Basic arithmetic
Richard Elwes reports on Harvey Friedman’s fascinating work on incompleteness in Boolean relation theory, but it is quite a stretch to extend his work to ask “are the rules of arithmetic… unsound?” (14 August, p 34). In logic, as in everyday language, the label “unsound” is generally reserved for arguments which prove false statements or contain faulty reasoning.
As mentioned in the article, Kurt G枚del showed that there are statements in any sufficiently complex theory of arithmetic that can neither be proved nor disproved within the theory. Friedman has found a concrete example. Such theorems are astonishing but they do not imply that our best theories of arithmetic are likely to be harbouring a falsehood or contradiction.
Furthermore, incompleteness does not lie as far away from elementary mathematics as the article suggests. If we extend Pythagoras’s theorem by allowing higher powers and more variables, and consider the set of such equations that cannot be solved in whole numbers, no formal arithmetical system can completely decide which equations are members of this set, no matter how many axioms are added to plug the gaps.
It is remarkable that such apparently straightforward questions can never be answered, even in principle, but this does not mean we should be unduly worried that the tenets of ordinary mathematics might turn out to be false. G枚del’s astounding theorems show that there are some things that mathematics can never prove, but they do not call into question those that it can.
鈥� It is certainly true that whether or not certain equations can be solved in the whole numbers is undecidable from Peano’s axioms 鈥� rules that underlie the logic of arithmetic. However, some arithmetical statements are more undecidable than others, so to speak.
Most known examples of arithmetical statements which are undecidable from Peano’s axioms can be decided when we move up to a stronger system, such as second-order arithmetic or Zermelo-Fraenkel set theory. The most undecidable ones need the use of large cardinal numbers. Friedman has not just found a concrete example of an undecidable statement, he has found a concrete example of a very, very, very undecidable statement: a brand new, and shocking, result.
Andrey Bovykin and Michiel de Smet have recently written on their work translating undecidable statements into statements involving polynomial equations (). They even manage to achieve this for Friedman’s recent work.
Richard Elwes asks if we can easily dismiss infinity from mathematics. If he simply means abstaining from setting an upper or positive lower limit to a set or function, then the answer is no. However, this does not mean that mathematical infinity actually exists, or is even a meaningful word within a realistic context.
Unfortunately, pure mathematicians tend to believe in their own, often fantastic, constructions. As far as we know, there is nothing that is infinite in the physical world: all processes involving energy exchange proceed by finite quanta and many physicists are beginning to think that the structure of space-time is grainy as well. As a scientific concept, infinity should be relegated to the has-been category, along with phlogiston and the philosopher’s stone.
Shaftesbury, Dorset, UK
No profit in fever
Your article on the role of fever in fighting infection (31 July, p 42) missed one of the most interesting chapters in this saga: in the late 1800s and early 1900s a doctor called William Coley attempted to cure cancer with fever, with some success. An earlier article in New 杏吧原创 (2 November 2002, p 54) described his work.
The trouble with using fever as a cancer cure is that it would not be patentable. It would be a repeat of the malaria-wormwood story: progress in the widespread use of artemisinin 鈥� the anti-malaria agent derived from wormwood 鈥� was stymied for many years by the lack of obvious profitability.
Helium hopes rise
Robert Richardson warns us of the perils of squandering our limited helium resources (14 August, p 29).
Schemes to make a lot of money by cornering the market in a supposedly irreplaceable commodity usually fail, because the market is ingenious at finding either new supplies or alternatives. However, Richardson suggests that the chance of either of these happening in the case of helium is very low indeed.
It would appear that any billionaire investor or fund manager is onto a winner in buying up as much as they can afford of the US helium stockpile. Their activities would trigger the price rise that Richardson fears 鈥� but the sooner the price rises, the sooner proper conservation measures will start, which will put a cap on maximum prices.
Could New 杏吧原创 persuade some bankers to organise a fund into which readers could put a few coppers with the dual aim of getting a little bit richer and saving an invaluable resource for science of the future?
Moving the Earth
Matt Kaplan writes about the effect that meteors could have on the Earth’s geology, suggesting that the Deccan Traps were caused by an impact somewhere else on the planet (5 June, p 38).
As a miner, I learned that rock is very strong under compression. All rock-breaking explosions had to be directed towards a nearby open area, which would reflect the compressive wave, making it a tension wave as it tore back through and broke up the rock. To be really effective the tension wave should be reflected back along the same path as the one that the compressive wave came in on, which produces an interference pattern that breaks up the rock.
The same rules should apply when considering an asteroid impact on Earth. The compressive shock wave travelling through Earth will glance diagonally off most of the surfaces of the sphere, causing little structural damage. The only place that major damage will occur is directly opposite the point of impact, where the planet’s surface is at right angles to the compressive shock wave.
Ancient acoustics
Trevor Cox’s discussion of the acoustics of ancient theatres reminded me of a trip to Greece with my medical student colleagues (21 August, p 44).
We tested the acoustics of the amphitheatre at Delphi by going to the top tier of seats and sending one of our number down to the stage. Not having memorised a speech from Aristophanes but being in the throes of learning anatomy, he intoned the longest Latin name for a muscle in the body, the levator labii superioris alaeque nasi 鈥� a facial muscle that lifts the upper lip and dilates the nostril.
The theatre acoustics were so impressive that all the other visitors stopped in their tracks and crossed themselves.
Defining species
Thomas Frost criticised using as a criterion for species membership the ability to interbreed to produce fertile offspring (31 July, p 27).
I was surprised that he did not cite the most widely accepted definition of species: the biological species concept. It was first proposed by Ernst Mayr, in his 1942 book Systematics and the Origin of Species from the Viewpoint of a Zoologist, as: “species are groups of actually or potentially interbreeding natural populations that are reproductively isolated from other such groups”.
Members of the genus Canis 鈥� dogs, wolves, coyotes and jackals 鈥� will interbreed to produce fertile offspring, as Frost points out, but their geographic isolation normally keeps them as separate species.
However, Frost also mentions that domestic sheep and goats, of the genera Ovis and Capra, can interbreed, but do not provide viable offspring unless the embryos are genetically manipulated.
Dino frills
In his article on morphing dinosaurs, Graham Lawton discusses possible functions of the triceratops’s neck frill, such as it being a signal of maturity. However, he surely implies its real function with the description: “it had numerous large blood vessels running over the surface” (31 July, p 6).
This suggests heat exchange: the frill was either a kind of solar panel, taking in solar energy for heating, or a radiator for cooling, similar to an African elephant’s ears, depending on the environment the animal lived in.
Evolving physics
Sebastian Hayes suggests in his letter that the laws of physics might evolve over time (26 June, p 30). Some versions of this idea have already been considered in fact, such as the possibility that the gravitational constant changes over time. Even if we don’t count that, we need to consider what would control the evolution of the laws of physics. If nothing does we’re back to a lawless universe, or if we find out what does control them, these meta laws will now serve as fundamental laws of physics.
There might be a final, awkward possibility of an infinite tower of laws, meta laws, and meta-meta laws which do not bottom out at all, but that’s rather unsatisfactory to anyone but the sillier sort of philosopher.
Elephant analogy
The simile comparing gravity to an elephant, unaware of quantum fluctuations, or “microbes”, on its skin (15 May, p 9) appeared in the same issue as Feedback’s illumination of the meaning of numbers via elephant-related metaphors. The amount of amusement arising from this concordance was equivalent to [number of elephants] [unlikely situation] [relevant units].
For the record
鈥� The correct reference for the paper by Rouzbeh Allahverdi and others on universal inflation (21 August, p 6), is .
鈥� Corsin M眉ller is at the University of Vienna in Austria, not the University of Zurich as we stated (21 August, p 12).