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¹ Interestingly, however, the discovery of the Pythagorean Theorem presented the problem of “incommensurables,” numbers that cannot be written as the ratio of two whole numbers (like 1/2, 1/3, 2/5). The diagonal of a triangle with two other sides of the length of 1 would be the square root of two. Such a square root cannot be written as such a ratio or as a finite series of digits. See Joseph Mazur, The Motion Paradox, p.18. Geometry happens to present such incommensurables with some frequency, such as the very important number Pi. These incommensurable numbers, by the way, are now referred to as irrational numbers.

² The historical study of the fossil record is clearly interesting. The computer simulation is less impressive. Where no advantage is granted for larger size, random variations appear not to lead to size increases; where there are advantages, the advantages tend to affect the evolution of the species. One wonders that computer simulations were necessary to generate that result. And, of course, the observed trends in real life could still be the results of totally random processes.

³ “If the amazing three volumes of Whitehead and Russell’s Principia Mathematica … (1910–13) had wholly succeeded, our seemingly naïve question would have a direct answer. Something is mathematics if it is logic!” Hacking, Why Is There A Philosophy of Mathematics At All?, p.54.

⁴ Roger Penrose does elsewhere write extensively about the relationships between mathematics and the human mind. E.g., Shadows of the Mind: A Search for the Missing Science of Human Consciousness (1994); The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics (1989). Some of his insights concerning human consciousness are discussed in the later chapter on The Mind. However, he continues to express the view that “major revolutions are required in our physical understanding … [before] much real progress can be made in understanding the actual nature of mental processes.” The Road to Reality, p.21.

⁵ In an earlier work, Penrose identified what he considered to be three deep mysteries that relate one to another the world of our consciousness, the physical world and the world of mathematical forms. Shadows of the Mind, pp.412–4. The first mystery is why mathematical laws play such significant roles in the behavior of the physical world. The second is how the physical world can give rise to entities that are conscious and able to perceive. The third is that that mental capability is able to develop the mathematical models that seem so accurately to describe the physical world.

⁶ Not to suggest that there have not been more detailed explorations of the question. See, e.g., Colyvan, An Introduction to the Philosophy of Mathematics, pp.99–117. One suggestion is that the question focuses on the notable successes and fails to note the instances of failure in the application of mathematics. Similarly, it is possible that “we tackle only the physical problems that are amenable to the mathematical methods we have at our disposal.” Id., p.105. These are, at best, only partial answers.

⁷ This fact may seem particularly strange given that the two branches of mathematics developed historically in different geographical regions and different cultures (geometry from the Greeks and algebra from India through Persia and Islam). Id., p.7.

⁸ One possibility is that we are living in an elaborate computer simulation constructed using the mathematics that we have been in the process of discovering. Thus, perhaps we are simply uncovering the elements of the programming code used in creating the simulation. If so, then it is easy to understand why the mathematics that we develop so accurately reflects the structure of the world in which we think that we live. See Edward Frenkel, “Is the Universe a Simulation?” The New York Times, Sunday Review, February 14, 2014.

⁹ Following a well-established philosophical tradition, Arkes identifies “necessary truths” that can be known independent of experience or as a priori facts. These propositions are things that must be so, because they are innate to the human cognitive process. They include space, time, causality and “the law of contradiction”. The objective of this discussion in the book is to develop Professor Arkes’ theory that the existence of “morals” is a similar “necessary truth.” For example, he asserts that the “skeptic” cannot advance an argument that morals do not exist without resorting to the “logic of morals,” thus falling into a contradiction. First Things, pp.51–84. Perhaps I do not fully understand his argument, but it seems to me that all that is established are that morals and the logic of morals are innate elements of the human being, not that morals exist independently. I have no problem with that more limited assertion. Similarly, I would argue, spirituality is an innate characteristic of the human species, or so I believe, but that does not prove that God exists independently of man.

¹⁰ Kant observed that arithmetic is about counting, which is a process taking place in time. Geometry is about space (spatial relationships). The experience of mathematicians suggests a unity or close analogy between time and space. Is that a characteristic of physical reality? Current neuroscience indicates that different packages of neurons constituting separate cognitive systems provide our understanding of numerosity and of spatial relationships. Yet, the results appear to be analogous. Id., pp.6–8.

¹¹ In 1854, George Boole published a book entitled An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probability. In that work, he undertook to set out in symbolic form Aristotelian logic. Every symbol had only two possible values: 0 for false and 1 for truth. He constructed rules of manipulation for the system. Multiplication is equivalent to “and”; addition is equivalent to “or”. So, for example, a false statement and a true statement is false (0 × 1 = 0); but a true statement and a true statement is true (1 × 1 = 1). In addition, a true statement or a false statement would be true (1 + 1 = 1), and a true statement or a true statement would be true (1 + 1 = 1).This last proposition would not be true for integers; but, for statements that can be either true or false, the logic is understandable and correct. Interestingly and importantly (for practical purposes), it happens that alternating current circuits have the properties of Boolean algebra. An open circuit and a closed circuit in sequence (“×” or “and”) will be open because the current will not flow (1 × 0 = 0). A closed circuit and a closed circuit in sequence will be closed because the current will flow (1 × 1 = 1). A closed circuit and an open circuit in parallel (“+” or “or”) will be closed because the current will flow through the closed alternative path (1 + 0 = 1), and a closed circuit or a closed circuit will also be closed (1 + 1 = 1). Using Boolean algebra, complicated circuits can be symbolically depicted and through the manipulation of the symbols, comparably operating circuits can be devised.

¹² Charles Seife is an experienced science writer and journalist. Several reviews of Seife’s book were quite complimentary, referring to his evident enthusiasm and the potential of his style of writing to engage and interest the lay person. See, e.g., Nicholas Lezard, “Explaining nothing, brilliantly,” The Guardian, 22 March 2003 (“one of the best-written popular science books to have come this way for quite a while…”). A review published by the American Mathematical Society, however, was sharply critical of his generalizations (and some of his history), asking “whether the assertions stem from misunderstanding or simply constitute absurd hyperbole.” Jeremy Gray, “Book Review,” Notices of the AMS, Vol. 47, No. 9, p.1080, October 2000.

¹³ Seife describes how the concept of zero was absent from the classical Western world—unacceptable to the Greek philosophy and not included in Roman numerals. Zero, pp.6–19.

¹⁴ The introduction of zero was of great practical significance in that it enabled Arabic numerical notation (as a place holder so that the same number could be used repeatedly in multiples of 10, depending on the column in which it was placed, as in 1, 10, 100, 1000, etc.). See Whitehead, An Introduction to Mathematics, pp.63–5. It also enabled the development of the algebraic form where a series of values can be expressed as equaling zero. Id., pp.66–8.

¹⁵ Aristotle’s view of infinity was actually somewhat similar to that expressed above. He distinguished between two types of infinity: potential infinity and actual infinity. Potential infinity is the mathematical concept of a series of numbers that never ends. That was acceptable. Actual infinity would exist in the physical world and that Aristotle believed was an impossibility. See, e.g., John D. Barrow, “Beyond numbers—the role of infinity in the understanding of the universe,” University of Cambridge Alumni News, May 2015.

¹⁶ Lee Smolin observes, “There is no more romantic notion than infinity, but in science the concept can easily lead to confusion.” Time Reborn, p.227.

¹⁷ The use of mathematical tricks to eliminate the infinities is widespread. See, e.g., Penrose, Cycles of Time, p.210; Hawking, A Brief History of Time, pp.157, 162. For example, “superstring theory” replaces particles or points with loops. One benefit is that the points would reduce to “singularities” or zeros, whereas the loops have added dimensions that avoid the unfortunate consequences for the mathematics of the zeros and resulting infinities. See Seife, Zero, pp.194–5. The extra dimensions mean “[no]thing, really. … They are simply mathematical constructs that make the mathematical operations in string theory work in the manner that they have to.” Id. p.197.

¹⁸ Barrows refers to this school as “Constructionism,” and he describes three other possible philosophies of mathematics as formalism (mathematics as purely logical constructs limited only by the requirement of consistency), “inventionism” (mathematics as a purely human invention) and realism or the Platonism (mathematical truths exist independently of mathematicians—i.e., mathematical truths are “discovered”). Id., pp.181–86. His usage does not conform to the generally recognized categorizations of Platonism, Logicism, Intuitionism, Formalism and Predictivism. See “Philosophy of Mathematics” (revised 2 May 2012), Stanford Encyclopedia of Philosophy. What Professor Barrows calls Constructionism appears to be more widely called Intuitionism.

¹⁹ The Theorem states that the probability that A wins given that B was excluded equals the probability that B is excluded if A won times the probability that A won divided by the sum of that number plus the results of two other multiplications: the probability that B is excluded even though B won times the probability that B won and the probability that B is excluded if C won times the probability that C won. The key to the result is that the last element, the probability that B is excluded if C won is 1.0, because the moderator is choosing between B and C: if C won, then the moderator must exclude B. (The probability that B is excluded if B won is zero.)

²⁰ Barrow asserted that “mathematics is a language that possesses a built-in logic which is unexpectedly attuned to reality.” Theories of Everything, p.174. I think that that comment could be rather misleading.

²¹ We argue that there must be regularities, otherwise we would not exist. Note that, as suggested above, that argument does not establish that the regularities or Laws must be or are the same everywhere or for all times. They need exist only for a region and a time sufficient to allow the emergence of a hospitable environment and the development of life. Do our Laws of Nature or the constant values incorporated therein (e.g., the speed of light) apply everywhere in the Universe? Have they always been the same and will they remain the same throughout infinity?

²² “All undecidable statements are, directly or indirectly, about infinite sets.” Id., p.185.

²³ The science of “mechanics” evolved from the Greek analyses of the mechanical advantage obtained through the use of lever and similar problems concerning the weights of bodies. Whitehead, An Introduction to Mathematics, pp.46–7. Later, the emphasis shifted to the motion of bodies, like planets or cannon balls, leading to “dynamics.”

²⁴ Leonard Susskind and George Hrabovsky wrote a book that purports to set out the minimum amount of mathematics that one needs to know in order to do physics. The Theoretical Minimum (2013). The mathematics they include goes well beyond trigonometry and calculus.

²⁵ The authors probably should have said “basic laws of mathematics that the Pythagorean School probably knew about.”