§1. Mathematics and the physical world. Genuine scientific knowledge cannot be certain, nor can it be justified a priori. Instead, it must be conjectured, and then tested by experiment, and this requires it to be expressed in a language appropriate for making precise, empirically testable predictions. That language is mathematics.This in turn constitutes a statement about what the physical world must be like if science, thus conceived, is to be possible. As Galileo put it, “the universe is written in the (...) language of mathematics”. Galileo's introduction of mathematically formulated, testable theories into physics marked the transition from the Aristotelian conception of physics, resting on supposedly necessary a priori principles, to its modern status as a theoretical, conjectural and empirical science. Instead of seeking an infallible universal mathematical design, Galilean science usesmathematics to express quantitative descriptions of an objective physical reality. Thus mathematics became the language in which we express our knowledge of the physical world — a language that is not only extraordinarily powerful and precise, but also effective in practice. Eugene Wigner referred to “the unreasonable effectiveness of mathematics in the physical sciences”. But is this effectiveness really unreasonable or miraculous?Numbers, sets, groups and algebras have an autonomous reality quite independent of what the laws of physics decree, and the properties of these mathematical structures can be just as objective as Plato believed they were. (shrink)

At a purely instrumental level, quantum theory is all about multiplication, addition and taking mod squares of complex numbers called probability amplitudes. The rules for combining amplitudes are deceptively simple. When two or more events are independent you multiply their respective probability amplitudes and when they are mutually exclusive you add them. Whenever you want to calculate probabilities you take mod squares of respective amplitudes. That’s it. If you are prepared to ignore the explanatory power of the theory (which you (...) should not) the rest is just a set of convenient mathematical tools developed for the purpose of book-keeping of amplitudes. Thus we tabulate amplitudes into state vectors and unitary matrices, and place them in Hilbert spaces. (shrink)

Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. In effect, they follow the same logical paradigm as (multi-particle) interferometers. We show how most known quantum algorithms for factorising and counting, may be cast in this manner. Quantum searching is described as inducing a desired relative phase between two eigenvectors to yield constructive interference on the sought elements and destructive interference on the remaining terms.

§1. Mathematics and the physical world. Genuine scientific knowledge cannot be certain, nor can it be justified a priori. Instead, it must be conjectured, and then tested by experiment, and this requires it to be expressed in a language appropriate for making precise, empirically testable predictions. That language is mathematics.This in turn constitutes a statement about what the physical world must be like if science, thus conceived, is to be possible. As Galileo put it, “the universe is written in the (...) language of mathematics”. Galileo's introduction of mathematically formulated, testable theories into physics marked the transition from the Aristotelian conception of physics, resting on supposedly necessary a priori principles, to its modern status as a theoretical, conjectural and empirical science. Instead of seeking an infallible universal mathematical design, Galilean science usesmathematics to express quantitative descriptions of an objective physical reality. Thus mathematics became the language in which we express our knowledge of the physical world — a language that is not only extraordinarily powerful and precise, but also effective in practice. Eugene Wigner referred to “the unreasonable effectiveness of mathematics in the physical sciences”. But is this effectiveness really unreasonable or miraculous?Numbers, sets, groups and algebras have an autonomous reality quite independent of what the laws of physics decree, and the properties of these mathematical structures can be just as objective as Plato believed they were (and as Roger Penrose now advocates). (shrink)