Sunday, December 10, 2017

HOW THE MARTIANS DISCOVERED ALGEBRA

How the Martians Discovered Algebra: Explorations in Induction and the Philosophy of Mathematics by Roger E. Bissell (Amazon here in book and Kindle formats) delivers an algorithm for generating Pythagorean Triples. Central to the thesis of the work, Bissell explains how he discovered this by means of induction, not deduction. From there, Bissell takes the reader into number theory in order to validate his new explanation of the proper understanding of multiplication, and to challenge widespread assumptions about the empty set and infinity.

The relationships between music and mathematics go back to Pythagoras. So, this set of essays by musician Roger Bissell enjoys a solid foundation. Bissell also dabbles at mathematics and has several philosophical explorations to his credit, published in The Reason Papers and the Journal of Ayn Rand Studies.

Objectivism is an integration of rationalism and empiricism. Objectivism rejects the false dichotomies of Descartes, Hume, James, and the myriad other philosophers before and after. Consequently, Bissell and other Objectivists provide logically consistent, reality-based and practicable methods for understanding the universe including our inner selves. Objectivism is what the scientific method was intended to be: a guide to living.

That said, this book failed to convince me on several points with which I was pre-disposed to agree. And I concede that the ultimate failure may be mine, not the author’s.

Bissell begins with some techniques in speed math. These discoveries from his senior year in high school demonstrate his inductive method.  They also provide an introduction to his algorithm for discovering Pythagorean Triples. That alone is worth the price of the book. It is easy enough to explain, though hard to show with the typesetting available here. Basically, you want three integers such that a^2 + b^2 = c^2. Easily, there must be some number, x to begin with. The other number must be some number added to x that can be expressed as x+a, and the result of adding their squares must be some (x+b)^2. It all follows from there.

But I had a hard time following it. I tend to read at bedtime. So, I filled my notebook with pages with arithmetic when I was tired. I told Roger that his algorithms did not work. He asked me to send him PDF scans. I did. He corrected my homework. So, I agree that the Bissell Algorithm will, indeed, generate Pythagorean Triples.

Moving forward, Chapter 5, “Mathematics as an Inductive Science”, adheres to the Objectivist interpretation from Ayn Rand, Leonard Peikoff, and David Harriman. The traditional view of induction is expressed as the problem of the black swan: no matter how many examples of anything you gather, the next test can disprove everything you thought you knew. Induction, the traditionalists say, gets stronger and stronger as more examples are revealed, but final proof and final truth are always denied to us. This view is found weakly in the narratives of Richard Feynman and strongly Stephen Hawking's popular work.  (See “Questions about A Brief History of Time” earlier here.) The Objectivist interpretation is that induction is valid if and to the extent that the generalization from specifics is properly based on the correct characteristics. The problem with the black swan is that nothing about “swanness” requires “whiteness.” Swans are not essentially white; swans are not intrinsically white. On the other hand, Bissell points out that water is essentially and intrinsically H2O, which was discovered to be essentially and intrinsically 6 up-quarks and 4 sideways-quarks. Our better understanding did not invalidate the previous truth to the extent that it was, indeed, true.

From acceptable (if not obvious) Chapter 6, “Equations as Propositions,” Bissell moves toward infinity by discussing “Zero as an Operation Stopper” (Chapter 7, which is also about the number One), and the “fiction” of the mathematical Empty Set. There, and approaching infinity (Chapter 8), he loses me; and I am wide awake.

To me, his complaints about the concept of nothing, zero, and the invalidity of infinity and orders of infinity echo the arguments of ancients against the existence of atoms. If atoms exist as the ultimate particles of the existents we perceive, what is between the atoms? Many ancient philosophers rejected the idea of the void: “nature abhors a vacuum.” But if the universe is a continuity of atoms, why are we not locked in atoms like flies in amber?

 http://www.rogerbissell.com
The ancients (some of them anyway) denied Zero. How can you have nothing? From that comes a denial of the validity of negative numbers, irrational numbers, imaginary numbers, dimensions more than 3 (or 4 or 10), or particles that act like waves.

The problem is language. Language allows us to construct nonsense: Mathematics is green. The Catholic Encyclopedia offers some insight on understanding difficult problems. When we say that a brave man is a lion or a clever man is a fox, we know that we are speaking in analogies. “… but no Theist of average intelligence ever thinks of understanding literally the metaphors he applies, or hears applied by others, to God, any more than he means to speak literally when he calls a brave man a lion, or a cunning one a fox.”  (“The Nature and Attributes of God” ) Mathematics is not green, but 5 can be squared. Language is contextual. Language is analogy. Mathematical symbology is highly abstract and compressed.

Words have meanings because language is based on reality. Whatever stars may be, intrinsically or essentially, categorically, provisionally, or contextually, stars exist and we perceive them. But we identify those perceptions in words not known 100 or 1000 or 10,000 years ago. In discussing the translations in their anthology of Aristotle’s works, Terence Irwin and Gail Fine offer this explanation about a single passage in Meteorologia.  Translated as nearly literally as possible, and allowing for alternative understandings [in square brackets]:
"Substance [essence] is said [spoken of], if not more-wise [several-wise], at any rate in four most; for indeed the essence and the universal and the genus [kind] seem [seems] to be substance [essence] of each, and fourth of these the subject." (See Aristotle: Selections, Irwin and Fine, editors, Hackett Publishing, 1995. Google Books here.)
I accept Roger Bissell’s point that the claim to “orders of infinity” is contrary to common sense. It is obvious that there are twice as many integers as even numbers. Nonetheless, you can, indeed map the “larger” set one-t0-one with elements of the smaller set and keep going forever… to infinity…

The central essay, “How the Martians Discovered Algebra” (Chapter 4) is a parable to demonstrate induction in mathematics as the doorway that opened to the world of algebra.

Bissell's original algorithm for generating Pythagorean Triples is worth the price of the book. If you have any interest in epistemology, mathematics, or the problem of induction, then Roger Bissell's book delivers more for the money.

Previously on Necessary Facts

Wednesday, December 6, 2017

More on The Forever War

Admiral James Stavridis and R. Manning Ancell created The Leader’s Bookshelf (reviewed here) by polling four-star generals and admirals on their recommended reading. Also in that volume are suggestions by junior officers. Among those books was Starship Troopers, but not The Forever War. Science fiction has no shortage of war stories but these two are often compared and contrasted.

As a writer, I like to think that I know good writing when I find it. In the stacks of my university library, I opened The Bluest Eye by Toni Morrison to a random page. No way could I ever write that well. Every word was perfect. Similarly, my mother passed Herzog by Saul Bellow to my sister who left it for me. Clearly, it was Nobel material. But neither did I actually read either one: the subject did not grip me. Forever War did. So, did Starship Troopers.

Forever War has an ineffable quality of first person narrative that opens the book with a  briefing and then puts you with the author in a field exercise in engineering, which is where Haldeman served. And that sense of experience continues, even though Joe Haldeman never jumped through a collapsar or wore a cybernetic fighting suit. Starship Troopers opens in an academy classroom, which was Heinlein’s personal experience at Annapolis. Again, the writer was never in servo-controlled armor.

Where Heinlein tells, Haldeman shows. The Forever War is the better read. Culturally, writing styles changed. Heinlein sounds more like Mark Twain and was intended for pulp magazines. It is cerebral. The Forever War was written from perceptions, reflections, and feelings.

Where Starship Troopers followed the formula of a John Wayne movie, The Forever War is closer in spirit to Catch-22 and M*A*S*H. The theme of Forever War is the senselessness of war. The plot is the story of a conscript who rises from private to major through no special talents, but who is lucky enough to survive a few pyrrhic victories. The theme of Starship Troopers is the necessity of military defense. The plot is the story of a volunteer whose training allows him to survive a series of engagements from which his leaders learn valuable lessons. Starship Troopers is romantic. The Forever War is naturalist.

I suspect but cannot prove that many young officers in today’s military recommended The Forever War, just as they recommended Atlas Shrugged. The editors of The Leader’s Bookshelf did not agree with the choice of Atlas Shrugged and therefore mischaracterized the book in their summary. The Forever War did not merit a mention.

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Monday, December 4, 2017

Reflections on Haldeman’s Forever War

“Inside each book is a man,” said Fireman Montag in Fahrenheit 451.  The Forever War (reviewed below) was the work of Joe Haldeman as he was 1974. The story was modified by Joe Haldeman of 1997. The current edition, with a Foreword by John Scalzi and endorsements by William Gibson, Cory Doctorow, Greg Bear, and Stephen King, came out in 2008. Although George Lucas’s frequent repairs to Star Wars: The New Hope may offer a door into revisionism, ultimately the author stands by the work even though time is unkind.

The apocryphal middle third (Chapters 23, 24, and 25) delivers a world impoverished by war. Early in the story, Pvt. William Mandella explains that their IQs of 150+ put him and his team in the draft, the Elite Conscription Act of 1996. They suffer horrible attrition. No wonder the world is getting worse. We are killing off our own best and brightest.

More to the point, Haldeman apparently accepted the premise that war could be good for the economy. At the end, Maj. William Mandella paraphrases his final debriefing: “The fact was, Earth’s economy needed a war, and this one was ideal. It gave a nice hole to throw buckets of money into, but would unify humanity rather than dividing it.” The truth is that power and market are dichotomous, contradictory, and mutually exclusive. If you make economic decisions for military or political reasons, the result must be less than optimal: you will lose money; you will become poorer.

Haldeman's assumption rests on deeper beliefs in the efficacy of a command economy. In order to re-establish order, the UN takes over food. Calories (kilocalories: K) become the unit of money. It is not perfect, of course. The system has safety valves in unregulated farms and tolerated black markets. And for all of the authority evident, gangs run rampant; decent people hire body guards for trips to the grocery. But centralized authorities control what Lenin called “the commanding heights of the economy.” It might not be optimal, but it is better than mass starvation. Or so they claim. But the command economy leads to starvation, with those illicit safeties as the only brakes on the inevitable slide. That the novel took both Nebula (1975) and Hugo (1976) awards speaks to the fact that Joe Haldeman is not alone in his universe.

When the book came out, it was “obviously” supposed to be “about Viet Nam.” True enough, Viet Nam was Haldeman’s war experience. But the story does not parallel that war. The Taurans are, indeed, genetic collectivists; and they copy technology rather than inventing it. But Earth is on its own: we have no corrupt locals to defend. While there are hints of resentment, the streets are not filled with anti-war protestors. And no one in uniform questions the purpose for the war or the strategies for prosecuting it. So, this was not really Viet Nam.
 https://www.va.gov/vetdata/docs/QuickFacts/Population_quickfacts.PDF
But the idea of the forever war resonates within the ranks of American armed forces today. For them, it is GWAT (“gee-watt”): the Global War Against Terrorism. It is an accepted truth, a clichÃ©, that Afghanistan is not a 15-year war, but 15 one-year wars as troops cycle in and out, never to establish and develop the accumulated personal experiences that become institutional memory. We never learn from our mistakes. We never get a feel for how to do things right. For 70 years, the United States had bases in over 50 nations. Today, that includes a division – 1000 soldiers – of combat troops who are “advisors” in Niger. They are not coming home anytime soon.

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Sunday, December 3, 2017

THE FOREVER WAR

The combat veterans who recommended Joe Haldeman’s The Forever War to me spoke at length about a passing detail and failed to identify the important elements of the plot, the theme, or the setting. It is true that relativistic travel will transform you into a visitor from the past. As a soldier, you will be fighting with archaic weapons. But my officers provided more detail there than was found in the novel itself. And they left out the part where everyone in the future is gay.

“In my work, it always comes back to the jungle. Walking through the stillness with fear at your back but, paradoxically, the power of life and death in the chunk of metal in your hands. I’ve never been so weirdly alive, and everything I write is refracted through that lens of experience.” -- “Answering the Call,” by Joe Haldeman, Proceedings of the USNI, April 2011.

Haldeman was a combat engineer in Viet Nam. So, from the first appearance as a series in Analog in 1974, and through its republications, the story has been offered as being “about Viet Nam” and consequently “anti-war.”  It is true that the work was shaped by the author’s experience of war. It is also true that some of the elements of the war between Humans and Taurans reflect that earlier time and place. The Taurans are effective collectivists, willing to sacrifice large numbers in concerted actions. They do not invent new technologies but copy ours. Nonetheless, there’s a lot missing: the “bright shining lies” that victory is around the corner, that our local allies share our values, that our local allies are not as bad as or worse than the enemy; the dark truth that many of the people at home, and also many of the conscripts themselves, are actively opposed to the war. The easy wrapper is that the strengths and attractions of this story are transcendent of time and place.

The weaknesses are those elements that are inextricable from the author's own lifetime. In this future, soldiers are required to sleep in male-female pairs drawn by lot. The burden of orgy falls on the females. Although the army is half female, the only failures we see up close are theirs: one grunt panicking while trapped in rockslide; one officer whose command goes to her head. It is also true that you cannot fault the author for the book he did not write; and it is easier to see the future of 1974 from 2017.

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Monday, November 27, 2017

Joy Hakim's "Aristotle Leads the Way"

Intended for children, The Story of Science: Aristotle Leads the Way by Joy Hakim as many small problems throughout but remains valuable for its sense of life. The author encourages understanding, exploration, discovery, and the integration of knowledge. You can find it remaindered online at prices low enough to gift an entire class of 5th graders, if you choose. Though intended for youngsters, nothing is dumbed down. So the book is enjoyable at an adult level.

 The Story of Science: Aristotle Leads the Way by Joy Hakim, Smithsonian Books, 2004.
The consistent problem is the lack of nuance and insight. Joy Hakim just repeats common claims about the ancient Greeks and science in the Middle Ages. She mentions Hypatia of Alexandria, but says nothing about her being the likely last and best editor of Ptolemy’s work. She never acknowledges Aspasia of Miletus. And like almost everyone else, she accepts and asserts that the ancient philosophers did not think it necessary to test theories but attempted all knowledge through pure logic.

Despite those problems and their consequences in presentations from cover to cover, the rich array of integrated facts should deliver years of engagement and encouragement to a young learner. Hakim does more than note the milestones; she reminds the reader of the road just traveled; and she looks to the next horizon.

 Detail of Raphael's School at Athens. Plato points to the sky, Aristotle reaches for the world.http://www.museivaticani.va/
Through lavish illustrations, and insightful narratives, she presents the mathematics needed at a level that can be grasped with arithmetic and geometry. Among the very many are the Pythagorean theorem (of course), how Aristarchus estimated the distance to the Moon, Democritus’s formula for the volume of a cone, the true nature of the cone as explained by Apollonius of Perga, and the work done by the lever of Archimedes. Occasional timelines remind us that knowledge is passed across generations to those who cared to learn or rediscover.

Writing about Thomas Aquinas, “An ‘Ox’ Who Bellowed” (Chapter 20), Hakim says:
“In the thirteenth century, Paris is the place to be, if you like tumult and activity. While most of Europe is still feudal, Paris is the center of an emerging market economy. Old ideas are being blown away. … Change is both energizing and upsetting. The feudal world was known. What a free, capitalist world might be like is unknown. It seems to offer little security or control. But there is no stopping the new forces. … In the monasteries, clerics are focused on saving their souls through prayer study and isolation. When it comes to science, they quote Pythagoras, Plato, and Augustine. That trio all concentrated, in one way or another, on ideal forms in nature, which often kept them from considering the real world. But at the budding universities, new scholars, inspired by the rediscovery of Greek science, are interested in understanding the forces of nature. Those new scholars are fascinated by Aristotle. Aristotle looked at the world about him and observed, made notes, and classified its inhabitants—plant and animal.” (page 229-230)
—NSTA Recommends
Teacher’s Guide

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Monday, November 20, 2017

COPERNICUS ON THE REVOLUTION OF HEAVENLY BODIES

It was not just that Copernicus put the sun at the center of the heavenly system. His book asserted modern ideas about the relativity of motion, the acceleration of falling bodies, and the nature of the planets. It is also true that many of his arguments are archaic, typical of the medieval scholasticism that rested on the ancient Greeks. Nonetheless, his heliocentric model changed not just how people viewed the universe, but how they thought about it.
That motion is relative was known to the Greeks. Copernicus cites the Aeneid on the point-of-view that the lands slips away as the ship moves past it. (page 519, EB “Great Books” edition). Fire burns brighter at lower altitudes; and when carried to higher altitudes, the fire dies down. (Book I Section; 8. 520, EB). Copernicus says as a matter of fact, not argument, that the planets are “dark bodies” that shine by reflected light. (Book I Section 10; Page 521 EB).

In our era, in school, we learn that Galileo discovered that bodies accelerate when they fall, contrary to Aristotle’s claim that they fall at a constant velocity, determined by their weight.  In fact, Copernicus acknowledged the acceleration of free fall.  in the original edition, Copernicus wrote: “Et quaecunquae decidunt a principio lentum facientia motu velocitatem augent cadendo.” That is rendered today as: "And those which fall downward possess a slow movement at the beginning but increase their velocity as they fall."(Book I Section 8. Page 520 EB)  See also Book III Section 3 where he mentions the motion of a pendulum being faster at the bottom of the arc and slower at the tops.

The acceleration of a body in free fall apparently was known to the “Oxford calculators” of the Merton School who worked about 1325-1350.  They developed a geometric solution for the “mean rate.” The “mean rate” is the average of the initial velocity and the final velocity. Galileo cited their results in The Two New Sciences, Third Day, Theorem I, Proposition I.

Most of On the Revolution of Heavenly Spheres is tedious for anyone not passionate about the geometric arguments. Copernicus had to prove his claims to the thinkers of his time in the language of their culture. The same was true of Newton’s Principia. Richard Feynman attempted to recreate a proof for his own lecture and found that he could not because he did not know enough geometry: we do it all with calculus now. Copernicus provides extensive ephemeris tables of his own measurements. But he also worked out trigonometric tables of “half-sines.” (We call it the “sine” in school, but it is actually the half.)  And he showed how his sun-centered circles explain all of the apparent motions in longitude, oppositions, retrograde, etc. And he often cited Ptolemy, who also made good measurements of the same events.

Copernicus’s On the Revolution of Heavenly Spheres is easily available in modern translations. Several archives have full scans of the original 1543 publication.
The ADS wishes to acknowledge the Lehigh University, Digital Library ("The Problem of the Planets") for providing the scans of the book. (

The Smithsonian also has one of the 1543 editions in its Dibner collection. (See https://airandspace.si.edu/exhibitions/explore-the-universe/online/kiosks/dibner/) The image title says: Nicholas Copernicus, De Revolutionibus Orbium Coelestium Libri VI (Basel, 1543), but the title page says “Norimbergae” i.e., “Nuremberg.”

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Sunday, November 19, 2017

Questions about “A Brief History of Time”

As I said earlier, the problem may be that Hawking is trying to explain in vernacular English concepts that can only be expressed in the equations of quantum mechanics and relativity. Nonetheless, I marked up my copy of A Brief History of Time with questions, problems, and internal contradictions.

An example of the limits of vernacular English (at least as Hawking uses it) is found on page 82: “A star is formed when a large amount of gas (mostly hydrogen) starts to collapse in on itself due to its gravitational attraction. … The heat released in this reaction, which is like a controlled hydrogen bomb explosion, is what makes the star shine.” What “controls” it? It is just an explosion that lasts millions of years. Hawking is careless with common language.

The worst failings are found in the discussions of action near or very near or at a black hole. Hawking believes that if you fell into one, you would be instantly destroyed (page 139, but also 89 and 112). Hawking does seem to get it right on page 87. The fact is that Einstein’s theory of relativity, and as offered by Hermann Minkowski (1864-1909) earlier, as you approach the speed of light, time slows down for you. The astronaut who falls into a black hole might never experience his own death.

Hawking’s presentation of the history of astronomy is shallow, with no evidence of scholarship, insight, or nuance. He leaves out Aristarchos of Samos (?310-230 BCE), Hipparchos of Nikea (190 to 120 BCE), and Archimedes. He also wrongly blames Aristotle for every error in ancient science.

The truth is that Aristarchus put the sun at the center of the universe. And he did that based on knowledge he inherited and built upon. Sir Thomas Heath wrote several books about the history of mathematics and astronomy. Among them was Aristarchus of Samos, the ancient Copernicus; a history of Greek astronomy to Aristarchus, together with Aristarchus's Treatise on the sizes and distances of the sun and moon: a new Greek text with translation and notes. London: Oxford University Press (1913). This is not an obscure reference: it has always been available. Other editions of Aristarchus’s work also were published. Hipparchus of Nicea (190 to 120 BCE) is famous in our time for his measurement of the precession of the equinoxes. The fact that the celestial north pole moves was always a significant problem to ancient astronomers. Precession of the equinoxes was cited by Copernicus as evidence that the Earth moves. Hawking mentions none of that. I understand that he did not want to clutter the book with asides, but presenting a proper history of science is important this book, especially.

If the Earth goes around the sun, then we ought to be able to measure the parallax, the apparent shift in position of the stars across the sky, say winter to summer. Archimedes tried to measure it but failed. He concluded that either the Earth is at the center, or the universe is unimaginably larger than their estimates. The point is that contrary to Hawking’s dismissive claim that the ancient Greeks did not value experiment, Archimedes measured to test a theory. Hipparchus measured. Eratosthenes measured. Ptolemy measured. They did not just spin airy fantasies.

Hawking attacks Aristotle several times. And he says that about Aristotle, that he placed the Earth at the center of the universe “for mystical reasons.” (page 2) Aristotle was not a mystic: he had good reasons for putting the Earth at the center. “The Aristotelian tradition also held that one could work out all the laws that govern the universe by pure thought: it was not necessary to check by observation.” (page 15). Despite the many errors we find his works, Aristotle argued for beginning with observation and experience. His description of the development of the chick embryo is still considered one of the greatest experiments in the history of science. Moreover, it was Plato who argued for beginning with ideas (eidos means "form"). Even so, in his dialog “Protagoras”, Plato has Socrates give clear examples of how empirical knowledge triumphs in a political debate. Hawking wrongly claims: “Aristotle believed that all the matter in the universe is made up of four basic elements: earth, air, fire and water.” (page 16; also 9). In fact, it was Empedocles of Akragas (490-430 BCE) who posited earth-water-air-fire. Aristotle added a fifth element: aether. (It became the “quintessence” of alchemy; and we still use “quintessential” rhetorically.) The ancient theory is not far from our modern ideas about solids, liquids, gases, and plasma.

As he opens with attacks on Aristotle, Hawking closes by disparaging Einstein. (He does acknowledge other contributions, such Einstein’s work on Brownian motion and the photo-electric effect.) On page 155, Hawking claims that “… Einstein refused to believe in the reality of quantum mechanics.” That is not true. Einstein, SchrÃ¶dinger, and others were not convinced of the “Copenhagen interpretation” offered by Bohr, Heisenberg, and others. SchrÃ¶dinger’s cat was offered as a reductio ad absurdum: the cat in the box is neither dead nor alive until you lift the lid and look.

While Hawking gives full credit to Galileo, he says nothing about the Oxford calculators of the Merton School (c. 1325-1350). Their work was cited by Galileo (Two New Sciences, Third Day, Theorem I, Proposition I: the mean speed theorem). Relative motion was known to the ancients. Copernicus cited Virgil’s Aeneid on the appearance of the land moving away from a ship.

Among the many weak analogies, Hawking explains Newton’s theory of gravity: “Thus the force between two bodies would be twice as strong if one of the bodies (say, body A) had its mass doubled. This is what you might expect because one could think for the new body A as being made of two bodies with the original mass. Each would attract body B with the original force. Thus the total force between A and B would be twice the original force” (page 16). That is approximately true. Two bodies cannot occupy the same space at the same time. So, A and A-prime would attract B with different forces. But Hawking never says “approximately.” He just moves on to give a poor explanation of why bodies of different weight fall at the same rate (page 17).

Among the many internal contradictions based on weak philosophy, Hawking says “… a theory is just a model of the universe, or a restricted part of it, and a set of rules that relate quantities in the model to observations we make. It exists only in our minds and does not have any other reality (whatever that might mean).” But on page 42: “Despite the success of his model and his prediction of Hubble’s observations, Friedmann’s work remained largely unknown in the West until similar models were discovered in 1935 by the American physicist Howard Robertson and the British mathematician Arthur Walker, in response to Hubble’s discovery of the uniform expansion of the universe.” If models were discovered then they can exist independent of our perception of them, like other laws of science, or mathematical truths. They are not arbitrary, existing only in our minds. Hawking equivocates without acknowledging the fundamental problem, or his assumption about it. Near the end (page 167), he says again: “As was explained in Chapter 1, we could never be quite sure that we indeed found the correct theory, since theories can’t be proved.” (See the same also on page 139.)

Hawking trips over philosophy on page 74: “The success of the unification of the electromagnetic and weak nuclear forces led to a number of attempts to combine these two forces with the strong nuclear force into what is called a grand unified theory (or GUT). This title is rather an exaggeration: the resultant theories are not all that grand, nor are they fully unified, as to do not include gravity. Nor are they really complete theories because they contain a number of parameters whose values cannot be predicted from the theory but have to be chosen to fit in with experiment.” What he means is that many constants are determined empirically.

Hawking has a preference for idealism and rationalism. Important as they are to epistemology, they are only half of the equation. Hawking prefers that all constants be derivable by theory, like Planck’s constant. That the gravitational constant G had to be measured (first by Henry Cavendish in 1798) is a fact of life. You could just set it equal to 1 and then adjust the meter and everything else to that. In the 19th century, mu-nought, the magnetic constant, was taken as 2 and made fundamental to the definition of the ampere, the unit of current. Today, electricians define the amp from the volt, and then magnetic permeability from those. The history and utility of dimensionless constants is beyond the book or this review. But it remains that Hawking is not comfortable with the empirical discovery of physical constants; and that is an error in his epistemology.

And it would seem to be interesting to Hawking and his readers that Maxwell’s equations can be manipulated to show that the speed of light squared is equal to the reciprocal of the product of the magnetic constant and the dielectric constant, but he does not mention it.  Again, he cannot say everything in under 190 pages. That limit, however, unlike the speed of light, was not written in the stars.

Despite Hawking’s claim (Chapter 4) that the Newtonian universe is deterministic, it is well-known (and has been since about 1801) that there are no easy solutions to a 3-body problem and no synthetic solutions to an N-body problem. In other words, from Newton’s famous equations, we can approximate well how the Earth and Moon interact. Add the Sun, and it gets difficult. Predictions about the relative positions of the eight planets, their moons, and the Sun are impossible, except by approximations using iterative numerical methods. Equations alone won’t do it.

Another example of Hawking’s shortfalls with Newtonian mechanics is found on pages 164-165 when he discusses counterfactual universes. He is attempting to outline the difficulties in constructing a consistent framework that includes relativity and quantum mechanics. He says: “There would also be problems with more than three space dimensions. The gravitational force between two bodies would decrease more rapidly than it does in three dimensions…. In fact the same behavior of gravity with distance in more than three space dimensions means that the sun would not be able to exist in a stable state with pressure balancing gravity.” In point of fact, it is a truth in Newtonian mechanics that motion in a conservative field takes place in a plane. Orbits are two dimensional. Perhaps the problem is that how or why F = G (Mm / r^2) would become some F* = G (Mm / r^3) or … r^1.5 cannot be explained in a book without equations – and that is not so much Hawking’s sin as his publisher’s, though he was complicit in the omission.  And, again, with the philosophy: the sun is not “stable.”  Whether some counterfactual physical laws could describe a sun with a longer or shorter life, the fact is that the sun is a process.

Long, long ago, I sang an ABC song for my daughter:
“These are the rules of astronomy.
We’re beset by entropy.
Things run down
And things burn out.
This I know without a doubt.
O B A F G K M.
Then they start all over again.”

Discussing the electromagnetic force (page 70) Hawking says: “A large body, such as the earth or sun, contains nearly equal numbers of positive and negative charges. Thus the attractive and repulsive forces between individual particles nearly cancel each other out, and there is very little net electromagnetic force.” OK… what about the fact that the Sun and the Earth both have magnetic fields, though Mars and the Moon do not? I realize that he cannot discuss everything in 182 pages, but neither would it have hurt to have said a little more to the millions of us who do know some science.

Hawking’s explanation of the diffraction slit paradox ignores the essence in the problem. He also assumes no common experience with waves. “That is to say, the crests of one set of waves may coincide with the troughs of the other set. The two sets of waves then cancel each other out, rather than adding up to a stronger wave as one might expect.” (page 57-58) Why would you expect that? In fact, he illustrates both cases on page 57. He goes back and forth calling electrons waves and particles, and that’s fine. But with the diffraction grate experiment, what happens if you try it with baseballs rather than photons (or electrons)?

His drawing on page 22 leads to the apparent conclusion that the farther you are from something, the faster light travels.

The explanation of arbitrary coordinate systems (page 23) offers a poor example of polar coordinates. When the Pythagorean theorem is applied to the illustration on page 25 the distance from the sun to Alpha Centauri comes out in years-miles, as if you could mix apples and oranges. The drawing on page 31 is greatly exaggerated without explanation. If starlight were bent so much by the sun’s gravity, that fact would have been obvious to the Babylonians.

Figure 5.2 on page 73 is labeled “A proton and an antiproton collide at high energy, producing a couple of almost free quarks.” Actually, the picture could be anything in a cloud chamber. It reminded me of the article in the Journal of Irreproducible Results on how to save money in research by using the same illustration, but changing the caption. What is an “almost free quark”? Did it come into existence almost without the exchange of energy? Are these “almost free quarks” almost bound together as almost-electrons? He never says.

Toward the close, discussing “The Arrow of Time” (Chapter 9). Hawking offers this: “Suppose the pieces of the jigsaw start off in a box in an ordered arrangement in which they form a picture. If you shake the box, the pieces will take up another arrangement. This will probably be a disordered arrangement in which the pieces don’t form a picture. … but the more you shake the box… the pieces will be in a completely jumbled state…” (page 146).  True enough.  But it is well known to mathematicians that if you want to pack marbles in the box efficiently, you can pour them in as a heap, and then shake the box to recursively find the optimal arrangement.

The bottom line is that this brief book full of golly and gee-whiz about our incomprehensible universe fails on close inspection.

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