Singh 1997 Fourth Estate
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|Singh Simon (1997) Fermat's last theorem. Fourth Estate, London 340 pp.|
• Bioblast editor: Gnaiger E
- The idea of a classic mathematical proof is to begin with a series of axioms, statements which can be assumed to be true or which are self-evidently true.Then by arguing logically, step by step, it is possible to arrive at a conclusion. If the axioms are correct and the logic is flawless, then the conclusion will be undeniable. This conclusion is the theorem. Mathematical theorems rely on this logical process and once proven are true until the end of time. Mathematical proofs are abolute.
- Like Pathagoras, Euclid believed in the search for mathematical truth for its own sake and did not look for applications in his work. One story tells of a student who questioned him about the use of the mathematics he was learning. Upon completing the lesson, Euclid truned to his slave and said, 'Give the boy a öenny since he desires to profit from all tha he learns.' The student was then expelled.
- The growth of any discipline depends on the ability to communicate and develop ideas, and this in turn relies on a language which is sufficiently detailed and flexible. The ideas of Pythagoras and Euclid were no less elegant for their awkward expression, but translated into the symbols of Arabia they would blossom and give fruit to newer and richer concepts.
- The history of numbers begins with the simple counting numbers (1, 2, 3, ...) otherwise known as natural numbers. These numbers are perfectly satisfacotry for adding together simple whole quantities, such as sheep or gold coins, to achieve a total number which is also a whole quantity. — Comment: this "total number" is not a number but a count.
- Number theorists consider prime numbers to be the most important numbers of all because they are the atoms of mathematics.
- Another of the great puzzlers of the Victorian Age was the Reverend Charles Dodgson, lecturer in mathematics at Christ Church, Oxford, and better known as the author Lewis Carroll.
- In mathematics a property which always holds true no matter what is done to the object is called an invariant.
- The art of number theory is so abstract that it is frighteningly easy to wander off the path of logic and be completely unaware that one has strayed into absurdity.
- Eventually the logicians found themselves dealing with a few essential statements which were so fundamental that they themselves could not be proved. These fundamental assumptions are the axioms of mathematics.
- One of Frege's key breakthroughs was to create the very definition of a number. For example, what do we actually mean by the number 3? It turns out that to define 3, Frege first had to define 'threeness'. 'Threeness' is the abstract quality which belongs to collections or sets of objects containing three objects.
- Russel recalled his own reaction to the dreaded realisation that mathematics might be inherently contradictory. .. Ironically Russel's contradiction grew out of Frege's much loved sets, or collections. .. 'It seemed to me that a class sometimes is, and sometimes is not, a member of itself. The class of teaspoons, for example, is not another teaspoon, but the class of things that are not teaspoons is one of the things that are not teaspoons.' .. 'Two must be two of something, and the proposition "2 and 2 are 4" is useless unless it can be applied. "Two entities and two entities are four entities." When you have told me what you means by "entity", we will resume the argument.'
- Infinity is unobtainable by the mere brute force of computerised number crunching.
- Isomorphic relationships: Relationships between apparently different subjects are as creatively important in mathematics as they are in any discipline. The relationship hints at some underlying truth which enriches both subjects.
- Barry Mazur: 'Mathematicians studying elliptic equations might not be well versed in things modular, and conversely. Then along comes the Taniyama-Shimura conjecture which is the grand surmise that there's a bridge between these two completely different worlds. Mathematicians love to build bridges.' The value of mathematical bridges is enormous. They enable communities of mathematicians who have been living on separate islands to exchange ideas and explore each other's creations. Mathematics consists of islands of knowledge in a sea of ignorance. For example, there is the island occupied by geometers who study shape and form, and then there is the island of probability where mathemtaticians discuss risk and chance. There are dozens of such islands, each one whith its own unique language, incomprehensible to the inhabitants of other islands. The language of geometry is quite different to the language of probability, and the slang of calculus is meaningless to those who speak only statistics.
- Andrew Wiles: 'Leading up to that kind of new idea there has to be a long period of tremendous focus on the problem without any distraction. You have to really think about nothing but that problem - just concentrate on it. Then you stop. Afterwards there seems to be a kind of period of relaxation during which the subconscious appears to take over and it's during that time that some new insight comes.'
- Gnaiger E (2020) Mitochondrial pathways and respiratory control. An introduction to OXPHOS analysis. 5th ed. Bioenerg Commun 2020.2. https://doi.org/10.26124/bec:2020-0002
Number, BEC 2020.2, X-mass Carol