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 has 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.
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


Monday, November 20, 2017


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 land 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.
Made available electronically by the NASAAstrophysics Data System (ADS) 
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 The image title says: Nicholas Copernicus, De Revolutionibus Orbium Coelestium Libri VI (Basel, 1543), but the title page says “Norimbergae” i.e., “Nuremberg.”


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 following 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), Eratosthenes (276-194 BCE), and Archimedes (?287-212 BCE). He also wrongly blames Aristotle for every error in ancient science.

1. 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. 

2. 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 to this book, especially.

3. Eratosthenes is famous for estimating the diameter of the Earth by measuring shadows on the first day of summer. (The story is a bit more complicated. He might not have done the actual work himself but only ordered it done by scholars working for him at the Great Library of Alexandria.  See Circumference by Nicholas Nicastro reviewed here.)

4. 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 Aristotle 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.

(See Aristotle Leads the Way by Joy Hakim, reviewed here.)

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, Schrödinger, and others were not convinced of the “Copenhagen interpretation” offered by Bohr, Heisenberg, and others. Schrödinger’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. To Einstein (and other modern physicists) the cat is either alive or dead, regardless of whether or not you look. In other words, Einstein was an objectivist or positivist who believed that reality exists independent of our perceptions: if we all were dead, everything else would still be here, including quantum packet wavicles, whatever they are.

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.  (See "Copernicus on the Revolution of Heavenly Bodies" reviewed here.)

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, are these:

1. 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).” 

2. 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. 

3. 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.)

4. 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 fails on close inspection.


Sunday, November 5, 2017

President Trump and the Military

A recent poll by the Military Times supports academic studies by staff officers: Commissioned officers are a self-selected elite while non-commissioned officers and regular troops tend to more closely reflect the civilian population. Commissioned officers more often disapprove of the President and his policies. That gulf is even wider among Air Force and Navy officers.

"Overall, about 44 percent of all troops surveyed in the Military Times poll have a favorable view of Trump, while roughly 40 percent have an unfavorable opinion of him. That’s a stark contrast to opinion polls of the general public, which have shown Trump’s popularity at less than 40 percent and an unfavorable rating as high as 56 percent.
"Yet, the poll of more than 1,100 active-duty troops, conducted in September, shows a deep divide over service members’ opinions of the commander in chief, whose first nine months in office have been marked by military policies that have drawn both praise and concern from Pentagon leaders.
"While almost 48 percent of enlisted troops approve of Trump, only about 30 percent of officers say the same, the poll shows
 Officers have college degrees. Some enlisteds do, also, but officers are required to have completed bachelor's degrees. In the higher ranks, master's degrees are common. Many colonels and generals earned doctorates. Again, granted, among non-commissioned officers, especially master sergeants or senior chief petty officers (E-8) some also complete college or university degrees, but more often, regular troops who earn advanced diplomas become commissioned officers. So, the final tally is that President Trump enjoys less support among educated citizens--even (or especially) those who are self-identified as conservative or Republican.

On this blog in July, I reviewed three studies of political attitudes within and about the military. (See "Do You Know Your Military?") Gen. James Mattis, Lt. Col. James K. Dempsey, and Col. Heidi Urben all found that enlisted soldiers and sailors more closely reflect the populace from which they come while commissioned officers, especially staff officers, do not.

More to the point, President Trump enjoys only grudging support from the conservatives within the Republican Party. His populist programs on trade and immigration set him against those who favor free enterprise.

Similarly, while those in military service often adhere to traditional opinions about gender and sexuality, the fact is that the historical trend has been for the inclusion of any and all who volunteer to serve. With some notable exceptions, the realities of war first brought women in as nurses (Army 1901; Navy 1908). Beyond that the U.S. Navy accepted its first female petty officer in 1917. (The Army waited until World War II.) A parallel history applies to race: For 200 years, African-Americans served with distinction, honor, and bravery while suffering discrimination. President Truman desegregated the military in 1948. Now the discussion is about transgendering. (No one seems to be trans-racing.) The President is out of step with America's military leaders.

The Military Times poll also showed political support for the President to be weaker within the Navy and Air Force than within the Army and Marines. The easy explanation is that the Navy and Air Force require more technical ability and critical thinking.

From Pearl Harbor to 9/11
Challenge Coins 
World War II Dinner and Dance
The Scientific Method