ACHILLES AND THE TORTOISE by GOTTHARD GÜNTHER 

From a philosophy a science is derived; from a science, an engineering system can be deduced. From engineering come the blueprints, and the machineshop directions. And the philosophy of presentday science, summated in Einstein's relativity, cannot lead to the interstellar cruiser. But another philosophy ¼ ?  
 part 3 of 3  

In the pioneer days of wireless telegraphy the wife of one of the
scientists engaged in developing the new invention was asked by her friends whether she
could explain to them what her husband was doing. "Of course," she said,
"my husband explained it to me just yesterday. Imagine a very long dog. His forelegs
are in Washington when his hindlegs are still in New York. If you pinch this dog in New
York, he will bark in Washington. Wireless telegraphy is exactly the same  but without
dog." Something similar could be said about interstellar spacetravel. It will be spacetravel all right  but without space. This seems to be a rather asinine statement. Nevertheless, we shall see later on that it expresses, in a somewhat cryptic fashion, the very secret of a possible spacetravel technique. We learned in Part 2 of "Achilles and the Tortoise" that neither Space nor Time are absolute data of our scientific experience, and that they exist only in an interdependent relation with Matter. But so far we have not yet discussed Matter. Well, what is Matter? Ancient Greek philosophy had an answer. One of its outstanding representatives (Democritus) said that Matter is the accumulation of tiny indivisible particles of an eternal and indestructible substance. All objects and phenomena of Nature can be explained by quantitative changes in the accumulations and configurations of these particles. This is the early theory of atoms. It has influenced scientific development for more than two thousand years. Recently, however, this theory has been modified, and modified to such an extent that the original idea is hardly recognizable. There are elementary particles, yes, but  there is something else as well. And, if somebody tells you that these particles are no particles at all, you won't be able to contradict him. At first sight the situation looks rather confusing, but this is only due to the fact that we are momentarily caught in a transition period where we are passing from Democritus' idea of absolute atoms and absolute Matter to the very novel discovery that Matter is as relative as Space, Time, color, motion et cetera. Just the same, in order to find out something about the modern concept of Matter the theory of elementary particles is still a good starting point. The following table enumerates the particles which were known to physics up to the year 1953, and also shows, us some of. their properties:
We are rather certain that this table is not exhaustive. According to present theory there should be an "antiproton". It is also improbable that the Mesongroup is complete with its eleven members. If the present system of counting particles is retained, a final table might enumerate twentyseven or twentynine particles. But this is just a hypothesis. However, these particles do not have the properties Democritus ascribed to them. Perhaps the best way to define them is to say, that each particle represents a "localized" manifestation of a quantum field. Each quantum field, on the other hand, fills the whole of Space and Time of our present Universe. But there is a difference between these fields. Only the gravitational and the electromagnetic field are genuine longrange fields. All the others are extremely shortranged in their observable effects. It might be said with a certain modicum of truth that the first two fields represent  partly at least  the longrange behavior of the shortrange quantum fields. The gravitational and the electromagnetic field are, therefore, called the classical fields. This abolishes completely the concept of empty Space. It looks as if Space is physically real only as an extension of the quantum fields beyond the existence of "solid" matter. This also implies that distance is a quantized property, produced by gradual accumulation of spacequants. And what goes for Space should have its analogy in Time. There is no experimental proof yet, but it is perfectly safe to predict that there is also a smallest physical unit length of Time, an irreducible timequant. The concept of the quant, first discovered to be a property of Matter in its fluid state as energy (Planck), is undoubtedly the general criterion of whether something is measurable or not. Space, Time and Matter  as far as they have measurable properties  are quantized. But, as far as they do not represent an accumulation of quants, they are not measurable. And what is not measurable does not existscientifically speaking. Or does it? Well, the equivalence between physical existence and measurability is certainly true for classical physics, but we shall obtain a very different  and most unexpected answer if we couch the available information in terms of quantum physics. However, before I attempt to formulate the quantumphysical answer  which, incidentally, will lead us straight down the road to the problem of spacetravel  permit me to recapitulate the salient features of our problem:
With the help of arguments (1) and (2) the theoretical feasibility of interstellar and even intergalactic spacetravel can be demonstrated. I shall show this by first analyzing the significance of these arguments. Achilles is capable of traveling, within the same time, a longer distance than the animal because by doing so he passes no more and no less real  nondenumerable  points than his competitor. Zeno assumed for the sake of simplicity that our hero travels exactly twice as far as the animal. But from what we know about the absolute equivalence of shorter and longer distances in terms of the transfinite number c [cf. argument (1)] Zeno's point would also be valid if his fast runner would travel a quintillion times as far as the slow one. The ratio of the distances is irrelevant. We therefore ask: What makes Achilles overtake the tortoise? The answer is trivial: He uses longer legs, and by doing so, he compensates for the fact that he has to pass more spacequants than his opponent. I'm afraid we should need awfully long legs to step from here to the Andromeda Nebula. Achilles´ personal method is not very practical for interstellar distances. But his example demonstrates a general principle. Distance per se does not mean a thing! The traversing of distance is "purely a matter of operational method" (Campbell). We have not yet reached the stars, because we use an extremely limited operational procedure for locomotion  a general procedure, by the way, which includes everything from the crawling of a toddler to the flight of a jet plane or a rocket. Our question is: Is there a basically different type of operational method? A method more adequate to traverse cosmic distances? The answer is Yes, and it is implied by arguments (3) and (4). If you have various forms of locomotion, some slower and some faster, but all unsatisfactory for a certain purpose, you may ask: What is their common characteristic which makes them all so inadequate? The answer in our case is simple. The toddler, the tortoise, Achilles or the jet plane, they all try to cover distances by passing spacequants. But in terms of spacequants there are always shorter and longer distances, and no matter how good your operational method is, there always comes a point where the number of spacequants becomes too much for your technique. And viewed from the order of magnitude of galactic distances there is hardly any difference in speed between the toddler and the jet. Both are equally outclassed in the race for the stars. The point, therefore, is, can we envisage an operational method of locomotion which does not try to cover distances in terms of spacequants? Because as long as we do so we have to distinguish between shorter and longer distances. Arguments (3) and (4) suggest that there should be such novel method. Let us find out, therefore, what these arguments really signify concerning the possibility of interstellar travel. According to (3) absolute empty Space does not exist in a strict physical sense. Consequently, absolute distances do not exist either. What does exist is "spatiality" (Einstein) as a measurable property of matteringeneral, i.e. of all physical states of nature. So the "absolute" distance between us and, let us say, the Andromeda Nebula, which has so far made it impossible for us to visit that distant galaxy, is not absolutely real. It is only relatively real with regard to our toddlerlocomotion, and it, would disappear at once if we were to discover the proper operational technique to deal with such distances. At this point I imagine I hear some of my readers mutter: "This sounds fishy! Even apart from the problem of spaceflight we know there is a stupendous remoteness between us and a distant galaxy." Surely, there is! But the point is: Do we have to interpret it in terms of spatial distance? A trip to the Andromeda Nebula performed with the velocity of light, would take approximately two million years  terrestrial time. There is nothing which could prevent our saying: There is a temporal interval between us and this galaxy. Our spacevessel rests motionless in Space, but a time interval of two million years will affect the immediate proximity of the Andromeda galaxy. Motion relative to empty Space is not observable! Motion relative to eventless Time is not observable either. It follows with inexorable logic that the two statements: I travel from here to the next, galaxy through Space or  exclusively  through Time are absolutely equivalent. For, if I travel through Space with the velocity of light, my docks will slow down to an absolute standstill. Obviously, Space and Time are interchangeable entities. But they are interchangeable only on the basis of Matter. When I said, we travel from here to the next galaxy two million years through Time but not through Space, I measured time outside the spaceship. When I stated that we travel the same way exclusively through space and not through Time the latter was measured by the clocks of the ship. The difference between the two sets of clocks is a material condition! But this means there are three interconnected interpretations by which the mutual relations of Earth and Andromeda Nebula may be defined. We may say:
We discovered that (T) and (S) were interchangeable, provided the material gradient (M) was represented by a constant. This constant is the velocity of light. It was assumed that our ship was traveling with that velocity. In fact, we are quite accustomed to the interchangeability of the three cosmic components of the Universe in our daily lives! But we never think about it, and we fail, therefore, to notice the general significance of the most familiar phenomena. 

In the preceding article I compared the MichelsonMorley
etherdrift experiment with the situation on two cars traveling on a highway, and I stated
that the difference in results as to the relative velocity of moving objects was due to
the fact that in the case of the two cars there existed an "absolute" constant:
the highway. Now, everybody who has ever driven a car, knows that his locomotion is
subject to the following laws:
(where T = time interval, S = spatial distance, M = velocity = material gradient). In the case of our highway travel (S) is always a constant, let us say, the distance between New York and Chicago. (S) is basic and cannot be changed. But (T) and (M) are variable, and to a certain degree interchangeable. We know from our experience that we can reduce the value of (T) by increasing the value of (M). In plain language: If we spend more gasoline, rubber, oil et cetera (M) we can save on time (T). But, if we want, to be economical with our engine, tires and fuel, we must pay for it in terms of longer duration. There are, of course, practical as well as theoretical limits to this experiment. Don't forget, not all traffic cops read Astounding, and are ready to accept our argument that we are only testing spacetravel theories as a novel excuse for speeding. There is, in fact, a very fundamental limitation to the operational procedure which affects the interchange between (T) and (M). This classical procedure can never be good enough to make one of the two components disappear completely, because in this case the value of the other would become infinite. But just the same, even the primitive level of terrestrial locomotion permits us to study the basic interchangerelation of (T) and (M) with (S) being an "absolute" constant. In the case of the MichelsonMorley experiment the interchangerelation of the cosmic components is calculated on a different basis. This time the absolute constant is (M), the material factor, and (M) is represented by the most general property of Matter, the velocity of electromagnetic waves. Consequently we are entitled to expect in this case an interchangerelation between (T) and (S). This is indeed confirmed by the transformation equations of H. A. Lorentz:
If we have two systems K and K´ being in rectilinear uniform motion relative to each other, then the following relation exists between temporal intervals (T) and spatial distances (S) according to Einstein's interpretation of the formulas /2/: The faster K moves relative to K´, the longer will be the time interval transferred from one to the other system, and the shorter will be the spatial distance in the same transfer. At the same ratio as time grows, Space tends to disappear. And vice versa. The basis of this SpaceTime interchangerelation is mass or Matter, because at the velocity of light the increase in mass for any moving body becomes infinite. So far we have obtained two very important results! If we use spatial distance (S) as a basis, we obtain an interchangerelation
If matter (M) is considered to be the constant we find that
holds. Having arrived at this point we can no more evade the question: Is there a third interchangerelation where (T) is the basic constant? My answer is a very positive Yes. There must be a third exchangerelation
because the mutual interdependence of (S), (M) and (T) is such that the first two interchange relations could never exist unless a third between (M) and (S) would balance them. (We shall later demonstrate this in terms of symbolic logic.) Interchangerelation (3) is the most interesting for us. We know (1) only to the limited extent of our daily experience in terrestrial travel. (2) Is at the moment a purely metric problem between the two continua Space and Time. But (3) represents the very core of modern quantummechanical physics. It is becoming more and more difficult nowadays to draw a distinctive line between what is purely "spatial" and what is "material" in any physical datum. The borderline between Space and Matter shows a sharp demarcation in classical physics only. It tends to disappear under the "boundary" conditions of microphysics as well as of astrophysics. It is only in the intermediate field of "terrestrial" macrophysics that we seem to know with some certainty what the difference between empty Space and solid Matter amounts to. In popular language: It is the difference between something and nothing. Matter is the sum total of "all things", and empty Space is the total absence of things. Every child can understand this. But not too long ago a most important formula was discovered: (Einstein)
(where E = the energy of a body at rest, m = its mass and c = velocity of light). Since then it has become general knowledge that "solid matter" (mass) may be transformed into energy (atom bomb), and that it is at least theoretically possible to retransform radiant energy (light) into Matter. Does our distinction between Something and Nothing still hold? Is energy a thing? We are told that Matter is an electromagnetic phenomenon. But the electromagnetic field extends through all of Space. Space itself appears to be a fieldphenomenon. That means, microphysically speaking, it becomes more and more impossible to draw a sharp line of demarcation between a thing and the space that surrounds it. An analogous intimate relation between Space and Matter exists in macrocosmic physics. A preliminary remark: Imagine you have a box partly filled with marbles. You take one after the other of these marbles out of the container till there is nothing left inside. Nobody will doubt that there is one thing left. That is the empty box. Its capacity of being a container has not been affected by the removal of the contents. This idea of the relation between Space and its contents is that of classical macrophysics. But now let us proceed to macrocosmic physics. Imagine yourself to be an entity with divine power, and located outside the Universe. You reach into the Universe and remove from it one galaxy and one nebula after the other, and you continue to do so till nothing material, not even the smallest meteorite or the most tenuous wisp of cosmic dust is left in the Universe. According to common sense two objects should then be left: the empty Space waiting to be filled again with things, and empty Time waiting for the event of a new creation. Common sense tells us further that the dimensions of our Universe should not be affected by the removal of Matter and of events. But common sense has failed us once before when we dealt with Cantor's theory of transfinite number, and it will fail us again in macrocosmic physics. We shall throw only a fleeting glance at what happens to Time in a Universe deprived of all Matter. In our present Universe Time has two directions. It stretches toward the past as well as toward the future. But in an empty Universe Time would have only one direction  toward the future. A past would not exist. The possibility of "passing" Time demands the presence of Matter. But what about Space? Some time ago the English physicist Sir Arthur Eddington tried to define the interdependency of Space and Matter in certain equations  in a somewhat similar way as the MaxwellHertz field theory connects the electromagnetic fields with charges or poles. Eddington´s equations permitted two interpretations which became known as, the Einstein and the de Sitteruniverse. Einstein's universe is "static"; de Sitter's is in constant expansion  and contains no Matter! Consider what that means! A space that "contains no matter" but is in constant expansion is Matter in some state of radiant energy. It was later discovered (Friedmann, Lemaitre, Robertson) that Eddington's equations allow of a series of solutions which define a connection between the extremes of the Einstein and the de Sitteruniverse. The gist of the theory can be described as follows: If you put some Matter into the de Sitteruniverse, then its gravitational energy will start to counteract the expansion. The expanding world will start to slow up. If you add more and more material, you will finally arrive at a point where expansional and gravitational forces balance each other. This is the static world of Einstein. But if you add even more Matter, then gravitation becomes stronger than expansion, and the Universe will start to shrink. It seems as if Matter has a "contracting" effect on Space, and a corresponding influence on the structure of Time. As far as Space is concerned one of the ways to describe its relation to Matter is represented by the formula:
(K = gravitational constant; M = mass of Universe; c = velocity of light; p = ratio of circumference to diameter; R = radius of curvature.) As (5) is at present a highly speculative formula I shall not deal in detail with it. But, even if it is only a rough approximation of the truth, it clearly implies that distance is a fieldeffect. By increasing or decreasing this effect spatial distances can be shortened or made longer. This seems probable anyhow. But distances in our universe are so enormous that even a considerable reduction of spatial dimensions would not help us much. What would be the use of pulling a galaxy which is about one billion lightyears away into our immediate "neighborhood" of fifty million parsecs. Moreover, we would need the energy of millions or billions of galaxies to do it. The very idea is the height of absurdity. Fortunately, our basic interchangerelations (1), (2) and (3) suggest something else. Some way back in these articles I stated that Space and Time are continua; but Matter has a discontinuous structure it is quantized. Then a very confusing thing happened: We found out that Space was quantized too, when we traveled through it. Zeno's paradox was produced by the conflict between denumerable (quantized) and nondenumerable numbers. How did that happen? The answer is simple: All classical forms of motion demand a partial "materialization" of Space. We travel through Space by means of a highway, for instance. And a highway, representing distance, as a functional part of an act of locomotion, is materialized space. And distance in form of a highway is, of course, quantized. For the airplane our atmosphere plays the corresponding part. And even the principle of rocket propulsion  which may suffice for the short hop to, the Moon  is still based on the idea of quantized motion. It is only a little more sophisticated  you take your "highway" along with you. This time it is your rocket tubes. But this convertibility of the spatial continuum into quantized material existence is only possible if there is a reversed process by which quantized material distance can be converted into a nonquantized state of continuity. We have at the moment not yet the slightest idea how this can be managed. Nevertheless we know two things with absolute certainty. We know that there can be no doubt that this inverse convertibility from quantized into nonquantized existence does exist. Because our three interchangerelations,
would not be possible if only: Space could turn up as a quantized form of Matter, but Matter never as a spatial continuum Matter has spatial extension in quantized form nobody ever doubted it. Our whole technology is based on that knowledge. But, if the inverse relation between Matter and Space did not exist we would have to assume that Matter  and with it Time  would constitute absolute data of nature. No modern scientist is ready to make this concession. The other thing we know with equal certainty is that in a technology no longer based on the discreteness of Matter but on the continuity of Space and Time distance, goes by the boards. There is no longer any question of shorter or longer distances. They disappear completely. A line segment one billionth part of a millimeter long is numerically equivalent to a line segment of any transcosmic length. Only a "little" question is now left. How can we develop a technique of locomotion which does no more use Space in its quantized material aspect, but which utilizes Matter in its, nonquantized spatial version? We are fortunate in so far as we are not totally ignorant in this respect. We have at least some negative knowledge  we are aware of the data that are missing. First: We shall have to discover the law which describes the physical interaction between the electromagnetic and the gravitational field. We need, further, detailed information about the cosmic "glue" that holds the atomic nucleus together (Meson theory). One scientist (H. Bethe) has recently expressed the opinion that we need more powerful mathematical tools to tackle the problem of the nucleus. It may well be that Cantor's theory of nondenumerable sets will finally provide the answer. It should not be forgotten that Cantor only discovered the existence of transfinite numbers. How to use them in physical science is still a mystery to us. In the preceding article I could only show that nondenumerable sets do apply to the problem of motion in Space, and that their application demonstrates that distance is a property of quantized Matter but not of a continuum like Space or Time. Thus these articles on the problem of interstellar and intergalactic spaceflight would have to conclude on a very unsatisfactory negative note, if it were not for a recent discovery in the field of applied symbolic logic. We are now well acquainted with the fact that symbolic logic can be used in analyzing electrical circuits and power patterns. I shall use this technique in order to demonstrate the basic power pattern all interstellar spaceships will have to apply. I know, of course, nothing about the details which will depend entirely on as yet unknown operational procedures of electromagnetic, graviton, and messianic character, but I know by logical analysis that they will have to conform to the following structural pattern. "M," "S" and "T" shall again be the symbols representing our three cosmic components of the Universe. Their interchangerelation shall be represented by « ´, « ´´ and « ´´´ respectively. We then obtain three elementary principles of interchange:
We know, on the other hand, that any physical event involves all three cosmic components; because, it is physical, and it happens in Space as well as in Time. That means: No isolated interchange relation of « , « ´, or « ´´ ever happens in the Universe. We, therefore, introduce two operational procedures: and with the intent to combine all three interchangerelations. The choice of these two procedures is by no means arbitrary. In fact they are the only possible operations, if we want to combine all three interchangerelations. We shall define as the combination of the upper line of all three relations of « ; that means in this case: of M to S, S to T and T to M. The second operation, , will accordingly combine the lower line of « relations /3/. In order to establish our operational procedures and we arrange the cosmic components M, S and T in all possible combinations in two horizontal lines X and Y. We then look for the values of in the upper lines of our « ´, « ´´ and « ´´´ tables, and we always take the second value. Thus, if our two horizontal lines contain the combination M S or we choose the value S from S M « ´ table. If the combination is or we choose again the second value of the upper line, this time of « ´´. Accordingly our value must be T, and so on. For we select the second value of the upper line, this time of the « tables. We thus obtain as definitions of the operations and the following table:
This arrangement may seem redundant, to the uninitiated. But it is not. As a basis for further calculations we need all logically possible combinations, even such seemingly redundant ones as between M and M, S and S, T and T. One thing is absolutely certain  no matter how interstellar and intergalactic spaceflight achieved, and what detailed technical arrangement may be used  all issues spaceships capable of traveling cosmic distances will have a switchboard based on the two operational patterns of and . The interesting point, of course, is what could be done with it. Well, many things. It is impossible to predict which of the individual technical procedure that fall into the pattern of and will finally be used. This depends entirely on what future discoveries will be made in nuclear and astrophysics. But we might as well  for the sake of the practical demonstration of our "switch board"  assume that the description of interstellar flight, given by one of our outstanding science fiction authors is approximately correct. Permit me to quote this description from one of Isaac Asimov´s novels, where one of the officers of a spaceship explains the principles of interstellar flight to the passengers /4/: "Ladies and gentlemen! We are ready for our first Jump ¼ The Jump is exactly what the name implies. In the fabric of spacetime itself, it is impossible to travel faster than the speed of light ¼ Therefore one leaves the spacetime fabric to enter the littleknown realm of hyperspace, where time and distance have no meaning. It is like traveling across a narrow isthmus to pass from one ocean to another, rather than remaining at sea and circling a continent to accomplish the same distance ¼ Great amounts of energy are required, of course, to enter this 'space within space' as some call it, and a great deal of ingenious calculation must be made to insure reentry into ordinary spacetime at the proper point. The result of the expenditure of this energy and intelligence is that immense distances can be transversed in zero time. It is only the Jump which makes interstellar travel possible." Asimov like many other science fiction writers sees quite clearly that the possibility of spacetravel depends on the elimination of spatial distance. We know now that distance must be interpreted as an accumulation of spacequants, and we travel distances by passing spacequants. I remarked in Part 2 of this series: No matter how good our travel methods might become, there is always a critical point where the accumulated number of spacequants becomes too much for our operational methods. The question, therefore, is: Will it be possible to devise a technique of locomotion where we "jump," as Asimov says. In other words: Where we do not pass spacequants when we cover distance. We know that what is quantized is the Mfactor in the Universe. Neither Time nor Space, being genuine continua, are quantized per se. It is only their matteraspect that gives them a quantized structure. It all boils down to the problem: Can we get rid of the Mfactor in our operations and . Our highway experience has already taught us that M and T might be traded against each other  at least to a limited degree. The same applies to S and T. But if there is general convertibility of all three cosmic components into each other, it should indeed be possible to eliminate one of the components operationally. Asimov's description suggests that the Mcomponent should disappear, because it is the matterfactor that produces distance. There is a very simple procedure by which the quantized Mfactor can be eliminated. For XY it is expressed by the formula:
By using (6) we transform our original sequence for XY MSMSSTMTT into the Mfree sequence /5/ SSSSSTSTT This new series which contains only S and T represents the symbolic meaning of (6): However, we are more interested in its practical significance for interstellar travel. Seen from this angle, (6) describes the basic logical pattern of a technical operation which has ceased to use quantized data. We remember that quantized data are always represented by M. But (6) does no longer contain an Mfactor. (Incidentally, (6) does not mean that Matter per se has been eliminated, but only its basic feature of quantization.) It follows that the mathematical theory of (6) is not based on denumerable arithmetics. Instead, it uses transfinite Cantorian numbers. And we remember that the concept of distance disappears entirely in transfinite arithmetics. In other words, (6) describes the energy pattern of a technical operation which eliminates distance. It can be said with absolute certainty that (6), its corresponding formula for the inverse operation XY
or any other expression of the types (6) and (7) will be the basic formulas of all future interstellar or even intergalactic spacetravel  provided, of course, that our general assumption of the universal convertibility of Space, Time and Matter is correct. If it isn't, we might as well say goodby to all our dreams of spacetravel outside our solar system. The distances are too great. The idea of interstellar voyages where space vessels travel for centuries, and where only the greatgrand children of the original travelers arrive, is perfectly absurd. Interstellar and intergalactic spacetravel will become a reasonable proposition only if we develop a technique which makes distance go by the boards  entirely. This, however, is absolutely predicated on the assumption of the primordial interchangeability of the three cosmic components of the Universe, and the additional assumption that one may be substituted by the other. There are two components which, are continua, and one which is quantized. Length of distance or length of interval means nothing in the continuum. It is significant only for the third (quantized) component. If we are capable of developing a technique which eliminates the quantized component by substituting nonquantized features, our problem how to cross the voids of Space is solved. Our formulas (6) and (7) demonstrate that such a technique is logically possible. What (6) and (7) imply practically can be put down in very simple words. Everything that exists has three, components. The three natural components of the Universe are M, S and T. But if Man introduces an artificial state of existence he likewise needs three components only. But, being artificial, this state of existence is based on an operation "op"  which, in its turn, acts as a basic component. Thus  introducing P for and  we have now four equally fundamental parameters: PMST This adds up to a redundancy, as far as human technique is concerned. We may omit one of the four components  or rather its properties. If we get rid of P, we have the Universe just as it is without Man's action. Of, course, P is still there as far as the objective world is concerned but it is distributed over MST. Process (P) appears in this case only as a natural event like the falling of rain, lightning, hunger, thirst, et cetera. But, if we introduce an independent parameter P as a fourth degree of freedom for action, we add human creative procedure to the natural events  as a means of producing something which has not existed before. In our special case spaceflight But the addition of P permits us to eliminate the properties of one of the other components by substitution. Our formulas (6) and (7) did this for the pattern: PST This is possible because all human technique and action is twovalued. In order to do something we require operational decisions and two alternative poles to decide between. Nothing else! Therefore, the pattern PMST is always redundant  as far as any individual technical procedure is concerned. M (or the properties of M) can be substituted as well as S,T or P. However, this redundancy concerns only any practical, i.e., limited, action or procedure, and not the general rational pattern on which our procedure is based. That is the reason why we cannot be satisfied that science, as it is known today, recognizes only three parameters of the Real: the objective parameter: Matter (M); the dimensional parameter: Space (S); the relational parameter: Time (T). It is necessary to add a fourth cosmical component: the operational parameter: Process (P). As long as we refuse to accept P as an independent parameter we can only build a technique based on the natural laws inherent in the relations between M, S and T. In this case we will never reach the stars. The velocity of light as upper limit for the propagation of physical events in space is such a natural law. And mind you, it will never be invalidated in a MSTuniverse. On the other hand, if we add a fourth parameter we gain a technical dimension in which the laws of a threeparameterworld are not abolished but capable of  modulation. I started my "Achilles And The Tortoise" series with an analysis of the problem of motion, as first conceived by Zeno. Permit me therefore, to conclude this final installment with a confrontation between the theory of motion in a three and four parameteruniverse. According to classical principles the phenomenon in question is based on the formulas. S/M=T, S/T=.M, MT=S. These are expressions of basic laws in a threeparameteruniverse (MST). Of course, P is also present in these three formulas, but as I remarked before, it is distributed over the other parameters, and, therefore, has no quotable value as an independent component. This situation is unavoidable, in classical science. This science can never provide us with an adequate theory of interstellar spaceflight because the distributed P cannot absorb the quantized character of M. And nothing can be done about it because, in a MSTuniverse P can never be taken out of its distribution. Why not? Well, according to our traditional ideas Matter is Something and empty Space and eventless Time are Nothing. But what is Process (P)? A very embarrassing question! It certainly is nothing! But is it a thing? No, we cannot admit that either. The science we know uses a strictly twovalued logic. In other words, there can be no basic Third between Something and Nothing. In this system there is no room for an independent P. Consequently the only method to, find an asylum for the fourth parameter was its distribution over the other components. Now we have finally discovered the original source of Zeno's trouble: He could not define motion because it was neither a thing nor a nothing. Motion is an event or process. To put it differently: Zeno's and our analysis of motion showed that this phenomenon could not be defined in terms of M, S and T. It demonstrated properties beyond the threeparameter reality of our traditional idea of Nature. We, therefore, introduced a fourth parameter P. This was done arithmetically by using Cantor's transfinite number c for the solution of Zeno's Paradox. This provides, us with a new interpretation of Cantorian transfinite sets. Cantor's Alephs are numbers which do not apply to a threeparameteruniverse. They represent the arithmetical order of the fourth, parameter. This gives us  this is at least the hope of the present writer  a "satisfactory correlation of Cantor's ideas with the real universe" (Campbell). The recognition of Process (P) as a new parameter beside Matter, Space and Time is equivalent to introducing in experiential terms that somuchtalkedabout "fourthdimension", through which the first three can be rotated at will. This method permits the substitution of the properties of one parameter by those of the three others. The development of a physical science which satisfies these conditions is now only a question of time. And when this time arrives  "Why, the stars are as near as we wish them." Footnotes: /1/ However it should not be forgotten that this concept of Time refers only to the temporal duration of our present Universe, where T is now of the approximate order of magnitude of 3× 10^{17} sec. /2/ In these equations v represents the velocity of K relative to K', c is the constant of light, x indicates the movement along the xaxis, and t represents time. /3/ A combination of a « relation of the upper line with one of the lower line is not possible because it would involve our operational procedures into contradictions. An example may demonstrate this. If we combine M to S with T to S in one operation we lose the distinction between M and T, and the « ´´´ relation becomes nonexistent. Because, if M is changing into S it cannot also change into T within the same operational procedure. The same holds for T. /4/ cf. Isaac Asimov, The Stars, Like Dust. N.Y., 1951 (Doubleday), pp.39/40. /5/ The transformation effected in (6) is quite elementary. It is done in two steps. First the interchangerelation to XopY. That means, wherever XopY has the value M it is changed to S, and where there is a S, it is replaced by M. Then the procedure op which was originally operative only between X and Y is applied between the established values of XopY and the interchanged values of (XopY).
THE END 

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