(In the Shadow of the Torah)
By Amiel Rossow
First posted on November 9, 2002
The Hebrew words B’Ohr Ha’Torah mean In the Light of the Torah. This is the title of a lavishly published journal whose Editor-in-Chief Herman Branover is a professor of physics at Ben-Gurion University in Beersheba, Israel. As is indicated on the title page, this journal (whose periodicity is not revealed) is published by an organization called SHAMIR, “the Israel Association of Religious Professionals from the former USSR.”
The announcement on that title page says that this is an “international forum for all Jews who want to understand how the Torah permeates everyday life, from personal behavior and social mores to scientific and artistic creativity.”
From the above statement it seems to follow that contributions from non-Jews are not welcome; likewise those Jews who do not accept the Torah as the depository of the ultimate truth apparently need not apply.
According to the above statement, I am not one of the potential contributors to the journal in question. Indeed, I tend to view it in a light different from its acceptable contributors. Hence the title of this essay; in my view many articles published in that journal show that their authors, often highly competent specialists in various fields, are actually benighted by the uncritical a priori acceptance of the Torah as a source of infinite wisdom and unlimited knowledge about everything, from what a Jew is allowed to eat and drink to the most intricate problems of the theory of relativity and of quantum mechanics. In my view their discourse is often characterized by myopia which is due to the deep shadow (Tsel in Hebrew) from the Torah story hovering over their minds.
I have no intention of arguing against the role the Torah played in the history of the Jewish people (while also having exerted enormous influence on many other nations, Christian, Muslim, and some others) or against the poetic power evident in that greatest bestseller of all times. I do not believe, though, that the Torah is a source of scientific knowledge and all attempts to reconcile its story with the data of science is, in my view, an exercise in futility.
I have chosen for my review issue number 13 of the journal, published in 2002 (no indication of the month of publication). The reason for this choice is that it contains that part of the Proceedings of a December 1999 conference that took place in Miami, Florida, devoted to the relation between modern science and the Torah story. The motto of that conference (which is conducted biannually) was “Absolute Standards in a World of Relativity.” The absolute standards, as the conference’s organizers believe, are given in the Torah, while scientific knowledge is relative.
Almost all of the authors of the articles in that issue have impressive scientific credentials. All of them are firm believers in the inerrancy of the Torah and try to show that modern science is fully compatible with the Torah. In that, issue 13 of the journal may be viewed as a sequel to or an updated version of the collection Challenge: Torah View on Science And Its Problems edited by Aryeh Carmell and Cyril Domb, published by the Association of Orthodox Jewish Scientists (via Feldheim Publishers of NY and Jerusalem) in 1976 and again in 1978 and critically reviewed at www.talkreason.org/articles/challenge.cfm .
I will not review each and every article in that issue, since it would be too long a discourse, and, moreover, some of the material in that issue lies beyond the subject of the relation between modern science and the Torah. I will, though, try to discuss the salient points of those articles wherein their authors endeavor to interpret both the data of science and the Torah story with a preconceived confidence of their perfect mutual compatibility.
On the journal’s cover we read the following lines, which apparently encapsulate what in the editor’s view is the most important thesis covered by the issue’s articles: “Adam & Eve collapse the quantum wave function? THE ROLE OF HUMAN OBSERVER IN NATURE.”
Indeed, several articles in the issue discuss in detail the above thesis, and I’ll devote a substantial portion of my review to its discussion.
The first paper in issue 13 is titled And G-d Said, “Let There Have Been a Big Bang.” Its author is a physicist, Dr. Avi Rabinowitz.
Rabinowitz starts his discourse by claiming the following “creation axiom”:
“In a free-willed act an all-powerful being designed and created a natural universe containing entities that are morally responsible for their choices.”
This beginning portends the gist of Rabinowitz’s article, where we cannot expect to see an objective analysis based on facts and logic since he adopts an “axiom” which predetermines conclusions forced by that “axiom.” For Rabinovitz there is no question whether his statement is true or not – he simply claims it to be true, thus effectively closing ways to an impartial discussion of any real problem of the universe’s and intelligent life’s existence.
Indeed, as could have been expected from the proclaimed “creation axiom,” the rest of Rabinowitz’s article contains little more than categorically proclaimed statements for which he does not offer a substantiation beyond his original “axiom.”
The sole merit I found in Rabinowitz’s paper is its relative brevity.
In the section titled The Common Ground of Science and Genesis, Rabinowitz correctly states that “Science does not deal with that which cannot be objectively and universally observed by scientists, and so does not deal with the soul.” Besides this obviously correct statement, the rest of the section offers nothing more than completely arbitrary suggestions regarding the “G-d’s infusion of a soul – and perhaps a mind as well – into a humanoid emerging from the ‘the dust of the earth,’ as detailed by evolutionary theory, in a universe which developed from a big bang created by G-d.”
While Rabinowitz’s acceptance of the evolutionary theory is fine, it can be noted that if by “evolutionary theory” he means either Darwin’s theory or the Neo-Darwinian synthesis, neither is really about the origin of life. Both are theories of the development of the variety of species from a common ancestor by means of descent with modification, led by natural selection. As to the origin of life, there are various hypotheses regarding the natural emergence of the first living replicators, but these hypotheses are not part of the theory of descent with modification. The main objection to the quoted statement by Rabinowitz is, though, related to his contention that the big bang was created by God. What is the source of Rabinowitz’s confidence that the big bang was indeed created by God? A skeptic would search Rabinowitz’s article in vain for any arguments in favor of his categorical assertion. If Rabinowitz wishes to believe that the big bang was “created by God” it is his privilege. Likewise, if he wishes he may believe that the moon is made of green cheese, and it is nobody’s business what his beliefs are. But for a skeptic who is not inclined to simply take Rabinowitz’s word without supporting evidence, his claim is hardly convincing.
The rest of Rabinowitz’s paper is in the same vein – unsubstantiated assertions not supported by evidence but simply evincing his beliefs. Some of his statements are plainly misleading. For example, in the section titled Designing the Big Bang: G-d’s Choice, we read, “According to scientific origin theory, in order to produce our universe, at some point a big bang would have to be created.”
It seems to be Rabinowitz’s secret which scientific theory he has in mind. No scientific theory asserts anything of the sort. The most widely accepted theory of the hot big bang (including its prevalent form - the inflationary theory by Guth – see Alan Guth, The Inflationary Universe: The Quest For a New Theory of Cosmic Origins, Addison-Wesley, Reading, MA, 1997) does not at all assert that the big bang “would have to be created.” Such an assertion would require derivation of the necessity of the big bang from certain underlying concepts. No such concepts are known. The big bang theory interprets the observed facts and concludes that they are best explained by the assumption of the big bang occurrence some 12 to 15 billion years ago. This is a far cry from the assertion, based on some more fundamental scientific concepts, that the big bang must have occurred to create our universe. The big bang theory says nothing at all about the “creation” of the big bang. The cause of the big bang is beyond science and is not discussed in science at all, leaving its discussion to philosophy, theology, and religion.
Rabinowitz continues, “Since a central purpose of the created being is its exercise of free-willed moral choice, the universe would have to be designed to contain morally meaningful situations and dilemmas.” Of course Rabinowitz is free to believe in anything his imagination may offer, including any wild hypotheses about the “central purpose of the created being,” but for a mind not encumbered by a priori notions, his suppositions have no evidentiary value.
In subsequent sections of his paper Rabinowitz continues to offer arbitrary assertions, never making the slightest attempt to substantiate them, but only referring to his self-published books, where he supposedly elaborates on the article in question.
Toward the end of his paper Rabinowitz turns to the question of the “collapse of the wave function” referred to on the journal’s cover. His idea is close to what is discussed in more detail in the article by Poltorak which immediately follows Rabinowitz’s. Since Poltorak’s discourse about this topic is more detailed, I will discuss the “collapse of the wave function” when reviewing Poltorak’s article.
The article by Dr. Alexander Poltorak is titled On The Age Of the Universe (pages 19-37). It is evident both from the article’s text and from the biographical note that Poltorak is a highly qualified theoretical physicist, an expert in the General Theory of Relativity. The main thesis of article in question is, however, in an area different from Poltorak’s scientific expertise. He tries to show that the biblical story, according to which the universe was created about 6,000 years ago, and the theories of the modern cosmology which hold that the universe exists for about 12 to 15 billion years can be reconciled by using certain concepts of quantum physics.
Unlike such writers as Schroeder (see, for example, www.talkreason.org/articles/schroeder.cfm) who suggested alternative explanations for the same discrepancy based on blatantly fallacious assumptions, Poltorak indeed knows his subject (in this case quantum-mechanical concepts) quite well. Nevertheless his explanation is also based on an arbitrary overextension of quantum-mechanical theory into the area where its application is at best doubtful and at worst plainly absurd. As I will show, in his endeavor Poltorak not only interprets concepts of the physical science in an illegitimate way, but offers ideas logically incompatible with the Torah story as well.
Poltorak’s article is preceded by the following sentence: “Readers with no physics background can skip equations and still follow the discussion.”
It seems reasonable to ask, if the equations are not really necessary for understanding Poltorak’s discussion, why are they shown in his article? Let us try to find the answer to that question in the text of Poltorak’s article.
Reviewing all those equations reveals several things:
(a) If all those equations were removed from the article, it would in no way affect Poltorak’s thesis (which will be discussed later). They play no role in Poltorak’s discourse except, perhaps, for demonstrating Poltorak’s bona fide credentials as a well-educated physicist. But there is no reason to doubt Poltorak’s qualifications and no reason for him to cite equations which can be found in multiple textbooks and monographs and to which he could simply refer if need be (although there seems to be no need for such references either, because these equations do not seem to be relevant to Poltorak’s main thesis).
(b) My opinion as expressed in item (a) seems to be even more justified if we note that the equations in question all are printed in Poltorak’s article in an unrecognizable form, with typos piled one upon another. Both subscripts and superscripts all are printed on the base line rather than above or under it, so they are indistinguishable from characters which are legitimately printed on the base line. Even a reader with a background in physics needs to decipher those formula based on his prior knowledge of those formulas. Therefore they play no useful role. It does not matter whether those typos are attributed to Poltorak, or to the editors, or to whoever else they may be; they make the formulas a meaningless addition to the article in question.
Actually, the entire first part (titled Cosmological Models) of Poltorak’s paper is nothing more than a synopsis-like tale about the data and theories related to the age of the universe and has no relation to his main thesis (which I’ll discuss shortly).
Poltorak’s main idea is the subject of the second part of the paper, titled The Torah View and the Role of the Observer. If the first part were completely excised from the paper, it would lose nothing of its essential content.
In Part 2 Poltorak still could not resist the temptation to write equations incomprehensible for ordinary readers and serving no useful role. Moreover, he devotes many words to the discussion of certain concepts of quantum physics wherein he offers no new ideas or interpretations but simply repeats the points which have been discussed many times before in much more detail, on various levels of sophistication. For example, he resurrects the worn-out discussion of the so-called Schroedinger cat gedanken experiment. Schroedinger was a brilliant theoretical physicist who contributed mightily to quantum theory. His famous example of a cat in a box was discussed an endless number of times in the literature. Since, however, Poltorak uses the Schroedinger cat gedanken experiment as a tool for his subsequent far-reaching conclusions, I have no choice but to briefly review that experiment here as well.
Imagine a box wherein a cat and a glass vial with a potent poison are placed. Above the vial a hammer hovers, supported by a hook. The hook is part of a simple mechanism that holds the hammer suspended. The hammer would be released, thus breaking the vial and killing the cat, if a pulse of electric voltage passed through the hook-holding mechanism. The voltage pulse is generated if a particle emanated from a lump of radioactive material passes a Geiger counter. This event – a particle passing through the counter – is unpredictable. Hence, the life or death of the cat depends on a random event. The probability of the Geiger counter triggering the breakdown of the vial can be calculated. Assume that it is ½ (or 50%).
Here is a quote from Poltorak’s article: “Prior to measurement, the state vector of the first atom to decay is a linear superposition of two possibilities, a decayed and a non-decayed atom. Accordingly, the state vector of the cat is also a linear superposition of two physical possibilities: a live cat and a dead cat. In other words, before a human being makes a measurement by looking into the box, the cat is simultaneously alive and dead at the same time! To be more precise, the cat is neither alive or dead but in a peculiar state of blurred combination of both possible states.”
I have no intention of delving into the discussion of Schroedinger’s famous example to which scores of articles and books have been devoted. I will say, though, that although Poltorak more or less faithfully repeats here notions offered more than once before by some prominent scientists, I dare submit that the suggestion of a cat that is both alive and dead or in the “blurred combination of both possible states” is void of meaning. The conclusion of the cat both alive and dead is the result of an unsubstantiated extension of certain formal mathematical concepts of quantum mechanics beyond their limitations. Regardless of whether or not an observer looks at the cat, the cat is either alive or dead at any moment of time. If the interpretation of quantum mechanical concepts says otherwise, it means that those concepts are either faulty or misinterpreted or misapplied.
The only statement legitimately applicable to the situation at hand is that until an observer has looked into the box, it is unknown whether the cat is alive or dead. The same can be said about thousands of other cats, within or without any boxes anywhere in the world.
Nothing in science, including quantum mechanics, is absolute. Science is a creation of human endeavors, and reflects reality only in an approximate way and always within certain limits, beyond which its application is just free play of the mind. The theoretical construct of the state vector as a linear superposition of several possibilities is a way we try to describe events in the subatomic world. Applying this concept to such situations as that with a cat in a box means inflating the meaning of the concept beyond its legitimate boundaries and leads to absurd conclusions like that of a cat being both alive and dead. Actually Schroedinger himself viewed the conclusion of a cat being both dead and alive as absurd, so that, in his opinion, testified to the incompleteness of quantum mechanics (an opinion shared also by many other scientists including Einstein).
Poltorak, besides uncritically repeating the meaningless inflation of quantum mechanical concepts to a cat in a box, inflates them even further, applying them to the universe as a whole and trying to connect them to the story told in the book of Genesis.
Poltorak’s main thesis is that the physical existence of our universe actually started at the moment when the first human (Adam) “looked at the universe” which occurred about 6,000 years ago. Until that moment the universe existed (for about 12 to 15 billion years) only “in a superposition of all possible states, including the states of existence and non-existence” (page 33).
To explain the meaning of the above statement, Poltorak turns to the concept of the wave function and of its “collapse” – concepts which have been discussed extensively in the literature on quantum mechanics. In order to analyze Poltorak’s argument, we need to make a few preliminary comments.
Wave function (traditionally denoted ψ) is a fundamental concept of quantum mechanics. It can be calculated by solving the famous Schroedinger equation for given boundary conditions. Since this review is for an ordinary reader, who may not have the background necessary to discuss the problem of the collapse of the wave function in a rigorously mathematical way, I will not delve into the discussion of the intricacies of quantum mechanics or its mathematical apparatus.
The name “wave function” is due to the fact that ψ, which is a function of coordinates (and time), from a formally mathematical viewpoint represents a wave. Of course the concept of a wave was well known long before the advent of quantum mechanics. For example, a widely known wave is that propagating in water (if a stone has been dropped into water, a wave will propagate radially away from the location of the stone’s fall). The simplest definition of a wave says that a wave is the propagation of a disturbance without the propagation of matter.
When Schroedinger suggested his equation, its solution – the function ψ -- turned out to be mathematically analogous to a function representing the previously known waves, such as waves in water, in air (sound waves), and electromagnetic waves (including visible light).
If we see a wave propagating in water, we realize that we observe the propagation of the oscillatory motion of water’s molecules. In the sound wave, what propagates is the alternating compressed and rarified clusters of the molecules of air (or any other medium wherein sound propagates). In an electromagnetic wave it is a combination of oscillating electric and magnetic fields that propagates. In all these cases it is an oscillatory motion of a certain entity which propagates from the source of disturbance. What is the entity whose oscillations are carried by function ψ?
One interpretation of ψ-function views it as describing the distribution of a particle’s mass (and charge) over a certain volume. For example, for an electron in an atom the function ψ2 , according to that interpretation, represents the distribution of the electron’s mass and charge over the atom’s volume. This interpretation can be followed in a formally consistent way and tied to a classic Newtonian dynamic via the Ehrenfest theorem (see, for example, G. Marx, Quantum Mechanics, Academiai Kiado pubslishers, Budapest, Hungary, 1962), whose discussion is beyond the scope of this essay. However, while formally consistent, this interpretation of ψ-function, as I’ll show in a few lines, contradicts empirical evidence which shows beyond a doubt that an electron is an indivisible (“elementary”) particle.
Attempts by the most prominent scientists, including Schroedinger himself, to interpret ψ-function as a real field analogous to electromagnetic or gravitational fields were rapidly realized to lead nowhere (as is correctly mentioned by Poltorak in his article).
As was first realized by Max Born in the late twenties, function ψ, although formally represented by the same mathematical expression for a wave as in all mentioned cases, does not represent oscillations of any material substance or any real field. Born’s interpretation, promptly supported by Niels Bohr and Pascual Jordan, was rapidly and overwhelmingly accepted as best fitting all the experimental evidence. According to Born’s idea, function ψ is just a mathematical tool reflecting the behavior of particles in a probabilistic way. For example, consider an electron about which we know that it is within a certain volume. However, we don’t know where exactly in that volume it is located. If we solve Schroedinger’s equation for that electron we obtain ψ as a function of different locations within the volume in question. If we square ψ, the quantity ψ 2dV (where dV is an element of volume calculated for every location within the volume in question), represents the probability that electron is within the element dV of volume at a given location. What distinguishes the quantum-mechanical solution from the Newtonian solution (which is excellent in the macroscopic world but fails in the microscopic one) is that in the Newtonian macroscopic world the location of a body can be calculated precisely, while the required precision of the calculation has in principle no theoretical limitations. In the subatomic world, where we can’t directly observe the behavior of particles, we are limited to a probabilistic approach wherein we obtain function ψ2 for every location within the volume in question. Any location where ψ2 is not zero is possible for the electron in question, albeit not with the same probability.
As mentioned earlier, the described situation led to an interpretation of the wave function according to which ψ2 represents not the probability of the particle being materially at a certain location but rather the distribution of the particle’s “density” over the volume in question. According to that interpretation, the particle is actually “spread” all over the volume, “more of it” being where ψ2 has larger value and “less of it” where the squared wave function has a smaller value. While formally that interpretation seems as consistent and valid as Born’s probabilistic interpretation, it actually has no foundation in any empirical data. Moreover, such an interpretation leads to absurd conclusions.
Imagine the following simple experiment which can easily be conducted. Generate a beam of electrons of a very small density, such that electrons are emitted from an electron gun and pass, one by one, through a tube leading to a box with walls impervious for electrons. The tube is equipped with a shutter. Open the shutter for a short time and close it immediately after a single electron has passed into the box. At the moment the electron was entering the box, its location was known with a precision determined by the diameter of the ingress aperture of the box. Upon entering the box, electron can happen to be anywhere in the box, and we don’t know its exact location except for the certainty that it is in the box. (In the often used quantum-mechanical terms, it is said that the electron is a “wave packet” which, upon entering the box, spreads over the width of the box). Let us provisionally accept the model of the particle’s matter being “spread” all over the box, wherein the squared wave function ψ2 represents the distribution of the density of electron’s mass and charge over the volume of the box. Now imagine that we insert into the box a partition which separates the two halves of the box. If the electron were “spread” over the box’s volume, then each half of the box would now hold ½ of the electron’s mass and charge. This is an obvious impossibility because an electron (as is every lepton) is a real elementary particle. In no experiment has a fraction of an electron ever been observed. It is always observed only as a whole. (For certain situations the last statement requires a subtle clarification, for example for an electron in a crystal lattice of solids, but although the electron’s behavior in crystals has some peculiar features, it does not negate the notion that electrons are indivisible particles.)
Therefore the only consistent interpretation of the wave function is that it describes the behavior of particles in probabilistic terms. There is neither a material substance nor a real field represented by function ψ.
Unfortunately, the interpretation of ψ2 as the spatial distribution of a particle’s mass and or charge, formally legitimate but contradicting empirical evidence, paved the way to viewing wave function as representing oscillations of some material entity, and this in turn led to the concept of the “collapse” of the wave function as of some real event in physical reality.
One more comment is in order. As the Ehrenfest theorem shows, quantum mechanics (at least for slowly changing fields) contains all classical mechanics as an approximation. In other words, equations of quantum mechanics are valid not only for subatomic particles, but also for every macroscopic body as well. For masses substantially exceeding those of subatomic particles, equations of quantum mechanics convert into equations of classical Newtonian mechanics. The peculiarities of the behavior of subatomic articles revealed by quantum mechanics become less and less noticeable as the mass of the bodies increases. The wave function is in principle a valid tool for the description of the behavior of not only subatomic particle, but also of whole atoms and even molecules, although for the molecules the uncertainty in their location as determined by the wave function becomes negligible. With the amazing advances in tunnelling electron microscopy, scientists are nowadays able to see (and even manipulate) individual atoms. What we see is entities which have a definite shape and distinctively occupying a definite volume each, by no means being spread over a large volume with a variable density. This provides another piece of strong evidence in favor of Born’s probabilistic interpretation of the wave function. There is practically no doubt that ψ does not represent any real material quantity or a real field. It is a mathematical construct reflecting, besides the objective behavior of subatomic particles, our inability to visualize subatomic particles which are unlike anything we interact with in our macroscopic world.
When we say that a particle possesses wave properties, this does not at all mean that an individual particle is indeed a wave. If particles pass one by one through the slits in a partition, no single particle ever generates a diffraction pattern. Each particle hits the screen at a definite location which can be identified, although with limited precision, but quite unequivocally. However, the location where the particle hits the screen is not arbitrary. For each location on the screen there is a definite probability of being hit by a particle. For some locations this probability is zero. No particle ever hits the screen at such locations. Wherever the probability is higher, more particles will hit such locations. The distribution of probabilities is such that when a large number of particles has hit the screen, either as high-density beam or one by one, all the points of encounters of the particles with the screen form a diffraction pattern which depends on the number of the open slits and follows the rules of diffraction for a wave mathematically described by the ψ 2 function.
The diffraction pattern appears only as a result of many particles hitting the screen, either in a high density beam, or one by one. What then is the meaning of the term “wave properties of particles” or “wave behavior of particles?” It means that each particle, when hitting the screen, never does it at a location where the corresponding wave would have a distractive interference. The behavior of each electron, although it moves through the slits individually and itself creates no diffraction pattern, follows (in a probabilistic way) only such paths which, when the entire collective of electrons hits the screen, would result in a diffraction pattern corresponding to a wave passing the given number of open slits.
Hence, while we definitely know that subatomic particles possess certain properties which make them follow the pattern of wave behavior, there is no way to visualize these particles. All we can do is to describe the particles’ behavior using a mathematical construct, and the ψ-function fits the bill very well.
Comment: There seems to be a difference between leptons and hadrons. Leptons (for example, electrons) are viewed as genuinely elementary particles having no constituent structural elements, so they can be viewed as having no volume at all (“material points”) while hadrons (for examples, protons and neutrons) are believed to contain quarks as their structural components, thus occupying certain volumes (which is definitely true for atoms). It is possible (although there is no empirical evidence for this) that the volume occupied by a particle is not constant but varies depending on its interaction with other particles and fields.
However, the above notions have no bearing on my thesis. When we say that a particle is at a definite location, this statement should be construed within the limitations imposed by the uncertainty principle, valid both for leptons and hadrons. The point is that wave function reflects the indeterminacy in the “precise” location of a particle (“precise” to the extent limited by the uncertainty principle) rather than reflecting a “spread” of matter constituting a particle over the volume where ψ is not zero.
With this in mind, we can now discuss the fallacy of the interpretation of the “collapse of the wave function” as exemplified by Poltorak’s article. Although Poltorak refers to von Neumann (John von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, NJ, 1955) and other prominent scientists who discussed various interpretations of the wave function and its “collapse,” my task in this paper is not the overall discussion of this difficult and important problem, but only a critical review of Poltorak’s presentation of the wave function collapse in the “light of the Torah.”
The notion of the collapse of the wave function can be explained as follows:
Imagine, for example, an electron beam directed toward a wall in which there are two narrow slits located close to each other. If the aperture of each slit and the distance between the slits are in a certain range, a diffraction pattern appears on a screen behind the wall. If one of the slits is closed, the diffraction pattern changes. If the closed slit is opened again, the original diffraction pattern reappears. This phenomenon is observed even if the electrons pass the slits one by one. It looks like an electron which is passing one of the slits somehow “knows” whether the second slit is open or closed. The explanation of this phenomenon is usually given by referring to the wave properties of the electron. If a wave passes through slits in a wall, it is always going through all opened slits simultaneously. For example, a wave front of sea water coming through opening in a breakwater passes through all openings simultaneously and generates a diffraction pattern which depends on the number of openings.
Now enter the wave function again. Its square value tells us the probability of the electron being, at this or that moment of time, at this or that location. Until the electron hits the screen its whereabouts are known only in terms of probabilities and are reflected in the wave function. At the moment the electron hits the screen its location immediately becomes determined by the spark of light on the screen. No probabilities any more, now the electron’s whereabouts become determined (albeit within certain limits of precision). This phenomenon of an instant revelation of the particle’s whereabouts is referred to as the “collapse of the wave function.”
This term is interpreted in quantum theory as the result of the interaction of the subatomic particle with a macroscopic device (a screen in the above example).
Poltorak, however, shares the view that the described collapse of the wave function happened only because the event was observed by a conscious observer. This idea was offered by some scientists but is by no means commonly accepted. In particular, proponents of that idea routinely refer to the prominent physicist John Archibald Wheeler as being allegedly one of the originators and a strong supporter of that idea. In fact, however, such references misuse a statement made by Wheeler in 1982, wherein he wrote, "No elementary quantum phenomenon is a phenomenon until it is a registered phenomenon... In some strange sense, this is a participatory universe." (In the collection Mind in Nature, edited by R. Q. Elvee, San Francisco, Harper and Row, 1982, page 17). Adherents of the idea of a participatory universe usually ignore other statements by Wheeler showing that he himself did not at all mean it in the sense supporting the concept of consciousness being a necessary component of quantum phenomena. In particular, Wheeler has also said, "Consciousness, we have been forced to recognize, has nothing whatsoever to do with quantum process." (In the same collection, page 21). Anyway, appeal to authority is not really a compelling argument.
Since Poltorak promotes the idea of consciousness as a necessary condition for the collapse of a wave function, a few words regarding that idea are in order.
First, remember that the wave function is a mathematical construct reflecting the impossibility of describing the behavior of subatomic particles in a visually comprehensible way, limiting us to a probabilistic approach. We just cannot in principle visualize an elementary particle and therefore can’t really describe what it means for a particle to be at a certain location in terms of our mundane experience.
Function ψ does not represent any material entity but only a mathematical phantom whose variations in space and time reflect the probabilities of a particle being in a certain state (in particular at a certain location). When the wave function “collapses,” nothing material does. (This notion can actually be traced to Bohr, and to Copenhagen interpretation in general.) All that collapses is a mathematical phantom, the way of description of the particle’s behavior. From the “viewpoint” of the particle itself, nothing “collapses.”
The collapse of the wave function would mean a collapse of a material entity only if the ψ-function indeed represented the physical “spread” of a particle’s density over a given volume. Although formally consistent, such a model contradicts empirical evidence, so the only proper interpretation of the “collapse” is that this term is only applicable to a mathematical construct, to such an event where the whereabouts of a particle (which before the “collapse” could only be determined in probabilistic terms) become revealed because of the interaction of the particle with a macroscopic device.
Although we do not really understand what the meaning of a particle being “located” here or there is, in a certain sense we may state that before the wave function collapsed, the particle was still actually “located” somewhere, rather than having its density distributed all over the volume in question according to the values of the squared wave function. We simply did not know its “whereabouts”; we knew only the probability of its being here or there. (It may be noted that there is still no consensus among scientists regarding what the precise meaning of the concept of a particle's whereabouts is.) All that happens when the particle hits the screen, is that its whereabouts become revealed to us. As always with probabilities, as soon as the event whose probability was estimated has actually taken place, all estimates of probability cease to be meaningful, being replaced by a certainty. The term “collapse” is an unfortunate one. All that “collapses” (if we choose at all to use such a term) is our mathematical construct, our best available tool for the description of imperfectly known reality. It is not needed any more because the location of the particle is revealed with a certain precision, which, although limited, is satisfactory from the cognitive viewpoint corresponding to our conventional macroscopic experience. No probabilistic approach is needed any longer.
Is a conscious observer necessary for the “collapse” of the wave function? Let us see. Imagine that we set up our experiment with an electron beam passing through slits and hitting a screen. Let us use a screen with a long afterglow, or a photographic plate instead of a screen. We turn on the setup and leave the lab until the next day. When, the next day, we come in we discover the photographic plate wherein the diffraction pattern has been recorded. The device was automatically turned off before we arrived, so no electrons are hitting the screen in our presence, but the photographic emulsion holds the diffraction pattern which formed in our absence. If we believe that a conscious observer is necessary for the wave function to collapse, then in our absence no diffraction pattern could have formed. This is an absurd supposition. We know that the pattern on the photo plate will be exactly the same regardless of whether a conscious observer was present or not in the lab during the experiment. The wave function which is just a mathematical construct would “collapse” (which simply means that the electrons’ whereabouts would be revealed for a potential future observer) as a result of their interaction with a macroscopic device (photographic emulsion) in our absence as well as if we were present in the lab. This “collapse” of the wave function would have nothing to do with any intervention by a conscious observer. So, quoting again the prominent physicist Wheeler, "consciousness has nothing whatsoever to do with quantum process." Poltorak's reference to Wheeler seems to be ill-conceived.
However, although the wave function does not represent any material entity, formally it can be handled as though it indeed were such a material entity. The mathematical treatment of the subatomic event is the same regardless of whether we interpret ψ as representing a material entity or a real field or just as reflecting the probability distribution.
The mathematical apparatus of quantum mechanics is very sophisticated, what with its Hermite’s operators, Hamiltonians, and other esoteric mathematical concepts which are incomprehensible to ordinary folks but are a source of enjoyment for the cognoscenti. The experts, enchanted by the mathematical beauty of their equations, are sometimes prone to forget the material reality which the equations so beautifully represent, and tend to view the mathematical abstraction as the reality itself. In this vein, the “collapse” of the wave function which actually occurs only on the formal mathematical plane, is sometimes attributed to the physical reality (while most often such an interpretation is done by philosophers rather than by scientists). In particular, the events which take place in von Neumann’s magnificent mathematical description of quantum processes are sometimes viewed as though they take place in the real physical world. In that, Poltorak goes much further than von Neumann himself suggested.
Poltorak's interpretation is not limited even to viewing the “collapse” of the wave function as though it happens in the real physical world rather than in its abstract mathematical, probabilistic reflection. He extends the application of the wave function well beyond its legitimate bounds.
The wave function which is found by solving Schroedinger’s equation describes in probabilistic terms the behavior of subatomic particles. As the mass of the articles increases, the indeterminacy in their behavior decreases; although Schroedinger’s equation may be formally written, say, even for a planet, it becomes of little meaning already on the molecular level.
The wavelength of the wave represented by the ψ –function is inversely proportional to the particle’s momentum, i.e. to its mass and velocity. When the mass of an object is on the order of a mass of a molecule, the wavelength becomes so miniscule that the wave behavior is practically beyond any observable limits. A billiard ball has a very definite location and simultaneously a very definite momentum, not because Schroedinger’s equation is principally inapplicable to the ball (it is) but because the mass of the ball is so large that the corresponding wave has an exceedingly small length, so the ball does not display any measurable wave behavior. The concept of location acquires a very definite meaning for bodies whose mass is much larger than it is for subatomic particles, and the use of the wave function for such relatively large bodies makes no sense. If this is true for bodies of a molecular size, it is much truer for larger bodies.
When we deal with macroscopic bodies, the very concept of the wave function and hence of its collapse deteriorates to a pure game of the mind with no practical content. Poltorak, however, applies the concept of the wave function and its collapse to the universe as a whole, which is an immensely exaggerated inflation of the concept’s reasonable applicability.
In Poltorak’s interpretation, which has no factual basis, the universe materialized only 6,000 years ago, when the first conscious observer, Adam, caused “the world wave function” to collapse.
Poltorak writes, “Prior to the first human, the universe existed in a superposition of all possible states, including the states of existence and non-existence. When the first human looked for the first time at the universe, the world wave function immediately collapsed and the world came into physical existence.”
The abject absent of elementary logic is coupled in that statement with an equally abject extension of the wave function concept far beyond its meaningful limits.
If the first human “looked the first time at the universe,” this means the universe he could look at must have already existed. Moreover, the first human himself must have been a part of an existent universe. Or did he exist by himself without anything else in existence?
If we believe the Torah, upon which Poltorak supposedly based his ideas, it tells us that the first human was created (according to Genesis 1) or formed (according to Genesis 2) on the last of the six days of creation, after everything else – the sun, the moon, the earth, water and firmament, etc., had been already in place. Hence, the first human, according to the Torah but contrary to Poltorak’s model, already had a universe available to look at, so this human hardly could cause the alleged “collapse of the world wave function” thus calling the universe into existence out of “both existence and non-existence.”
Either the universe existed before the first human or it did not. There is no tertiary quid. If the universe could have been called into existence only by the first human looking at it and thus causing the “collapse of the world wave function,” this means the first human existed before the universe and therefore could not “look” at the yet non-existent universe.
If, though, a universe existed before the first human could look at it, then its existence could not be caused by the as yet non-existent human looking at it.
Poltorak’s model makes no sense both from the viewpoint of the Torah and that of science.
Poltorak’s is a model based on groundless suppositions having nothing to do either with quantum mechanics or any other science or with the Torah’s story. It seems to stem from an overriding desire to reconcile, by whatever means, the biblical legendary narrative with the achievements of science.
Poltorak is entitled to believe whatever he likes. However, his attempts to substantiate his beliefs by using the concepts of quantum theory which have no relation to Poltorak’s theological exercise is far from convincing.
FERBER AND BRANOVER
The article by Professors Ruvin Ferber and Herman Branover (from now on to be referred as FB) is titled The Observer and the System of Reference: A Unified View.
FB’s presentation consists of two very different parts. The second part, titled The Changing Views of Science, could be a separate article written on a good (although popular) scientific level, wherein the main thesis, although it could be disputed (and I intend to dispute it in this review) is a legitimate subject of discussion wherein FB are entitled to their interpretation of the relativity of motion. I believe his interpretation is wrong, but it is far cry from unsubstantiated ruminations we find in the articles by Poltorak, Rabinowitz, or some other authors. At least FB talk in that part of their article only about subjects relevant to their topic – the interpretation of relativity of motion.
Unfortunately, the first part of the article looks like a ransom paid to a religious agenda. Take, for example, the sections titled The Rebbe’s view and The Wager wherein BF refer to the opinion of the seventh Lubavitcher Rebbe Menachem Mendel Schneerson in regard to relativity of motion and the alleged equivalence of Ptolemy’s and Copernicus’s models of the solar system.
Viewing the esteemed late Rebbe as an authority on scientific problems sounds like a joke. Schneerson never took part in any scientific activity and did not even have sufficient education in any area of science, but apparently liked to indulge in discussing scientific questions in all areas of knowledge. The amateurish level of his judgments was, for example, illustrated at www.talkreason.org/articles/challenge.cfm .
It is hard to comprehend how a well qualified physicist like Professor Ferber could feel proper to lend his signature to these ridiculous sections of the article in question.
Let us talk now about the real thesis of FB’s presentation.
On page 45 we read, “One of the conclusions of the theory of relativity is that when there are two systems, or planets, in motion relative to each other – such as the sun and the earth in our case – either view, namely the sun rotating around the earth, or the earth rotating around the sun, has equal validity.”
Professors Ferber and Branover, you should know better.
First, when you mention the theory of relativity, it is advisable to specify whether you mean the special theory of relativity (STR) or the general theory of relativity (GTR).
However, whichever theory of relativity you have in mind, neither asserts the “equal validity” of the two mentioned models of the solar system. FB’s statement is nothing more than their personal interpretation of both STR and GTR rather than a straight conclusion of either of the two theories.
Before discussing in detail FB’s arguments, let us briefly review in layman’s terms what STR and GTR indeed say about the problem at hand.
I’ll show that from the viewpoint of STR Ferber-Branover’s position is wrong. From the viewpoint of GTR, while not plainly wrong, their position is just their preferred interpretation of the GTR rather than an uncontroversial conclusion.
Start with STR. This theory asserts that all inertial frames of reference are equivalent. The term inertial denotes such systems (i.e. physical bodies) which move without acceleration (i.e. with a constant velocity; velocity itself has no absolute value -- it depends on which frame of reference is chosen as being “at rest” -- so the velocity of the system in question is measured relative to the chosen frame of reference). The choice of the frame of reference (i.e. of a body which is agreed to be viewed as being at rest) is arbitrary. Hence, within the framework of STR only inertial systems are equivalent, while systems (i.e. bodies) which have acceleration (i.e. are non-inertial) are not equivalent and can be distinguished unequivocally from inertial systems and from each other by detecting their accelerations. Acceleration, unlike velocity, is, within the framework of STR, absolute and does not depend on the frame of reference.
Apply the above to the solar system. The central body (the sun) has no acceleration. Every planet moves on an elliptic orbit wherein it has acceleration. The main component of the acceleration is the centripetal acceleration caused by the gravitational attraction of the planet to the sun. This component of acceleration makes the planet’s velocity turn toward the sun, thus forcing the planet to circle the sun on an elliptical path. Planets also have a longitudinal acceleration due to the deviation of their elliptical orbits from a perfect circle. Hence planets are not inertial systems and therefore not equivalent to the central body -- the sun. Therefore, if we limit ourselves to the tenets of STR, there is no equivalence between Ptolemy’s and Copernicus’s models. The sun is the only body in the solar system which has no acceleration, all planets do have one. The sun occupies a unique position in the solar system. Copernicus was right and Ptolemy (and with him the Judaic, Christian, and Muslim religions) was wrong.
So from the viewpoint of STR the question about the alleged equal validity of two models is answered unequivocally and easily in favor of Copernicus’s heliocentric system.
From the standpoint of GTR, however, the answer is not that simple and plain and FB take advantage of the complexity of the position based on GTR to offer their interpretation according to which the two models are, after all, equally valid. Let us see if their argument is convincing enough to assert that the model of the sun rotating around the earth is as valid as the heliocentric model is.
GTR is a complex scientific theory which can be rendered in various forms. For the purpose of this discussion (and having in mind that it is for an ordinary reader rather than for the mathematically sophisticate) it seems convenient to present the main point of GTR as follows:
While STR is the mechanics of inertial systems, GTR is the mechanics of non-inertial systems. This is, though, only one aspect of GTR. It also is a theory of gravitation and as such it asserts equivalence of gravitational and inertial forces. Based on that concept, FB announce the equivalence of heliocentric and geocentric models of the solar system, which serves their purpose of substantiating the Torah’s assertion that the earth is the center of the universe and the sun (and the rest of the universe) is rotating around the earth. To be sure, FB only suggest that the heliocentric system is no more reasonable from a scientific viewpoint than the geocentric system of the Torah, so there are no scientific reasons to reject the Torah’s model. They do not insist that science directly supports the Torah’s model.
However, it can be shown that GTR actually provides reason to prefer Copernicus’s heliocentric system to Ptolemy’s geocentric one (supported by the Torah). To this end let us consider another rendition of GTR which can be offered without resorting to mathematical equations. One of the easily visualized renditions of GTR tells us that space does not exist by itself but must be viewed as an attribute of matter. Wherever there is a clump of matter it makes space around it curved. The sun makes space around it curved, and so does the earth. However, since the sun has an immensely larger mass than the earth (or than any other planet in the solar system) the curvature of space caused by the sun is immensely stronger than the curvature caused by the earth. There is no equivalence between the effects of space curving caused by the sun and by the earth. The model according to which the planets move on elliptical orbits, their path determined by the curvature of space caused by the immense mass of the sun, is plainly preferable to the model in which the sun, with its much larger mass moves on a path curved by the much smaller mass of the earth (or of the other planets).
Furthermore, if we consider the motion of the sun and of the earth relative to the rest of our galaxy, it is much more reasonable to view the solar system as a whole as such wherein there is a central body (the sun) and a family of planets orbiting the sun on simple elliptical paths rather than the earth as the center of that system and everything else rotating around the earth on fancy convoluted paths.
The theory of relativity, either special or general, provides no support whatsoever for the Torah’s geocentric picture, regardless of how strongly FB and their co-believers would like us to believe.
(Besides their main thesis of the equivalence of Ptolemy’s and Copernicus’s models, the article contains a number of sections dealing with quantum mechanical theory of measurement including von Neumann’s ideas of the role of an observer, as well as their further discussion by Wheeler and Wigner; the Copenhagen interpretation of the quantum mechanics; Mach’s principle attributing the inertial forces to the overall effect of the galaxies, etc. All this material, interesting in itself, would be appropriate for a semi-popular article for laymen interested in learning about some theories discussed in modern science, or for a teaching aid for a freshman class in physics, but it does not offer any material which would be the authors’ original contribution to the subject. It is not clear why such a brief narrative about well known problems of science was thought to be a proper contribution to a conference where the relation between science and the Torah was to be discussed. The only point which went beyond just one more rendition of the stuff elucidated many times before was FB’s assertion of the equal validity of Ptolemy’s and Copernicus’s models of the solar system wherein the two professors strived to show that the Torah’s story does not contradict scientific data.)
The article by professor of physics Nathan Aviezer is titled Richard Dawkins and Darwinian Fundamentalism.
The very title of that article portends its highly polemical and controversial nature. Everybody familiar with the real situation in contemporary biological science knows that there is no such thing as Darwinian fundamentalism (although individual scientists may sometimes feel reluctant to subject the established theories to a major overhaul). This derogatory term implies a dogmatic adherence to a set of preconceived notions impervious to reason and critique. It may be justifiably applied to the position of the authors and editors of the B’Ohr Ha’Torah journal who base all of their arguments and opinions on an unshakeable belief in the Torah’s inerrancy, a belief indeed impervious to reason and seeking no reason. Biological science is a different matter. Although the overwhelming majority of biologists are firmly convinced of the validity of the principal tenets of the Neo-Darwinian synthesis, there are constantly lively discussions and disputes among biologists regarding various aspects of their science, and no subject or principle, however fundamental to Darwinism, is immune to rebuttals and modifications. It is sufficient to point out the heated dispute between the proponents of the “punctuated equilibrium” concept and its detractors (discussed in more detail below), with both sides adhering to Darwin’s evolution theory but fervently disputing its various details and interpretations. There are no sacred cows in biology. Aviezer’s application of the term fundamentalism to the Darwinian theory (as well as his use of another pejorative term, “ultra-Darwinism”) betrays his prejudice.
The punctuated equilibrium theory was suggested by the late Stephen Jay Gould and Niles Eldredge (see for example S. J. Gould and N. Eldredge, Punctuated Equilibrium Comes of Age, Nature, v. 366, p. 223, 1993). Since the prominent writers and biologists Richard Dawkins and Daniel Dennet were among the most outspoken detractors of the “punctuated equilibrium” hypothesis, Gould (whose acerbic style of dispute, often displaying arrogant self-confidence, was his trademark) singled out these two colleagues cum adversaries for his most energetic rebuttals. However, while Gould and Eldredge on the one side, and Dawkins, Dennet, and some other biologists on the other side, vigorously defended their respective position in regard to “punctuated equilibrium,” both sides adhered to the Neo-Darwinian synthesis in general.
Aviezer’s position is different. His goal in the article in question is to smear his targets – three prominent writers and scientists, Richard Dawkins, Daniel Dennet, and Peter Atkins, who happened not to share Aviezer’s religious beliefs.
Instead of discussing the substance of the arguments offered by Dawkins, Dennet, and Atkins, Aviezer resorts to ad hominem remarks and innuendos, accusing these three writers of various kinds of misconduct, dishonesty, and other shameful acts. In doing so, Aviezer does not shy away from quoting out of context and misrepresenting the views of his targets thus himself committing those untoward actions he attributes to the subjects of his assault.
The main thrust of Aviezer’s attack is against Richard Dawkins.
Referring to Dawkins’s popular book The Blind Watchmaker (Norton, NY, 1986) Aviezer exaggerates the animosity between Dawkins and Gould as it is allegedly expressed in Dawkins’s book. Contrary to Aviezer’s story, Dawkins does not depict Gould as a “villain.” Dawkins offers arguments against Gould’s “punctuated equilibrium” hypothesis, which is a normal occurrence in scientific disputes.
In his acclaimed essay Nonoverlapping Magisteria (Natural History, March 1997, p. 16) Gould, while professing his respect for religious faiths, clearly indicated that he himself did not share any such faith. Aviezer’s attempt to enroll Gould for his attack on Dawkins is obviously caused by the abject paucity of allies having a reasonable scientific status to whom Aviezer could refer in his crusade. Therefore Aviezer resorted to exploiting the dispute between Dawkins and Gould which was about a certain specific hypothesis and not about the fundamentals of the Neo-Darwinian synthesis, as if it provided an indication of the deep rift within the scientific community and wherein Dawkins allegedly committed grievous sins.
The entire section in Aviezer’s paper titled Ultra-Darwinian Tactics is a bunch of innuendos, ad hominem remarks and inventions aimed at smearing Dawkins and distorting Dawkins’s position to the extent of making it unrecognizable. Although Aviezer derides Dawkins’s assertion of passionately caring “about what is true,” what is clear for an unbiased reader familiar both with Dawkins’s and Aviezer’s writing is that it is Aviezer who is ready, willing, and able to distort the plain meaning of Dawkins’s words to suit his agenda.
Aviezer devotes two whole sections of his paper to the discussion of Gould and Eldredge’s “punctuated equilibrium” hypothesis. His goal is to show that this hypothesis is really a firmly established theory which cardinally undermines Darwinian evolution theory. In fact, this is far from true.
First, in the course of several years after Gould and Eldredge suggested their hypothesis their original concept underwent substantial modifications. Some of Gould’s and Eldredge’s notions became a part of Neo-Darwinian theory, while some other parts were rejected by mainstream biology, as often happen with scientific theories. It is a common situation that the authors of an original theory tend to overextend its supposed area of applicability and the theory in its initial form contains both valid elements and excessive generalizations. The critique by biologists such as Dawkins, Miller, and others played a constructive role in defining the limits of the theory’s applicability. Within these limits, the original notions of the theory’s authors were clarified. In particular, the consensus among the scientist is that “punctuated equilibrium” theory, as it is legitimately interpreted, is perfectly compatible with the foundations of Darwin’s evolution theory. The “rapid” evolution of new species between the long period of “stasis” is actually not really “rapid” since, according to the paleontological data used by Gould and Eldredge, this “rapid” evolution took millions of years and looks “rapid” only on the background of the much longer periods of time paleontology usually deals with. There are many other fine arguments partially rejecting certain elements of Gould-Eldredge’s original theory and partially interpreting them in a way compatible with the Darwinian position. Aviezer, using selective quotations, paints a distorted picture.
Contrary to Aviezer’s assertions, the Darwinian gradualism can be reconciled with properly interpreted punctuated equilibrium, and the majority of biologists are comfortable with such a reconciliation.
Consider, for example, how Aviezer distorts the exchange of views between Gould and Dawkins regarding the evolution of the eye. He quotes both Gould and Dawkins, wherein Gould used the expression “What good is 5 percent of an eye?” Dawkins, however, used instead an expression “Vision that is 5 percent as good as yours or mine is very much worth having in comparison with no vision at all.”
In Aviezer’s presentation, Dawkins’s statement means “ridiculing” Gould. To an unbiased reader it is obvious that there is not even a shadow of ridicule in Dawkins’s sentence but only the expression of a legitimate disagreement on some nuances of the matter, on whose principal features Gould agrees with Dawkins to a much larger extent than Gould could ever support Aviezer’s general position.
As Aviezer continues, he misrepresents the gist of the disagreement between Dawkins and Gould by making the unsubstantiated claim that Dawkins “transformed” Gould’s statement about “5 percent of an eye… into a statement about ‘5 percent vision.’” However, reading both Gould’s and Dawkins’s texts reveals that no distortion has taken place. The expressions “part of an eye” and “part of vision” were used to refer to exactly the same thing. To see that it is indeed so, it is sufficient to look at Dawkins’s example of a light-sensitive cell which is both “a small percentage of an eye” and provides “a small percentage of vision.” Dawkins did not distort Gould’s notions. It is Aviezer who misused the inconsequential variations in the language used by Dawkins and Gould. In doing so, it is Aviezer who actually ridicules Dawkins and thus creates deception aimed at misrepresenting the gist of the Dawkins vs. Gould discussion, for the sake of Aviezer’s agenda.
There is little doubt that were Gould still alive, he would reject the dubious “help” from Aviezer in his dispute with Dawkins. Gould used to be a powerful defender of the evolution theory and simultaneously an iconoclast in the interpretation of some of its features. In that he had many detractors within the ranks of his fellow evolutionists. However, Aviezer’s attempt to enroll Gould as an ally in his fight against the so-called “ultra-Darwinists” is an insult both to Gould and to the idea of objectivity in a dispute.
In a similar way, Aviezer refers to other renowned scientists who allegedly share his views regarding the “ultra-Darwinists.” When assaulting Daniel Dennet, Aviezer again tries to create the impression that Gould and another respected biologist, Allen Orr, share Aviezer’s position. The actual situation is different. H. Allen Orr is as firm a Darwinist as they come. His position has nothing in common with that of Aviezer. To see that this is indeed so, it is sufficient to read Orr’s critical reviews of the books by latter-day creationists Behe (H. Allen Orr Responds, Boston Review, Feb-March 1997, p. 35) and Dembski (Boston Review, Summer 2002; also available at http://bostonreview.mit.edu/BR27.3/orr.html), as well as his essay (H. Allen Orr, The Descent of Gould: How a paleontologist sought to revolutionize evolution. The New Yorker, Sept. 30, 2002, p. 132) on life and work of Gould. In the latter essay Orr demonstrates that according to the overwhelming consensus among the biologists the original form of Gould’s and Eldredge’s punctuation equilibrium theory did not survive the test of impartial verification; while some elements of punctuated equilibrium theory have been successfully incorporated into Darwinian evolution theory, its many other features have been rejected as contrary to evidence. The efforts of Gould’s opponents like Dawkins have served as catalyzing agents in defining the limits of that theory’s applicability. Despite the sometimes acrimonious tenor of the dispute (not at all uncommon in science) it did not trespass the boundaries of a normal process of refining a good (but never perfect) theory as the Darwinian theory of evolution is.
Neither Gould nor Orr are Aviezer’s allies, all his attempts to prove otherwise notwithstanding.
At the end of his article Aviezer shows his disregard for plain logic. Discussing the book by British chemist Peter Atkins, Aviezer writes, “Note the striking difference between Atkins’s assertion that it is undeniable that the evolution of human beings was inevitable, and the exactly opposite opinion expressed by world class scientists like Alvarez and Gould. It follows, of course, that if Atkins’s scientific assertions are incorrect, then his conclusions about religion based on them are equally erroneous.”
Is this indeed so? First, if Atkins on the one hand and Gould or Alvarez on the other hand, have different opinion on whatever subject, this does not mean that Atkins is necessarily wrong. Although Gould and Alvarez are both fine scientists and their view deserves respect, they have no monopoly on truth. The greatest scientists all have made mistakes, sometimes very serious ones. Regarding the particular point Aviezer addresses, there is no consensus among the scientists regarding the question of whether life’s emergence was inevitable or it was a random occurrence which may never occur again anywhere in the universe. Both views have their proponents. However, whatever side one takes in that discussion has little to do with Atkins’s views on religion. Atkins’s views on religion have been shared by scientists both agreeing and disagreeing with him in regard to the inevitability of life’s spontaneous emergence; in particular, Gould’s view of religion was certainly closer to that of Atkins than that of Aviezer, although Gould disagreed with Atkins on the inevitability of life’s emergence. There are other scientists who disagree with Atkins’s view of religion but share his view regarding the spontaneous emergence of life. Therefore Aviezer’s “conclusion” that Atkins’s views on religion are erroneous is nothing more than Aviezer’s personal opinion, not supported by objective arguments. Aviezer’s attempt to enroll prominent scientists as allegedly sharing his worldview is hardly convincing.
Overall, Aviezer’s article is a misleading piece by a religious propagandist who does not shy away from using whatever means available to drive his beliefs into the minds of his readers and using for this end some not quite kosher means.
On pages 108-111 we find an article by Professor Edward Simon wherein he reviews Dr. Lee Spetner’s book Not By Chance: Shattering the Modern Theory of Evolution. Spetner’s book (published in 1998 by the Judaica Press) has been reviewed many times, including a very critical and well substantiated review by Dr. C. Feit in Jewish Action magazine (Spring 1999 issue). This magazine serves the Orthodox Jewish community and Dr. Feit criticizes Spetner from the position of an observant Jew.
Unlike his co-believer Feit, Professor Simon has a very positive opinion of Spetner’s book. Although in Simon’s view Spetner’s alternative to Neo-Darwinism is not substantiated well enough, he asserts that Spetner has successfully debunked the Neo-Darwinian paradigm. In Simon’s words, Spetner did an “excellent job in pointing out the problems associated with attributing evolution to random mutations and selection (the new Darwinian theory).” Continuing, Simon writes that Darwinian evolution, “for all its strengths and attractiveness, fails to explain how life arose and developed.”
To put it mildly, this is a rather odd statement for a professor of biology. It does not take an expert to know that evolution theory is not about “how life arose.” Evolution theory is about the evolution of the variety of living organisms from a common ancestor. As to the origin of that common ancestor, the first replicator, this question is beyond evolution theory. Since Spetner discusses only the mechanism of evolution, his assault on Neo-Darwinism has little to do with the question of “how life arose.”
Spetner’s critique of Neo-Darwinian theory is based on three points: (1) Estimates of probabilities; (2) Discussion of randomness; and (3) Discussion of information. Professor Simon is a biologist. Perhaps this is the reason he does not address the above three points in his review. However, without addressing the validity of Spetner’s treatment of these three concepts no proper evaluation of Spetner’s arguments can be done.
In his review in Jewish Action Dr. Feit (who also is a biologist) does not write about probabilities or information, but he rejects Spetner’s treatment of randomness, and this point alone makes Feit’s review much more reasonable than that of Simon. Feit, though, criticizes Spetner’s treatment of randomness not as much from a mathematical as from a religious viewpoint.
A consistent critical review of all three main points of Spetner’s assault on Neo-Darwinism is given in an article on this site (www.talkreason.org/articles/spetner.cfm). As shown in that article, Spetner’s treatment of probabilities, randomness, and information is largely fallacious. Simon does not seem to be familiar with that critique. However, without answering the mentioned critique, assertions of an “excellent job” allegedly done by Spetner sound hollow. Simon’s opinion that Spetner’s book indeed shattered the Neo-Darwinian evolution theory is nothing more than attributing to the desired the status of the actual.
A detailed and substantial critique of Spetner’s book from a biological viewpoint is found in a number of publications at www.talkorigins.org . Simon does not seem to be familiar with this critique either, although as a biologist he should be expected to be aware of it. Another critical discussion of Spetner’s book which is much better substantiated than Simon’s, is found at Gert Korthof’s website (http://home.planet.nl/~gkorthof/ ). The conclusion made in all the listed sources is different from the one suggested by Simon. All the mentioned reviewers from Feit to Perakh to a number of authors on talkorigins.org and to Korthof, find many faults and fallacies in Spetner’s book which Simon seems to miss and which show that Spetner’s opus makes no real contribution to biology.
The Neo-Darwinian synthesis is nowadays as strong as ever and is supported by an ever growing and ever better body of evidence. Spetner’s book is a feeble (albeit popular) attempt to overturn a solid theory, strongly supported by an immense amount of evidence, by using poorly substantiated arguments.
Contrary to Simon’s assertion at the end of his review, the critique of Spetner’s book in the mentioned sources was by no means offered in a dogmatic way. In fact, this critique pointed to specific weaknesses and errors in Spetner’s work and that is why, contrary to professor Simon’s view, the overwhelming majority of both biologists and mathematicians rejected Spetner’s critique of Neo-Darwinism as inconsequential.
Overall, Simon’s evaluation of Spetner’s book is without merit as it leaves without real analysis the actual weaknesses of Spetner’s book.
The paper by Dr. Tsvi Victor Saks is titled Different Levels of Infinity in Torah and Mathematics.
After an obligatory reference to the Lubavitcher Rebbe Menahem M. Schneerson, as to an allegedly great thinker of our time with the status of a prophet, Saks offers a brief explanation of the concept of mathematical infinity, limiting his discourse mainly to the ideas of the prominent German mathematician Georg Cantor (1845-1918). Saks emphasizes that Cantor was Jewish, “had an intimate relationship with G-d” (page 118), and received his concepts of infinite sets directly from God. Saks does not share with readers the source of his uncanny knowledge of how Cantor received such a revelation from God except for mentioning (without a specific reference) Mittag-Leffler to whom Cantor reportedly revealed that he received a direct message from God. Similar statements are heard day in and day out by physicians in mental hospitals as well.
While the story of Cantor’s mathematical work as told by Saks is more or less factually accurate, it is not the whole story. Cantor was indeed a brilliant mathematician and his contribution to the set theory in general and to the concept of mathematical infinity in particular was very significant. However, Cantor was not alone in the development of the above mentioned concepts. As Saks remarks, Galileo Galilei (1564-1642) (in his famous Dialogue Concerning the Two Chief World Systems (1632)) had already discussed certain ideas related to infinite sets. A substantial contribution was also made by the German mathematician Richard Dedekind (1831-1916), the already mentioned Swedish mathematician Gösta Mittag-Leffler (1846-1927), British mathematicians Augustus de Morgan (1806-1871), and Bertrand Russel (1872-1970) and others. I don’t know whether or not any of them had an “intimate relationship with God” (Russel was known as a militant atheist) but none of them was Jewish. Furthermore, if God had indeed revealed to Cantor the esoteric essence of mathematical infinity, why did he not go all the way? Why did he instead abandon Cantor halfway to the completion of his theory? Indeed, Cantor tried hard to prove the so-called continuum hypothesis, which would be a real crowning achievement for his set theory. (For the sake of the most curious readers, briefly, the continuum hypothesis asserts that an infinite set of real numbers either is denumerable, that is, in one-to-one correspondence with the set of natural numbers, or can be put in one-to-one correspondence with the set that contains all real numbers between 0 and 1; the meaning of the terms in this definition can be found in textbooks on mathematics; an excellent semi-popular explanation is given in Mathematics From the Birth Of Numbers by Jan Gullberg, W.W. Norton, New York, 1997.) Despite Cantor’s persistent effort and his alleged direct communication with God, he failed to prove his hypothesis. In 1940, German-American mathematician Kurt Gödel showed that Cantor’s hypothesis cannot be disproved. In 1963, American mathematician Paul Cohen showed that the continuum hypothesis is undecidable. Although Cohen’s surname probably indicates his Jewish (or at least partly Jewish) origin, as far as I know he never claimed to have received a revelation about the continuum hypothesis directly from God.
In view of the above, it seems evident that Saks’s paper is largely irrelevant to the question of the Torah vs. science relationship. Its mathematical part belongs in an introductory mathematical textbook, and its non-mathematical parts is a semi-philosophical treatise designed to give vent to Saks’s personal views and religious preferences. This paper is of little interest to readers who would like to gain even a modest insight into the Torah vs. science controversy.
The article by Professor Shimon Silman is titled A System of Logic for Messianic Phenomena. As are many other articles in the same issue, this paper is long on explanation of matters which are irrelevant to the relationship between the Torah and science, in this case certain concepts of mathematical logic. The sole reason for that detailed excursion into the esoteric depths of modern mathematics seems to be the author’s desire to demonstrate his erudition and intelligence (as is also the case with the articles by Belenkiy, Poltorak, Farber & Branover, Rabinowitz and others). Since Silman’s credentials are provided in the summary preceding the article, and they are impressive, readers do not need an additional proof of Silman’s qualifications to be given by the way of explaining the 3-valued logic which, in such a brief rendition, will anyway remain largely mysterious for a lay reader.
The level of the essential part of Silman’s discourse can perhaps be demonstrated by referring to just two assertions found in his paper.
One is a secondary, albeit telltale point found on page 127. Silman discusses here a question which used to be a topic of discourse in the late twenties and early thirties of the last century, but later practically disappeared from scientific publications, probably because of its more philosophical rather than scientific character – why the volume occupied by an atom is so immensely larger than the volume of the particles which are the atom’s constituent parts, so the overwhelming part of the atom’s volume is empty. Mentioning the famous Danish physicist Niels Bohr, Silman irrelevantly refers to him as a “Jewish physicist.”
Perhaps Silman wanted to stress the disproportional contribution to 20th century physics by scientists who happened to be Jews, either observant or not, thus providing an outlet for his pride in belonging to the same ancient tribe. However, while such a desire may be understood, the way Silman indulged in it can only cause derision, and the editors did not do their job as they failed to excise that ridiculous epithet from Silman’s text.
Reportedly, Niels Bohr did indeed have some fraction of “Jewish blood,” if such a concept has any meaning to reasonable people and not only to visceral anti-Semites. However, he was not of the Jewish faith, never expressed any interest in specifically Jewish affairs or religion, was reportedly inclined to share religious concepts of Hinduism, so to characterize him as a “Jewish physicist” is a display of a non-scientifically biased attitude by Silman.
Another point speaking volumes about Silman’s far-from-scientific position is found on page 135. Silman writes, “In 1966, a team of physicists working with Dr. C. Monroe at the National Institute of Standards and Technology in Boulder, Colorado, succeeded in getting an entire atom to exist simultaneously in two widely separated places.” Silman does not provide a direct reference to Monroe’s publications, referring instead to a report in New York Times.
Silman should have been a little more cautious in his references. Note first that if Monroe’s assertion were true, it would drastically contradict the assertions by Silman’s colleagues in the same issue such as Branover, Poltorak, Rabinowitz, and Ferber, who all wrote a lot about the “collapse of the wave function” which occurs in the act of measurement by a conscious observer. Monroe and his collaborators were certainly conscious observers. Their measurement of the atom’s location, according to the statements by Silman’s colleagues, must have resulted in a collapse of the wave function of the atom, and as soon as the wave function collapses, the atom’s whereabouts become instantly determined; upon the measurement such an atom is located at a certain point. If, though, the atom is indeed simultaneously in more than one location, the wave function has not collapsed. You can’t have it both ways, gentlemen of B’Ohr Ha’Torah.
Of course we know that the collapse of a wave function does not occur in the material sense; only a mathematical construct reflecting the probabilities of the atom’s whereabouts “collapses” in the sense that probability is replaced with certainty.
The amazing development of the tunnel electron microscopy nowadays enables physicists to see atoms. What they see are entities which have a definite shape, each at a definite location, not at two or more locations at the same time. Physicists (for example, at the IBM Research Center) can move individual atoms from location to location as whole entities. Of course, atoms have a complex structure, so their constituent parts (such as electrons, nuclei, and constituent particles of the nucleus such as protons and neutrons), can be separated and thus turn out to be at different locations. This has been known for a century and has nothing to do with the reported sensational discovery by Monroe et al.
More than six years have elapsed since the report in New York Times of May 28, 1996. If Monroe’s finding were true, no doubt the sensation would not have died as it did -- practically without further development.
The story reported by Silman belongs to the genre loved by journalists, although sometimes scientists in pursuit of fame give journalists a perfect alibi by acclaiming and interpreting their data beyond their legitimate boundaries. Here is an example.
In July 2002 the prestigious peer-reviewed journal Physical Review Letters printed an article by a group of Australian scientists (G.M. Wang, E.M. Sevick, Emil Mittag, Debra J. Searles, and Denis J. Evans) titled “Experimental Demonstration of Violation of the Second Law of Thermodynamics for Small Systems and Short Time Scale.”
Very soon a popular journal, New Scientist, printed a brief report about the article by the five Aussies. This report did not describe the Australians’ experiment in detail and may have created in the minds of gullible readers an impression that the Second Law of thermodynamics has been shown to be not as universal a law as it was meant to be. A couple of weeks later some websites (for example http://compulrenta.ru/2002/7/23/32402/ ) announced the sensational news: the Second Law has been overturned!
Nothing of the sort actually happened. The five Australians had conducted a sophisticated and ingenious experiment. Their results do not cause doubts (although, as always in science, the final acceptance of these data can occur only after other researchers reproduce these data). However, they did not discover anything sensational or contradicting the Second Law of thermodynamics. On the contrary, their data, albeit interesting, were fully expected based on the fundamentals of statistical thermodynamics known since the last quarter of the 19th century. The crucial words in the title of their article were “For Small Systems and Short Time Scale.” The Second Law of thermodynamics has been formulated only for macroscopic systems. Even in macroscopic systems the laws of thermodynamics are realized via fluctuations around the most probable state. For small systems and short periods of time fluctuations are expected to occur as local and temporary deviations from the general trend which is the subject of the statistically valid law. The Australians’ ingenious experiment confirmed the prediction of statistical thermodynamics, as they probably realized themselves. Unfortunately, they did not withstand the temptation of giving their report a sensational sounding and easily misconstrued title. This provided journalists, and along with them people like Silman who apparently are prone to use any source if it seems to support their preconceived position, with a pretext to publicize an allegedly sensational discovery.
The article by Dr. Menachem Kovacs is titled Finding G-d in the Study of the Sociology of Religion. This paper is full of imprecise statements making Kovacs’s entire exercise largely irrelevant. Here are a few examples.
On page 164 we read that “the typical treatment of religion in the introductory sociology texts today is usually cursory and highly critical. It is often based on the Marxist view that religion is an illusion, a deception to comfort the ignorant masses with the promise of a better life in a non-existent afterlife.”
Marxism indeed entails an anti-religious trend. However, there are scores of thinkers who reject Marxism but also reject religious beliefs. The negative view of religions is by no means a monopoly of Marxism. In fact, Marxism is itself a form of religion, since like any religion it is based not on evidence but on a hypothetical theory utilizing only a limited subset of data out of the much larger entirety of all data. Like religions Marxism is dogmatic as it requires from its faithful an uncritical acceptance of every word uttered by its prophets Marx and Engels, and in its most tested (and failed) form also by Lenin and/or Stalin. Marxism’s hostility to other religions is akin to the mutual hostility among various religions which led to mass murders committed by Chrisitan on Mulsims, by Muslims on Christians, by both Muslims and Christians on Jews, by Catholics on Protestants and vice versa, exactly as the followers of Marxism did to their alleged enemies, who were believed to be all those who did not share the Marxist beliefs.
On page 165 Kovacs provides a classification of religions. The very first point in that classification lists theistic religions in the following words: “(1) theism – religions that believe in G-d, such as Judaism (monotheism), Christianity and Islam; or in gods, such as Hinduism (polytheism).“
Ignoring the odd wording of a passage which asserts that “religions believe” in this or that (actually it is people who believe or disbelieve this or that; religions are systems of beliefs and as such cannot be said to believe in anything), the gist of that item is not quite accurate.
First, it sounds strange that among the three monotheistic religions only Judaism is accorded by Kovacs the status of monotheism. Furthermore, the difference between the monotheistic religions on the one hand and the polytheistic on the other, however great in many respects, is not really in what Kovacs says. Take, for example, Hinduism. Yes, in that religion there are scores of deities known under various names – Shiva, Krishna, Vishnu, Mahesvara, Shakti, etc. However, one of the most fundamental principles of Hinduism is that in their essence all these deities are just different forms assumed by one and only one real God (see, for example, Louis Renou, ed., Hinduism, George Brazilier publ., New York, 1962).
Likewise, Judaism, Islam, and Christianity all entail similar beliefs. In Catholicism and in the Russian and Greek Orthodox religions the concept of Trinity is paramount and is just another, truncated version of polytheistic beliefs. In Judaism, Islam and Christianity belief in angels is very similar to the Hindus’ belief in the scores of lesser gods subordinated to the “God of gods,” the supreme being whose attributes are similar in all of these religions, although with a number of variations. Therefore the classification suggested by Kovacs seems to be rather shallow and not really very useful for the proclaimed goal of his discourse.
Overall, Kovacs’s article has little relation to the proclaimed subject of the conference in Miami and of issue 13 of B’Ohr Ha’Torah.
The article by professor of mechanical engineering and mathematical sciences Isaac Elishakoff is titled Probabilistic Analysis of the ‘Torah Codes’: A False Premise? The main thesis of that article is that both proponents and opponents of the Torah “codes” are in error in that both base their argument on a false premise. This false premise, according to Elishakoff, is the assumption that the appearance of the equidistant letter sequences (ELS) in the text of Genesis can be analyzed using calculations of probabilities. This, maintains Elishakoff, implies that there are (or may be) multiple “Torahs” in existence rather than the unique Torah revealed to Moses on Sinai. In the abstract preceding his paper, Elishakoff writes, “… in order to apply probabilistic methods the existence of multiple ‘Torahs’ would have to be assumed – inadvertently or deliberately.” Elishakoff asks, “Can one randomize Torah, as the supporters of the Torah ‘codes’ in effect suggest, and to which the opponents of the Torah ‘codes’ seemingly do not object?”
Reviewing the above thesis, a comment which immediately comes to mind is that Elishakoff has mingled two different questions, to wit: (a) Is it legitimate to apply a probabilistic estimate to a unique object (in this case the Torah)? (b) Is it legitimate to randomize the text of the Torah for the sake of probabilistic analysis?
The answers to the above questions are different, which is contrary to Elishakoff’s assertion denying the legitimacy of both (a) and (b).
In regard to (a), the answer is that it is indeed illegitimate to calculate the probability when only a single (unique) event is available for analysis. However, asserting that both codes proponents and opponents erroneously agree that a probabilistic analysis is legitimate in the case in question, Elishakoff seems to show a lack of sufficient familiarity with the literature in question. This sounds especially odd since Elishakoff refers to several publications by a prominent expert in mathematical statistics and an opponent of the Torah “codes,” professor A. Michael Hasofer. In a paper which, for example, can be seen at http://www.talkreasonlorg/mkarep/hasofer.htm, Hasofer has addressed that point. Unlike the nebulous assertion by Elishakoff, Hasofer’s statement is very clear when he writes that a probabilistic estimate in the case of a unique event “has no frequentist interpretation.” To my knowledge, the code proponents have never responded to Hasofer’s critique of that point.
The answer to (b) is, however, different. There is nothing illegitimate in randomizing the text of Genesis thus generating a multitude of texts, the overwhelming majority of them meaningless, in order to compare the behavior of clusters of ELSs in the original text of Genesis with their behavior in randomized texts. Contrary to Elishakoff’s contention, such randomization (for example, by randomly permuting the letters of the text of Genesis) is not tantamount to creating multiple Torah’s. The permuted texts each contain the same set of letters as the original text of Genesis, but this does not make them alternatives Torah’s but only alternative strings of characters mostly lacking semantic meaning.
While the described randomization is a legitimate procedure for the sake of a probabilistic analysis, this does not mean such a method will be a reliable way to discriminate between chance and design as a source of the ELSs clusters in texts. A detailed analysis of the pitfalls such a method entails is given in the appendix to a paper at http://www.talkreason.org/mkarep/witzcode.html . The main shortcoming of the text randomization method stems from the substantial degree of order which characterizes meaningful texts. Indeed, the specific types of order have been shown to distinguish meaningful texts from strings of gibberish (see http://www.talkreason.org/marperak/Texts ). Among the variety of texts obtained by permuting the letters (or words) of a meaningful text, the original non-permuted text has a lower entropy than the overwhelming majority of permuted texts. On the other hand, any mathematical function reflecting properties of a text is necessarily tied in this or that way to the text’s entropy. Therefore any such function calculated for the original meaningful text is expected to have a value close to either minimum or maximum. The function used by the original proponents of the “codes,” Witzum, Rips and Rosenberg (WRR), which estimates the “distance” between any two ELSs in the text is no exception. Hence, if the “distance” between any two semantically related ELSs is found to be close to a minimal value in the original text of Genesis as compared with permuted texts, this does not point to design as the source of the “code” but quite reasonably may be attributed simply to the existence of order (i.e. of low entropy) in semantically meaningful texts.
All this said, the main fault in Elishakoff’s discourse is, however, that it is irrelevant to the dispute between proponents and opponents of the “code.” Indeed, contrary to Elishakoff’s apparent impression, in their work published in Statistical Science, WRR did not apply randomization to the Genesis text. Their method (reportedly suggested by Professor Persi Diaconis who served as a referee for Statistical Science) was different. They randomized not the text of the Torah, as Elishakoff erroneously thought, but the list containing the appellations of 66 “famous rabbis” and the dates of birth and/or deaths of these rabbis which happened to be found as ELSs in the text of Genesis. This procedure did not entail an assumption of the existence of multiple “Torahs.” WRR’s procedure was strongly criticized by a number of authors, who pointed to various faults in WRR’s procedure, but none of this critique required any assumption of multiple Torahs either (references to most of the publications critical of WRR’s work can be found either at http://cs.anu.edu.au/bdm/dilugim/ or at http://www.talkreason.org/mkarep/ ).
Elishakoff’s discourse, even we ignore its inconsistencies, is simply irrelevant to the dispute in question.
Besides the articles reviewed in the preceding sections of this paper, issue 13 of B’Ohr Ha’Torah contains several more articles which I am not going to discuss in detail, mainly because most of them have little relation to the main topic of that issue – the relationship between science and the Torah (for example, a paper about “lending without interest in a Jewish state” by Raphael Yehezkiel or a paper titled “Lactational Amenorrhea and Mesuleket Damim – a Medical Halakhic Analysis” by Dr. Deena R. Zimmerman). Some other articles deal with the main thesis but in a largely theological or philosophical way (for example, a paper titled “Creation and the Symbiosis of Science and Judasim” by Professor Norbert Samuelson or a paper titled “Genesis 1 Speaks about the Creation of Prophecy, Not the Creation of World” by Dr. Russell Jay Hendel). One paper titled “Mathematical Drash” by Dr. Ari Belenkiy, contains a number of topological diagrams hardly comprehensible to an ordinary reader, whose relevance to the theme of the issue is at best fuzzy. For example, Belenkiy asserts (page 152) that “whether the Jewish sages thought the surface of the Earth to be like a plane or like a sphere” is unimportant, “because in both cases the fundamental group is trivial.” Such application of abstract mathematical concepts to a simple question of whether the Earth is flat or it is a sphere seems to me a display of a prejudice forcing its author to perform mental acrobatics in order to support his irrational beliefs.
Overall, the reviewed issue of the B’Ohr Ha’Torah with its seeming sophistication and a list of authors with advanced scientific degrees, strikes an unbiased reader as a collection of quasi-scientific exercises by a bunch of people for whom a blind faith in the Torah’s inerrancy precludes a rational discussion of objective arguments because in each case the conclusion is reached before the discussion starts. Overall, this journal seems to be an example of a wasted effort by a number of intelligent and educated people who should have known better and directed their effort to a more useful end.