Chapter 14: Evolution and intelligence

Synopsis

We consider ourselves intelligent and we know that we have evolved from simple one celled creatures that appeared on Earth about three billion years ago. Darwin popularized evolution by natural selection and we now know how it works. Billions of variant forms of life have arisen by random mutation and in the lottery of life those forms that can reproduce themselves survive. In the absence of an omniscient and omnipotent divinity to design and create the world, evolution must be the source of our intelligence and it must go back to the beginning. We must therefore attribute it to gravitation and quantum mechanics, the first movers of our world.

14.1: Particles or waves?

14.2: Louis de Broglie

14.3: From simple functions to multi-dimensional operators

14.4: The eigenvalue problem

14.5: Where is mathematics in the world?

14.6: Entropy, control and survival

14.1 Particles or waves?

The most exciting discovery in the classical theory of electrodynamics was Maxwell’s realization in 1865 that electricity and magnetism they could be coupled to one another to make a harmonic oscillator, like a pendulum. A moving magnetic field creates an electric field. A moving electric field creates a magnetic field. Due to the electromagnetic properties of space-time, this coupling moves through space at the speed of light. Maxwell's equations - Wikipedia

In 1860 Gustav Kirhoff used a thermodynamic argument to show that every body emitted radiation whose frequency spectrum depended only on its temperature. In 1900 Max Planck found the equation that links temperature to spectral irradiance and quantum mechanics was born. To get his equation he had to assume that radiation was emitted in discrete quanta whose energy E was related to their frequency f by the first equation of quantum the theory E = hf where h is a new universal constant, extraordinarily small. h is the angular momentum of the quantum of action, the smallest event on the Universe. Kirchoff's law of thermal radiation - Wikipedia, Planck's Law - Wikipedia

In 1905 Einstein proposed that Planck’s quantum of energy is a real particle that we now call the photon. This work started a long debate, still with us: is light particles or waves?

14.2: Louis de Broglie

An answer came from Louis de Broglie in 1924. He agreed with Einstein: electrons are both. Standing waves explain the presence of fixed structures like atomic orbitals.

De Broglie was awarded the 1929 Nobel Prize and describes his ideas very succinctly in his lecture The Wave Nature of the Electron.

Now a purely corpuscular theory does not contain any element permitting the definition of frequency. This also renders it necessary in the case of light to introduce simultaneously the corpuscle concept and the concept of periodicity.

On the other hand the determination of the stable motions of the electrons in the atom involves whole numbers, and so far the only phenomena in which whole numbers were involved in physics were those of interference and of eigenvibrations. That suggested the idea to me that electrons themselves could not be represented as simple corpuscles either, but that a periodicity had also to be assigned to them too.

In other words the existence of corpuscles accompanied by waves has to be assumed in all cases. However, since corpuscles and waves cannot be independent . . . it must be possible to establish a certain parallelism between the motion of a corpuscle and the propagation of the associated wave. . . .

This prompted the thought that classical mechanics is also only an approximation relative to a vaster wave mechanics. . . . This new mechanics has since been developed, thanks mainly to the fine work done by Schrödinger. . . ..

Thus to describe the properties of matter as well as those of light, waves and corpuscles have to be referred to at one and the same time. The electron can no longer be conceived as a single, small granule of electricity; it must be associated with a wave and this wave is no myth; its wavelength can be measured and its interferences predicted. Louis de Broglie (1929): Nobel Lecture: The Wave Nature of the Electron

The mathematical foundations of quantum mechanics, as written by von Neumann, is in essence the mathematics of standing waves.

14.3: From simple functions to operators

Rene Descartes greatly clarified mathematics by inventing coordinate geometry, so connecting arithmetic and geometry which ultimately led to Einstein's general relativity. By establishing axes,x andy on the Euclidean plane he enabled us to visualize simple functions like y = f (x). Linear functions appear as straight lines. Function of higher powers as curves. Cartesian coordinate system - Wikipedia

Hilbert space, the home of quantum mechanics, is a the space of vectors described in Chapter 11: The axioms of abstract Hilbert space. Functions in Cartesian space map x points onto y points. The equivalent in quantum mechanics are operators which map vectors onto vectors. We construct complicated vectors in Hilbert space by adding basis vectors in various proportion in a process called superposition. Hilbert space - Wikipedia

We can visualize the addition of moving waves by throwing stones into smooth water. When two stones fall simultaneously close together we see that where the expanding circles of waves spreading from each impact meet they add and subtract from one another to form a complex patterns which seem to pass through one another. We represent waves like this mathematically with vectors.

14.4: The eigenvalue problem

The term eigenvalue reflects the German history of quantum mechanics. In English we could write characteristic equation where characteristic means special or selected. The eigenvalue equation selects stationary points out of the perpetual motion in Hilbert space. Eigenvalues and eigenvectors - Wikipedia

In a vector space of n dimension the operator A is written as a square array of n × n rational or complex numbers. An eigenvector x is a one dimensional string of these numbers and the eigenvalue λ is a rational number. The problem is then to find an operator A and a set of n eigenvectors x and eigenvalues λthat have the relationship

Ax = λx.

λ and the corresponding x are called an eigenpair and thexs are called the spectrum of the operator A. Since λis simply a number, the effect of the matrix A is to change the length of the vector x but it does not change its direction. Since in Hilbert space the information about a state is carried by direction, this result means that an observable feature of a particular state is λ These values appear to be defined in nature to a very high degree of precision, so that the rational numbers representing observable features like the spectrum of a certain photon can be measured and calculated to ten or more decimal digits of precision.

Physicists studying quantum mechanics are continually faced with the eigenvalue problem. The information that they start with is quite sparse with respect to the problem. All they have is experimental measurements of λs, for instance the frequencies of the photons emitted by an atom. From this data they have to work out plausible pairs of operators and eigenvectors to to give the observed results. In general, this is not easy but, it seems, nature does it every time elementary particles interact. Eigenvalue algorithm - Wikipedia

The history of physics since 1900 has been the gradual discovery of the eigenpairs observable in the Universe and the search for operators like the electronic structure of an atom explain them.

The reason for calling this site cognitive cosmology is that I feel that the ability of the Universe to solve what quantum mechanics understands to be the eigenvalue problem suggests that it is intelligent. A solution to an eigenvalue problem seems to be equivalent to an intellectual insight. It is a puzzle rather than a calculation and in the general case we approach a solution by trial and error.

My intuitive feeling about the eigenvalue problem arises is the following scenario:

On introspection I begin with a blank mind, no sharp imagery. Then, in a Cartesian moment which is analogous to the construction and solution of the equation above, suddenly an operator, an eigenfunction and a spectrum of eigenvalues associated with the eigenfunction appear flowing out of the tip of my pen as this real time observation of an idea that occurred about three minutes ago. Manley, D. B., & Taylor, C. S. (1996): Descartes Meditations - Trilingual Edition

This set up and solution of the eigenvalue problem is the act of quantum mechanical insight, formulating the problem and the answer simultaneously. I have noted in Chapter 10 that our brains also work by superposition rather like quantum mechanics. In quantum mechanics, the superposition is strictly linear; in our brains the superposition evolves non-linearly in time by the adjustment of synaptic weights.

Nature, it seems must also work by trial and error, finding stable eigenvectors amid the noise of Hilbert space. These vectors are in effect the genes of stable particles which will derive energy from gravitation as described in chapters 15 and 16 to become the real elementary particles from which the Universe is built.

4.5: Where is the mathematics in the world?

Where is Hilbert space? It is a mathematical space which, I claim, exits prior to and independent of the familiar space in which we live. From this point of view, it is nowhere. Does a bag of beans contain arithmetic? Georg Cantor might say yes. He laid a foundation for arithmetic and formal mathematics by inventing set theory which is an imaginative fiction (which we can represent in writing) which explains arithmetic by putting things (elements) in boxes (sets). Set theory turned out to be a very powerful foundation for mathematics in general. Set theory - Wikipedia

Plato is alleged to have had a sign on the doorway of his Academy saying No-one ignorant of mathematics may enter here. Two thousand years later Galileo drew attention to the fact that the Universe speaks mathematics:

Philosophy is written in this grand book - the universe, which stands continually open before our gaze. But the book cannot be understood unless one first learns to comprehend the language and to read the alphabet in which it is composed. It is written in the language of mathematics. Galileo Galilei (1610, 1957): Discoveries and Opinions of Galileo: Including the Starry Messenger (1610 Letter to the Grand Duchess Christina)

There has long been a question about whether mathematics is created by mathematicians or discovered. The Platonic view is that it exists independently of mathematicians, and is therefore discovered. Given the fact that the Universe began in a state of complete ignorance as a structureless initial singularity, the mathematics which we discover must first have been created by the Universe itself in the process of its evolution.

We say that quantum mechanics is kinematic. It is a puppet, like all mathematics. It does not do itself, it has to be embedded in a dynamic system like a student's brain, a computer or reality to make it work. In the first instance this dynamic system is the initial singularity. I have called it naked gravitation since quantum mechanics has not yet developed the Minkowski space whose metric provides the foundation for general relativity.

Like waves and particles, formal mathematics and actual things are intrinsically united and come into being together. This book is a fictional story of how this might happen, adding flesh to the ancient story of creation which ultimately brought us Jesus of Nazareth who summarized creation in love, the first eigenvalue.

24.6: Controlling chaos

Science proceeds by measurement and the simplest form of measurement is counting, the subject of statistics. Since the dawn of time we have used fire to give us light and warmth and to cook our food but life totally changed when we invented heat engines, mechanisms that turn the energy of fire into mechanical energy capable of driving everything from looms to bulldozers. This discovery powered the industrial revolution which has brought is to our present overheated epoch.

In 1824 Sadi Carnot founded the theory of heat engines, thermodynamics, before people know much about energy or atoms. He devised an imaginary machine, the carnot cycle, which shows how to get mechanical power out of heat. This is as significant a development as the machine Turing imagined a century later to execute computation. Thermodynamics - Wikipedia, Carnot cycle - Wikipedia, Turing machine - Wikipedia

The carnot cycle conserves energy and entropy and performs a kinematic task very similar to the quantum mechanical solution of the eigenvalue problem. Entropy is the ratio of energy to temperature. The carnot cycle operates between a hot source and a cold sink.

Entropy is simply a count of states. Just as many states (say atoms) go into the carnot cycle as come out, so entropy is conserved. The effect of the cycle is to take some of the kinetic energy from each microscopic hot atom (cooling it) and deliver this energy in a single state, an energetic macroscopic motion with entropy zero.

Quantum mechanics and heat engines are examples of creative control producing stable systems out of chaotic motion. The problem we face is that we have burnt the world down using heat engines to twist the world into our own image. The world built itself using quantum mechanics. Our salvation lies in living on sunlight like all other living creatures lest we drive the atmospheric carnot cycle to the point where the mechanical energy of storms destroys us and our built world.

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Notes and references

Galilei (1610, 1957), Galileo, and Stillman Drake (translator), Discoveries and Opinions of Galileo: Including the Starry Messenger (1610 Letter to the Grand Duchess Christina), Doubleday Anchor 1957 Amazon: 'Although the introductory sections are a bit dated, this book contains some of the best translations available of Galileo's works in English. It includes a broad range of his theories (both those we recognize as "correct" and those in which he was "in error"). Both types indicate his creativity. The reproductions of his sketches of the moons of Jupiter (in "The Starry Messenger") are accurate enough to match to modern computer programs which show the positions of the moons for any date in history. The appendix with a chronological summary of Galileo's life is very useful in placing the readings in context.' A Reader.
Amazon back

Links

Carnot cycle - Wikipedia, Carnot cycle - Wikipedia, the free encyclopedia, ' The Carnot cycle is a theoretical thermodynamic cycle proposed by Nicolas Léonard Sadi Carnot in 1824 and expanded by Benoit Paul Émile Clapeyron in the 1830s and 40s. It can be shown that it is the most efficient cycle for converting a given amount of thermal energy into work, or conversely, creating a temperature difference (e.g. refrigeration) by doing a given amount of work.' back

Cartesian coordinate system - Wikipedia, Cartesian coordinate system - Wikipedia, the free encyclopedia, ' A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0, 0).' back

Eigenvalue algorithm - Wikipedia, Eigenvalue algorithm - Wikipedia, the free encyclopedia, ' In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors.' back

Eigenvalues and eigenvectors - Wikipedia, Eigenvalues and eigenvectors - Wikipedia, the free encyclopedia, ' In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by λ, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated.' 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 a vector space equipped with an inner product, an operation that allows defining lengths and angles. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used.' back

Kirchoff's law of thermal radiation - Wikipedia, Kirchoff's law of thermal radiation - Wikipedia, the free encyclopedia, 'Kirchhoff's law states that: For a body of any arbitrary material, emitting and absorbing thermal electromagnetic radiation at every wavelength in thermodynamic equilibrium, the ratio of its emissive power to its dimensionless coefficient of absorption is equal to a universal function only of radiative wavelength and temperature, the perfect black-body emissive power. back

Louis de Broglie (1929), Nobel Lecture: The Wave Nature of the Electron, ' Nevertheless, it was still necessary to adopt the wave theory to account for interference and diffraction phenomena and no way whatsoever of reconciling the wave theory with the existence of light corpuscles could be visualized. The necessity of assuming for light two contradictory theories-that of waves and that of corpuscles - and the inability to understand why, among the infinity of motions which an electron ought to be able to have in the atom according to classical concepts, only certain ones were possible: such were the enigmas confronting physicists at the time I resumed my studies of theoretical physics.Now a purely corpuscular theory does not contain any element permitting the definition of frequency. This also renders it necessary in the case of light to introduce simultaneously the corpuscle concept and the concept of periodicity. On the other hand the determination of the stable motions of the electrons in the atom involves whole numbers, and so far the only phenomena in which whole numbers were involved in physics were those of interference and of eigenvibrations. That suggested the idea to me that electrons themselves could not be represented as simple corpuscles either, but that a periodicity had also to be assigned to them too. In other words the existence of corpuscles accompanied by waves has to be assumed in all cases. However, since corpuscles and waves cannot be independent because, according to Bohr's expression, it must be possible to establish a certain parallelism between the motion of a corpuscle and the propagation of the associated wave. . . .. They showed clearly that it was possible to establish a correspondence between waves and corpuscles such that the laws of mechanics correspond to the laws of geometrical optics. . . .. This prompted the thought that classical mechanics is also only an approximation relative to a vaster wave mechanics. I stated as much almost at the outset of my studies, i.e. "A new mechanics must be developed which is to classical mechanics what wave optics is to geometrical optics". This new mechanics has since been developed, thanks mainly to the fine work done by Schrödinger. . . .. I cannot attempt even briefly to sum up here the development of the new mechanics. I merely wish to say that on examination it proved to be identical with a mechanics independently developed, first by Heisenberg, then by Born, Jordan, Pauli, Dirac, etc quantum mechanics. The two mechanics, wave and quantum, are equivalent from the mathematical point of view. . . .. Since the wavelength of the electron waves is of the order of that of X-rays, it must be expected that crystals can cause diffraction of these waves completely analogous to the Laue phenomenon. . . . Thus to describe the properties of matter as well as those of light, waves and corpuscles have to be referred to at one and the same time. The electron can no longer be conceived as a single, small granule of electricity; it must be associated with a wave and this wave is no myth; its wavelength can be measured and its interferences predicted. It has thus been possible to predict a whole group of phenomena without their actually having been discovered. And it is on this concept of the duality of waves and corpuscles in Nature, expressed in a more or less abstract form, that the whole recent development of theoretical physics has been founded and that all future development of this science will apparently have to be founded.' back

Manley, D. B., & Taylor, C. S. (1996), Descartes Meditations - Trilingual Edition, ' The publication of this English-Latin-French edition of Descartes' Meditations on First Philosophy is quite simply an experiment in electronic scholarship. We decided to make this edition available and to encourage its free distribution for scholarly purposes. The idea behind the experiment is to see how others involved in electronic scholarship might put these texts to use. We have no predetermined ideas of what such use may be when transformed from this origin. The texts have no hypertext annotations except for those used for navigation. We invite others to download this edition and to create their own hypertext annotated editions and then to publish those additions on their own Web servers for everyone to use.' back

Maxwell's equations - Wikipedia, Maxwell's equations - Wikipedia, the free encyclopedia, ' Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields.' back

Planck's Law - Wikipedia, Planck's Law - Wikipedia, the free encyclopedia, 'In physics, Planck's law describes the spectral radiance of electromagnetic radiation at all wavelengths from a black body at temperature T. As a function of frequency ν. back

Set theory - Wikipedia, Set theory - Wikipedia, the free encyclopedia, 'Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.' back

Thermodynamics - Wikipedia, Thermodynamics - Wikipedia, the free encyclopedia, 'Thermodynamics is a branch of physics concerned with heat and temperature and their relation to energy and work. It defines macroscopic variables, such as internal energy, entropy, and pressure, that partly describe a body of matter or radiation. It states that the behavior of those variables is subject to general constraints, that are common to all materials, not the peculiar properties of particular materials.' back

Turing machine - Wikipedia, Turing machine - Wikipedia, the free encyclopedia, ' A Turing machine is a hypothetical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a computer. The "machine" was invented in 1936 by Alan Turingwho called it an "a-machine" (automatic machine). The Turing machine is not intended as practical computing technology, but rather as a hypothetical device representing a computing machine. Turing machines help computer scientists understand the limits of mechanical computation.' back