by Melvyn Bragg
LESLIE GOLDBERG: However, if we look at the history of mathematics, some problems take a lot longer to solve. For example, if you look at Fermat’s Theorem, the famous thing about the margin, that was 1637. When was it solved by Andrew Wiles? 1994.
Sometimes in mathematics, problems stay around for a very long time but that doesn’t mean they cannot be solved; there is always the chance that they might be solved.
Melvyn was keen to know what the practical implications would be if it transpired that P is equal to NP – namely everything that can be checked can also be found. Colva Roney-Dougal was hedging her bets on this, saying it depended on how P and NP were found to be equal, but we might imagine a new fast algorithm that would solve all of these problems. She offered one bad consequence and then a good one.
COLVA RONEY-DOUGAL: The one bad thing – we’ve already mentioned this: all internet security breaks instantly …
MELVYN BRAGG: Why?
COLVA RONEY-DOUGAL: … because we can now factorise numbers that are products of two very big primes and that’s the basis for all of our current cryptography, so we would need a different way of exchanging information secretly. The good thing is that almost every single thing you buy becomes cheaper, because some of the problems that are in NP that are difficult to deal with at the moment are scheduling tasks in factories, transporting goods between factories.
Tim Gowers said he thought it was incredibly unlikely that P would equal NP. Besides, the problem of finding proof, which is what mathematicians do for a living, is in NP. So, if P turned out to equal NP with the qualifications Colva Roney-Dougal mentioned, then mathematicians would be out of a job, because computers could very, very easily do all the work that mathematicians at the moment sweat away doing in their offices. As she put it, if P were equal to NP, you could claim all the Clay Mathematics Prizes because you would be able to run the code and generate short proofs for all of the remaining ones. One famous computer scientist, Donald Knuth, has said that he thinks it is so unlikely that P is equal to NP that, to the first person who shows this, he will award the prize of one live turkey in addition to the $1 million. Conversely, if we were to find conclusively that P is not equal to NP, Leslie Goldberg said that would be a good first step in our understanding of the difficulty of problems. The uncertainty may be solved, and the prize claimed, by showing that P is not equal to NP, rather than showing that they are the same.
In the studio immediately afterwards, Melvyn asked what else was on his guests’ minds. Colva Roney-Dougal wanted to clarify that the word ‘algorithm’ came from the mathematician al-Khwarizmi, as he was doing equation solving and was giving step-by-step instructions for solving those sets of equations. Louise Goldberg mentioned that there is a whole field of study about how difficult NP-complete problems are to solve not fully but approximately. Melvyn checked that there was no sense in which mathematicians were going to give up trying to solve the P vs NP problem, and was reassured that the search would continue.
TIM GOWERS: Some of the experts say, ‘Well, I can’t afford to spend too much of my time on this problem because it’s such a hard problem and I don’t want to end up with no publications whatsoever.’ Most theoretical computer scientists ration the time they spend on the problem and spend some of their time working on slightly less ambitious …
MELVYN BRAGG: I’d have thought so, like [how] I finish a novel instead of sitting trying to write the greatest novel ever written and never writing anything.
TIM GOWERS: It’s a little bit like that.
MELVYN BRAGG: Ah, here’s Simon. I think a double brandy and tea!
ABSOLUTE ZERO
The coldest natural temperature ever known on earth was recorded in July 1983 at a Soviet research base in the Antarctic. At quarter to three in the morning, the thermometer registered -89.2°C. Beyond our atmosphere, it can be dramatically colder even than this. Astronomers believe that interstellar space has a temperature of around -270°C. But the coldest temperatures yet known, colder even than space, have been created artificially in the laboratory. Scientists have been creeping ever closer to the lowest possible temperature, known as absolute zero. The idea that temperature had a lower limit was first suggested in the seventeenth century. The race for ever-colder temperatures began 200 years later and has resulted in some of the strangest, most important and useful discoveries of modern science. But, although physicists can get within a billionth of a degree of absolute zero, they will never quite get there.
A worker cold-shock tests copper vessels used in producing liquid oxygen.
With Melvyn to discuss absolute zero were: Simon Schaffer, professor of the history of science at the University of Cambridge; Stephen Blundell, professor of physics and fellow of Mansfield College, Oxford; and Nicola Wilkin, professor in the school of physics and astronomy and director of education at the College of Engineering and Physical Sciences, University of Birmingham.
Since we were looking at temperature, Melvyn started by asking Simon Schaffer what the early ideas were about this and what ‘temperature’ was. The Greeks lived in a culture in which all energy sources were either human, mechanical or thermal and understanding heat and temperature was fundamental to their cosmology. For Aristotle, for example, temperature, in the sense of possessing heat, was an essential quality of all bodies and, in his natural philosophy, he had proposed the possibility of what came to be called, in Latin, the primum frigidum, the fundamental cold.
SIMON SCHAFFER: It was even supposed by many scholastics, followers of Aristotle, that cold was, as it were, a substance equivalent to heat. When substances became colder it was because they were absorbing cold and when they became hotter it was because they were absorbing heat.
This idea was challenged in the seventeenth century with the development of new technologies such as the thermoscope with a sealed bulb full of air, plunged into water, where the height of the water would measure, in some sense, the temperature of the bodies surrounding the bulb. Experimenters such as Robert Boyle were satisfied that there was no such thing as the primum frigidum, that there was no fundamentally cold substance. Boyle thought, following Francis Bacon, that cold was a deprivation of motion and heat was a form of motion. That made possible the idea of an absolute limit to cold.
This was explored further by a Frenchman, Guillaume Amontons (1663–1705), who was an entrepreneur who wanted to drive a pump by the expansive force of air using what he called a ‘fire engine’. He worked out ways of measuring what he, following Boyle, thought of as the spring of the air and designed a fairly accurate thermometer with which he could work out that, as temperatures drop, the springiness, what we might think of as the pressure of the air, also drops.
SIMON SCHAFFER: And he projected the idea that there must be a point at which the air loses all its spring and that would be the limit of temperature, the absolute zero, in that sense, of temperature. He didn’t pursue the speculation but, after this work, that idea of an absolute limit became thinkable.
The next advancement came when Michael Faraday (1791–1867) began to explore lower temperatures at the Royal Institution in the 1830s. Stephen Blundell told us that Faraday started to do some experiments that followed on from one of the great achievements of his mentor Humphry Davy, which was to show that chlorine was an element. Faraday experimented with crystals of what was called chlorine hydrate at the time, basically ice with chlorine dissolved inside it.
STEPHEN BLUNDELL: He put chlorine hydrate in a sealed glass tube and heated it up. Now the problem was that’s quite a dangerous thing to do because the chlorine is released and goes to very high pressure and the glass tubes frequently exploded. So, after many of these experiments, Faraday found himself with glass all over his face and had to pick them out of his eyes, the shards.
Faraday wanted to raise the pressure inside the glass tubes by heating them up. By putting chlorine under very high pressure, he made a little oily liquid on the inside of the glass, which was liquid chlorine. If the pressure
of a liquid increases, the boiling temperature increases. Chlorine would normally go from liquid to gas at about -30°C but, by going to high pressure, Faraday could raise that point up to the temperature of the Royal Institution in February and, when he reduced the pressure, he had some very very cold liquid chlorine in his laboratory. He went on to do the same with hydrogen sulphide, nitrogen dioxide and sulphur dioxide.
William Thomson (1824–1907), later Lord Kelvin, was a theoretical physicist rather than an experimenter as Faraday was. He was very much concerned by the issue of thermometry because one of the problems was that any material that might be used to make a thermometer, such as mercury or alcohol, allows measurement by the expansion of that liquid, but that assumes the expansion gives a well-defined temperature, something that was not known. Messrs Fahrenheit and Celsius offered a particular scale for the expansion of liquid in a tube.
STEPHEN BLUNDELL: Celsius, for example, set zero degrees at the boiling point of water and 100 at the freezing point, later inverted to give the more familiar Celsius scale. But essentially what they were looking for was fixed points on a scale but they were always reduced to thinking about how some material property behaves, whether it’s mercury or alcohol or water or gas. And Kelvin wants to take us away from thinking about a material substance: ‘Is there an absolute definition of temperature?’
When Kelvin did look for a definition, the net result of his work was to see that absolute zero, the absolute zero of Amontons, -273°C, which we now call zero Kelvin, was the right zero of the scale.
Melvyn wanted more clarity on what was actually meant by temperature. Nicola Wilkin asked him to think of two cups of coffee that can be measured to see if they are equally hot or cold.
NICOLA WILKIN: [The thermometer] is a bit like a speedometer, it’s looking at the average speed of the atoms in your cup of coffee. So your cup of coffee looks like it’s completely still but, inside it, all the atoms are moving around, they’re bouncing off each other, they’re bouncing off the walls, they’re vibrating, they’re rotating and that average speed is what’s giving you the temperature.
As the coffee cools, it looks much the same but the atoms have slowed down very slightly and that is what is picked up on the temperature scale. At the end of the nineteenth century, Nicola Wilkin said, before quantum mechanics, it was thought that everything was just slowing down with no other source of energy in there, which, at absolute zero, would mean that everything was pinned down, the electrons as well as the atoms. If that were the case, it could be conjectured that electrical resistance would become infinite at absolute zero as nothing would be able to move.
Among scientists in the nineteenth century, there was, as Melvyn put it, almost a race to the South Pole to be the first person to the lowest temperature. This, as Simon Schaffer pointed out, was a time when what were seen as the heroic triumphs of European imperial geography were on everybody’s minds. James Dewar (1842–1923) was one of the British protagonists at the Royal Institution and his rival, William Ramsay, was nearby at University College, London.
SIMON SCHAFFER: Dewar and Ramsay loathed each other. This was, in a certain sense, tragic for the development of London physics, London chemistry, the science of cold in London. What these two men had, if they’d been put together, was an absolutely unrivalled combination of experimental technique, theoretical understanding and high-powered engineering equipment.
Dewar saw the move towards absolute zero as his destiny and, by assembling in the basement of the Royal Institution an almost unrivalled group of lab technicians and equipment, he was able to liquefy hydrogen before the end of the century, which was an extraordinary achievement. While Faraday had liquefied one range of gases, there had been another range of gases, including oxygen and hydrogen, which he had called the permanent gases as he thought they could not be liquefied. Meanwhile, Ramsay and his team were producing a range of hitherto unknown gases called the noble gases, argon, neon and ultimately helium, which also seemed to challenge liquefaction and offer the possibility of moving towards absolute zero. The rivalry was exacerbated by the lack of access to raw materials, as Ramsay had control over most of the helium in Britain, which happened to be in Bath, which made it difficult for his rival to work with it and make it liquid. As a side note: Dewar did, though, invent the vacuum flask.
Picking up from here, Stephen Blundell observed that one of the reasons that helium had not yet been liquefied was that it had only recently been discovered, and, even then, the discovery was in the solar spectra before it was found on earth. It is also essentially part of alpha particles from radioactivity, and the earth’s radioactive compounds give rise to helium all the time, which means it is found in traces in some minerals.
STEPHEN BLUNDELL: It was so special because, once liquid nitrogen, liquid hydrogen, liquid oxygen had all been made, it was discovered that, when you put helium gas in contact with those liquids, it wouldn’t itself liquefy. Which means that helium had to liquefy closer to absolute zero than anything else. So it was going to be the gas that gave you the closest approach to absolute zero.
Helium was therefore the final part in the race for the Pole. There was also another scientist of interest, Kamerlingh Onnes (1853–1926) at the University of Leiden, who had many highly trained technicians and, at that time, the most equipped and the best organised laboratory in the world. Onnes, Nicola Wilkin said, investigated what might happen to other materials if they were subjected to the same low temperatures. He chose mercury and cooled it down, anticipating that the electrical resistance would reach a constant and nothing would flow. Instead, to his surprise, he found that there was no electrical resistance.
NICOLA WILKIN: He managed to set up, within a ring of mercury, a current flowing. And he let it stay there. In fact, he let it stay there for a year. So he set this current flowing without any batteries, left it there undisturbed for a year, [and the] current was still flowing. Because there was no resistance, he had this persistent current and hence it became a superconductor.
This was a completely new phenomenon. Onnes could see extraordinary technological possibilities for this, such as building the huge magnets we have today.
Taking this further, Stephen Blundell turned to the quantum mechanics that began in the twentieth century, which showed that electrons would go around an atom for ever, with no need for a battery. The superconductivity spotted in the ring of mercury was similar to what happened to an atom, where a current of electricity would go round a loop of wire for ever.
STEPHEN BLUNDELL: It was a scientist called Fritz London, working in Oxford in the 1930s, who made the connection that superconductivity was essentially like a giant atom, a phenomenon that would normally only be operative in the microscopic world of atoms, electrons going around and around for ever. [It was] now seen in a macroscopic way, seen in large objects.
Noticing that quantum mechanics was manifest in this way was a crucial observation. It turned out that these kinds of phenomena were disrupted by the thermal vibrations that we heard about earlier, the jiggling around at high temperatures, so they became low temperature phenomena.
These enquiries were driven by curiosity, Simon Schaffer said, but the scientists were also helping to develop machines such as fridges, which went on to become so essential to civilisation.
SIMON SCHAFFER: There were two key techniques that Dewar and Kamerlingh Onnes principally drew on. On the one hand, the Joule–Kelvin effect – when a gas escapes from a containment at very high pressure through a nozzle or syphon into an area of very low pressure, you get an enormous, dramatic and very sudden cooling.
Engineers such as one of Simon Schaffer’s heroes, William Hampson (1854–1926), turned that process into an automatic air-liquefying machine, leading to the establishment of the British Oxygen Company. The second process that Dewar and Onnes drew on was called the cascade technique, which was to liquefy a gas and then use that liquid to liquefy a second gas and so on, so that there is a kind of regenerati
ve feedback loop in which the ambient temperature of the gas and then liquid can be driven down fast and very efficiently.
Scottish chemist and physicist James Dewar was the first to liquefy hydrogen.
SIMON SCHAFFER: Combine the two, the Joule–Kelvin effect and the cascade effect, as Dewar, to a certain extent, did and Kamerlingh Onnes did amazingly, brilliantly, and you can reach the temperatures of liquid helium and below, that’s to say 4K.
When studying physics, Nicola Wilkin cautioned, just as you think you have understood it and you are going to prove what is going on, nature throws something in your way. For scientists at the turn of the twentieth century, that was quantum mechanics. One of the first points that arose from that, relating to absolute zero, was what became called the Heisenberg uncertainty principle. That tells us that we are precluded from knowing precisely where a particle is and how fast it is going.
NICOLA WILKIN: You can immediately see that there’s a problem with the physics describing absolute zero earlier, where we were going to have particles nailed down and absolutely no speed associated with them, because quantum mechanics says we can’t do that. There has to be some sort of energy still left at absolute zero.
As Melvyn was interested in the idea of superfluids, Nicola Wilkin went on to a discovery published in 1938 that came out of investigations of helium. Liquefied, helium had no viscosity so, if you could imagine wading through it, you would feel no resistance to your strides. That is a very strange liquid.
NICOLA WILKIN: If you put it, for instance, in a beaker, it can climb the edges of that beaker. I did a back-of-the-envelope calculation: it’s something like (depending upon how accurately you do this) about 30km it could keep climbing up.
