Everything in the world around us, including ourselves, is made up of identical material particles with regular properties which obey physical laws. Second, these physical laws can be described well using mathematical equations, giving us the ability to make accurate predictions of the way things will be in the future and the way they were in the past.
How do we know this? We know it through mathemathics.
Reed on but only if you are a dedicated masochist.
One of the deep features of mathematics is that most ideas can be represented either geometrically, and so comprehended visually, or else analytically, i.e. represented through equations and inequalities. Many branches of mathematics are essentially ways of bridging between these two approaches.
All branches of mathematics involve abstraction, which entails a formalisation of representation and argument enabling the ‘proof’ of mathematical propositions. Abstraction occurs, for example, in the way the study of solutions of equations inexorably leads to the need to introduce negative numbers, irrational numbers such as pi (which can’t be represented as fractions), and complex numbers, involving the square root of –1 (which can be usefully represented in geometrical ways).
Roger Penrose marvels at the nature of complex numbers, claiming that ‘there is not only a special magic in the mathematics of these numbers, but . . . nature herself appears to harness this magic in weaving her universe at its deepest levels.’ Abstraction enables mathematics to deal with ever more general concepts: from numbers to variables to functions and on to ‘tensors’ and ‘spinors’, each associated with a suitable symbolic representation and a set of rules for manipulating those symbols.
In the last part of the book, Penrose comments on three crucial issues in present-day theoretical physics, providing a good corrective to much recent writing that suggests we already have the answers in these areas. The key cosmological problem Penrose focuses on is the origin of the arrow of time. The fundamental physical laws are time-symmetric: for each solution of the equations, there is an equal and opposite solution in which the direction of time is reversed. For example, Maxwell’s equations for electromagnetism allow a radio signal to be received before it is transmitted, and in principle people can grow younger. The ‘arrow of time problem’ is that these things don’t happen: what is it that disallows the time-reversed solution? Penrose’s elegant discussion of this issue leads to the conclusion that there must have been special conditions in place at the start of the universe.
Penrose’s next target is the measurement problem in quantum theory. There is at present no consistent picture of the process of measurement that takes quantum physics into account. Usually, it is assumed that the measurement apparatus does not obey the rules of quantum theory, but this contradicts the presupposition that all matter is at its foundation quantum mechanical in nature. Penrose gives a clear description of the various ways this paradox can be handled, and how its resolution depends on what one believes about the nature of reality. Many physicists do not believe this is a question one should ask (‘I don’t demand that a theory correspond to reality because I don’t know what reality is,’ Stephen Hawking says), but Penrose believes that such matters are crucial to quantum mechanics and are far from being settled. He explains why standard approaches fail to solve the problem, and proposes instead that quantum measurement processes are associated with quantum gravity. This idea has not gained wide acceptance, but is undeniably deeply thought through.
As for the nature of what is real, Penrose argues that many mathematical phenomena (the numerical value of pi, for example, or the irrationality of the square root of two, or the existence of Mandelbrot figures) are discovered rather than invented. Consequently, he proposes that there is a Platonic reality to mathematics: that it exists in an abstract sense, independent of the human mind, and that we must recognise three different kinds of ‘world’ representing three different forms of existence – the physical, the mental and the mathematical. His discussion of the nature of these different kinds of realities and the relations between them is one of the highlights of this book.
Does he provide ‘A Complete Guide to the Laws of the Universe’? In discussing the fundamental physical laws and their underlying mathematics, he comes close. With regard to how complexity and life arise out of this physics, however, the answer is no; in that sense, a theory of everything remains elusive.
The Road to Reality: A Complete Guide to the Laws of the Universe by Roger Penrose
Vintage, 1099 pp, £15.00, February 2006, ISBN 0 09 944068 7
How do we know this? We know it through mathemathics.
Reed on but only if you are a dedicated masochist.
One of the deep features of mathematics is that most ideas can be represented either geometrically, and so comprehended visually, or else analytically, i.e. represented through equations and inequalities. Many branches of mathematics are essentially ways of bridging between these two approaches.
All branches of mathematics involve abstraction, which entails a formalisation of representation and argument enabling the ‘proof’ of mathematical propositions. Abstraction occurs, for example, in the way the study of solutions of equations inexorably leads to the need to introduce negative numbers, irrational numbers such as pi (which can’t be represented as fractions), and complex numbers, involving the square root of –1 (which can be usefully represented in geometrical ways).
Roger Penrose marvels at the nature of complex numbers, claiming that ‘there is not only a special magic in the mathematics of these numbers, but . . . nature herself appears to harness this magic in weaving her universe at its deepest levels.’ Abstraction enables mathematics to deal with ever more general concepts: from numbers to variables to functions and on to ‘tensors’ and ‘spinors’, each associated with a suitable symbolic representation and a set of rules for manipulating those symbols.
In the last part of the book, Penrose comments on three crucial issues in present-day theoretical physics, providing a good corrective to much recent writing that suggests we already have the answers in these areas. The key cosmological problem Penrose focuses on is the origin of the arrow of time. The fundamental physical laws are time-symmetric: for each solution of the equations, there is an equal and opposite solution in which the direction of time is reversed. For example, Maxwell’s equations for electromagnetism allow a radio signal to be received before it is transmitted, and in principle people can grow younger. The ‘arrow of time problem’ is that these things don’t happen: what is it that disallows the time-reversed solution? Penrose’s elegant discussion of this issue leads to the conclusion that there must have been special conditions in place at the start of the universe.
Penrose’s next target is the measurement problem in quantum theory. There is at present no consistent picture of the process of measurement that takes quantum physics into account. Usually, it is assumed that the measurement apparatus does not obey the rules of quantum theory, but this contradicts the presupposition that all matter is at its foundation quantum mechanical in nature. Penrose gives a clear description of the various ways this paradox can be handled, and how its resolution depends on what one believes about the nature of reality. Many physicists do not believe this is a question one should ask (‘I don’t demand that a theory correspond to reality because I don’t know what reality is,’ Stephen Hawking says), but Penrose believes that such matters are crucial to quantum mechanics and are far from being settled. He explains why standard approaches fail to solve the problem, and proposes instead that quantum measurement processes are associated with quantum gravity. This idea has not gained wide acceptance, but is undeniably deeply thought through.
As for the nature of what is real, Penrose argues that many mathematical phenomena (the numerical value of pi, for example, or the irrationality of the square root of two, or the existence of Mandelbrot figures) are discovered rather than invented. Consequently, he proposes that there is a Platonic reality to mathematics: that it exists in an abstract sense, independent of the human mind, and that we must recognise three different kinds of ‘world’ representing three different forms of existence – the physical, the mental and the mathematical. His discussion of the nature of these different kinds of realities and the relations between them is one of the highlights of this book.
Does he provide ‘A Complete Guide to the Laws of the Universe’? In discussing the fundamental physical laws and their underlying mathematics, he comes close. With regard to how complexity and life arise out of this physics, however, the answer is no; in that sense, a theory of everything remains elusive.
The Road to Reality: A Complete Guide to the Laws of the Universe by Roger Penrose
Vintage, 1099 pp, £15.00, February 2006, ISBN 0 09 944068 7
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