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Simon Singh

Fermat’s Last Theorem

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    Valentyna Brusenkoцитує5 років тому
    I know it’s a rare privilege, but if you can tackle something in adult life that means that much to you, then it’s more rewarding than anything imaginable.
    Valentyna Brusenkoцитує5 років тому
    Basically it’s just a matter of thinking. Often you write something down to clarify your thoughts, but not necessarily. In particular when you’ve reached a real impasse, when there’s a real problem that you want to overcome, then the routine kind of mathematical thinking is of no use to you. 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.’
    Valentyna Brusenkoцитує5 років тому
    An expert problem solver must be endowed with two incompatible qualities – a restless imagination and a patient pertinacity.
    Valentyna Brusenkoцитує5 років тому
    An expert problem solver must be endowed with two incompatible qualities – a restless imagination and a patient pertinacity.
    Howard W. Eves
    Valentyna Brusenkoцитує5 років тому
    Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove this conjecture.’
    Valentyna Brusenkoцитує5 років тому
    Any insoluble problem in one area of mathematics could be transformed into an analogous problem in another area, where a whole new arsenal of techniques could be brought to bear on it. If a solution was still elusive, the problem could be transformed and transported to yet another area of mathematics, and so on, until it was solved.
    Valentyna Brusenkoцитує5 років тому
    There are dozens of such islands, each one with 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.
    Valentyna Brusenkoцитує5 років тому
    For example, there is the island occupied by geometers who study shape and form, and then there is the island of probability where mathematicians discuss risk and chance.
    Valentyna Brusenkoцитує5 років тому
    Mathematics consists of islands of knowledge in a sea of ignorance.
    Valentyna Brusenkoцитує5 років тому
    Just as the E-series is the DNA for elliptic equations, the M-series is the DNA for modular forms. The amount of each ingredient listed in the M-series is critical. Depending how you change the amount of, say, the first ingredient you might generate a completely different, but equally symmetrical, modular form, or you might destroy the symmetry altogether and generate a new object which is not a modular form. If the quantity of each ingredient is arbitrarily chosen, then the result will probably be an object with little or no symmetry.
    Valentyna Brusenkoцитує5 років тому
    The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.
    G.H. Hardy
    Valentyna Brusenkoцитує5 років тому
    The definition of a good mathematical problem is the mathematics it generates rather than the problem itself.’
    Valentyna Brusenkoцитує5 років тому
    In 1944 John von Neumann co-wrote the book The Theory of Games and Economic Behavior,
    Valentyna Brusenkoцитує5 років тому
    If the Last Theorem were in fact false, then it would be possible to prove this by identifying a solution (a counter-example). Therefore the Last Theorem would be decidable. Being false would be inconsistent with being undecidable. However, if the Last Theorem were true, there would not necessarily be such an unequivocal way of proving it so, i.e. it could be undecidable. In conclusion, Fermat’s Last Theorem might be true, but there may be no way of proving it.
    Valentyna Brusenkoцитує5 років тому
    After all Gödel had only said that these statements existed; he could not actually point to one. Then in 1963 Gödel’s theoretical nightmare became a full-blooded reality.
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    One approach was to create an additional axiom which forbade any class from being a member of itself.
    Valentyna Brusenkoцитує5 років тому
    Mathematics cannot tolerate inconsistencies, paradoxes or contradictions
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    Appendix 8 defines the set of arithmetic axioms and gives an idea of how logicians set about building the rest of mathematics.
    Valentyna Brusenkoцитує5 років тому
    Loyd’s puzzle and the disorder parameter demonstrate the power of an invariant. Invariants provide mathematicians with an important strategy to prove that it is impossible to transform one object into another
    Valentyna Brusenkoцитує5 років тому
    In mathematics a property which always holds true no matter what is done to the object is called an invariant.
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