First posted on April 11, 1999. Updated in December 2001.
The arguments related to the beginning of the Bronze age.
The arguments related to archeological data about times earlier than 5760 years ago
Schroeder estimates probabilities
Some specific faults and errors in Schroeder's first book
Schroeder's second book
Change of dates in the second book compared to the first one
Zero time interval in a light beam
Diffraction of waves according to Schroeder
Diluted heat and masers
Schroeder's third book
In this article I will discuss three books by Gerald L. Schroeder [1,2,3]. which have gained a substantial popularity among readers and have often been acclaimed as a very successful clarification of how to reconcile the biblical story with scientific data.
The first book  titled Big Bang and Genesis is subtitled "The discovery of harmony between modern science and the Bible." This subtitle is a concise expression of the main thrust of the book. Schroeder set out on an ambitious road aimed at demonstrating to both believers and skeptics that the contradictions between the biblical revelations and the claims of science simply stem from incorrect interpretations and that every word of the Bible is really in complete agreement with the results of scientific exploration.
Schroeder starts his book  with a rather lengthy narrative about his participation in the underground tests of atomic bombs. This narrative has little to do with the book's subject, but was apparently designed to demonstrate to the readers that Schroeder is eminently qualified to do the job. We learn that Schroeder is not some ignoramus in science blindly defending the biblical stories, but, on the contrary, an experienced physicist with a Ph.D. degree. Indeed, we find in his book multiple respectful references to facts established by science, from archeological data to Einstein's theory of relativity. Overall, Schroeder's idea is that every word in the Bible is literally true and also absolutely compatible with scientific data. No contradiction whatsoever exists between the revelations of the Bible and the results of modern science - or so says Dr. Schroeder.
Who could object to such a statement if it were proven in a perfectly logical way without any distortion of known facts?
In Chapter 1 of his first book  Schroeder tells us that this book was a result of his efforts to answer his children's questions, particularly those of his son Joshua, who had studied the Bible from the age of three, and at the age of eight started discovering what seemed to be contradictions between the Bible and science.
Schroeder does not explicitly tell us if his explanations satisfied his eight-year old son, but such a notion seems to be implied. What is of interest though, is whether or not Schroeder's explanation can satisfy people of a more advanced age than eight.
The discussion of Schroeder's book  will include the following sections:
1) Schroeder's chronology
2) Schroeder's estimate of probabilities.
3) Some particular details and examples from Schroeder's book.
On page 12 of his book  Schroeder writes: "..the discord between archeology and theology is neither necessary nor valid." A few lines later he continues: "My goal in this book is to explain this compatibility to expert and layperson alike."
Hence, as we see, Schroeder has formulated the ambitious intention of explaining to both experts and laypersons, all of whom had wandered in dark until his book, the actual meaning of both scientific data and biblical revelations. It is, surely, a very commendable goal. An important milestone on the road to his goal is Schroeder's analysis of chronological data, which until his book seemed to be irreconcilably different in the Bible and in science.
Schroeder's chronological exploration consists of two parts. The first part deals with the biblical account of the creation of the world in six days and the second with the period of time between the end of those six days and our time. Let us consider each part of Schroeder's chronology in the above order.
Since the general thrust of Schroeder's book was to prove that the biblical account and scientific data do not contradict each other, one of his important tasks was to reconcile the biblical story about the creation of the world in six days with scientific evidence which asserts that the universe is about 15 billion years old.
Before his book, this contradiction between the Bible and science had been discussed an uncounted number of times. A notion that was often suggested was that the word "day" in the Bible was not meant to literally denote "day" as we understand it, i.e. as approximately 1/365 of the duration of earth's revolution around the sun. What is a billion years for us humans, may be one day for God - that is how that interpretation went.
Despite Schroeder's assertions to the contrary, his explanation is actually in the same vein, but with one difference. Schroeder turns to his background in physics, in particular to Einstein's theory of relativity, in order to provide a specific clarification of the six-day creation story, which, in his view, is compatible with science.
As Schroeder explains, the theory of relativity has established, among other things, that there is no absolute time. Of course, this is true. In different frames of reference, continues Schroeder, the time interval between two events may be quite different. This, of course, is also true. What lasted six days in one frame of reference may very well last 15 billion years in some other frame of reference. This is true as well.
Schroeder proceeds to give an example of "time dilation." This is a well known experimental result regarding the behavior of mu-mesons (muons). As Schroeder tells the readers, while 200 microseconds elapse in the frame of reference attached to the ground, only 4.5 microseconds elapse in the frame of reference of the moving muon itself. Experts would gladly agree with that statement, while laypersons have to rely on Schroeder's scientific background and accept his statements.
From this point on the paths of experts and of laypersons necessarily will go in different directions. Laypersons, impressed by Schroeder's scientific credentials, may continue accepting his further explanations. The experts, however, would rather shrug off the next step in Schroeder's line of thought. In that next step, Schroeder makes a leap from the case of fast moving muons to the case of creation of the universe in six days. According to Schroeder, before creating the first man, Adam, God acted in his own frame of reference, vastly different from the frame of reference which would be chosen by him at the moment of Adam's creation. In the pre-Adam frame of reference - the frame of the Creator - what would become billions of years on the future post-Adam clock were just days. At the moment of Adam's creation, God chose to switch to the post-Adam frame of reference, which then became the same for God and men. What had been only six days in the pre-Adam frame of reference, in the new frame of reference common for both the Creator and the creation, would become billions of years. That is what Schroeder tells us.
Can we assert that the above idea is false? We can't. On the other hand, can we assert, based on rational considerations, that the above idea is true? Again, we can't.
Schroeder's explanation requires a leap of faith. It is fine as long as it is not suggested to be an explanation based on science. There is nothing scientific in the notion that God's frame of reference may be vastly different from men's frame of reference. As far as faith is considered, the above assertion is not a new one, and is simply beyond any discussion in rational, scientific terms. Schroeder, though, wants readers to believe that the described explanation is somehow based on the theory of relativity. It is not.
What the special theory of relativity (STR) has established is indeed that time flows at different rate in different "inertial frames of reference." What the STR meant by different inertial frames of reference was quite rigorously defined. The rate of time flow is different in two such frames of reference which mechanically move relative to each other with certain constant speeds. To make a period of time that is billions of years long in one frame of reference, last only six days in the other frame of reference, these two frames of reference must move relative to each other with an extremely high speed.
Of course, to apply this rigorously defined situation to the creation of the universe according to the Bible requires a considerable stretch of imagination. To satisfy the requirements of the special theory of relativity, as per Schroeder's explanation, we have to accept that, first, God is a physical body, second, that it is a body which occupies a certain localized volume in space, and third, to imagine that, during the six days of creation, the Creator was rushing at an enormous speed past the universe he was creating. What would then remain from the concept of the omnipresent non-material God? God performing a marathon - this picture might have been satisfactory for Schroeder's eight-year old son. For those over eight years of age, a better choice seems to be relegating the six-days creation story back to the realm of faith.
In his second book  as well as in some more recent postings on the Internet, Schroeder has modified his interpretation of the biblical story by referring, instead of the special theory of relativity, to Einstein's general theory of relativity (GTR).
It has been shown in the GTR that gravitational fields also affect the rate of time flow. In his second book  and the above paper on the Internet, Schroeder builds his alleged rational explanation of the six days of creation by calculating in detail the duration of every of the six days in question in terms of the men's calendar, using the concept of the time's dependence on gravitation.
The essence of the alleged explanation, given by Schroeder, is as follows: At the moment of the big bang, a "fireball" was instantly created, which was pure energy. This embryo of the emerging universe did not yet contain any mass, and therefore, according to Schroeder, there was no gravitation. Along with the expansion of the emerging universe, energy was transforming into mass (in accordance with Einstein's famous equation E=mc2). Accompanying the emergence of masses, gravitational forces took hold, increasing along with the appearance of the ever larger amount of mass. As the gravitational forces grew, the time flow slowed down. Schroeder suggests that the decrease of the rate of time flow followed a certain mathematical law. What lasted just one (the first) day of creation, on the scale of our conventional calendar is eight billion years. What lasted one more (the second) day of creation, on the scale of our conventional calendar was four billion years, etc. The total duration of creation was exactly six days as per the scale of time which corresponded to the levels of gravitational forces at each step of the universe's expansion, whereas on our conventional time scale it lasted about 15 billion years, in full agreement with the scientific data.
It is easy to see that we can represent Schroeder’s implicit calculation by the following formula:
Tn = 1.6 x 1010/2n,
where n =1 for the First day of creation, n=2 for the Second day, etc., and Tn is the duration of the "day" of creation number n on our time scale, expressed in the number of conventional years. Schroeder provides no reason why the rate of the time flow should have gradually decreased according to that rather than to any other regularity except for his desire to reconcile the biblical story with scientific data.
It is easy to imagine how laymen may be impressed by the above explanation, which looks so elegant on its face. Let us avoid discussion of some minor dubious points in Schroeder's exercise, concentrating instead on the most egregious misunderstanding of the matter by Schroeder, which renders his alleged explanation meaningless.
While supposedly utilizing the concept of relativity of time, Schroeder actually has based his discussion on the concept of an absolute time, thus contradicting the very premise of his discourse.
To explain Schroeder's distortion of the theory of relativity, let us first again recall certain concepts of the special theory of relativity using them as an analogy. The STR had established relativity of velocity. It taught us that there is no such thing as absolute velocity. The concept of velocity has meaning only in terms of a comparison between two (or more) frames of reference. If we considered the entire universe as a single frame of reference, the concept of velocity would have no meaning. Consider an even simpler example. Imagine that we are in a plane, which is flying with a constant velocity relative to the ground. The windows of the plane are curtained, so we cannot see the ground or clouds. In such a situation there is no way to assert whether the plane is moving or is stationary. From the viewpoint of physics, there is no difference between the states of motion with a constant velocity and of being at rest. No physical experiment performed inside that plane would allow us to distinguish between the states of motion with a constant velocity and remaining at rest. Therefore, within the confines of that plane, the concept of its velocity has no meaning. It would acquire meaning though if we had a way to relate the motion of that plane to some other body (i.e. to some other frame of reference) such as, for example, the ground. If we consider now the entire universe as a single frame of reference, there is no meaning which can be attributed to the velocity of the universe's motion, unless there is another universe, which could serve as a reference.
There is some analogy between the above situation and the model suggested by Schroeder. The concept of different rates of time flow depending on gravitation has a meaningful interpretation only in terms of at least two different bodies subjected to different gravitational forces. In such two systems, clocks would tick at different rates. However, to discover the difference between the mentioned clocks, there must be a way to compare them to each other.
Schroeder actually considers the universe as a whole, and suggests that along with the increase in the amount of mass, and, hence, in the level of gravitation all over the universe, the time flow is slowing down. This consideration has a number of serious flaws. I will discuss here only his most egregious misinterpretation of the GTR.
As the amount of masses increased, as postulated by Schroeder, this affected all clocks in the universe equally. (There are differences between the clocks located in various parts of the universe, which is though irrelevant to Schroeder's model). The universal clock suggested by Schroeder, namely the frequency of Penzias-Wilson radiation, is subjected to that effect as well as any other imaginable clock. Since all the clocks are subjected to the same effect of increasing gravitation, there is no way to discover the alleged deceleration of time's flow. (The extrapolation of the red shift into the past can be utilized to estimate the length of the universe's existence, but not to discover or measure the time dilation, not any more than it could be possible to lift oneself out of a pit by grabbing one's hair with one's own hand and pulling it up, without any independent point of support). To reveal any difference in the rate of time flow, one would need an independent clock, which is not affected by the change of gravitation, and which could be used as a reference. In other words, one would need a reference clock, which would be nothing less than a clock showing absolute time. Such a clock is not known. Since the alleged deceleration of time's flow cannot be measured, or even discovered, it has no physical meaning, and has no effect on any physical processes, unless an absolute time, independent of frames of reference, is postulated.
Therefore, what lasted six days 15 billions years ago, lasts exactly six days now. If there were available some other observable universe, which could be utilized as an independent frame of reference, then it could be possible to find out if one day in our century is different from one day 15 billion years ago. As it is, the length of a day in our century is to all intents and purposes exactly the same as it was 15 billion years ago. This length of the day is necessarily measured in arbitrary units, and has no definable absolute ("actual") magnitude.
In other words, again, the difference in the rate of time flow between our time and that fifteen billions years ago could be attributed some meaning only if there existed absolute time. Since GTR rejects the concept of absolute time, the idea of Schroeder contradicts the very essence of the theory of relativity, on which he has allegedly based his model.
Conclusion: Schroeder's attempt to reconcile the biblical tale about six days of creation with scientific data which assert that the age of the universe is about 15 billion years, failed. It was based on a misapplication of the theory of relativity.
As Schroeder tells us, at the moment the first man was created, God instantly switched from his previous frame of reference, in which the creation took six days, to a new frame of reference, this time the same for God and men. Since then that frame of reference is the one the mankind lives in. Hence, everything the Bible tells us about events that occurred starting with Adam's appearance, is chronologically precise. If the Bible says, for example, that Adam was 130 years old when his son Seth was born, then it is precisely 130 years as we understand them, i.e. the period of time it took the Earth to make 130 revolutions about the sun. If the Bible says that Adam was created by God 5,750 years before the publication of his book , it means precisely that – the time the earth circled the sun 5,750 times, counted in the same years we conventionally mean. Schroeder tells us that the above number – 5,750 years since Adam - is precisely compatible with the scientifically established chronology of human history. Here is the pertinent quote from Schroeder's book (page 31): "In the year 1990, all the generations since Adam have a cumulative age of 5,750 years. This biblical date for the dawn of recorded history is closely matched by the archeological finds of the last two centuries."
This is a very impressive statement. The average reader would be eagerly waiting for proof based on scientific data. Schroeder proceeds to provide such a proof.
His argument in support of his thesis is twofold. One part of his argument deals with the beginning of the Bronze age and the other with the nature of those human-like creatures, who, according to archeological data, lived already thousands years earlier than about 6,000 years ago.
Let us consider both parts of Schroeder's argumentation.
a) The arguments related to the beginning of the Bronze age.
Schroeder juxtaposes the generations of the descendants of Adam's two sons, Seth and Cain. The Bible provides the ages of all the descendants of Seth, starting with Seth's son Enosh, all the way to Noah, who, at the time of Flood, was 600 years old. There were nine generations between Seth and Noah. Then Schroeder counts nine generations of descendants of Cain, ending with Tuval-Cain. The ages of Cain's descendants are not provided in the Bible, so Schroeder assumes that Tuval-Cain was a contemporary of Noah and from that he concludes that the time interval between Adam and Tuval-Cain was the same as between Adam and Noah, i.e. according to Schroeder's calculations, some 1350 years. Subtracting 1350 from 5750, Schroeder concludes that Tuval-Cain lived about 4400 years ago.
The above calculation is however wrong, and, as we will see later (when discussing Schroeder's second book ) he himself was forced to change the above numbers. Schroeder does not say a single word as to why he has changed the described data in his second book. It is sufficient, though, to read the pertinent verses (4:20 to 4:22) in the Book of Genesis to see the source of Schroeder's error. While counting generations between Cain and Tuval-Cain, Schroeder includes in his count Iaval (the seventh generation) and Yuval (the eighth generation), although it is clearly said in those verses that Yaval (also sometimes transliterated as Jabal) and Yuval (Jubal) were not the grandfather and the father of Tuval-Cain, but rather his half-brothers, as they all were sons of Lemach. Hence, Tuval-Cain, according to the Bible, belonged to the seventh generation after Cain rather than to the ninth. This shifts Tuval-Cain's lifetime back several hundred years from Schroeder's assertion in his first book.
However, since we are now discussing the first book where Schroeder has based important and far-reaching conclusions on the above numbers, let us accept those numbers for further discussion, especially since in his second book Schroeder did not change his conclusions based on the above dates.
As the next step, Schroeder tells us that Tuval-Cain, according to the Bible, was the inventor of bronze. Hence, according to Schroeder, the Bible informs us that the Bronze age started about 4400 years ago. This, asserts Schroeder, is precisely what the science tells us! What a proof of complete compatibility of the Bible and science!
Is it indeed?
Let us check precisely what the Bible says about that. Schroeder refers to Genesis 4:22. Let us look at that reference. Here is the literal translation into English of the text of the book of Genesis, 4:22: "Also Zillah gave birth to Tuval-Cain, smith of all cutting tools of copper and iron..."
Note that the last two words (copper and iron) are, in the Hebrew text, "nekhoshet ubarzel." Schroeder tells us that in the early Hebrew the word "nekhoshet" (which he transliterated as "nhoshet") meant both bronze and brass.
Of course, this can only be guessed, since the literal meaning of that word is copper, whereas the Hebrew word for bronze is arad and for brass it is pliz. However, the words arad and pliz are absent in the text of the Torah, which may indicate that these words appeared in the Hebrew vocabulary later than the Torah was written. Therefore it is possible that the word nekhoshet (Nun-Khet-Shin-Tav) was indeed used in that context to denote not just pure copper but also copper-based alloys like bronze and brass.
However, it is hard not to notice that Schroeder, in his quest for the chronological coincidence between the Biblical account and scientific data, conveniently pretends not to notice in the same passage about Tuval-Cain the word "barzel," which means iron. If Tuval-Cain made tools not only of copper or bronze, but also of iron, then we must place his lifetime at a much later date than Schroeder wants us to believe. The use of iron started, roughly, some fifteen centuries after that of bronze. Hence, just one word omitted by Schroeder in his reference to the biblical text makes his chronological exercise ingloriously collapse. Moreover, the glaring omission of that reference to iron, which obviously could be only deliberate, undermines all of Schroeder's arguments, making his attitude suspect of being not quite impartial. There goes all Schroeder's post-Adam chronology.
As we will see, in his second book, where Schroeder changes the dates (without any explanation for the reason) the described discrepancy is only exacerbated.
b) The arguments related to archeological data about times earlier than 5760 years ago
Having successfully, in his view, dealt with the beginning of the Bronze age, Schroeder faced another controversy between science and the Bible. According to the Bible, the first man, Adam, was created less than 6,000 years ago. On the other hand, archeological data indicate that creatures possessing many characteristics of humans lived at much earlier time. To reconcile these two viewpoints Schroeder advances some rather clever explanations. According to his suggestion, the human-like creatures that lived before Adam were not really human. They were, Schroeder tells us, animals which had some characteristics in common with post-Adam humans, but did not possess a human soul (in Hebrew - neshamah). Adam, according to that explanation. was the first real man, the first to receive from God his neshamah, whereas pre-Adam humanlike animals did not have one.
Before discussing in detail Schroeder's notion about Adam being the first real human possessing the "neshamah" let us make note of the following historical evidence. If we believe Schroeder, at a certain moment in the history of men, namely less than 6,000 years ego, a revolutionary event occurred which radically changed the nature of the humanlike animals converting them into real humans, who were given the "neshama," thus separating them in a crucial manner from all other animals. There is, though, historic evidence that, for example in the country of Egypt, a well organized state existed earlier than 6,000 years ago. There is plenty of information about that state. If at a certain moment, about 6,000 years ago, a revolutionary change in the nature of men took place, wouldn't we expect that such an event would be somehow reflected in the records pertaining to that time? There is no evidence whatsoever in the scientifically established history of Egypt which would indicate any radical change in the nature of men at the above mentioned time.
To further discuss Schroeder's clever explanation we have to define what distinguishes a man from an animal. In that endeavor I will not argue against Schroeder's contention that pre-Adam humanlike creatures did not possess a neshamah, because this notion cannot be either proven or rejected based on any rationally acceptable evidence. It is a matter of faith, and in this article I do not argue either in favor or against faith. We can, though, try to establish some verifiable criteria which would enable us to distinguish between animals and men.
It seems reasonable to assume that the following features are usual characteristics of humans while absent in animals:
Humans possess language, necessarily spoken and more often than not also written. Each human language has a well defined grammatical structure. No animals have a real language (unless the word language is applied to the primitive systems of sounds used by some animals, like dolphins or chimpanzees, which lack any grammatical structure and have an extremely limited "vocabulary"). Of course, no animal uses a written language.
Humans often form societies comprising thousands of members, structured vertically with many layers of hierarchy. The closest form of a hierarchy in the animal world is that in a family, say, of wolves, apes, or lions. The human society often forms a government and a law system. There is no equivalent to a government in the animal world (where the formations closest to a society are cattle herds and wolf packs).
Humans usually develop technologies aimed at improving their standard of living. These technologies involve building living quarters, making tools for acquiring food and clothing, etc. Humans possess an obvious capacity to improve the design of their creations and to invent new forms of technology. Some animals also build living quarters or even construct dams (beavers) and the like, but they never change the scheme of those constructions and have no evident capacity to improve and to invent.
Humans typically develop arts, including music, dance, and art of images (like painting). No animals are known to have any form of art.
Finally, humans often develop some form of religion, whereas no animals are known to have any religious concepts or any notion of a deity (unless you believe your dog considers you to be God). I assume the development of religion can very decisively distinguish real man from any other humanlike or non-humanlike creature.
If we account for the above points, then, according to Schroeder's concept, the humanlike predecessors of Adam who had not really been humans, but just humanlike animals, did not have a written language, did not have a law system, did not have any government, did not have any art and did not have any religion.
Archeological data strongly contradict the above points.
In this article I do not try to prove either the biblical or the scientific viewpoint, leaving it to each reader to decide for him/herself whether to accept the biblical or the scientific side of the story. My goal in this article is only to test Schroeder's arguments in regard to the absence of contradiction between the above two views. Since Schroeder accepts equally both world outlooks, we have no need to judge the merits of either of the two views, but only to see if these two views can be logically reconciled.
What is the Bible's story? It says that the first man walked on the earth roughly 6,000 years ago.
What is the scientific story? It is rather different.
Archeological data indicate that humanlike creatures made tools of stone as early as about 40,000 years ago. Is any animal known to shape stones into tools?
The color pictures of animals had been drawn on rocks as early as about 27,000 years ago (for example, the image of a horse in Pech Merle, France). What animal is known to engage in art?
As archeological data indicate, as early as 23,000 years ago humanlike creatures used to embellish their looks by means of beads attached to their clothing. Is there any animal known to use artificially made clothing and, moreover, to embellish them in any way?
Archeological data indicate that as early as 18,000 years ago humanlike creatures used needles made of bones to sew clothing. Is there any animal known to sew clothing?
Archeological data indicate that the bow and arrow was already in use about 11,000 years ago. Is there any animal capable of inventing and using sophisticated tools such as the bow and arrow?
Archeological data indicate that as early as about 9,000 years ago humanlike creatures already used pottery. To make the pottery, those humanlike creatures used kilns where the temperature reached about 1000 degrees. Is there any animal capable of constructing kilns and, moreover, of making and utilizing pottery?
At about the same time, some 9,000 years ago, the humanlike creatures built settlements occupying over two hectares each, where buildings were used as living quarters and as storage sheds. About 8,000 years ago, the size of such settlements was sometimes up to 15 hectares. The inhabitants of those settlements used artificial irrigation and grew crops. They used seals, which proves the existence of some form of a writing system and of some form of documentation. Are there any animals known for building villages, using artificial irrigation, growing crops, and using a writing system and a documentation?
Finally, one more thing archeology tells us about is as follows: As early as about 8,000 years ago, that is some 2,000 years before the date, when, according to the Bible, the first man, Adam, appeared, humanlike creatures (as, for example, inhabitants of the Sumerian settlement at Tepe Gawra in Mesopotamia) systematically built, in the centers of their villages, religious shrines and temples. Is there an animal, however intelligent, that is known to have any religious concepts?
What we can say with a reasonable confidence, is that the arguments by Schroeder aimed at proving the compatibility of the biblical and the scientific accounts, fall apart at even a perfunctory glance.
In the chapters dealing with probabilities, Schroeder's attitude is different from that in the chapters dealing with chronology. In the latter, Schroeder fully accepted scientific data and tried to reconcile them with the biblical story. That attitude was due to Schroeder's opinion that the archeological data are so firmly established that refuting them would be surely a losing proposition. Hence, to salvage his belief in the biblical story Schroeder had no other resort but to look for a way to reconcile the biblical and the archeological stories. As the previous sections of this paper have demonstrated, Schroeder failed in his attempt.
In the chapters dealing with probabilistic estimates in regard to the origin of life, Schroeder felt more comfortable in adhering to the biblical story because the origin of life, as suggested in science, has not been proven scientifically to the same irrefutable extent as the archeological data. Therefore, when dealing with the origin of life, Schroeder seems to be confident in his ability to refute the scientific hypothesis in favor of the biblical story. There are no more attempts to reconcile the scientific and the biblical views, but rather an unequivocal rejection of the scientific hypothesis, and an equally unequivocal acceptance of the biblical story.
The scientific hypothesis is based on the assumption that life emerged spontaneously as a result of chance interactions between chemicals in the primeval atmosphere of the Earth several billions years ago. Another version of that hypothesis suggests that life could have been brought to the Earth from some other worlds, where it originated spontaneously via the above mentioned, largely stochastic process.
Schroeder has based his refutation of the above hypothesis on certain probabilistic considerations. For example, on page 111 of his book  he wrote about the scientists who adhere to the above scientific hypothesis: "These scientists were making assumptions without any attempt to rigorously investigate the probability of such events." Obviously, what Schroeder wants us to believe is that unlike those less-than-rigorous scientists, he has rigorously investigated the probability of a spontaneous emergence of life via chance occurrence of chemical interactions. The conclusion Schroeder extracted from his rigorous investigation was that spontaneous emergence of life was impossible because the time needed for such an event to occur is so immense as to exceed the entire duration of the existence of the universe. This conclusion is far from being new and has been advanced an uncounted number of times by the defenders of the biblical story.
The above consideration in turn is based on a certain interpretation of the concept of probability, which Schroeder shares and tries to explain. The essence of the interpretation in question can be best illustrated by quoting again from Schroeder's book (page 113): "The probability of duplicating, by chance, two identical protein chains, each with 100 amino-acids, is 1 chance in 20100, which equals the digit 1 followed by 130 zeroes..."
While the arithmetic in the quoted segment is not exactly correct (the actual expression contains many digits other than zeroes), in principle Schroeder's statement is correct in that the probability in question is indeed an extremely small number. However, as Schroeder continues, his further reasoning becomes contrary to the correct concept of probability.
This is what follows on page 113: "To reach the probable condition that a single protein might have developed by chance, we would need 10110 trials to have been completed each second since the start of the time. To carry out these concurrent trials, the feed stock of the reactions would require 1090 grams of carbon. But the entire mass of the Earth (all elements combined) is only 6x1027 grams!"
What a triumph for the opponents of the hypothesis of the spontaneous emergence of life! What a nice calculation! What an impressive statement so categorically demolishing the hypothesis of the spontaneous emergence of life! If only these statements and arguments were correct beyond the sheer arithmetic!
In reality, the above argumentation is wrong. It misinterprets the meaning of probability. If that argumentation were correct, then we would have to accept that to win big in a California lottery, where the chance of winning the jackpot is about 1 in 16,000,000, one necessarily needs to play the lottery 16,000,000 times. We know, though, that some people have won big, having bought just one ticket the first time in their life. Such a smile by the Lady luck happened more than once, defying the 16 million-to-one odds.
The concept of probability and its application is discussed in detail in my article Improbable Probabilities. That discussion demonstrates the lack of substantiation in Schroeder's position.
The value of probability does not provide any information in regard to what will occur in any individual trial. If the probability of an event is one out of trillion of trillions, there is nothing surprising if that event actually occurs in a particular trial, since its appearance is as likely as of any other equally probable event. From this standpoint, the arguments by Schroeder based on the very small calculated probability of life emerging as a result of random chemical reactions are meaningless, since a small calculated probability in no way means the impossibility of an event.
Moreover, there are many situations where the possible event are not equally probable. As it is shown in my article on probabilities, referred to above, the calculated probability in a certain sense reflects the level of information available about the object whose behavior is being guessed, rather than the objective likelihood of the event in question.
In regard to the random chemical reactions in the primordial atmosphere, the assumption of equiprobability has no foundation. Some chemical interactions must have been much more likely than some others. Certain conditions in the primordial atmosphere could have been conducive to certain interactions while inhibiting some other interactions. Under certain conditions, powerful catalytic effects could have emerged, greatly enhancing the likelihood of interactions which would become steps toward the emergence of life. Schroeder's calculations of probabilities completely ignore such alternatives and are therefore nowhere near a reasonable proof.
Furthermore, the theories of the life emergence do not suggest that it was a single act wherein all the components of the living matter came together at once by sheer chance. On the contrary, the scientific theories suggest that life emerged as a result of many small steps each requiring a chance combination of only a very few components and hence having a reasonably large probability. This consideration alone makes Shroeder's calculations irrelevant.
Moreover, even if the probability of a spontaneous life emergence is assumed to have been very low, it is not a valid argument against its possibility. Life emerged at some time in the past. Whatever the steps were this process included, it was just one among a very large number N of possible (but by no means equally probable) events. Its calculated probability, however small it happens to be, has by itself no cognitive value. Any other of N possible events could have occurred, but did not. Maybe some of those N-1 events which did not happen would be as complex and amazing as life is. We will never know the answer to that. Hence the argument by Schroeder based on the calculation of probabilities of life emerging through some sequence of randomly occurring chemical reactions is meaningless. Equally unfounded are his calculations of the time allegedly necessary for such reactions to take place. If certain interaction were catalytically enhanced, with each step increasing the likelihood of the next consecutive step toward the emergence of life, the time necessary for the emergence of proteins was actually shorter by many orders of magnitude than that calculated by Schroeder.
Can the above considerations serve as proof that life emerged spontaneously rather than having been created by a divine design? Admittedly, it cannot. However, it shows that the hypothesis of the spontaneous emergence of life does not contradict any probabilistic considerations. The consideration of probabilities by Schroeder cannot serve as a proof to the contrary.
Contrary to his proclaimed intention, Schroeder failed to demonstrate the compatibility of the biblical account with the scientific view in regard to the origin of life, whereas his probabilistic considerations are irrelevant.
Besides the above discussed general weaknesses of Schroeder's case, his book contains a plethora of small and not so small specific erroneous statements and arguments. Some of those faults will be discussed in this section.
1) The Bible code example.
On page 182 of Schroeder's book, we read: "I offer here only two examples of the thousands of subtleties found in the Bible."
The first example offered by Schroeder is that of the so-called Bible code. The Bible code controversy is discussed in detail, for example, at B-Codes Page where the lack of substantiation of the claims of the Bible code proponents is shown. Schroeder's discourse is limited to a few primitive examples which lack any statistical significance and testify to Schroeder's amateurish level of familiarity with the subject he endeavored to discuss
2) Is weight and mass the same? Is kinetic energy proportional to velocity? Are the laws of physics "nothing more" than laws of nature?
The above questions sound silly if one discusses statements by a physicist, especially one with a Ph.D. degree. Unfortunately, Schroeder provides reason to raise the above questions. For example, on page 40 of his book  Schroeder writes : "The mass (or weight) of the object while at rest is called, in technical terms, its rest mass." I find it hard to believe that a Ph.D. in physics could indeed think that mass and weight are the same. I prefer to interpret the above sentence as a display of sloppiness in style rather than of ignorance. This interpretation is reinforced by the use of the words "in technical terms" which do not seem to convey any meaning. In which non-technical terms is the mass "while at rest" called something else rather then rest mass?
Unfortunately for the above relatively benign interpretation, Schroeder expresses himself in the same way more than once. The same expression "mass (or weight)" appears, for example, on page 37, giving rise to suspicion that Schroeder may actually believe that mass is the same as weight. Such a statement made by an undergraduate student on an exam in general physics would result in an immediate F grade. The rest mass is a body's property, a constant independent of the frame of reference, whereas the total (or relativistic) mass is a function of velocity, and, as such, depends on the choice of frame of reference. The body's weight is a completely different quantity, reflecting the interaction of the body in question with a planet. It depends on both the mass of the body in question and the mass of the planet and, in the first approximation, on the squared distance from that planet's center (assuming the body in question is much smaller than the planet or has spherical shape itself). I am sure Schroeder had studied these facts as a student.
On the same page Schroeder writes: "It acquires velocity and in so doing acquires kinetic energy proportional to the velocity." Come on, Dr. Schroeder! Don't you know that kinetic energy is proportional to squared velocity? Is it sloppiness again?
On page 41 we find one more expression which, to put it mildly, sounds strange from a Ph.D. in physics. I quote: "...laws of Physics (which are no more than laws of nature)...." Are they indeed? Schroeder seems to be unaware that laws of physics are postulates based on interpretation of experimental evidence. At the best, the laws of physics can be considered our guesses as to what the reasonable approximation of laws of nature can be. Laws of nature supposedly did not change from, say, the 17th to the 20th century, whereas the laws of physics have gone through drastic modifications and many amendments.
There are more examples of very dubious statements by Schroeder, and it is immaterial whether they stemmed from a lack of knowledge or a lack of meticulousness in writing a book which supposedly sheds light on important matters.
Just a couple more of examples of less than accurate ways Dr. Schroeder expresses himself. On page 100 we read: "We see randomness of entropy increase in every observable system." This is gobbledygook. Entropy itself is a measure of disorder, i.e. of randomness. The expression "randomness of entropy" is devoid of any meaning.
Page 117: "The centrifugal force of the spin flattened the cloud into a disk." This is one more display of either a lack of sufficient understanding or of stylistic sloppiness. Centrifugal force is what is called in physics force of inertia. It is considered to be a fictional force. The real force in this case is centripetal force, caused by gravitation. It is convenient in certain cases to use the concept of centrifugal force when writing certain equations. (It is sometimes referred to as D'Alembert's principle). However, saying that centrifugal force "flattened the cloud" means veiling the essence of the matter in a cloud of a meaningless phraseology.
Further examples of less than reliable statements could be given.
Now let us discuss the second book by Schroeder, The Science of God, subtitled "The Convergence of Scientific and Biblical Wisdom" . It is considerably larger than the first book. Judging by the size and the title, one might expect that the second book would continue and expand on the subject of the first book. Indeed, to some extent it turns out to be true, as Schroeder repeats in his second book a number of points discussed in the first book, partially modifying and adding to them.
On the other hand, the second book differs rather substantially from the first one. One difference is that in the first book Schroeder limited himself to a relatively narrow circle of topics, while in the second book he tries to cover a much larger scope of problems. One of the results is a loss of the sharpness of the book's focus. Various chapters discuss questions sometimes little related to each other, hence, instead of the relatively clear plan in the first book, in the second one we see a farrago of topics and subjects, each discussed separately. This situation dictates as well the form of my review. Unlike with the first book, this time I will discuss various aspects of Schroeder's second book one by one, without subordinating the discussion to any logical sequence.
Another peculiarity of Schroeder's second book is its uneven character. One can indicate some well written passages in that book, where the subject is explained correctly and in an easily comprehensible form. On the other hand, many sections contain factual errors, sometimes of an elementary nature and sounding quite odd, if we remember that the writer claims to be an experienced physicist with a Ph.D. degree. We will discuss examples of both types.
Finally, one more feature of the book in question: in this book Schroeder provides data which contradict the data given in his own first book, without a word of explanation. We start with the discussion of precisely this feature.
One of the differences between the first and the second books by Schroeder relates to the chronological data in regard to Noah, the Flood, Tuval-Cain, and the onset of the Bronze age.
In his first book, Schroeder calculated that the interval of time between Adam and Tuval-Cain was about 1350 years, that Tuval-Cain was a contemporary of Noah, and that the Bronze age started some 4400 years ago (pages 31-32 and Table 2 in the first book ). In the second book  the onset of the Bronze age is said to have happened about 5000 years ago, i.e. about 600 years earlier than in the first book (see, for example, page 131 in ). Furthermore, on page 130 of  Schroeder maintains that the time interval between Adam and Tuval-Cain was 700 years, instead of the 1350 years he indicated in . On the other hand, in  (page 131) the date of the Flood is said to be about 4100 years ago, which is about 900 years later than that date of the advent of Bronze which is given in that book. According to these altered dates, Tuval-Cain is no longer considered to be Noah's contemporary, since the invention of bronze, attributed by Schroeder to Tuval-Cain, is now said to have happened 900 years earlier than in . Schroeder does not provide a single word of explanation as to why he changed the dates after his first book.
As was indicated in the previous sections of this article, the above alteration of dates was apparently done because somebody had shown to Schroeder the error in his count of generations between Cain and Tuval-Cain in his first book.
If a writer changes his views, are the readers not entitled to at least some explanation?
Despite the change of dates, Schroeder still asserts in his second book that the advent of the Bronze age, according to the archeological data, precisely coincides with the biblical story (as per his calculations). Like in his first book, Schroeder again pretends not to notice the statement in Genesis 4.22 indicating that Tuval-Cain also made tools of iron. Since in the second book the lifetime of Tuval-Cain is shifted back by some 900 years, it makes it even farther in time from the iron age as determined by archeology. This completely undermines Schroeder's chronological exercise.
The section in question (pages 161-164) describes the "shrinking of time" in fast moving frames of reference according to the theory of relativity. As that theory asserts, the time interval between two events which occur at the same location within some frame of reference and measured in that frame of reference is always shorter than the time interval between the same events, measured in another frame of reference which moves with a certain speed relative to the first frame of reference, so in that second frame of reference the events in question occur at different locations. The larger the speed of motion of one frame of reference relative to the other one, the larger the difference between the time intervals. If the speed approaches speed of light, the "local" time interval approaches zero.
Schroeder considers the example of a light signal which carries information about the explosion of a supernova that occurred 170 thousand years ago (measured in the earth's time). Since the supernova (1978A) is located 170 thousand light years from the earth, the light signal took exactly that long to reach the earth. Since, though, the light signal moves with the speed of light, the flow of time in the frame of reference attached to the signal "stopped." If there existed an observer "living" in the frame of reference of the signal, (which is impossible as no physical body can move with the speed of light) for "him" the explosion of the supernova and the arrival of the signal on Earth would have happened simultaneously.
Schroeder's explanation entails a serious flaw. A frame of reference which can be attached to photons simply does not exist. If it existed, the photons would be at rest in such a system. However, photons cannot be at rest. According to the special theory of relativity, photons always move with the same speed (in vacuum) in every frame of reference. If a frame of reference wherein photons are at rest existed, time would stop in such a frame. Since, though, no such frames of reference are possible, Schroeder's concept is meaningless.
Of course, in accordance with his agenda, Schroeder tries to prove the analogy between the described alleged "paradox" of the theory of relativity and the concept of God being "outside time." The concept of God being "outside time" belongs to the realm of faith and has nothing in common with the non-existing effect of "time stopping" in systems moving with the speed of light. Schroeder's attempt, inadvertently invoking the image of God running with speed of light past stars and planets in order to satisfy the conditions of the theory of relativity, can only discredit Schroeder's approach, and, with it, the concepts of faith themselves.
On pages 150-151 in  Schroeder explains the diffraction of waves. Like his example about the light signal from a supernova, these paragraphs have been well written. Up to a certain point, Schroeder's explanations are correct and explain the diffraction, using the example of sea waves, in an easily comprehensible form. Unfortunately, in this case too, Schroeder failed to maintain the competent level of discourse throughout his presentation. As long as Schroeder considers diffraction on an opening whose size is close to the wavelength, his explanation is correct. However, it becomes wrong when he talks about openings whose size exceeds the wavelength. Indeed, on page 150 he asserts that diffraction does not take place if the opening's size is larger than the wavelength.
If that assertion were correct, it would make it possible to build optical microscopes with an unlimited magnification. The ingress aperture of a microscope is much larger than the wavelength of visible light. If there were no diffraction on that aperture, then, utilizing a large number of consecutive lenses, one could reach a magnification of many millions without resorting to electron microscopy. Alas, diffraction puts a limit on the resolving power of any optical device, and with it on the useful magnification.
In the simplest case of diffraction on a single opening, increasing the opening's size does not eliminate diffraction but rather widens the central diffraction maximum, pushing the maxima of higher order to the opening's edges, and squeezing them closer to each other. Diffraction is inevitable even if one of the opening's edges lies infinitely far, so actually rather than to speak about an opening, in this case one has to consider the wave's motion past the corner of an opaque wall. Diffraction always takes place if there is any constraint imposed on the free propagation of a wave.
The above is one of many examples of imprecision typical of Schroeder's style, casting shadow on all of his arguments.
In this section I will discuss a number of the particular errors which are abundant in Schroeder's second book.
1) On page 152, at the beginning of a section titled "The discovery of wave-particle duality" Schroeder tells the story of the discovery and interpretation of the photoelectric effect. Here, within a half-page, he manages to accumulate a long list of errors.
Schroeder starts with the statement that in 1905 Einstein published the results of experiments that demonstrated what has become known as the "photoelectric effect." In fact, Einstein did not publish any experimental data on photo effect because he never performed any such experiments. The photoelectric effect was discovered and partially studied by Heinrich Hertz more than 25 years earlier. Einstein offered in his paper a theory of that effect.
Schroeder writes further: "Light, shining on certain metals, knocks free a stream of electrons..." In fact, not "certain metals," but every metal or semiconductor.
Schroeder continues: "Einstein demonstrated that the rate at which electrons are emitted from metal is related not only to the intensity of the light beam but also to the color of the light." This is an incorrect statement. The rate of electron emission, i.e. the number of electrons emitted per second, does depend on the light intensity but does not depend on the "color" of the light. What indeed depends on the "color" of the light, i.e. on the wavelength, is the kinetic energy of the emitted electrons. Also, Einstein did not demonstrate this, as the dependence of electron's kinetic energy, measured via the so- called "stopping voltage, " on the wavelength of light, had already been discovered by Hertz.
2) On page 154 Schroeder describes experiments with a beam of atoms directed toward a plate with slits in it. He wrote: "Here we use a maser, a gun that can fire one atom at a time." Really?
A reader familiar at least with some rudimentary information in physics or electronics, must wonder whether Schroeder has less understanding of atoms or of masers. It is known that word "maser" is the abbreviation for the expression "Microwave Amplification by Stimulated Emission of Radiation." No maser is capable of firing atoms, either one by one or in groups. Most commonly masers are used as sources of coherent electromagnetic radiation in the microwave range. A version of a maser that works in the range of visible light is sometimes called the optical maser, but more often, laser, which has by now become a ubiquitous component of many appliances, including, for example, CD players. Schroeder could acquire that information from any technical handbook or textbook.
It would be possible to continue a discussion of many erroneous and unsubstantiated statements in Schroeder's second book.I feel, however, that I have already spent much more time and effort on the analysis of that opus than it deserves.
Therefore I will make only one more comment.
On page 180 we read: "The cooling effect of an expansion is logical. It is the dilution of a given amount of heat in an ever larger volume."
From the viewpoint of elementary thermodynamics, Schroeder's statement is nothing short of a bad joke. If Schroeder opened a textbook on physics, the chapter on thermodynamics, he would learn or recall that heat cannot be "diluted" in whatever volume. The expression used by Schroeder gives rise to an impression that, however hard it is to believe, he still adheres to the theory of caloric fluid, abandoned by science as erroneous since Rumford (1796). It is a trivial knowledge in physics that heat is not a substance which is contained within a volume.
On page 183 Schroeder repeats his meaningless statement about "heat dilution."
Heat is defined in thermodynamics as a quantity analogous to work and serving as a measure of energy transformation, provided such a transformation occurs via random molecular interactions. This quantity cannot experience a dilution either at expansion or in any other process. Of course, it is possible that Schroeder improperly used the term heat, actually having in mind a component of the internal energy which is actually the kinetic energy of molecular motion, and which often is referred to as thermal energy. If this is the case, it still does not make Schroeder's notion about the reason for a temperature drop at expansion correct. According to the first law of thermodynamics, the temperature drop at expansion occurs because the work of expansion is done at the expense of the internal energy, and the temperature is the measure of that energy. Expansion causes not a "dilution" of thermal energy, but the decrease of the root-mean-square speed of molecular motion.
For example, expansion into vacuum is an isothermal process. This means that if a system expands into vacuum, its temperature remains constant (because no work is done in such a process). Expansion itself cannot cause a system's temperature drop, which occurs only if, in the course of expansion, some work is done by the system.
It should be noted that the application of thermodynamics to the universe as a whole is uncertain (which in no way justifies the ridiculous invention of a "heat dilution" by Schroeder). The theory of universe's expansion after the big bang does not explain where the universe was expanding to. Actually this theory seems to imply that the expansion occurred into "nothing," whatever the meaning of that word can be. This "nothing" is of course not what within the universe is referred to as vacuum.
The above uncertainty calls for a question, why the temperature of the universe dropped along with its expansion? Thermodynamics per se cannot provide a direct and detailed answer to that question. However, we may reasonably expect that at least the most fundamental laws of physics, including the energy conservation law, retain validity when we discuss the expansion of the universe. In view of that, the cooling down of the universe can be hypothesized as the result of the endothermic nature of the processes in which stars, galaxies, chemical elements, etc, had been created out of the primordial "hot soup," postulated by the theory. Enormous amounts of thermal energy of the primordial "fireball" converted into other forms of energy, including the energy potentially contained in the created masses, as per Einstein's equation (E=mc2).
Obviously, to address the complex question of the universe's cooling by inventing the meaningless "dilution of heat in ever increasing volume" tells much more about Schroeder's scientific competence than about the subject of the discussion.
In his latest book  Schroeder mainly discusses various aspects of molecular biology, while the sections dealing with physics occupy only a small part of the book. Since I am not an expert in molecular biology, I will not argue against Schroeder's excursion into that marvelous science. Schroeder is a physicist by education hence he is as much a dilettante in biology as I am. Any dispute between him and me in regard to biology would therefore be like a dispute between two blind men regarding the quality of a beautiful landscape painted by a famous artist. However, I am qualified to argue against Schroeder's errors when he endeavors to discuss physics. Given the sad fact that he was awarded a PhD degree in physics by a prestigious institution, the elementary errors in Schroeder's discourse are simply stunning.
Here is an example.
On page 38 of his book Schroeder suggested the following equation:
where h is Planck's constant, f is the frequency of deBroglie's wave for a particle, m is the particle's mass and c is speed of light.
Whereas equation (1) is absurd, it is easy to figure out how Schroeder derived it. He read somewhere about the following correct equations :
and 2) E=mc2.......................(3)
Equation (2) was originally suggested by Planck (in 1900) for the quantum of energy emitted by a black body. In 1905, Einstein applied that equation to the energy of photons regardless of whether they are emitted, traveling or absorbed by a material. In 1923 de Broglie suggested to expand the application of that equation to all particles, either massless as a photon or having a rest mass m. As to equation (3), it is probably the most widely known equation of science derived by Einstein in 1905 as a part of his special theory of relativity.
In both equations (2) and (3) E denotes energy of a particle. Obviously lacking proper understanding of these two equations, and seeing the same letter E on the left side of both, Schroeder mechanically combined the equations (2) and (3) into one equation (1).
Unfortunately for Schroeder, he obviously did not know that E in equation (2) and E in equation (3), while both denoting the energy of a particle, actually denote two different energies. E in equation (2) denotes the variable energy of a moving particle, related to that particle's momentum. E in equation (3) is a constant for a given particle, which denotes the so-called rest energy. These two types of particle's energy have little to do with each other. The absurdity of Schroeder's equation (1) is immediately obvious when we notice that it equalizes a variable quantity to a constant. Indeed, the frequency f of de Broglie wave for a particle is not a fixed constant but depends on the particle's momentum, i.e. on its velocity.
Schroeder could have easily realized the senselessness of his equation (1) if he thought for a minute what are the values of the quantities in that equation.
Planck's constant equals about h=6.626.10-34 J.s while the speed of light equals c=2.997.108 m/s. Let us apply Schroeder's equation (1), for example, to the electron. The mass of an electron is close to 9.1x10-31kg. Then the right side of equation (1), i.e. the electron's rest energy is about 8.17x10-14 Joule, or about 5.1x105 eV (electron-volt). Hence, if Schroeder's equation (1) were correct, all electrons in the world would always have the same energy of about 5.1x105 eV. To have such level of energy, free electrons must be accelerated by a voltage a little more than half a million volt. Of course, different electrons (as well as any other particles) actually possess different levels of energy in a wide range rather than all having the same energy of about half a million electron-volt.
In fact, an equation formally looking like equation (1) can be legitimately written for the process of particle-antiparticle annihilation. For example, for an electron-positron annihilation an equation looking like (1) would be correct if m denoted the electron's (or, equally, the positron's) mass and hf the energy of one of the two gamma-photons emerging as a result of those particles' mutual annihilation. The imprecision would in this case be only due to neglecting those constituents of the particle's energy which do not originate from its rest mass (such as kinetic energy of the particle's motion etc). Anyway, though, this case has nothing to do with Schroeder's discourse and with de Broglie's wave.
The equation (1) is not the only error in Schroeder's new book. However, it seems sufficient to limit the demonstration of the inaccuracies in Schroeder's literary production to the above examples.
Since Schroeder's insufficient competence in physics, which is his professional field, is obvious, what credibility can be given to his lengthy discourse on molecular biology which is not his professional field? Moreover, what credibility can be given to his general thesis asserting the alleged harmony between the Bible and science?
 Gerald L. Schroeder, Genesis and the Big Bang, The discovery of the harmony between modern science and the Bible. Bantam Books, 1992.
 Gerald L. Schroeder, The Science of God. The Convergence of Scientific and Biblical Wisdom, The Free Press, 1997.
 Gerald L. Schroeder, The Hidden Face of God, How Science Reveals the Ultimate Truth, The Free Press, 2001.
 David Halliday, Robert Resnick and Jearl Walker, Fundamentals of Physics, John Wiley & Son, Inc., 1993.
Mark Perakh's home page.