The Vastness of Hyperspace
In the decade 1960-1969 humankind began to conquer space. The first cosmonaut, Yuri Alekseyevich Gagarin (Russian cosmonaut, 1934–1968), flew in space on 12 April 1961. Before the end of the decade, on 20 July 1969, Neil Alden Armstrong (American astronaut, 1930–2012) and Edwin Eugene Aldrin, Jr. (American astronaut, 1930–1974) walked on the Moon. They and Michael Collins (American astronaut, 1930–2021) returned safely to Earth on 24 July. Another moonwalk in November closed out the first decade of human space exploration.
Automatic design began and failed in the same decade. In more than three decades since, computer technology has advanced much more rapidly than space exploration technology. Nevertheless, automatic design is still stuck in square one. What is the difference between automatic design and space exploration?
Perhaps we don’t know all the differences, but some have begun to emerge. Space exploration is a journey in time through the three well-known dimensions of space. Automatic design explores a vast hyperspace of many dimensions.
Space is transparent. We are equipped with eyes that can see. With the aid of telescopes, we can explore outer space, deep space, and beyond.
In NASA’s vocabulary, outer space extends from the limits of the Earth’s atmosphere to the orbit of the Moon. Deep space starts there and goes as far as space probes can be tracked. So far that distance is about 60 or 70 astronomical units, that is, 60 or 70 times the distance from the Earth to the Sun, about twice the distance from Pluto to the Sun. Beyond deep space is the visible universe. We use the speed of light to express dimensions in the visible universe. It takes light 8 minutes to reach the Earth from the Sun. That distance is one astronomical unit. Pioneer 10 reached about 60 astronomical units before its signal became so erratic NASA had to give up tracking it. That distance is 8 light hours. The visible universe ends at 13 820 million light years. The knowable universe adds only a relatively thin shell around the visible universe, only 380 000 light years thick. Time sets the limit, not space. We can’t know what was before the beginning. The concept of time before the beginning is meaningless.
The hyperspace for designing a lens, as we saw earlier, has perhaps 50 dimensions or more. A space of so many dimensions is very difficult to explore. Furthermore, we have no sense like vision to explore it. We may calculate the location of points in the space, but we can only visualize them two or three dimensions at a time, using ordinary or three-dimensional graphs. It would take too much computer time to produce a detailed map of this hyperspace.
Each dimension has an axis, with a scale of values. The values may be discrete or continuous. For example, a lens may consist of various optical elements, like curved mirrors or polished glass parts. The number of surfaces a ray encounters from entrance to exit is a discrete variable. It may be 1, 2, 3, or more. The distance between one surface and the next along the center line of the lens is a continuous variable. It may be nearly zero if two solid optical parts are in close contact and held together with transparent cement. Other distances are positive, but none is negative. The radius of curvature of a surface may be negative, infinite, or positive. If a surface curves back toward the entrance to the lens then the radius is positive, but if the surface curves forward toward the exit from the lens then the radius is negative. A flat surface has infinite radius. An infinite radius may be considered as either positive infinity or negative infinity without changing the results of the analysis. Usually the possible values along an axis of the hyperspace are in numerical order, but some axes may have an arbitrary order. There are various kinds of optical glass with different properties. One can arrange the choice of glass for an element along an axis according to the speed of yellow light in the glass, for example. Light slows down in solid transparent materials. A material’s refractive index is the ratio of the speed of light in vacuum to the speed of light in the material. Using refractive index to order materials along an axis would put fused silica at 1.45, borosilicate crown glass at perhaps 1.51, and extra dense flint glass at 1.71. However, the same materials have different refractive indices for other colors of light. The variation of refractive index with light color is called optical dispersion. Dispersion is positive if the material slows blue light more than red light. The glasses mentioned above all have positive dispersion. Some optical materials, like polished rock salt, have negative dispersion for certain infrared wavelengths.
All of the above is a mere taste of the complications of lens design. It is, of course, possible to write a computer program that takes all the above details into account, and many more. If one then starts with a good optical design, the computer can refine the design and optimize it. This ability to refine a good design is called “homing in” on the ideal design.
We can now begin to see why Darwinism and lens design are similar, and what they both lack for completeness. Both can “home in” on an ideal, starting with a design that is not too far from the ideal. Darwin did not discover the homing mechanism. What he called “natural selection” was what some of his predecessors called “natural government.” Mutant, weak, or sickly offspring of a species do not usually survive to adulthood and so do not usually reproduce. They therefore do not dilute the strength of the species. This happens naturally through predation and the competition for food in most species. Among a few “social” species like bees or ants there is a kind of “eugenic police.” Worker bees or ants search through the nursery and simply throw out any mutant, weak, or sickly eggs or larvae. Such species are “social” because the individuals live in large communities, but their behavior is not what we call social. We do social work to help those who fall behind, not to eliminate them. Our ideals obviously do not come from observing nature and imitating it. They come from religious leaders who have called us to follow a better way.
Natural selection or natural government is a stabilizing mechanism, a homing method. Normally it homes in on the stable characteristics of the species, because each species already has its ecological niche and is well adapted to it. Small variations within limits can appear if the environment changes. In a time of drought, for example, birds may develop harder and sharper beaks which enable them to feed on dried and hardened seeds and nuts. When years of normal rainfall return, the bird beaks also return to their normal forms, which are better adapted for eating more plentiful, softer kinds of food.
The great need, both of automatic design and Darwinism, is for an effective exploration method for hyperspace far from “home,” that is, far from the stable point an ideal design occupies, or far from the stable characteristics of a viable species. Random exploration near the stable point only produces a return to that point. That is, the homing mechanism keeps everything on square one, unless the search for a new stable point starts relatively far afield.
The idea of searching for a distant stable point in hyperspace is easy to understand if we think of a group of hikers trying to climb the highest peak in a given area. Suppose that on the only day the hikers have free for hiking, there is thick fog that keeps anyone from seeing more than a short distance in any direction. If they are determined to find a peak anyway, all they have to do is follow the direction of steepest ascent on the ground they can see before them. Eventually they will reach a point from which the land slopes down in all directions. They can then rightly conclude that they have reached a peak. Now suppose that the fog dissipates, and suddenly they can see many peaks near and far, some much higher than theirs. The method of steepest ascent only ensures finding a local peak. It is a homing mechanism, not an effective search method.
If the hikers had seen the landscape beforehand, on a clear day, they would have known where to start their climb and so would have reached the highest peak in spite of the fog. Experienced designers have in their minds a more or less complete map of their design hyperspace. They know of successful designs that have solved various problems before. But the process Darwinists propose is blind, purposeless, and never transfers experience from one problem to the next.
But what if the hikers encounter a landscape like the profile of a city, with tall pinnacles rising vertically from a plain? Human explorers have conquered many landscapes like that. People can climb vertical walls. They have methods of following the lines of steepest ascent even when those lines are discontinuous. It is even harder, but still possible, to climb up overhanging cliffs. Small steps following lines of steepest ascent cannot lead to a peak surrounded by overhanging cliffs. All around such a peak is a region where the direction of steepest ascent points away from the peak. Such features exist in design hyperspaces, too. Small-step methods can’t explore rugged spaces. With one small step Armstrong finally stood on the Moon, but only after a great leap over the immense space between worlds.
Human designers look for new solutions on the conceptual level. They say, “Who has solved a similar problem before, and how did they do it?” They put keywords into search engines, and the engines point them to relevant material. Nothing like these methods is available to mindless processes like those of Darwinism. Random mutations are simply a method of striking out blindly in all directions on the chance that one out of a thousand shots might be close to a new stable point.
Someone may say, “All right, Darwin never said his process was efficient. It is only necessary that it work enough times to produce all the variety of living species on Earth after many millions of years. So, we are all the products of blind evolution. Darwinism has at least produced human brains capable of speech and conceptual searches, so now all design processes can speed up, and they have.”
So runs the argument. Likewise, a computer trying to design a new kind of lens might use methods that are horribly inefficient, methods so tedious that human designers refuse to use them. Nevertheless, a programmer might think that the computer program would produce something new far faster than humans can with their more efficient methods, because computers excel at doing drudgery rapidly. That is exactly the argument that I presented to my research colleagues many years ago. They knew then, and they still know, that artificial Darwinism on computers doesn’t work. Designers sitting at computers have, over the years, come up with many new designs. Designers search the hyperspace of possible lens designs on a conceptual level and leave to the computer the task of refining their concepts.
In the foregoing we made a distinction between efficient search methods and effective search methods. If there is no effective method, it doesn’t matter how efficient some of the methods are. It doesn’t matter how fast you can drive around in a circle if there is no road to your destination. No high-performance race car can drive from America to Europe.
Darwinism lacks an effective search method. It won’t do to say, “OK, we’ll add a search method.” Darwin didn’t just say, “We need a homing mechanism,” or “We need a natural selection mechanism,” He explained how the mechanism he identified could work naturally. Competition for food naturally selects the fittest to survive. Darwinists will have to explain how a blind, unintelligent, natural process can search like a search engine, associatively, on a conceptual level, bringing in accumulated experience from all over the Earth. Darwin identified the homing mechanism, but he didn’t know he needed a search mechanism. Trying blindly in all directions is not an effective search mechanism even if the search space is limited to only a few dimensions.
In our day the people who know the most about searching are those who design search engines for the Internet. How does one search? One enters keywords, and the engine finds online materials that mention those words. In short, searching is done at a conceptual level. This is something humans can do, because of all animals, we are the best at using words. However, effective searching was needed at all stages of the evolution of species from one common ancestor. Surely amoebas could not initiate searches at the conceptual level when they wanted to become better amoebas!
There is another argument Darwinist popularizers frequently bring out. This is the myth of the smooth pathway between the characteristics of one species and those of the most similar species. “A journey of a thousand miles begins with one step,” reads the proverb. Even Darwin recognized that massive, coordinated changes involving many genes are very improbable. The first eye did not suddenly appear one fine, bright, lucky day hundreds of millions of years ago. Such specialized organs must have been the final product of a long series of less effective versions. Rather, say the popularizers, a primitive organism may first have developed light-sensitive spots on its skin that gave it some advantage. Then, through a series of changes, the spots became differentiated and concentrated in greater numbers, to the point where crude imaging was possible. The spots evolved into cavities because they were better protected that way, and then the cavities were protected with a clear window that eventually became a lens that focuses on a retina. So runs the story.
If the species separated from one another by small steps over a continuous pathway, the Earth should be littered with the fossils of the transitional forms. When Darwin first proposed his ideas, he could say that the transitional fossils had not yet been found. That initiated a century-long world-wide search. Now we know that the “missing links” are really missing. Species persist in the fossil record for long periods with very little change in characteristics. This fact by itself should have discredited Darwinism long ago, but Darwinists are clever. They accept the deficiencies of their conjectures, give them a fancy name, and call them new theories. According to some contemporary Darwinists, what we see in the fossil record is “punctuated equilibrium.”
The idea runs as follows. Most individual members of a species live among the great herds, unaware of the hidden valley over the pass where a few of their kind are mutating very rapidly in response to extreme conditions. The few leave few bones behind, but one day their progeny come back over the pass as a new species with a tremendous survival advantage that makes the great herd obsolete and drives it rapidly to extinction.
None of the punctuated-equilibrium proponents have adequately explained why beneficial mutations occur much more rapidly in a small, isolated population than in a great herd. If the mutations occur randomly their incidence is in proportion to the size of the population. The great herd will always have many more new mutations than the small population in the hidden valley. Darwinists apparently confuse this scientifically observable process with the effects of close interbreeding. It is true that close interbreeding produces many birth defects. A great herd may have a number of defective genes in its gene pool. If the defective genes are recessive they may persist for generations without showing up, because the enzymes copy instructions from the dominant gene. Two close descendants of an ancestor with a defective gene may both pass the gene to their offspring, producing a birth defect.[i] It is true that a small population promotes close interbreeding, but a rash of birth defects is not likely to produce a survival advantage. Even if individuals with the same birth defect passed it on to their offspring and the defect became dominant in the population, there would be no Darwinist progress, because the defective gene was pre-existing.
Some people think that punctuated equilibrium is merely an elaborate excuse for not doing fieldwork. Regardless of what anyone thinks, punctuated equilibrium is a phenomenon that the fossil record clearly reveals. Recognizing it is a frank admission that the normal action of natural selection is to stabilize species, not to drift toward new ones.
Systems engineers, scientific researchers, and mathematicians can easily grasp the points I am making. The points are not beyond an intelligent layperson without specialized training, either. However, there are certain popularizers of evolution who persist in saying that Darwinism is the only plausible explanation for the existence of life in all its forms. Many of evolution’s present-day popularizers have conceded that they are “mathematically challenged,” though some have disguised their intellectual handicap with showy computer programs that turn evolution into a kind of video game. We will show that Darwinism is not a plausible explanation, using arguments that are mathematically rigorous but still as understandable as the rules of a simple word game.
[i] This is the scientific basis for the Mosaic prohibition against incest.
Automatic design began and failed in the same decade. In more than three decades since, computer technology has advanced much more rapidly than space exploration technology. Nevertheless, automatic design is still stuck in square one. What is the difference between automatic design and space exploration?
Perhaps we don’t know all the differences, but some have begun to emerge. Space exploration is a journey in time through the three well-known dimensions of space. Automatic design explores a vast hyperspace of many dimensions.
Space is transparent. We are equipped with eyes that can see. With the aid of telescopes, we can explore outer space, deep space, and beyond.
In NASA’s vocabulary, outer space extends from the limits of the Earth’s atmosphere to the orbit of the Moon. Deep space starts there and goes as far as space probes can be tracked. So far that distance is about 60 or 70 astronomical units, that is, 60 or 70 times the distance from the Earth to the Sun, about twice the distance from Pluto to the Sun. Beyond deep space is the visible universe. We use the speed of light to express dimensions in the visible universe. It takes light 8 minutes to reach the Earth from the Sun. That distance is one astronomical unit. Pioneer 10 reached about 60 astronomical units before its signal became so erratic NASA had to give up tracking it. That distance is 8 light hours. The visible universe ends at 13 820 million light years. The knowable universe adds only a relatively thin shell around the visible universe, only 380 000 light years thick. Time sets the limit, not space. We can’t know what was before the beginning. The concept of time before the beginning is meaningless.
The hyperspace for designing a lens, as we saw earlier, has perhaps 50 dimensions or more. A space of so many dimensions is very difficult to explore. Furthermore, we have no sense like vision to explore it. We may calculate the location of points in the space, but we can only visualize them two or three dimensions at a time, using ordinary or three-dimensional graphs. It would take too much computer time to produce a detailed map of this hyperspace.
Each dimension has an axis, with a scale of values. The values may be discrete or continuous. For example, a lens may consist of various optical elements, like curved mirrors or polished glass parts. The number of surfaces a ray encounters from entrance to exit is a discrete variable. It may be 1, 2, 3, or more. The distance between one surface and the next along the center line of the lens is a continuous variable. It may be nearly zero if two solid optical parts are in close contact and held together with transparent cement. Other distances are positive, but none is negative. The radius of curvature of a surface may be negative, infinite, or positive. If a surface curves back toward the entrance to the lens then the radius is positive, but if the surface curves forward toward the exit from the lens then the radius is negative. A flat surface has infinite radius. An infinite radius may be considered as either positive infinity or negative infinity without changing the results of the analysis. Usually the possible values along an axis of the hyperspace are in numerical order, but some axes may have an arbitrary order. There are various kinds of optical glass with different properties. One can arrange the choice of glass for an element along an axis according to the speed of yellow light in the glass, for example. Light slows down in solid transparent materials. A material’s refractive index is the ratio of the speed of light in vacuum to the speed of light in the material. Using refractive index to order materials along an axis would put fused silica at 1.45, borosilicate crown glass at perhaps 1.51, and extra dense flint glass at 1.71. However, the same materials have different refractive indices for other colors of light. The variation of refractive index with light color is called optical dispersion. Dispersion is positive if the material slows blue light more than red light. The glasses mentioned above all have positive dispersion. Some optical materials, like polished rock salt, have negative dispersion for certain infrared wavelengths.
All of the above is a mere taste of the complications of lens design. It is, of course, possible to write a computer program that takes all the above details into account, and many more. If one then starts with a good optical design, the computer can refine the design and optimize it. This ability to refine a good design is called “homing in” on the ideal design.
We can now begin to see why Darwinism and lens design are similar, and what they both lack for completeness. Both can “home in” on an ideal, starting with a design that is not too far from the ideal. Darwin did not discover the homing mechanism. What he called “natural selection” was what some of his predecessors called “natural government.” Mutant, weak, or sickly offspring of a species do not usually survive to adulthood and so do not usually reproduce. They therefore do not dilute the strength of the species. This happens naturally through predation and the competition for food in most species. Among a few “social” species like bees or ants there is a kind of “eugenic police.” Worker bees or ants search through the nursery and simply throw out any mutant, weak, or sickly eggs or larvae. Such species are “social” because the individuals live in large communities, but their behavior is not what we call social. We do social work to help those who fall behind, not to eliminate them. Our ideals obviously do not come from observing nature and imitating it. They come from religious leaders who have called us to follow a better way.
Natural selection or natural government is a stabilizing mechanism, a homing method. Normally it homes in on the stable characteristics of the species, because each species already has its ecological niche and is well adapted to it. Small variations within limits can appear if the environment changes. In a time of drought, for example, birds may develop harder and sharper beaks which enable them to feed on dried and hardened seeds and nuts. When years of normal rainfall return, the bird beaks also return to their normal forms, which are better adapted for eating more plentiful, softer kinds of food.
The great need, both of automatic design and Darwinism, is for an effective exploration method for hyperspace far from “home,” that is, far from the stable point an ideal design occupies, or far from the stable characteristics of a viable species. Random exploration near the stable point only produces a return to that point. That is, the homing mechanism keeps everything on square one, unless the search for a new stable point starts relatively far afield.
The idea of searching for a distant stable point in hyperspace is easy to understand if we think of a group of hikers trying to climb the highest peak in a given area. Suppose that on the only day the hikers have free for hiking, there is thick fog that keeps anyone from seeing more than a short distance in any direction. If they are determined to find a peak anyway, all they have to do is follow the direction of steepest ascent on the ground they can see before them. Eventually they will reach a point from which the land slopes down in all directions. They can then rightly conclude that they have reached a peak. Now suppose that the fog dissipates, and suddenly they can see many peaks near and far, some much higher than theirs. The method of steepest ascent only ensures finding a local peak. It is a homing mechanism, not an effective search method.
If the hikers had seen the landscape beforehand, on a clear day, they would have known where to start their climb and so would have reached the highest peak in spite of the fog. Experienced designers have in their minds a more or less complete map of their design hyperspace. They know of successful designs that have solved various problems before. But the process Darwinists propose is blind, purposeless, and never transfers experience from one problem to the next.
But what if the hikers encounter a landscape like the profile of a city, with tall pinnacles rising vertically from a plain? Human explorers have conquered many landscapes like that. People can climb vertical walls. They have methods of following the lines of steepest ascent even when those lines are discontinuous. It is even harder, but still possible, to climb up overhanging cliffs. Small steps following lines of steepest ascent cannot lead to a peak surrounded by overhanging cliffs. All around such a peak is a region where the direction of steepest ascent points away from the peak. Such features exist in design hyperspaces, too. Small-step methods can’t explore rugged spaces. With one small step Armstrong finally stood on the Moon, but only after a great leap over the immense space between worlds.
Human designers look for new solutions on the conceptual level. They say, “Who has solved a similar problem before, and how did they do it?” They put keywords into search engines, and the engines point them to relevant material. Nothing like these methods is available to mindless processes like those of Darwinism. Random mutations are simply a method of striking out blindly in all directions on the chance that one out of a thousand shots might be close to a new stable point.
Someone may say, “All right, Darwin never said his process was efficient. It is only necessary that it work enough times to produce all the variety of living species on Earth after many millions of years. So, we are all the products of blind evolution. Darwinism has at least produced human brains capable of speech and conceptual searches, so now all design processes can speed up, and they have.”
So runs the argument. Likewise, a computer trying to design a new kind of lens might use methods that are horribly inefficient, methods so tedious that human designers refuse to use them. Nevertheless, a programmer might think that the computer program would produce something new far faster than humans can with their more efficient methods, because computers excel at doing drudgery rapidly. That is exactly the argument that I presented to my research colleagues many years ago. They knew then, and they still know, that artificial Darwinism on computers doesn’t work. Designers sitting at computers have, over the years, come up with many new designs. Designers search the hyperspace of possible lens designs on a conceptual level and leave to the computer the task of refining their concepts.
In the foregoing we made a distinction between efficient search methods and effective search methods. If there is no effective method, it doesn’t matter how efficient some of the methods are. It doesn’t matter how fast you can drive around in a circle if there is no road to your destination. No high-performance race car can drive from America to Europe.
Darwinism lacks an effective search method. It won’t do to say, “OK, we’ll add a search method.” Darwin didn’t just say, “We need a homing mechanism,” or “We need a natural selection mechanism,” He explained how the mechanism he identified could work naturally. Competition for food naturally selects the fittest to survive. Darwinists will have to explain how a blind, unintelligent, natural process can search like a search engine, associatively, on a conceptual level, bringing in accumulated experience from all over the Earth. Darwin identified the homing mechanism, but he didn’t know he needed a search mechanism. Trying blindly in all directions is not an effective search mechanism even if the search space is limited to only a few dimensions.
In our day the people who know the most about searching are those who design search engines for the Internet. How does one search? One enters keywords, and the engine finds online materials that mention those words. In short, searching is done at a conceptual level. This is something humans can do, because of all animals, we are the best at using words. However, effective searching was needed at all stages of the evolution of species from one common ancestor. Surely amoebas could not initiate searches at the conceptual level when they wanted to become better amoebas!
There is another argument Darwinist popularizers frequently bring out. This is the myth of the smooth pathway between the characteristics of one species and those of the most similar species. “A journey of a thousand miles begins with one step,” reads the proverb. Even Darwin recognized that massive, coordinated changes involving many genes are very improbable. The first eye did not suddenly appear one fine, bright, lucky day hundreds of millions of years ago. Such specialized organs must have been the final product of a long series of less effective versions. Rather, say the popularizers, a primitive organism may first have developed light-sensitive spots on its skin that gave it some advantage. Then, through a series of changes, the spots became differentiated and concentrated in greater numbers, to the point where crude imaging was possible. The spots evolved into cavities because they were better protected that way, and then the cavities were protected with a clear window that eventually became a lens that focuses on a retina. So runs the story.
If the species separated from one another by small steps over a continuous pathway, the Earth should be littered with the fossils of the transitional forms. When Darwin first proposed his ideas, he could say that the transitional fossils had not yet been found. That initiated a century-long world-wide search. Now we know that the “missing links” are really missing. Species persist in the fossil record for long periods with very little change in characteristics. This fact by itself should have discredited Darwinism long ago, but Darwinists are clever. They accept the deficiencies of their conjectures, give them a fancy name, and call them new theories. According to some contemporary Darwinists, what we see in the fossil record is “punctuated equilibrium.”
The idea runs as follows. Most individual members of a species live among the great herds, unaware of the hidden valley over the pass where a few of their kind are mutating very rapidly in response to extreme conditions. The few leave few bones behind, but one day their progeny come back over the pass as a new species with a tremendous survival advantage that makes the great herd obsolete and drives it rapidly to extinction.
None of the punctuated-equilibrium proponents have adequately explained why beneficial mutations occur much more rapidly in a small, isolated population than in a great herd. If the mutations occur randomly their incidence is in proportion to the size of the population. The great herd will always have many more new mutations than the small population in the hidden valley. Darwinists apparently confuse this scientifically observable process with the effects of close interbreeding. It is true that close interbreeding produces many birth defects. A great herd may have a number of defective genes in its gene pool. If the defective genes are recessive they may persist for generations without showing up, because the enzymes copy instructions from the dominant gene. Two close descendants of an ancestor with a defective gene may both pass the gene to their offspring, producing a birth defect.[i] It is true that a small population promotes close interbreeding, but a rash of birth defects is not likely to produce a survival advantage. Even if individuals with the same birth defect passed it on to their offspring and the defect became dominant in the population, there would be no Darwinist progress, because the defective gene was pre-existing.
Some people think that punctuated equilibrium is merely an elaborate excuse for not doing fieldwork. Regardless of what anyone thinks, punctuated equilibrium is a phenomenon that the fossil record clearly reveals. Recognizing it is a frank admission that the normal action of natural selection is to stabilize species, not to drift toward new ones.
Systems engineers, scientific researchers, and mathematicians can easily grasp the points I am making. The points are not beyond an intelligent layperson without specialized training, either. However, there are certain popularizers of evolution who persist in saying that Darwinism is the only plausible explanation for the existence of life in all its forms. Many of evolution’s present-day popularizers have conceded that they are “mathematically challenged,” though some have disguised their intellectual handicap with showy computer programs that turn evolution into a kind of video game. We will show that Darwinism is not a plausible explanation, using arguments that are mathematically rigorous but still as understandable as the rules of a simple word game.
[i] This is the scientific basis for the Mosaic prohibition against incest.