28th
The Concept of Laws. The special status of the laws of mathematics and physics
Platonic mathematical world illustrated in Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, Jonathan Cape Ltd
“We have a closed circle of consistency here: the laws of physics produce complex systems, and these complex systems lead to consciousness, which then produces mathematics, which can then encode in a succinct and inspiring way the very underlying laws of physics that gave rise to it.”
“Plato made it clear that the mathematical propositions – the things that could be regarded as unassailably true – referred not to actual physical objects (like approximate squares, triangles, circles, spheres, and cubes that might be constructed from marks in the sand, or from wood or stone) but to certain idealized entities. He envisaged that these ideal entities inhabited a different world, distinct from the physical world. Today we might refer to this world as the Platonic world of mathematical forms.”
— Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe
“There are some, King Gelon, who think that the number of the sands is infinite in multitude; and I mean by sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited. Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its multitude. And it is clear that they who hold this view, if they imagined a mass made up of sand in other respects as large as the mass of the earth, including in it all the seas and the hollows of the earth filled up to the height equal to that of the highest mountains, would be many times further still from recognizing that any number could be expressed which exceeded the multitude of the sand so taken. But I will try to show you, by means of geometrical proofs which you will be able to follow, that, of the numbers named by me and given in the work which I sent to Zeuxippus, some exceed not only the number of the mass of sand equal in size to the earth filled up in the way described, but also that of a mass equal in size to the universe.”
— Archimedes, The Sand Reckoner
“If we go back to our checker game, the fundamental laws are rules by which the checkers move. Mathematics may be applied in the complex situation to figure out what in given circumstances is a good move to make. But very little mathematics is needed for the simple fundamental character of the basic laws. They can be simple stated in English for checkers.”
— Richard Feynman, American physicist, Nobel Laureate in Physics (1918-1988), The Character of Physical Law
„The fact that the physical world conforms to mathematical laws led Galileo to make a famous remark. “The great book of nature – he wrote – can be read only by those who know the language in which it was written. And this language is mathematics.” (…)
It is the mathematical aspect that makes possible what physicists mean by the much-musunderstood word theory. Theoretical physics entails writing down equations that capture (or model, as scientists say) the real world of experience in a mathematical world of numbers and algebraic formulas. Then, by manipulating the mathematical symbols, one can work out what will happen in the real world, without actually carrying out the observation. (…) For example, by using Newton’s laws of motion and gravitation, engineers can figure out when a spacecraft launched from Earth will reach Mars. They can also calculate the required mass of fuel, the most favorable orbit, and a host of other factors in advance of the mission. And it works! The mathematical model faithfully describes what actually happens in the real world. (…) How can you possibly know what a ball will do by writing things on a sheet of paper? (…) Why is nature shadowed by a mathematical reality? Why does theoretical physics work?”
How Many Laws Are There?
As scientists have probed deeper and deeper into the workings of nature, all sorts of laws have come to light that are not at all obvious from a casual inspection of the world, for example, laws that regulate the internal components of atoms or the structure of stars. The multiplicity of laws raises another challenging question: How long would a complete list of laws be? Would it include ten? twenty? two hundred? Might the list even be infinitely long?
Not all the laws are independent of one another. It wasn’t long after Galileo, Kepler, Newton, and Boyle began discovering laws of physics that scientists found links between them. For example, Newton’s laws of gravitation and motion explain Kepler’s three laws of planetary motion and so are in some sense deeper and more powerful. Newton’s laws of motion also explain Boyle’s law of gases when they are applied in statistical way to a large collection of chaotically moving molecules.
In the four centuries that have passed since the first laws of physics were discovered, more and more have come to light, but more and more links have been spotted too. The laws of magnetism, which in turn explained the laws of light. These interconnections led to a certain amount of confusion about which laws were “primary” and which could be derived from others. Physicists began talking about “fundamental” laws and “secondary” laws, with the implication that the letter were formulated for convenience only. (…) The Great Rule Book of Nature (at least as it is currently understood) would fit comfortably onto a single page. This streamlining and repackaging process – finding links between laws and reducing them to ever more fundamental laws – continues apace, and it’s tempting to believe that, at rock bottom, there is just a handful of truly fundamental laws, possible even a single superlaw, from which all the other laws derive.”
— Paul Davies, Cosmic jackpot: why our universe is just right for life, Houghton Mifflin Harcourt, 2007, p. 9-11.
What it is that mathematicians study?
“But, as the mathematician speculates from abstraction (for he contemplates by abstracting all sensible natures, as, for instance, gravity and levity, hardness and its contrary, and besides these, heat and cold, and other sensible contrarieties), but alone leaves quantity and the continuous, of which some pertain to one, others to two, and others to three [dimensions].”
— Aristotle, The Metaphysics of Aristotle, translated by Thomas Taylor. London: Davis, Wilks, and Taylor, Chancery-Lane, 1801.
“It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.”
“Since the Renaissance mathematicians have been concerned with the laws of cause and effect. That is, they have sought to employ the mathematical expression of these laws so that for any particular phenomenon there exists a one-to-one relationship between each cause and each effect. Their goal has been to solve the following simple-sounding problem: Given a unique cause, predict the unique effect. Many phenomena are well suited to this type of mathematical analysis, but there are also situations in which this approach fails. When this happens scientists have learned to rephrase the problem in the language of probability. One modern version of this idea can be described as follows: Given a unique cause, predict the most probable effect. A more general version of the problem is: Given the most probable cause, predict the most probable effect. These types of probabilistic problems are now a fundamental part of science, but this is a fairly recent development.”
— John Tabak, Mathematics and the Laws of Nature. Developing the Language of Science , Facts On File, Library of Congress Cataloging in Publication Data, 2004, p. 153.
Does mathematics help explain the physical world or does it actually hinder a grasp of the physical mechanisms that explain the how and why of natural phenomena?
“Mathematics – our creation – is stunningly effective in explaining and limiting the physical world, even very elementary mathematics, mere counting. Why should mathematics, this human invention, be so effective, so relevant to nature, so controlling of it? Perhaps the mathematics that we create is forces by nature and that is why it can describe its world. Mathematics is not arbitrary, it was developed to describe our universe (at least that part that actually does) so is able to. And this seems more and more true but more and more incomplete, more and more puzzling.”
— R. Mirman, Our Almost Impossible Universe: Why the Laws of Nature Make the Existence of Humans Extraordinarily Unlikely, iUniverse, 2006
By the time we reach the seventeenth century and the Newtonian revolution in physics, the problem reappears in the context of a change of criteria of explanation and intelligibility. This has been beautifully described in an article by Y. Gingras (2001). Gingras argues that “the use of mathematics in dynamics (as distinct from its use in kinematics) had the effect of transforming the very meaning of the term ‘explanation’ as it was used by philosophers in the seventeenth century”. What Gingras describes, among other things, is how the mathematical treatment of force espoused by Newton and his followers — a treatment that ignored the mechanisms that could explain why and how this force operated—became an accepted standard for explanation during the eighteenth century. After referring to the seventeenth and eighteenth centuries discussions on the mechanical explanation of gravity, he remarks:
This episode shows that the evaluation criteria for what was to count as an acceptable ‘explanation’ (of gravitation in this case) were shifting towards mathematics and away from mechanical explanations. Confronted with a mathematical formulation of a phenomenon for which there was no mechanical explanation, more and more actors chose the former even at the price of not finding the latter. This was something new. For the whole of the seventeenth century and most of the eighteenth, to ‘explain’ a physical phenomenon meant to give a physical mechanism involved in its production….The publication of Newton’s Principia marks the beginning of this shift where mathematical explanations came to be preferred to mechanical explanations when the latter did not conform to calculations. (Gingras 2001, 398)
(…) Another challenge has been raised by Sorin Bangu 2008, who claims that mathematical language is essential to the formulation of the question to be answered (“why is the life cycle period prime?”) and thus that the argument begs the question against the nominalist. The existence of numbers and properties of numbers is already assumed in the acceptance of the statement “the life cycle period is prime”.
A similar objection to any attempt to use mathematical explanations in physics for inferring the existence of the mathematical entities involved in the explanation had already been raised in 1978 by Mark Steiner, who had discarded such arguments with the observation that what needed explanation could not even be described without use of the mathematical language. Thus, the existence of mathematical explanations of empirical phenomena could not be used to infer the existence of mathematical entities, for this very existence was presupposed in the description of the fact to be explained. Indeed, he endorsed a line of argument originating from Willard Van Orman Quine and Nelson Goodman according to which “we cannot say what the world would be like without numbers, because describing any thinkable experience (except for utter emptiness) presupposes their existence.” (1978b, 20)
— Explanation in Mathematics, Stanford Encyclopedia of Philosophy
Mathematics: Invented or Discovered?
Mario Livio discusses the complementary processes of mathematical invention and discovery. While we invent some mathematical concepts—such as prime and imaginary numbers—by deciding how to define them, these concepts can lead to a plethora of mathematical discoveries.
— Mathematics: Invented or Discovered?, World Science Festival, Jan 14, 2011.
Mario Livio: Platonism vs. Formalism
Platonists believe that there is a universal truth underlying all of mathematics. Formalists believe all of mathematics can be defined by a set of predefined rules. Ever wonder about the deeper significance of these two critical mathematical philosophies? Using thought experiments like the Allegory of the Cave and the Barber’s Paradox Mario Livio, untangles these two didactic ways of viewing the world and the very nature of human knowledge.
— Mario Livio (Astrophysicist), Platonism vs. Formalism, World Science Festival, Jan 11, 2011.
[This note will be gradually expanded…]
See also:
☞ The Limits of Understanding - debate between Mario Livio (Astrophysicist), Gregory Chaitin (Mathematician), Rebecca Goldstein (Novelist, Philosopher), Marvin Minsky Cognitive Scientist), Sir Paul Nurse (Nobel Laureate, Medicine), World Science Festival video, Jan 11, 2011.
☞ Cosmologist Paul Davies: Faith in the Mathematical Order, World Science Festival, 2011.04.15
☞ Gerald B. Folland, Speaking with the Natives: Reflections on Mathematical Communication (pdf)
☞ Vlatko Vedral: Decoding Reality: the universe as quantum information
☞ Geoffrey West on Why Cities Keep Growing, Corporations and People Always Die, and Life Gets Faster
☞ Mario Livio, Why Math Works - Is math invented or discovered? A leading astrophysicist suggests that the answer to the millennia-old question is both, Scientific American, August 2, 2011
☞ Greg Chaitin on the limits of Reason (pdf) - Ideas on complexity and randomness originally suggested by Gottfried W. Leibniz in 1686, combined with modern information theory, imply that there can never be a “theory of everything” for all of mathematics.
☞ Steven Weinberg, Symmetry: A ‘Key to Nature’s Secrets’, The New York Review of Books, Oct 27, 2011
☞ Tim Maudlin, What Happened Before the Big Bang? The New Philosophy of Cosmology, The Atlantic, Jan 2012.
☞ Is mathematics invented or discovered?, answers on Quora
