Thought Laboratory

To almost everyone this would seem a ridiculous question. Even to most mathematicians the existence of Pi (defined here as the ratio of a circle’s circumference to it’s diameter) is taken for granted. But the question of the existence of the class of number to which Pi belongs is a key example of a fundamental question in the philosophical understanding of Mathematics, and indeed of a broader issue in metaphysics and epistemology.

To understand the meaning of Pi, let’s start with a quick review of the types of numbers we encounter in elementary mathematics. We start with the “natural numbers” – the numbers 1,2,3 and so on. The very fact that these are the numbers we first learn and first teach to our children in their infancy is ultimately the key to understanding the issue in front of us. In the natural numbers, there are no negative numbers, and no zero.  The extension of the number system to include negative numbers I’ve discussed previously.  The concept of zero I’ll defer yet again, though I’ll hint that I believe it represents balance. 

 Both natural numbers and integers are “countable” – meaning that they can be ordered and indicated in order.  Considered as sets, they both contain a “countable infinity” of elements. 

The next extension of the number system is to include all fractions – the division of natural numbers by natural numbers, and while we’re at it we throw in division of signed integers by signed integers other than zero.  This new number system we designate as the “rational numbers”.  The natural number or the signed integer “n” is represented in the rational number system as ratios n/1, 2n/2, 3n/3, etc.  (Note that I say the signed integers are “represented” – I maintain that the signed integer n is different than the rational number n/1, 2n/2, …).   The nature in which rational numbers exist is again the topic for another discussion, though I’ll note that their nature is related to the epistemological process of measurement, and the mathematical concept of continuity.  I’ll also note that the rational numbers can be so ordered as to be countable, and so are “countably infinite” in number.

The next step is a bit tougher to follow.  The “algebraic numbers” are defined as all numbers which are solutions to polynomial equations whose coefficients and exponents are natural numbers.  In a simpler form, it is this number system that sparked the first significant philosophical debate about the meaning of numbers – back in Greece in the time of Pythagoras.  What we now refer to as the “Pythagorean Theorem” states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the other sides, or a^2+b^2=c^2, where c is the hypotenuse.  In its simplest form, if we are computing the length of the diagonal of a square with unit length sides (a=b=1), then c*c=2.  That the value of c, the square root of 2, cannot be expressed as a fraction of natural numbers was a major discovery of Greek mathematics. (This fact was proven by members of the Pythagorean school, possibly by one Hippasus who may have been executed for revealing the proof because it had been held as a state secret).  A good deal of debate occurred through the centuries over the existence of these “irrational numbers”.  I have not spent enough time considering the nature of these numbers to offer an opinion as to what they represent, but I will venture that this number system will be found to have an acceptable basis in reality.

As with the rational numbers, the algebraic numbers can be ordered in such a manner as to be countable.  There are therefore a countable infinity of algebraic numbers.

Both the rational and algebraic numbers can be shown to be “dense” on the real number line.  Think of the real number line as an “infinitely long” straight line, as you did in high school.  If we take any interval of this line, there will be contained in that interval an infinite number of rational numbers, no matter how small an interval is chosen.  This is the density property of a set of numbers.  This observation, in combination with the fact that all of these number classes are “countably infinite” leads to some confusing conclusions.  In the sense that we can order and count the elements of these number classes, we can establish a “one to one” relationship between all rational numbers with the natural numbers.  “In a sense”, this implies there are “as many” rational numbers as natural numbers – despite the fact that between any two natural numbers there are an infinite number of rational numbers!  The resolution of this confusion lies in the use made here of the concept “infinite” – which ultimately is not a valid concept.  But that is another digression.

The last development in this trail of number classes is the discovery of transcendental numbers.  The historical development of this number class derived from the problem of “squaring the circle”, which was a phrase used to describe the problem of computing the area of a circle with a rational-valued diameter in terms of algebraic numbers.  It was shown in 1882 that this is impossible, because Pi is not the solution of any polynomial with natural number coefficients.  Pi then belongs to yet another class of number.  Rather than deal with the area problem, let us define Pi in the simplest possible manner.  Pi is the ratio of the circumference of a circle to its diameter.  Hence, if a circle has a diameter equal to a natural number of units (or a rational, or even algebraic number of units), its  circumference will be a transcendental number. 

Using this definition, we have for the first time a number tied to a perfect geometrical figure.  A circle is an ideal figure that can never be found in the concrete world of objects.  Any “circle” you encounter in the physical world will be imperfect at some level.  “Circle” is a idealization of these experienced real-world circles.  It can be thought of as the limiting perfection of all physical circles.  And so, the question of whether Pi “exists” will become the question of whether this idealization can be said to exist.  More generally, it is the question as to whether a conceptual idealized abstraction can be said to exist.  Such an abstraction can never be exhibited in the physical world.

The answer to this question lies in the fundamental philosophical principles that are applied.  The empiricist will require that to “exist”, a thing must be present in the physical world, and furthermore, must be experienced.  Such an approach will then conclude that ideal circles do not exist, and therefore Pi does not exist.  The rationalist/Platonist will be happy to place the idealizations in their own realm (ultimately a realm occupied by a God), and declare that they in fact exist in this realm, but can only be approximated in the denigrated physical realm.

Finally, the Objectivist will answer with clarity.  The concept circle – as with all other human concepts – exists as a relationship between the human mind and the physical world.  To exist, the concept must have physical representations from which the details specific to each (such as imperfections) have been abstracted away (omitted as measurements) to arrive at the concept in human consciousness.   The concept circle then exists as an epistemological linkage between human understanding of the physical world, and the real world itself.  And since the circle exists, it is in the same sense that the number Pi (and ultimately all transcendental numbers) exist.

The acceptance of transcendental numbers leads to additional confusions involving the invalid concept of infinity.  Unlike the other number classes, it can be proven that there is an uncountable infinity of transcendental numbers.  A one to one relationship between the transcendental numbers and the natural numbers cannot be established.  Indeed between any two rational numbers (or even algebraic numbers) there is an uncountable infinity of transcendental numbers.  (Intriguingly, a one-to-one relationship can be established between transcendental numbers and the set of all subsets of natural numbers, which in turn implies that the set of all subsets of natural numbers is uncountably infinite as well). 

The further discussion of how these “infinities” and the concept of continuity and “infinite divisibility” should be approached to allow a resolution of the apparent contradictions will need to wait for another time.  Interested parties should consult writings and lectures given by Pat Corvini for excellent discussions of these topics.

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I’ve just completed “reading” (listening to) The Black Cloud, by astronomer Fred Hoyle. It has been refreshing to discover a book – science fiction at that – which I can enthusiastically recommend to almost any reader. Hoyle was a successful astronomer who turned to writing science fiction later in life. Hoyle’s science fiction emphasizes the science more so than most authors of the genre, and in particular almost all modern authors. The Black Cloud describes the proper process by which scientific discovery and inquiry advances. In particular, for those interested and knowledgable in astronomy, it is a wonderfully accurate depiction of the science as it was practiced in the 1950s. My fascination with the book also derives, in part, from being old enough to remember at least some of the technology that Hoyle describes, though I was experiencing it much later, in the 1980s, not the 50s.

The story in the book (which I won’t spoil here) does get sufficiently unusual to easily classify as science fiction, but continues to bring in fascinating philosophical sidelights. Unlike most science fiction, I’d even go so far as to say there was no obvious lack of plausibility in the story line – though I’m sure if I thought about it harder there would be plenty of mud to throw, it is fiction afterall. For the faint of heart, I’ll warn you that many people die, though the description of the calamity is not particularly detailed – I might even classify it as “callous”. But for Hoyle, the story is only a vehicle to make some very deep points about the philosophy of science, the nature of information, the nature of life, and even voice some frustrations about government.

What I want to discuss here are the thoughts he provides on the nature of science – I may very well come back to other thoughts he expresses in this wonderful book in later discussions. The point he makes (only stated openly twice in the book, but demonstrated continuously) is that observations of correlations or coincidences are not a proper basis for science, and do not represent causality. This is certainly not a new sentiment presented by Hoyle, but it is one that is essential to grasp, and which is of increasing relevance in what passes today as “science”.

The purpose of science is to determine the causes of phenomena. By a cause we mean an antecedent particular entity or event whose existence results in the existence of the phenomenon. The determination of cause is a structured activity, involving observation, formulating a hypothesis, prediction, and verification.

The correct progression in the development of a scientific theory starts with observation of the phenomenon under study. The researcher may then find correlation between the phenomenon and other entities or events. This must lead to the formation of a hypothesis which explains the connection between these antecedents and the phenomenon, and this explanation must be formulated within the context of existing knowledge. That is, the hypothesis cannot be arbitrary, whimsical, or rely upon unknown forces or entities. The hypothesis must arise from inductive reasoning from the observations made of the phenomenon and its antecedents. Any valid hypothesis must further be capable of making predictions which can be subsequently verified.

Hoyle’s commentary specifically deals with the critical step of when a hypothesis becomes a theory – when it is accepted as an explanation, and the cause of a phenomenon established. Hoyle points out that the mere observation of correlation is never sufficient to establish a connection; furthermore, that attempting to “prove” the hypothesis by deductive reasoning using the correlation and hypothesis itself as a starting point leads to completely invalid chains of logic, and cannot bring additional validity to the hypothesis. The only way to validate a hypothesis is to use it, in conjunction with other observations if necessary, to make predictions of future observations, and then to make those observations and confirm the predictions.

Let’s have a look now at the actual text from The Black Cloud. It is additionally interesting how Hoyle manages to present his ideas in very short passages. He further shortens them by using the Russian character “Alexandrov”, who speaks very tersely. After a few comments from Alexandrov that only later become interesting (and to which I’ll return), the first example of Hoyle’s attack on poorly constructed science arises after the astronomers observe a strange behavior in how the Cloud reacts to block radio transmissions. After reviewing the actual observations, the scientists begin hypothesizing a feedback mechanism, but without explaining how it would work. Here is the relevant passage:

“Let’s go into this in a bit more detail,” … “It seems to me that this hypothetical ionising agency must have pretty good judgment. Suppose we switch on a ten centimetre transmission. Then according to your idea, Chris, the agency, whatever it is, drives the ionisation up until the ten centimetre waves remain trapped inside the Earth’s atmosphere. And — here’s my point — the ionisation goes no higher than that. It’s all got to be very nicely adjusted. The agency has to know just how far to go and no further.”

“Which doesn’t make it seem very plausible,” said Weichart.

“And there are other difficulties. Why were we able to go on so long with the twenty-five centimetre communication? That lasted for quite a number of days, not for only half an hour. And why doesn’t the same thing happen — your pattern A as you call it — when we use a one centimetre wave-length?”

“Bloody bad philosophy,” grunted Alexandrov. “Waste of breath. Hypothesis judged by prediction. Only sound method.”

The next outburst by Alexandrov follows a comment about ESP experiments:

“I know this is rather a red herring, but I thought these extra-sensory people had established some rather remarkable correlations,” Parkinson persisted.

“Bloody bad science,” growled Alexandrov. “Correlations obtained after experiments done is bloody bad. Only prediction in science.”

…“What Alexis means is that only predictions really count in science … It’s no good doing a lot of experiments first and then discovering a lot of correlations afterwards, not unless the correlations can be used for making new predictions. Otherwise it’s like betting on a race after it’s been run.”

The final passage uses an example that can easily be remembered. This both reinforces the theme discussed so far, and suggests another variant of the theme. The situation leading to this passage requires some explanation. The governments of Earth had launched nuclear missiles into the cloud. These had been redirected by the Cloud to return to their points of origin, with “random perturbations”. Three major cities had been destroyed.

“It looks to me as if those perturbations of the rockets must have been deliberately engineered,” began Weichart.

“Why do you say that, Dave?” asked Marlowe.

“Well, the probability of three cities being hit by a hundred odd rockets moving at random is obviously very small. Therefore I conclude that the rockets were not perturbed at random. I think they must have been deliberately guided to give direct hits.”

“There’s something of an objection to that,” argued McNeil. “If the rockets were deliberately guided, how is it that only three of ‘em found their targets?”

“Maybe only three were guided, or maybe the guiding wasn’t all that good. I wouldn’t know.”

There was a derisive laugh from Alexandrov.

“Bloody argument,” he asserted.

“What d’you mean ‘bloody argument’?”

“Invent bloody argument, like this. Golfer hits ball. Ball lands on tuft of grass — so. Probability ball landed on tuft very small, very very small. Million other tufts for ball to land on. Probability very small, very very very small. So golfer did not hit ball, ball deliberately guided on tuft. Is bloody argument. Yes? Like Weichart’s argument.”

“What Alexis means I think,” explained Kingsley, “is that we are not justified in supposing that there were any particular targets. The fallacy in the argument about the golfer lies in choosing a particular tuft of grass as a target, when obviously the golfer didn’t think of it in those terms before he made his shot.”

The Russian nodded.

“Must say what dam’ target is before shoot, not after shoot. Put shirt on before, not after event.”

“Because only prediction is important in science?”

“Dam’ right. Weichart predicted rockets guided. All right, ask Cloud. Only way decide. Cannot be decided by argument.”

I will return to discuss the new aspect which I believe Hoyle has introduced in this particular passage in my next post.

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Over this winter vacation, I’ve been intellectually focused on some background mathematics supporting my work in detection. One of my projects is reading through a rather introductory text on probability by Athanasios Papoulis (1965). I’ve found the opening sections of this book very philosophically relevant, in addition to being a promising text book – if I can ignore the philosophy being presented.

Papoulis opens the text with a very blunt statement, which instantly alerts me to the conceptual framework in which this mind is operating: “Scientific theories deal with concepts, never with reality”. What follows in his introduction is a justification for approaching the subject purely from a deductive process starting with a set of axioms, and not worrying – much – about the relationship between the theory and “the real world, whatever that means”. Among the blunt statements in the introduction is this:

To conclude, we repeat that the probability P(A) of an event A must be interpreted as a number assigned to this event, as mass is assigned to a body. In the development of the theory, one should not worry about the “physical meaning” of P(A). This is what is done in all theories.

After this very open confession of intellectual sterility, the author gets down to work by asking how probability – which he has just declared to be arbitrarily defined numbers – should be defined. He offers three strawmen definitions, and then settles on a fourth. The strawmen are of some interest as well:

(1) Relative Frequency Definition. This is probably the definition most commonly assumed – that probability is the ratio of the number of outcomes of interest, divided by the total number of trials. This needs to be more carefully constructed as the limit of this ratio as the number of trials “goes to infinity”, or, as I’ll even more carefully say it, as the number of trials becomes arbitrarily large. Papoulis rejects this approach as too cumbersome, because no matter how many actual trials are made, we never approach “infinity”, and therefore can never say that we have a sufficient number of trials to establish meaning for this ratio.

(2) Classical Definition. Here the probability of an event is determined by a-priori reasoning about the situation at hand. For example (Papoulis’ example) a six-sided (fair) die has six equally possible outcomes, so the probability of getting a one on a die roll is 1/6. No experiment need be made to reach this (rational) conclusion. This strawman is knocked down by stating that this really only can work for simple cases, and works most easily only when the outcomes are of equal likelihood. A couple examples are given to show that determining the proper probability in this manner is very error prone. For example, one could attempt to determine the likelihood of rolling a 2 with two die by saying the number of possible outcomes is 11 (2,3,…,12), and we are interested in the value 2, so the probability is 1/11, which is of course wrong.

(3) Measure of Belief. This one seems to be thrown in as an easily dismissed psychologically based argument.

Papoulis chooses to define probability from three axioms, and claims to then proceed to develop the entire theory of probability from these axioms (plus one more minor extension). The axioms seem ridiculously primitive:

The probability of an event A is a number P(A) assigned to this event. This number obeys the following three postulates, but is otherwise unspecified:
I. P(A) is positive or zero
II. The probability of the certain event is 1 [the certain event always occurs]
III. If A and B are mutually exclusive [they both cannot happen in the same trial] then P(A+B)=P(A)+P(B). [The probability that A or B happens is equal to the sum of the probability that A happens and the probability that B happens].

And that’s it! That’s his “definition” for probability, which clearly lacks any meaningful tie to the “real world, whatever that means”. So he can proceed in philosophical comfort.

Or can he?

Interestingly, in order to start his progression from these axioms, he needs to introduce a large segment of set theory – otherwise he cannot define what an “event” is, which is contained in this definition of probability. Without belaboring set theory here, an “outcome” is one possible result of a trial of the process we are trying to test. The set of all possible outcomes is the “certain event” mentioned in the definition – one element of the certain event will always be the outcome of any trial, so the probability of the certain event is 1. An “event” is then any subset of the set of all possible outcomes.

Next, we encounter a very bizarre twist in this “axiomatic” probability theory. Not all events, says Papoulis, can have a probability assigned to them. That is, not all sets of possible outcomes can be given a number that will meet the axiomatic conditions from which the theory of probability will be developed. Papoulis does not confront this problem directly, treating it as merely an annoyance, and refering the reader to measure theory for a better explanation, but he does give a major example to indicate where the problem lies. Suppose the outcome of a process that we are interested in could be any real number (real numbers are all of the common numbers from “negative infinity” to “postive infinity”, including all rational and irrational numbers). Then the “certain event” is the set of all real numbers. But consider the event which is the set consisting of a single number, say the set {3}. If every set of this type is given a probability, there will be an “infinite” number of these probabilities, and in order for axiom III to remain true, each of these probabilities would need to be zero. Then for any specific outcome of a trial of the process – A -, the event {A} would have a probability zero – which is a clear contradiction.

The escape from this contradiction is itself quite bizarre, and only partially explained (we are referred to measure theory for a more complete explanation). For this example, only events that can be formed from the union or intersection of a countable number of continuous intervals or isolated points will be given a probability. There are sets that cannot be so formed (we are told they are complicated to construct, and I recall similar constructions from my days of formal math training, and I agree with him), and these will not be given probabilities.

If this escape seems to make little sense – and Papoulis seems to understand that the reader will not be able to make sense of this – he offers a better escape: “…one can construct certain pathological sets that are not countable intersections or unions of intervals. Sets of this kind have no probabilities, but are of no importance in applications, and we can forget them“.

Now this is a simply amazing demonstration of a rationalist getting boxed into a corner, and then escaping inelegantly by refering us back to the real world (whatever that means) which he has already denied can be consulted in developing a proper mathematical “theory”.

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This Fall my reading has centered on David Harriman’s recently published “The Logical Leap: Induction in Physics”. I started off listening to the book on Audible, then realized that there was too much detail I was missing, so I went ahead and purchased a hard copy. After a great deal of reflection on the book, I’ve reached the conclusion that this is a very significant philosophical advance, largely derivative from Ayn Rand’s epistemology. The main thesis in the work consists of several observations regarding how the proper process of induction is performed, and how following this process leads to the attainment of truth. The presentation of these ideas is accomplished through relatively brief explanatory material in the opening and closing chapters of the book, and through a lengthy series of examples from the history of science through the central mass of the work.

There has been much commentary on the accuracy of the presented history, and whether the errors in this history result in an invalidation of the theory of induction Harriman presents. This lead me to review original (translated) material from Galileo in particular to judge the accuracy for myself. Indeed, I found clear errors in Harriman’s account of some of Galileo’s work on falling bodies, matching the factual criticisms by John P. McCaskey. I am also generally skeptical of anyone who attempts to describe the thought processes that lead a researcher to perform various experiments and reach conclusions. In the case of Galileo, my skepticism is raised because of the stylized nature of Galileo’s writings (in artificial dialogues), where the actual thought process is not likely to be the thought process presented in the work. However, Galileo was very explicit about his theory of knowledge and support for proper scientific method in other writings. Though Harriman may take some license with his portrayal of the details of Galileo’s thought process, he does portray Galileo’s underlying theory of knowledge perfectly.

The errors Harriman makes in describing Galileo’s falling body experiments center on indicating that Galileo had not conducted experiments in an aqueous medium, and that if he had he would not have been lead to the induction of universal gravitational acceleration. However, Galileo describes such experiments in detail, and in fact used the results of these experiments to inductively conclude that the acceleration of falling bodies in a vacuum is independent of the falling body’s mass. Although there is a curious confusion here in Harriman’s account (given the breadth and depth of Harriman’s research into the history of science), the error is not essential to Harriman’s evidence for the nature of the inductive process. I conclude therefore that his inaccuracies do not affect the validity of Harriman’s theory of induction.

I still have not read through the original works of the other scientists (and proto-scientists) that Harriman uses as examples, though I do have selections from most of them (Ptolemy, Newton, Lavoisier) and I will eventually read through this material. I am willing in the meantime to accept Harriman’s accounts as representing at least the “essence” of their thinking.

Regarding Galileo himself, I have greatly enjoyed reading both of his Dialogues – On the Two New Sciences, On the Two World Systems, as well as Letters on Sunspots, The Assayer, and some of his other letters dealing with the relationship between science and the Church. Throughout these, we see Galileo declaring the proper source of scientific truth – induction from observation – and disdaining the peripathetic argument from the “authority” of Aristotle. In several places, Galileo states that Aristotle himself would change his conclusions if he were presented with the observational evidence available to Galileo. Since I am also a great proponent of Aristotelian logic (though not his “science”), I found these statements gratifying.

Galileo’s struggle with the Church is fascinating. Far from being an atheist, he defends his “freethinking” by relying upon other ecclesiastic authorities to make his argument, in particular Augustine. Augustine I had written off as the worst of the Christian “philosophers” (and such he remains, from an ethics viewpoint); however, he offers at least a partial defense of science by separating matters of faith from matters of fact. In matters of fact, says Augustine, the Bible should not be interpreted literally, and as we discover new explanations for phenomena through the use of logic and observation that are at apparent odds with scripture, it is our interpretation of the scripture that should be questioned and changed, not the use of logic that should be abandoned. Another form of this argument suggests that since our rational faculty is a given to us by an perfect God, both its use and scripture must lead to Truth. When those truths conflict, it is our faulty understanding of scripture (which is the word of the unfathomable) which is in err. Of course both of these approaches to balancing faith and reason are ultimately fatally flawed, as no definition of the boundaries of the arbitrary field of “faith” can be described.

I would enjoy reading the entire body of Galileo’s work – of which an immense quantity apparently exists, filling many volumes. But, amazingly and disturbingly, the vast majority has never been translated into English. There is an apparently comprehensive translation from the Italian dialects in which he wrote to French, but no one has found a reason to translate most of the work to English. Given the astounding range of Galileo’s contributions (astronomy, mathematics, mechanics, dynamics, fluid dynamics, meteorology, not to mention the philosophy of science) this is a disturbing fact.

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It’s been far too long since this space has been active, though you’ve heard that sad tale before. The best I can do for tonight is list some recent reading and provide shallow commentary. Much of my free time has gone into teaching Astronomy, and the rest into developing the hobby of astrophotography. I’ve still maintained a reading schedule, but without much reflection on content.

After completing Durant’s Age of Faith this spring, I read some minor fiction and then moved into a reading of an old textbook, The Use of Force, which is a large collection of essays regarding foreign policy, culminating in the Cold War tactics of the 1980s. This reading was mostly of historical interest, bringing out tidbits such as the dreadnought arms race prior to WW1, and some fascinating history of the development and very slow adoption of technology in armed forces. It also pointed out my lack of knowledge of the Cold War era.

I also read a history of Spain, though that was largely forgettable – the book itself was too superficial, and written to long ago to cover the Fall of Franco.

More recently, I read most of “Conceptual Foundations of Scientific Thought” by Marx Wartofsky. This work was of some interest, though the author focused on pragmatic interpretations of the philosophy of science, with an emphasis on linguistics. Eventually I stopped reading at the final section dealing with “modern physics” as it was sure to become intolerably painful.

I’ve also read through a few science fiction works, including Stranger in a Strange Land, which has been entertaining up until the final parts of the story, and the Mote in Gods Eye pair of books by Pournelle, which were a bit more entertaining.

One obscure book I just read this weekend was “Glide Path” by Arthur C Clarke. Not quite an SF story, this is a fictionalized account of the development of ground tracking radar in WW2. Mildly interesting, but really only for real nerds. Not much of a plot, nor character development.

I am planning to return to the Objectivist canon next, though I’m currently entertaining myself with Dashiel Hammett.

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I do accept comments on this blog; however, the frequency of spam comments is simply astounding. Despite the use of a spam blocker, I am still getting a couple obviously spam comments each month. The spammers are improving their techniques, using algorithms that produce comments that can look legitimate. I’ve just cleaned out the queue once again, and it’s possible that I’ve thrown out two or three actual valid comments. If you wish to place a comment in the future, either contact me through email (those of you who know me) or refer to something specific in the post that you are commenting upon. Otherwise you’ll most likely end up in the trash.

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We recently re-joined Audible.com, and the first “new” book I purchased was Basic Economics, by Thomas Sowell.  I had read many of Sowell’s articles posted on Capitalism Magazine (www.capmag.com) and found them to be very clearly written and always in agreement with free market principles.  Sowell has published a couple dozen books, mostly on economics – which actually was a concern for me in selecting to read (alright, listen to) his work.  I often worry that a prolific author may either be poorly edited, or repetitive.  I had also worried that a book entitled “Basic Economics” may have little to add to my reasonable knowledge of the subject.

The book is ponderous – in print it is 640 pages; as an audio book it ran over 18 hours.  I had mistakenly thought it was shorter because of how Audible had structured the downloads, but I was pleasantly surprised when the book did not end after the first 12 hours – and for a book on economics, being listened to in a car, that’s saying alot!  Sowell accomplishes a quite thorough review of major elements of economics at an introductory level, while making the material accessible and just barely entertaining.  In every instance where I was beginning to grow impatient with the length of the discussion on a topic, he either brought the topic to an end, or threw in some intriguing real-world case study.   I have only a couple minor complaints about the structure of the book.  There are the odd “Overview” chapters, occuring at the end of each major section, and which appear to contain more than mere summaries, might be misleading, and seem awfully long.  There are a few instances of straight out repetition of the text, which seem to be accidental – the kind of thing any editor who read the entire book would find and correct.

Sowell’s overall theme of the book is that the principles of economics are really quite simple, but become confusing in the popular mind when mixed with emotion, psychology, and politics.   He clearly defines economics as “the efficient allocation of sparse resources which have alternative uses” – and if you haven’t memorized this after he repeats it at least 50 times throughout the text, then you haven’t read the book.  He does an excellent job of boiling each element of economics down to fundamental principles – supply and demand as the fundamental of the value to be exchanged for an item, the difference between value and price as determined by the money supply, the nature of profit and loss and their effects on business, the fact that labor is just another commodity to be traded.  His coverage of banking and the financial system is a bit light, but accurate, and probably as deep as he can go without causing confusion in his target reader.

The most interesting sections for my advancement in understanding were in his treatment of risks and insurance, and his discussion of international trade.  He clearly describes the difference between an insurance policy – run by a profitable business – and the so-called government insurance programs, which he rightly identifies as merely a form of forced redistribution of wealth from the younger to the older generations.   In an insurance company, the study of risk is paramount, and premiums can be computed scientifically, based on the statistics of claims of various sorts for the various classes of clients.  In the government programs, where the insurance is an “entitlement”, risk is irrelevant, premiums are independent of class (other than being assigned as a percentage of income), and the funds collected as “premiums” are intentionally confused with general tax collection funds and spent as the current government sees fit.

In the international trade section, Sowell provides outstanding descriptions of how the fallacy of the “zero-sum game”, wherein any wealth transfered between countries is seen as a loss for debtor and a gain for the creditor, can be easily refuted, by noting that wealth is constantly being created through investments.  The conclusion is that with very rare exception any trade occuring between countries, regardless of the balance of exports and imports (in goods or funds) is greatly beneficial to both countries involved.

Equally as strong as his general themes are his selections of examples.  In explaining the economics of big business, he provides a lengthy description of the history, and change in market positions, of such companies as Sears and Roebuck, Montgomery Ward, JC Penny, McDonalds, White Castle, A&P, and Walmart.  These are fascinating histories in and of themselves, and a separate book just discussing these and similar histories would make extraordinarily interesting reading.

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This is more of a “status” post than a topical one.  My time for “thinking” has become spread out over a broad space of activities, with a particular focus on my astronomy hobby and the course I’m running for children, as well as a significant increase in the level of thought devoted to my occupation.  This has reduced the intensity with which I’ve been pursuing big-picture thinking; I’ve fallen into a sort-of autopilot mode.  This phase will end, but for now my stimulus remains the material I’m reading.

My current “large” reading project is the fourth volume of Durant’s History of Civilization – “The Age of Faith”.  I’m finding this particular volume to be really slow going, in part because my reading time has shrunk a bit, but that in turn is in part because this reading is, well, boring.  What is of some interest (and may become a more complete topic later on) is the history and core philosophy of Islam.  This isn’t new information for me, but in the current world context it is more relevant now than the last time I exposed myself to this history (in some other work).  My interest this time around centers on these themes:

1. Islam in its original form nominally supported a laissez-faire economy – what caused this to change?
2. What caused the fall of the “golden age” of Islam, when most of what we know of the classical world was retained only through the translation of works by the Islamic society?  Just when the Islamic world was beginning to make an impact on scientific thought, the progress ceased.

In parallel with this reading, I’ve started reading e-books once again.  The recent acquisition (Christmas) of an ITouch ensures that I’ll be doing quite a bit more with e-books in the short term, as the Kindle reader is SO much more useable than Microsoft Reader on the Dell Axim (which is still a fine PDA, just becoming dated and starting to fail).  In the past month, I’ve read Billy Budd (Melville), All Around the Moon (Verne), Journey to Other Worlds (Astor), and currently Michael Strogoff, Courier of the Czar (Verne).

Each of these contained surprises.  Billy Budd remains a great timeless story of the balance between ethics and military discipline.  An exceptional English merchant sailor is impressed into naval service, runs afoul of an evil officer of the Navy who dislikes him because of his exceptional character, and ends up killing the officer with a single blow when the officer accuses him of assisting in plotting a non-existant mutiny.  At a time following a recent mutiny elsewhere in the Navy, the military law is strictly enforced, which makes striking an officer (no less killing him) a capital offense, regardless of cause or circumstance.  The required sentence is carried out.  Is this an ethical outcome? I believe the answer to be unclear.

Jules Verne has always been a fascinatingly confusing author, and these two works just add to that confusion.  “All Around the Moon” was published some 5 years after the more famous work “From the Earth to the Moon”.  In the earlier work, preparations for a launch of a spacecraft, using a ballistic cannon, are completed after a lengthy development of technology and inter-personal politics (note that I have NOT read this earlier book).  Apparently, at the very end of “From the Earth to the Moon”, the launch occurs successfully, but nothing is said about the fate of the voyage.  “All Around the Moon” recounts the voyage itself.  This is a fascinatingly boring book – Verne takes the story as a stage on which to narrate at exhausting length on the physics of spaceflight.  He is amazingly accurate on many of the topics that he covers, especially because he is completely mistaken about some of the fundamental physics involved.  He uses a ballistic approach to achieving escape velocity, even makes approximately the correct calculation, but has the occupants of the spacecraft continually experiencing the force of gravity, holding their feet to the floor toward the Earth until they reach a “neutral point”, then flipping over to have the base of the craft, and the gravitational force, pointed toward the Moon.  Very surprising that he did not understand that during ballistic flight one always experiences “free fall” (weightlessness).  To make the story even less appealing, the spacecraft misses the Moon due to the gravitational influence of a “comet”, and winds up in the Pacific Ocean after circumnavigating the Moon.

In Michael Strogoff, Courier of the Czar, we get the “other” Verne.  This is an adventure story, along the lines of “20000 Leagues Under the Sea”, while “All Around the Moon” is in line with “Around the World in 80 Days”, which I recall to be another fascinatingly boring book.  But Verne’s adventure stories are truly excellent, and Strogoff is a wonderful book to read.  A combination of a predictable plot outline – Strogoff needs to travel from Moscow to Irkust as quickly and quietly as possible, and we know this within the first 5 pages – and a good dose of mystery and suspense – there is a Tartar rebellion that threatens the Czar’s brother in Irkust, and the rebel leader is traveling in disguise – add together intricately to keep the reader glued to the story (or in my case, the screen).

What I can’t understand is how one author can have created both of these streams of work – and honestly, how monstrousities like Around the World in 80 Days can be considered great works.  I understand that Verne’s publisher (Hetzel) had a great influence on his writing, but both All Around the Moon (as well as the earlier part of that story) and Strogoff were published by Hetzel, so this is not the explanation.

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I recently read William Gilbert’s renaissance masterpiece “On the Lodestone”, and found his work to be extremely intriguing.  Gilbert lived at the dawn of the scientific era, from 1544-1603, predating Francis Bacon – originator of the modern scientific method – by about 20 years.  He is considered by some to be the father of electromagnetism, and indeed is the first to use the term “electricity” in describing what we now know as static electricity (the primary source of static electricity in Gilbert’s experience was amber; the Greek word for amber is elektron). 

Gilbert is a staunch defender of the experimental basis for Truth – at least most of the time.  He attacks earlier writings concerning magnetism as simple repetitions of prior writings which generally are based in nonsensical assertions, which could be easily discounted if anyone bothered to acquire a magnet and observe its behavior.  In the sixteenth century, the primary source of magnetism was the naturally occuring mineral magnetite (ferrous-ferric oxide), or lodestone.  It should be understood that the best (purest) lodestones are barely capable of lifting iron objects of their own weight – very unlike modern “magnets” which generally can lift objects much heavier than themselves, particularly in the case of rare earth magnets which can lift several thousand times their own weight.

Gilbert attacks many myths about the magnet – that coating a lodestone with garlic oil removes the magnetism; that a diamond placed near a lodestone similarly destroys its power; that electrical attraction and magnetism are the same force.  In each case he makes strong derogatory statements about earlier authors who never even saw, let alone tested, a lodestone.  Gilbert proceeds to build a set of facts and observations of his own, each supported by experiment.  Many of his observations I had never considered before (as a trained physicist).  For example:

• To determine the north and south poles of a magnet, allow the magnet to rotate freely in Earth’s magnetic field.  Mark the end that points to geologic north as the south pole, and the end that points south the north pole.  Which of course makes perfect sense, since opposing poles attract.
• Apply a magnet’s pole to the center an iron bar, thereby magnetizing the bar.  If you use the north pole of the magnet, this will create north poles at both ends of the bar.   If the bar is curved into a C shape, there will be a repelling force between the cusps of the C.
• Cut a magnet in half, holding one piece firmly. The cut ends will immediately repel each other, causing the free magnet to rotate rapidly to bring the opposite end to face the cut end.  This implies a continual stress present in  the material near the poles of any magnet.

Gilbert uses spherical lodestones, which he calls “terrellas” for “little Earths” for many of his demonstrations.  Using a device of his invention, the “versorium” – basically a compass needle mounted on a very free-turning point – he maps out the magnetic field lines of the terrella, and demonstrates their equivalence to the directions in which a compass points as it travels over the Earth.  Furthermore, he demonstrates the equivalence of the “dip” of the versorium at high “latitudes” on the terrella with the corresponding subtle dip of an accurate three-dimensional compass observed by navigators as they sail in higher latitudes.  The dip is caused again by the attraction of the pole, which is both north and “under” the compass increasingly as we reach higher latitudes.

There is a wealth of additional experimental and empirical information Gilbert conveys in this work, about not only magnetism, but static electricity as well – I am only remembering the highlights as I write this, some 3 months after finishing it.  And so Gilbert would appear to be a solid hero of scientific reasoning, living at the very end of the middle ages, and opening the door to the coming scientific revolution.  And, as far as the material above, this is certainly the case.

The first inkling we have that Gilbert may not be consistent in his scientific thinking is when he begins describing the relationship between the lodestone and the Earth.  He accurately shows that iron ore and lodestone are related – the one is attracted to the other; the iron can take on weak magnetic properties after exposure to lodestone.  But then he makes a large leap – which just happens to be true – in asserting that the Earth is mostly made from magnetic materials (iron and lodestone), and that what we experience on the surface – bodies of water, various soils, mountains and canyons – are but aberrations of the Earth that exist only on the relatively small surface in comparison to the bulk of the planet.  He, of course, has no experimental evidence for this claim (our experimental evidence came hundreds of years later in mapping how earthquake tremors penetrate the planet).

However, his entire thesis for the work is to explain magnetism, not merely describe its effects and laws.  And this is where he turns shockingly away from reason.  The magnet is aligning itself to the Soul of the Earth – so he asserts without demonstration.  Further, the Earth is a living Being, and this Soul is not a literary euphemism – it is asserted to be real.   After building up a large assortment of truly impressive scientifically-verified facts, and teasing the reader along the way, indicating that his studies have lead to a determination of the true Nature of the lodestone, he quite suddenly moves from demonstration to dogmatic assertion.  In addition to the Earth as a living entity, he goes on to assert that all celestial bodies are alive, each with its own Soul, and that each will exert a force on the material from which it is made, just as the Earth exerts a force on lodestone and iron.  He winds up this strange path through the irrational with an appeal to astrology – that these same forces affect the development of humans born under the various stars and constellations.  If it weren’t so tragic, it would almost be comical.

Gilbert is a fascinating example of an intellectual genius caught between two radically different philosophical worlds, with one foot planted in each.

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In reading modern discussions of evolution, and in various discussions, it has become clear to me that there is a difference between what I view as evolution theory, and what that label has come to mean in popular usage.  Reducing evolution to its fundamental and necessary principles, it requires the following conditions:

(1) Organisms that are capable of reproduction, and of passing traits to their offspring
(2) Organisms that may cease to exist (“die”) prior to their reproduction
(3) Competition among organisms for resources or conditions required for survival (at least until reproduction)
(4) Variability in traits between individual organisms, due both to inheritance and random occurence (“mutation”)

Evolution is loosely defined as “survival of the fittest”.  More carefully defined, it is the progression of the probability distribution of traits among a reproducing population of organisms which meet the four conditions described above.  The key to the qualitative progression of the distribution is the adversity to survival implied in condition (3), though each of the other conditions is necessary.  Using this definition, there can be no debate about whether evolution occurs – it is a mathematical consequence of this combination of conditions. 

Note that in this definition of evolution, the concept of “genes” does not occur directly, only the transmission of traits from generation to generation.  This is not to say that the modern theory of DNA genetics is invalid – it is a scientifically established fact that this is the mechanism for trait transmission in known life forms – but it is not an essential characteristic of evolution.  Also, note the absence of the concept of species.  The identification of species as a class of organisms that can mutually reproduce remains a useful definition, but it is not a dominant factor in evolution.

Any discussion of evolution as defined above is going to center on the progression of the probability distribution of traits in the population of individual organisms.  However, it is very easy to lose sight of the fact that this probability distribution is not a primary entity – it is derived only from a group of individual organisms, and has no existance or behavior apart from that of the individuals.  Nonetheless, efficiency in discussion is going to lead to the use of words such as species and genes to represent the aspects of the distribution of traits, and attributes will come to be assigned to these collective concepts.   This can, and almost always will, lead to a confusion of language in which the genes and species will be understood as primary actors in the evolutionary progression.

Current discussions of evolution often start at this point of confusion.  Species are said to evolve, genes are said to be acting in their best interest, to ensure the survival of the “gene pool”.  No attempt is (usually) made to tie these abstractions back to the actions of individual organisms.  The collective concepts become the entities to which the theory is applied, not only as a efficiency in discussion, but in the meaning to be projected in the discussion.  It is my assertion that this results – intentionally results – in a very different set of logical consequences than is implied by the original theory of evolution. 

The modern discussion of evolution is a discussion of the behavior of collectives, in which the value to be preserved is the perpetuation of the existance of the collective (gene pool or species).  Anthropomorphic terms are applied to both the genes and species – the genes have a “goal” they are “moving toward”.  When discussing symbiotic relationships between animals (for example, aphids and ants), the participants are said to be co-operating, or (even) acting altruistically(!).  It even seems to be recognized that the individual organisms cannot cooperate, because they lack intelligence, but somehow the species can. 

The advantage that this kind of misuse of evolution brings to environmentalism is only a minor example of the damage that is done philosophically by casting evolution in this form.  Species survival becomes over-valued.  Loss of “diversity” in the gene pool is a fundamental concern if the genes are the value that is to be preserved in evolution.  And certainly, any human action is at odds with the “natural” progression of evolution.

The larger issue with the collectivist form of evolution theory is that it creates an opportunity for critical attack.  Looking at the progression of traits in a historical population, any apparent sudden change can be questioned – “what caused the genes to ‘do’ this?”, or “what happened to this species – where did it come from; where did it go?”.  In the proper consideration of the theory, these are seen to be questions focussed on invalid concepts; however, in the modern discussion, these are weak points to defend.

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