3.10. Make a translation of the texts.

Text A. Mathematics is about patterns (2, 000 characters).

The random workings of chance seem to be about as far from patterns as you can get. In fact, one of the current definitions of ‘random’ boils down to ‘lacking any discernible pattern’. Mathematicians had been investigating patterns in geometry, algebra, and analysis for centuries before they realised that even randomness has its own patterns. But the patterns of chance do not conflict with the idea that random events have no pattern, because the regularities of random events are statistical. They are features of a whole series of events, such as the average behaviour over a long run of trials. They tell us nothing about which event occurs at which instant. For example, if you throw a dice1 repeatedly, then about one sixth of the time you will roll 1, and the same holds for 2, 3, 4, 5, and 6 – a clear statistical pattern. But this tells you nothing about which number will turn up on the next throw.

Only in the nineteenth century did mathematicians and scientists realise the importance of statistical patterns in chance events. Even human actions, such as suicide and divorce, are subject to quantitative laws, on average and in the long run. It took time to get used to what seemed at first to contradict free will. But today these statistical regularities form the basis of medical trials, social policy, insurance premiums, risk assessments, and professional sports.

And gambling, which is where it all began. Appropriately, it was all started by the gambling scholar, Girolamo Cardano. Being something of a wastrel, Cardano brought in much needed cash by taking wagers on games of chess and games of chance. He applied his powerful intellect to both. Chess does not depend on chance: winning depends on a good memory for standard positions and moves, and an intuitive sense of the overall flow of the game. In a game of chance, however, the player is subject to the whims of Lady Luck. Cardano realised that he could apply his mathematical talents to good effect even in this tempestuous relationship. He could improve his performance at games of chance by having a better grasp of the odds – the likelihood of winning or losing – than his opponents did. He put together a book on the topic, Liber de Ludo Aleae (‘Book on Games of Chance’). It remained unpublished until 1633. Its scholarly content is the first systematic treatment of the mathematics of probability. Its less reputable content is a chapter on how to cheat and get away with it.

Text B. Mathematical physics (2, 000 characters).

The term "mathematical physics" is sometimes used to denote research aimed at studying and solving problems inspired by physics or thought experimentswithin a mathematicallyrigorousframework. In this sense, mathematical physics covers a very broad academic realm distinguished only by the blending of puremathematics and physics. Although related totheoretical physics, mathematical physics in this sense emphasizes the mathematical rigour of the same type as found in mathematics.

On the other hand, theoretical physics emphasizes the links to observations and experimental physics, which often requires theoretical physicists (and mathematical physicists in the more general sense) to useheuristic,intuitive, and approximate arguments. Such arguments are not considered rigorous by mathematicians. Arguably, rigorous mathematical physics is closer to mathematics, and theoretical physics is closer to physics. This is reflected institutionally: mathematical physicists are often members of the mathematics department.

Such mathematical physicists primarily expand and elucidate physical theories. Because of the required level of mathematical rigour, these researchers often deal with questions that theoretical physicists have considered to already be solved. However, they can sometimes show (but neither commonly nor easily) that the previous solution was incomplete, incorrect, or simply, too naive. Issues about attempts to infer the second law of thermodynamics from statistical mechanics are examples. Other examples concerns all the subtleties involved with synchronisation procedures in special and general relativity (Sagnac effectandEinstein synchronisation).

The effort to put physical theories on a mathematically rigorous footing has inspired many mathematical developments. For example, the development of quantum mechanics and some aspects of functional analysisparallel each other in many ways. The mathematical study of quantum mechanics, quantum field theory and quantum statistical mechanics has motivated results inoperator algebras. The attempt to construct a rigorous quantum field theory has also brought about progress in fields such asrepresentation theory. Use ofgeometryandtopologyplays an important role instring theory.

Text C. Quantum Informatics and the Relations between Informatics, Physics and Mathematics ( 2, 500 characters).

A century ago hardly anyone would consider information an important concept for physics. One can even say that at that time one could hardly see information as a scientific concept at all. A significant breakthrough in the views on the relation between physics and information came, actually only after Landauer’s observation that information is physical: physical carriers are needed to store, transform and transmit information and therefore the laws and limitations of physics determine also the laws and limitations of information processing. An additional breakthrough came later with the view usually attributed to John Wheeler: physics is informational. Information processing phenomena, as well as their laws and limitations, are of key importance for understanding the laws and limitations of physics.

What are actually the main contributions of quantum information processing and communication (QIPC) to (quantum) physics?

They are numerous. On a very general level, QIPC should be seen as both an attempt to develop a new, more powerful, information processing technology, and as a new way to get deeper insights into the physical world, into its laws and limitations. QIPC gave rise to quantum informatics as the area of science combining goals and tools of both physics and informatics. QIPC is an area that brings new paradigms, goals, value systems, concepts, methods and tools to explore the physical world and its potential for information processing and communication is significant. On a more particular level, QIPC provides (quantum) physics with new concepts, models, tools, paradigms, images, analogies and makes many old concepts quantitative and more precise. In some cases, this even allowed to solve quite easily old problems. For example, an application of computability and complexity theories allowed to see that some old ideas about the physical world are wrong and that various proposals of modification of quantum mechanics are unlikely to work because they would allow the physical world to “easily” compute what is very likely beyond the classes BPP and BQP. Moreover, QIPC helped the understanding of phenomena that were considered for years strange and even mysterious, such as entanglement and non-locality. QIPC brought new measures allowing to quantify the power of various quantum resources and a deeper understanding of what can be and what cannot be distinguished and measured either exactly or approximately.

Moreover, as recently Barnum pointed out, the nature of information, its flow and processing, as seen from various operational perspectives, is likely to be the key to a unified view of the physical world in which quantum mechanics is its appropriate description, at least from certain points of view.

Finally, QIPC results brought new ways to use quantum phenomena for QIPC, not only through unitary operations, but also through adiabatic computations and, very surprisingly, through measurements only.

(Adapted from http://www.fi.muni.cz/usr/gruska/cris8.pdf)

Text D. Good Proofs and Bad Proofs

In a purely technical sense, a mathematical proof is verification of a proposition by a chain of logical deductions from a base set of axioms. But the purpose of a proof is to provide readers with compelling evidence for the truth of an assertion. Here are some tips on writing good proofs:

State your game plan. A good proof begins by explaining the general line of reasoning, e.g. “We use induction” or “We argue by contradiction”. This creates a rough mental picture into which the reader can fit the subsequent details.

Keep a linear flow. We sometimes see proofs that are like mathematical mosaics, with juicy tidbits of reasoning sprinkled judiciously across the page. This is not good. The steps of your argument should follow one another in a clear, sequential order.

Explain your reasoning. Many students initially write proofs the way they compute integrals. The result is a long sequence of expressions without explanation. This is bad. A good proof usually looks like an essay with some equations thrown in. Use complete sentences.

Avoid excessive symbolism. Your reader is probably good at understanding words, but much less skilled at reading arcane mathematical symbols. So use words where you reasonably can.

Simplify. Long, complicated proofs take the reader more time and effort to understand and can more easily conceal errors. So a proof with fewer logical steps is a better proof.

Introduce notation thoughtfully. Sometimes an argument can be greatly simplified by introducing a variable, devising a special notation, or defining a new term. But do this sparingly, since you’re requiring the reader to remember all this new stuff. And remember to actually define the meanings of new variables, terms, or notations; don’t just start using them.

Structure long proofs. Long program are usually broken into a heirarchy of smaller procedures. Long proofs are much the same. Facts needed in your proof that are easily stated, but not readily proved are best pulled out and proved in a preliminary lemmas. Also, if you are repeating essentially the same argument over and over, try to capture that argument in a general lemma and then repeatedly cite that instead.

Finish. At some point in a proof, you’ll have established all the essential facts you need. Resist the temptation to quit and leave the reader to draw the right conclusions. Instead, tie everything together yourself and explain why the original claim follows. The analogy between good proofs and good programs extends beyond structure. The same rigorous thinking needed for proofs is essential in the design of critical computer system. When algorithms and protocols only “mostly work” due to reliance on handwaving arguments, the results can range from problematic to catastrophic. An early example was the Therac 25, a machine that provided radiation therapy to cancer victims, but occasionally killed them with massive overdoses due to a software race condition.

More recently, a buggy electronic voting system credited presidential candidate Al Gore with negative 16,022 votes in one county. In August 2004, a single faulty command to a computer system used by United and American Airlines grounded the entire fleet of both companies — and all their passengers.

It is a certainty that we’ll all one day be at the mercy of critical computer systems designed by you and your classmates. So we really hope that you’ll develop the ability to formulate rock-solid logical arguments that a system actually does what you think it does.

ADDITIONAL EXERCISES