – in the work, just before the second volume was published. With Alfred North Whitehead, Russell was able to repair the flaw, in a three-volume work called Principia Mathematica (published in 1910, 1912, and 1913).

After 379 pages, Whitehead and Russell are well on the way to proving that 1 + 1 = 2.

The logicist programme, however, is not free of problems. First, it is extremely complex. It took Whitehead and Russell hundreds of pages of complicated logic to prove that 1 + 1 = 2. Normally, we try to explain complex things in terms of simple ones. It seems a little perverse to give such a complicated explanation of numerical facts that we understood in kindergarten. And, by including set theory as part of the basis, it isn’t really “pure logic” any more.

Second, there is more than one way of defining numbers as sets, and none of them is obviously “right.” This has led to the suggestion that sets are “what numbers could not be” (the title of an article by Paul Benacerraf in The Philosophical Review, 74, Jan 1965, pp. 47–73), and that numbers must be fundamentally different in nature from sets – if not Platonic objects satisfying certain axioms, then something else which exists in a non-contingent way.

Third, it is unclear whether logicism has actually gained anything. The starting assumption was that logic was simple and obvious, raising no philosophical problems. But if all of mathematics is hidden deep inside the structure of logic, then perhaps logic is not as simple as it first seemed. Mathematics and logic may in fact be different aspects of the same thing, but this may not make the fundamental questions about mathematical existence go away.

Personally, I still see Platonism as the best answer. How about you?

Alternatives to mathematical Platonism (1)

In my last post, I outlined the view of mathematical Platonism taken by Roger Penrose and other mathematicians. Briefly, in the words of Joel Spencer, “Mathematics is there. It’s beautiful. It’s this jewel we uncover” (quoted in The Man Who Loved Only Numbers, p 27).

Uncovering hidden jewels (James Tissot).

However, some modern mathematicians feel that the time for such ideas is past. In the words of Brian Davies, “It is about time that we … ditched the last remnant of this ancient religion” (in his article “Let Platonism die,” European Mathematical Society Newsletter, June 2007).

One alternative commonly presented is Empiricism. Santa Claus and the Tooth Fairy do not exist, and neither do infinite decimals, or perfect circles, or the set of all natural numbers. Only the physical universe exists. In the words of astronomer Carl Sagan, “The Cosmos is all that is or ever was or ever will be” (the opening sentence of his book Cosmos). Our only truly certain knowledge is physics, and mathematics is in fact a branch of physics. When we say that 2 + 2 = 4, we are not talking about a relationship between Platonic number-objects. Instead, what we mean – and all that we mean – is the empirical truth that two atoms plus two atoms gives four atoms, and likewise for stars, rocks, or people. “Four” is not a noun, it’s an adjective.

The empiricist point of view seems to solve the mystery of the “unreasonable effectiveness of mathematics.” Mathematics is just part of physics, and it isn’t surprising that different branches of physics agree with each other. Empiricism also avoids the need to postulate a “soul” or some other mechanism for peering into an ethereal Platonic world.

Conic sections in theory and practice.

There are two problems with the empiricist point of view, however. First, it isn’t true to the history of mathematics. Galileo used parabolas to describe the motion of falling objects, but the ancient Greeks had originally described parabolas in a quite different context, that of conic sections. Similarly, imaginary numbers were originally discussed without the slightest idea that centuries later they would become a fundamental part of quantum theory. There’s still a mystery there: why should mathematics from one context work so well in another?

The second problem is that scientific truths are contingent. We could, for example, live in a world where plants were purple. The fundamental forces of physics could be different from the way they are (although that could imply a lifeless universe). We can even imagine a universe containing no matter at all, only empty space. The laws of mathematics, however, could not have been different – they are necessary. It is impossible to imagine – at least, to imagine consistently – a universe in which 2 + 2 = 5. Even God cannot make 2 + 2 = 5. This indicates that there must be more to mathematical truth than just physics. Statements about mathematics, such as “2 + 2 = 4” are in a different category to statements about the universe, such as “light travels at 299,792,458 metres per second.” But if that is the case, then mathematical truth must in some sense lie outside the universe – which brings us back to Platonism.

Three Worlds

Roger Penrose, in his book Shadows of the Mind, outlines an idea adapted from Karl Popper – that there are “three worlds.” The physical universe needs no explanation, except perhaps to Bishop Berkeley, while the subjective world of our own conscious perceptions is one we each know well. The third world is the Platonic world of mathematical objects.

Penrose says of the third world: “What right do we have to say that the Platonic world is actually a ‘world,’ that can ‘exist’ in the same kind of sense in which the other two worlds exist? It may well seem to the reader to be just a rag-bag of abstract concepts that mathematicians have come up with from time to time. Yet its existence rests on the profound, timeless, and universal nature of these concepts, and on the fact that their laws are independent of those who discover them. The rag-bag – if indeed that is what it is – was not of our creation. The natural numbers were there before there were human beings, or indeed any other creature here on earth, and they will remain after all life has perished.” (Shadows of the Mind, p. 413)

Edward Everett, whose dedication speech at Gettysburg was so famously upstaged by Abraham Lincoln, put it more poetically: “In the pure mathematics we contemplate absolute truths, which existed in the Divine Mind before the morning stars sang together, and which will continue to exist there, when the last of their radiant host shall have fallen from heaven.” G. H. Hardy was ambivalent about the Divine, but like most mathematicians he believed “that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations.”

This trilogy of worlds raises some questions, of course. The first is what Eugene Wigner called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” As William Newton-Smith asks, “if mathematics is about this independently existing reality, how come it is useful for dealing with the world?” Why does the world follow the dictates of eternal Reason? Or, as Einstein put it, “how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?”

The second question is the mind-body problem. How are we conscious of the universe, and how do our decisions to act affect it? Does even our perception have strange quantum effects?

Finally, how do we become aware of the Platonic world? Elsewhere, Penrose says “When one ‘sees’ a mathematical truth, one’s consciousness breaks through into this world of ideas, and makes direct contact with it… When mathematicians communicate, this is made possible by each one having a direct route to truth” (The Emperor’s New Mind, p. 554). But what exactly does that mean? Does one’s soul go on some kind of “spirit journey”?

Doubling the square

Plato, in the story of Socrates and Meno’s slave, tells how an uneducated slave is prompted to discover how to double a square. Plato saw this as evidence of memory from a past life, but it provides an example of mathematical intuition that all (successful) students of mathematics will recognise. As Saint Augustine said, “The man who knows them [mathematical lines] does so without any cogitation of physical objects whatever, but intuits them within himself.” Yet Plato’s (and Augustine’s) belief in such an intuitive soul makes the mind-body problem more acute. How do the three worlds tie together? It seems a mystery.

Three Worlds by M. C. Escher

Theatres of Glass: a book review

Theatres of Glass by Rebecca Stott (2003)

I recently read Theatres of Glass: The Woman who Brought the Sea to the City by Rebecca Stott. The book tells the story of Anna Thynne, wife of the Reverend Lord John Thynne, who was Sub-Dean of Westminster Abbey from 1831 to 1881.

Anna with her daughters Selina and Emily

Beginning in 1846, when she took her children on a holiday to Devonshire, Anna Thynne developed the marine aquarium (realising that constant aeration of the water was required) and studied stony corals such as Caryophyllia smithii, publishing an article with the deeply religious zoologist Philip Gosse.

The Devonshire Cup Coral, Caryophyllia smithii (photo: National Museums Northern Ireland)

Stott quotes Tennyson’s The Princess as an indication of the Victorian mania for collection:

“And me that morning Walter showed the house,
Greek, set with busts: from vases in the hall
Flowers of all heavens, and lovelier than their names,
Grew side by side; and on the pavement lay
Carved stones of the Abbey-ruin in the park,
Huge Ammonites, and the first bones of Time;
And on the tables every clime and age
Jumbled together; celts and calumets,
Claymore and snowshoe, toys in lava, fans
Of sandal, amber, ancient rosaries,
Laborious orient ivory sphere in sphere,
The cursed Malayan crease, and battle-clubs
From the isles of palm: and higher on the walls,
Betwixt the monstrous horns of elk and deer,
His own forefathers’ arms and armour hung.”

Indeed, as Stott explains, Anna Thynne’s work (together with the book The Aquarium: an unveiling of the wonders of the deep sea by Gosse) helped extend that mania to aquaria as well. My understanding of the chronology is this:

1806: Anna born (April 1)
1824: Anna marries Lord John Thynne (age 17, almost 18)
1841: Ward experiments with freshwater aquaria
1846: Anna begins her marine aquarium (age 40)
1849: Anna moves to Tenby in Pembrokeshire
1850: Anna conducts detailed investigations of the “Madrepores” in her aquarium
1850: Warrington experiments with freshwater aquaria
1852: Anna abandons her aquarium and moves to Hawnes Park, Bedfordshire
1852: Warrington experiments with marine aquaria; the London Zoo establishes an aquarium
1854: Gosse publishes The Aquarium
1856: Second edition of The Aquarium
1859: Anna (age 53) and Gosse publish “On the increase of Madrepores” in The Annals and Magazine of Natural History
1866: Anna dies (age 60)

An illustration from Gosse’s book, which appears on the dust cover of Stott’s

Overall, a very enjoyable book, about a forgotten heroine of science. I didn’t quite feel I’d gotten inside Anna’s head, but that is probably because the limited range of source material often forces Stott to speculate. Anna’s role is weakened a little by the fact that Gosse was unaware of her when he wrote his book in 1854, although he lists her as one of three pioneers in the second edition (and quotes the sentence “The individual to whom is due the merit of having introduced marine vivaria into London is Mrs. Thynne,” which justifies the carefully chosen subtitle of Stott’s book).

Also, I was perhaps subconsciously expecting some of the flavour of Barchester Towers, in a book about the wife of a Sub-Dean. No doubt that is the wrong way to think about Anna Thynne, but Stott does not give a strong alternative. I can’t help wish that more source material existed. I would have liked some colour plates in the book too.

Theatres of Glass by Rebecca Stott: 3.5 stars