volume II: Synopsis
part III: Modern Physics
page 19: John von Neumann
(1903-1957)
When George Cantor first announced the transfinite numbers, some theologians objected on the ground that the only actual infinity in existence is God. Their objection was based on confusion of 'Platonic' mathematical existence and real existence. Hilbert pioneered the 'formalist' approach to mathematics which sees mathematics as a game played with symbols whose only constraint is consistency. In Hilbert's view, the only requirement for mathematical existence is self consistency. The transfinite numbers are self-consistent and so, mathematically, they exist. Cohen, David Hilbert - Wikipedia
Hilbert showed that Cantor's new infinities fitted easily into mathematics and described new class of infinite spaces known as function spaces. John von Neumann used the function space now called Hilbert space to resolve the apparent conflict between the particle and wave (discrete and continuous) descriptions of the world, opening the way for the consistent development of quantum theory. John von Neumann - Wikipedia, Hilbert space - Wikipedia, von Neumann
Hilbert space is a function space. This means that each point in the space represents a function. A function is a mapping between a set of elements called the domain of the function (say x) to a set called the range of the function (say y). We represent this mapping by the expression y = f(x). Some functions can be represented very simply as an algebraic expression, eg y = x2. Then if x = 1, y = 1; x = 2, y = 4; x = 3, y = 9; and so on.
This may be a function whose range and domain are the natural numbers. When no law exists that enables the succinct expression of a function, we must represent it by a table of values. We note here that there are n! ways of mapping n things onto themselves, ie permutations of n objects. Since there are ℵ0 natural numbers, there are ℵ0! = ℵ1 functions whose range and domain are the natural numbers. Each of these functions is represented by a single point in a function space whose cardinal is ℵ1. The information carried by a point in a space is the same as the complexity or entropy of the space. Following Cantor, we may now imagine the space of permutations of the ℵ1 elements of this space to obtain a space with ℵ2 elements, and so on.
In Hilbert space, functions are expressed as ordered lists of values called vectors. For a given application, the dimension of the appropriate Hilbert space must be equal to complexity of the state represented, ranging from 2 for the spin states of an electron to a countable infinity for the energy states of a hydrogen atom. In this respect, Hilbert space is a natural extension of ordinary three dimensional space.
Quantum mechanics represents physical motion, that is changes of state, by operators which transform state vectors into one another. Since a change of state in the quantum world is generally accompanied by the emission or absorption of observable particles, quantum mechanics uses operators to represent observables. Different operators acting on state vectors yield the energy, momentum and angular momentum of the particles observed. Dirac
When we come to consider two or more particles, the Hilbert space we need is the 'tensor product' of the Hilbert spaces for the original particles. The size of the resulting Hilbert space grows exponentially with the number of particles represented, in the same way as the size of the number represented by an 'Arabic' numeral grows exponentially with the length of the numeral and the cardinal numbers of transfinite sets grow as we move up the Cantor hierarchy. Tensor product - Wikipedia
It is this exponential growth that leads us to suspect that the 'state vectors of the Universe' are so large that they require the formal Cantor Universe for their adequate representation.
(revised 5 April 2020)
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Further readingBooks
Cohen, Paul J, Set Theory and the Continuum Hypothesis, Benjamin/Cummings 1966-1980 Preface: 'The notes that follow are based on a course given at Harvard University, Spring 1965. The main objective was to give the proof of the independence of the continuum hypothesis [from the Zermelo-Fraenkel axioms for set theory with the axiom of choice included]. To keep the course as self contained as possible we included background materials in logic and axiomatic set theory as well as an account of Gödel's proof of the consistency of the continuum hypothesis. . . .'
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Dirac, P A M, The Principles of Quantum Mechanics (4th ed), Oxford UP/Clarendon 1983 Jacket: '[this] is the standard work in the fundamental principles of quantum mechanics, indispensible both to the advanced student and the mature research worker, who will always find it a fresh source of knowledge and stimulation.' (Nature)
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Hilbert, David, and Leon Unger (translator, from the tenth German edition). Revised and Enlarged by Paul Bernays, Foundations of Geometry (Grundlagen der Geometrie), Open Court 1999 Jacket: 'Along with the writings of Hilbert's friend and correspondent Frege, Hilbert's Grundlagen der Geometrie is the major prop that set the stage for Russell and Whitehead's Principa Mathematica. Hilbert presents a new axiomatization of geometry, the reduction of geometry to algebra, and introduces the distinction between mathematics and metamathematics, with a new theory of proof.
This edition is translated from the tenth German edition, including all the improvements which Hilbert derived from his own reflections and the contributions of other writers.
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Pais, Abraham, 'Subtle is the Lord...': The Science and Life of Albert Einstein, Oxford UP 1982 Jacket: In this . . . major work Abraham Pais, himself an eminent physicist who worked alongside Einstein in the post-war years, traces the development of Einstein's entire ouvre. . . . Running through the book is a completely non-scientific biography . . . including many letters which appear in English for the first time, as well as other information not published before.'
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Reid, Constance, Hilbert-Courant, Springer Verlag 1986 Jacket: '[Hilbert] is woven out of three distinct themes. It presents a sensitive portrait of a great human being. It describes accurately and intelligibly on a non-technical level the world of mathematical ideas in which Hilbert created his masterpieces. And it illuminates the background of German social history against which the drama of Hilbert's life was played. ... Beyond this, it is a poem in praise of mathematics.' Science
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von Neumann, John, and Robert T Beyer (translator), Mathematical Foundations of Quantum Mechanics, Princeton University Press 1983 Jacket: '. . . a revolutionary book that caused a sea change in theoretical physics. . . . JvN begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which JvN regards as the definitive form of quantum mechanics. . . . Regarded as a tour de force at the time of its publication, this book is still indispensable for those interested in the fundamental issues of quantum mechanics.'
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Links
David Hilbert - Wikipedia, David Hilbert - Wikipedia, the free encyclopedia, 'David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He invented or developed a broad range of fundamental ideas, in invariant theory, the axiomatization of geometry, and with the notion of Hilbert space, one of the foundations of functional analysis.
Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century.
Hilbert and his students supplied significant portions of the mathematical infrastructure required for quantum mechanics and general relativity. He is also known as one of the founders of proof theory, mathematical logic and the distinction between mathematics and metamathematics.' back |
Formalism (mathematics) - Wikipedia, Formalism (mathematics) - Wikipedia, the free encyclopedia, 'In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules.
For example, Euclidean geometry can be seen as a game whose play consists in moving around certain strings of symbols called axioms according to a set of rules called "rules of inference" to generate new strings. In playing this game one can "prove" that the Pythagorean theorem is valid because the string representing the Pythagorean theorem can be constructed using only the stated rules.' back |
Hilbert space - Wikipedia, Hilbert space - Wikipedia, the free encyclopedia, 'The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.' back |
John von Neumann - Wikipedia, John von Neumann - Wikipedia, the free encyclopedia, ' John von Neumann (December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, and polymath. He made major contributions to a number of fields, including mathematics (foundations of mathematics, functional analysis, ergodic theory, representation theory, operator algebras, geometry, topology, and numerical analysis), physics (quantum mechanics, hydrodynamics, and quantum statistical mechanics), economics (game theory), computing (Von Neumann architecture, linear programming, self-replicating machines, stochastic computing), and statistics.' back |
Tensor product - Wikipedia, Tensor product - Wikipedia, the free encyclopedia, ' n mathematics, the tensor product V ⊗ W of two vector spaces V and W (over the same field) is itself a vector space, endowed with the operation of bilinear composition, denoted by ⊗, from ordered pairs in the Cartesian product V × W to V ⊗ W in a way that generalizes the outer product.
Essentially the difference between a tensor product of two vectors and an ordered pair of vectors is that if one vector is multiplied by a nonzero scalar and the other is multiplied by the reciprocal of that scalar, the result is a different ordered pair of vectors, but the same tensor product of two vectors.
The tensor product of V and W is the vector space generated by the symbols v ⊗ w, with v ∈ V and w ∈ W, in which the relations of bilinearity are imposed for the product operation ⊗, and no other relations are assumed to hold. The tensor product space is thus the "freest" (or most general) such vector space, in the sense of having the fewest constraints. ' back |
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