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In what follows, knowledge of the Aristotelian-Thomistic Metaphysics is presupposed. It can be found all over in First Part of Website (Back to Homepage), especially as to the notion of Substance (in the metaphysical sense). And also presupposed is knowledge of the theory of the subdivision of Reality into the Explicate and Implicate Orders, a theory so far developed in our noëtic theory of organic evolution, especially as it has been expounded in the two-document Theoretic Intermezzo after Part VIII of present Part of Website.
Our assumption of the fact that there is, in addition to the material order of Being, something like an immaterial order of Being (which we have called the Implicate Order) has followed from an evaluation of the organic world, and especially had followed from the presence in that world of highly adaptive and functional structures (in the form of sophisticated morphological structures, and of goal-oriented instincts such as care for the offspring and socialism (as we saw in Hymenoptera)). The conclusion from all this to the existence of an immaterial order of Being was drawn by us as a result of our conviction that such structures cannot be produced in and by a material order all by itself. They must have been produced in an order of being where no spatial distance exists, no contingency, and no individuality. Such an order is the Implicate Order.
Now having thus concluded from the nature of organisms to a duality of Reality (s.l.) -- (and this duality) consisting of two interacting (i.e. not transcendent) orders or domains of being, viz., the material, spatially extensive, Explicate Order and the immaterial, intensive, 'noëtic', Implicate Order -- we must see how things are in this respect in the basic inorganic features making up the general framework of the Explicate Order and forming the non-living context of organisms and their evolution (as it is taken in our noëtic theory of organic evolution), and how these features relate to the Implicate Order. That is to say, we must investigate the nature of intensive qualities (such as heat, color, magnetism, etc.), especially as to 'where' their intensive aspect resides, i.e. in what order of being it has its seat. We must also investigate the nature of the alleged holism of, especially, certain inorganic beings such as atoms, molecules and crystals. And this brings in the Aristotelian-Thomistic metaphysical notion of substance and of potency and act. Further we must investigate the nature of space, spatiality, and place (by which things become located), and also the nature of motion and change of inorganic beings.
Many of these things and problems, as regards inorganic entities, have already been discussed in especially First and Fourth Parts of Website, but still not yet in the context of our more m a t u r e theory of the Explicate and Implicate Orders. We are especially interested in what role precisely is played by the Implicate Order in allegedly non-mechanical processes, i.e. how it comes that so many processes and things, as seen from the Explicate Order, seem to be mechanical and reductionistic, although in other respects they oppose against such a characterization. And do intensive qualities, in so far as to their purely intensive aspect, reside in the Implicate Order only, although especially they do seem to account for the observability of things in the Explicate Order in contrast to pure spatial extensivity? And further, is a true notion of substance (in the metaphysical, i.e. ontological, sense) possible if, and only if, we take into account the existence, not only of the Explicate Order, but also of that of the Implicate Order? Is only the latter Order fully responsible for the phenomenon of holism in things, not only in organic beings but also in inorganic entities, or does the Explicate Order play a role in it too? And it is so that in particular these features (non-mechanical things and processes, intensive qualities, substance, and holism) are essential tenets of the Aristotelian-Thomistic Metaphysics. So it is clear that we must investigate this metaphysical doctrine again, but now in the context of our theory of the Implicate and Explicate Orders so far developed. And when this has all been accomplished we will finally possess the complete theory of the Explicate and Implicate Orders and their interconnection.
Especially it will be demonstrated that the import into physics and chemistry of (1) the existence of intensivity (existence of intensive qualities, not only in heat and light, but even -- as impetus -- in local motion), of (2) the holistic constitution of certain inorganic beings like atoms, molecules and crystals (and with it the virtual existence of their constituent particles as particles, virtual, in my view, as 'seen' from the Implicate Order, but still actual as seen from the Explicate Order), and, therefore, also (the import into physics and chemistry of ) the existence of true substances (in the metaphysical sense) even in the inorganic world, will not have any effect on the content and truth of (1) classical atomic theory (i.e. the theory of chemical elements and their compounds, in short, the theory of atoms and molecules), of (2) conventional crystallography, and, finally, (on the content and truth) of (3) classical mechanics (still applicable to macroscopic objects) and quantum mechanics. And that means that modern physics and chemistry can be viewed as specifications, not of any form of democritean mechanical and reductionistic metaphysics, but of the non-mechanical and holistic metaphysics of Aristotle and St Thomas (of course with amendments and additions where necessary). That is, if, in my view, we consider the whole of Reality, i.e. if we consider the Explicate Order and Implicate Order together, we see that the general metaphysics of Reality is largely that of Aristotle and St Thomas.
And then, when we've done all this, we shall continue our investigation of data on fossil and recent insects in subsequent documents, for its own sake as well as to further refine our noëtic theory of evolution.
Before systematically, extensively, and philosophically investigating -- mainly following HOENEN, 1947, and Van MELSEN, 1955 -- all this, we first present some of the key issues, in order to give the reader some idea of what is going to be so investigated.
Many properties of natural things such as heat, color, etc., are qualities (intensive qualities) : Heat is the kinetic energy of the thing's particles (where energy -- capacity for doing work -- is also a quality, and where the average kinetic energy of the particles in the thing is the latter's temperature, measuring the intensity of its heat), and this kinetic energy is an aspect (or effect) of the impetus present in each of these particles, which (impetus) is itself a quality. Further, a physical field (gravitational, electrical, magnetic, electromagnetic, etc.) may be viewed as a qualitative determination of the "aether of Lorenz" (which is itself a (imponderable) substance, and general medium of localization of (extensive) objects). Also physical mass (in and of physical bodies), which may be defined as "degree of bondage (adherence) to the aether", is a quality. And in all this, we here mean : intensive qualities (as opposed to, for instance, a graphic figure -- a constitutive pattern -- which may be viewed as an "extensive quality"). The study of the nature of quality will furnish us with more insight into the (by us presupposed) exclusively qualitative nature of the Implicate Order.
The problem of the existence of intensive qualities does not consist so much in the question whether they do exist at all, they certainly do, but (the problem is) 'where' they exist (i.e. in which domain of being or of cognitive apprehension they naturally reside). They may have their seat :
Everywhere where local motion is involved we have to do with the impetus, which is an intensive quality of the objects being moved. And this is why, for instance, heat, although it has extensive effects through which we can actually measure (thermometer) its intensity, is a true intensive quality, because motion of particles is involved here.
A comparable state of things, as to If-Then constants, we may expect in all other intensive qualities that have extensive effects (and these effects can only manifest themselves in the Explicate Order).
Space and time
In what follows we will comprehensively talk about the following items :
Metaphysical analysis of the continuum
The extensional aspect.
The first material property, revealed to us by considering Nature, is surely the extensional aspect : natural bodies have extension, they are spatial. The first property indeed, because, apparently, it is through the extensional aspect that qualities determine the bodily substances [The extensional aspect guarantees a possible distribution (and application) of qualities in one and the same bodily substance]. The first (property) it is especially, because the extensional aspect is better known to human reason than are all other material properties. Therefore it is the subject of a grandiose science, geometry. And this is the cognitive-theoretic ground why the mechanists, abandoning qualities as objectively existing, viewing them as mere appearances, stuck to quantity, as it is known through direct observation, as being real.
Consideration of spatial being, the ens extensum, precisely as spatial, as extensional, as extensum, takes place in geometry, and thus falls outside the scope of natural philosophy. But we can consider the ens extensum also as ens, being, as we do in metaphysics. And this metaphysical analysis is the task of the natural philosopher, who, of course, must use the data obtained by geometry as well.
The concept of extensum
What in fact the extensional aspect, or rather extensum really is, we will not attempt to define in a formulation or statement. Not because of a lack of insight, but precisely because the immediate clarity of our insight in it. Would anyone question what we are supposed to understand by "an extensional thing", an ens extensum, a line segment? Any attempt to explain this concept to someone not yet having it, by reducing it to more primitive concepts would fail miserably. More clear-cut and primitive concepts than that of extensum, which we could immediately abstract from observation, we don't have. It is not possible to define everything with a formula, like we cannot prove everything, because every formulated definition presupposes prior -- ultimately first -- concepts, precisely like every proof presupposes -- ultimately first -- truths. And the notion of ens extensum belongs to such first concepts.
Let us, for now, give two "definitions" that one has given. Does one take them as mere descriptions telling the reader or hearer about what we're talking about, by presenting him with something illustrative, from which he immediately obtains the concept, then we admit them without objection. But as strict definitions they are improper and even dangerous.
The first one is the follwing : ens extensum is "quod habet partes extra partes", i.e. "what has parts outside one another". As a definition it is improper for two reasons. To understand the definition and for it to be a true statement, the parts must be known as extensional. Moreover -- and this is far worse -- the concept "outside" already presupposes the extensional, and is unintelligible without this notion. So the concept to be defined is already presupposed to be known in the "definition" twice. And the second time (presupposed to be known) even in a derived concept, viz., the concept "outside"
The second "definition" is no less incorrect. It is the following : extensional is "that which occupies space". Except for the fact that "to occupy" does not make sense to someone who doesn't already know what extensionality is, and moreover that the concept "to occupy" is derived from the concept "extensum", also the notion of "space" presupposes the concept of "extensionality". "Space" is here something as a recipient having extension, and so cannot serve to define the concept of extensionality. It might be added (more so than with respect to the concept "outside") that "space" is a concept that is formed only as a result of prolonged digestion (we will elaborate on this later, in the Section on "Place and Space") of the, in these matters, first given : the ens extensum. Attempts to give priority to the space concept are, as we see immediately, false and, moreover, result -- and this is more serious -- in philosophical disasters.
We distinguish extensa of different measures, dimensions. Direct abstraction from the observable gives us an extensum of three dimensions. This follows from a process that at the same time also gives extensa with a smaller number of dimensions. We know the originally given extensionality as divisible. The boundary of two such parts is a plane. This is known as again divisible and the boundary between two parts is a line. And the latter is again divisible into parts having as their boundary a point. We immediately see that we cannot proceed further. The process has terminated. A point is indivisible, it has no dimensions. But it still belongs to the world of extensionality where it is present as a boundary. The ancient Greeks expressed this fact by saying that the point is the indivisible entity that has position, "thesis". Accordingly, we know the point, which has no parts, but has position, further, the line, having one dimension, the plane, having two of them, and the solid, having three dimensions. The reader will have noticed that we did not attempt to define the concept of dimension in one single formula, but instead having traced it (and what goes along with it) in our insight with sharpness. And this certainly is the way along which natural reason arrives at these concepts.
The concept of continuum
The concept of extensum is, accordingly, in the above described sense, not definable. It may, of course, be investigated as to its properties and essence. The former is done by geometry, the latter is the task of philosophy. It must give an answer to the question : in what way is the extensional ens an ens? Problems pop up in trying to formulate an answer. But the solution was already given by Aristotle with his notion of potency. This is in concreto the case in his application of this concept (potency) in his analysis of the continuum. With "continuum" we here do not mean that what is called continuum in modern mathematical analysis, namely the arithmetic continuum, the "continuum of real numbers". We here mean the observable (s.l.) (geometric or physical) continuum, from which originally the arithmetical has been derived (Also we believe, that mathematical analysis must return to that original derivation if it desires to solve the modern "crisis of mathematics"). For a continuum, so taken, we can, with Aristotle, make up a definition.
He distinguishes, namely, three groups (types) of spatial systems. Let us take as an example a system consisting of two (physical) cubes (from which we can construct these three types). These two bodies may be at a certain distance from each other, without there being a third similar one between them. They are then, in a sense, "immediately following upon each other". This system is of no significance to us further, except as point of departure of our consideration. Let now the cubes approach each other until they touch. Then they are, using a classical term, contigua ("haptomena" in Aristotle). They are "immediately following upon each other" (now) in the strict sense. Between them no similar body is placed. By touching one another they, in a sense, form a unity, be it a purely extrinsic, accidental, unity. If they are perfect cubes, vision alone cannot distinguish this system from even a parallelepipedum having a length twice that of the cubes. Nevertheless, they remain only extrinsically a unity as a result of their mutual position and orientation. They cannot approach each other any further (we might assume penetrability, but this is not significant to us now). But the unity may be more intrinsic (profound). They may indeed become, from two, one single body, or (if one wants to avoid this supposition) we may compare with the system of two cubes the system consisting of one single body only, the parallelepipedum with double length, which as such is not distinguishable by merely seeing it. Most profoundly there is now a great difference : While the first system is an accidental unity, the second is an intrinsic unity, it is just one single body, one single ens, one extensional substance, in which we can, at least mentally, distinguish two parts each one of which is congruent with the two bodies of the first system. And precisely that is what we call a continuum.
The consideration of Aristotle is very simple indeed, and the difference between this continuum and the two contigua forming a same geometric figure is very easy to understand. Nevertheless, later this insight into this difference, more than once will turn out to be of great significance. Neglect of it has resulted in philosophical disasters. And now we arrive, then, at the definition : a continuum is an "extensum that intrinsically is one". This, then, is the concept that must be subjected to metaphysical analysis. The attentive reader has, of course, already noticed that the first two systems, considered above, consist of two such continua (in the first system : at a distance from each other, in the second : touching each other). But then, only the continuum ((of) the third system, as well as those in the first and second systems) is the true "ens extensum", the extensional thing, for the other systems were not an ens, they were each one of them a set of more than one ens. But our consideration, also later still having much significance, has now directed our attention to the first demand of metaphysics on the ens extensum, namely that it must possess the fundamental property of a true ens, namely that it must be intrinsically one (as opposed to many). It may seem silly to insist upon it, but it will turn out that it is of the greatest significance. Much the better then, that it is in itself so simple.
Further analysis of the continuum
In this first analysis we continually had in mind the principal case, namely that of one single extensional substance. Our considerations do, however, apply, with some, easy to find, qualifications, to other continua as well : to purely mathematical solids, to a line, and to a plane. They do also apply to qualitative continua such as one extended color, etc. All these are the so-called permanent static continua. But our considerations also apply to so-called fluent dynamic continua : motion and time.
So the continuum is, as to its very concept, profoundly one. Yet it is at the same time multitudinous. It may be divided into parts (it is after all an extensum) and must "consist of " these parts. On this "consists of " we must focus our attention. This is an instance of the problem of "the one and the many".
First of all we note that these parts are : "integral parts", i.e. that they are of the same kind as is the whole. Of course not as to their quantity. Then they are smaller and is only their sum equal to the whole. But they are of the same kind as the whole insofar as they are considered to be ens, namely in this sense : the integral parts of a physical body are themselves physical bodies, those of a mathematical solid are mathematical solids, those of a plane are planes, those of a line are lines, those of a color are colors, and those of a motion are motions, and those of a time are times. But this precisely accentuate the antinomy : the continuum seems in the same respect to be at the same time one and many. There can be no objection against the fact that someting is one in one respect and many in another : The one human being can, at the same time, think, see, and hear, and do other things. He is one in substance and many as to various determinations, accidents. In this way one body may have many colors without loosing its substantial unity. Unity and multitude only exclude each other, and are thus impossible (as together existing) if they are this at the same time and in the same respect. But in a continuum the integral parts are of the same nature as the whole : the solid "consists of " parts which are themselves solids, the line of lines, etc. Here unity and multitude seem to be taken to be present in the same respect, and yet one single ens cannot be at the same time many entia. So the mathematical consideration, seeing the continuum as extensum and thus as divisible, as many, seems to collide with the metaphysical consideration of it, demanding the oneness of ens.
Here the solution of this antinomy is not difficult. The continuum, because it is an intrinsic unity, is not divided, but only divisible. It is not a mere aggregate, not a collection of parts lying adjacent to (and touching) one another, but a totality. Otherwise we would not have to do with a continuum at all, but with a series of contigua, which merely form just an accidental, not an intrinsic (profound) unity. The parts of a true continuum, accordingly, do not exist actually, but are only potentially present in the one continuum itself existing actually. So we do not have, as we tacitly assumed above, the simple contraposition : one and many in the same respect (which would imply a contradiction), but we have the contraposition : actual unity and potential multitude. And these two may, also when they must be taken in the same respect, go together without contradiction. So the continuum is actually one, potentially many. We arrive at the conclusion : The parts of which the continuum "consists" ("is made up") are, here in the continuum, not things actually existing, but only potentially so. Division develops, actualizes, that potency, but then the continuum no longer as such exists. For all this it must be realized that, to be able to provide this in itself clear and definitive solution, metaphysics must have at its disposal the conceps of real potency, i.e. potential being. A metaphysics not having this notion already cannot justify the simple concept of the continuum, an extensum, a "divisible something that is one". The problems of such a metaphysics become, upon further analysis, greater still, but they are already there, and unsolvable for it.
It may be useful, before we adress these problems, to elucidate the above by a concrete example. In biology one has succeeded to divide a frog-embryo, for instance in its two-cell state, such that each of the separated cells subsequently develops into (eventually) an adult individual. Without this intervention only one such individual would have been developed. So from one living being one has, by division, made two of the same species. This one single living being was not a sum, an aggregate of two frogs, it was only one, a totality. So the intervention of the biologist has at least produced one new living being. It did not exist in that first living being, at least not actually so, but it did potentially, a potency that could immediately be actualized as a result of a proper, but simple, operative intervention. In this instance the above derived statement will directly be clear, saying that these parts of the whole in this whole are present not actually but only potentially. And so it is in every continuum, because this is one, one ens.
Think again of the difference between the accidental, extrinsic, unity of two contigua (= two things or entities which each for themselves may or may not be a continuum, but, together do not form one), and the intrinsic, essential, unity of the continuum. Partition (division) of the first system is merely separation of what already were two individuals. Partition (division) of the continuum, on the other hand, is production of at least one new individual, in the [case of the] embryo as well as in every extensum which is one substance. Not a particular problem is created by the fact that a simple intervention such as a division may produce a new individual, and even (in the case of the embryo) a new living substance. What is effected, not only depends on the efficient cause but also on the material cause. The greater the part attributed to the latter (it is potency more or less close to its actualization) the smaller it is for the efficient cause. Later we will return to this "principle of distribution of influence between material and efficient cause". Here we see, in a concrete instance, its truth immediately.
We spoke about at least one new individual being produced. Because of the following reason. Maybe both individuals are new, while the first one -- the two-cell being -- has ceased to exist. But maybe the origial individual is lost in one of the two resulting individuals, so that only one new individual is produced. It is not easy to settle this question. In another instance, however, such a question may well be answered : Alcibiades cut off his dog's tail, and thus divided this one being into two (not into two dogs). It will be clear that the original individual now lives on (mutilated) in one of the two parts.
Infinite divisibility of the continuum
As an example we take a simple line segment. Already a most elementary geometric consideration teaches us that it, however small it may be, is divisible all the way up to infinity. More precisely stated : the continued process of successive division has no natural end (that's what "infinity" means). In antiquity the so-called dichotomy was, in a certain sense, celebrated. A line may be divided into two. Each of them may again be so divided, and so on, and so on. But the dichotomy will never result in indivisible parts, i.e. never exhaust divisibility. That the two parts need not be equal to each other is clear. Also is clear that instead of dichotomy one may apply any other division into proportional (as they were called) parts.
We can obtain yet a sharper result as was the case in this consideration. The line may be partitioned into precisely those parts from which it can be constructed. Well, a line cannot be constructed from points, because these are not extensional, and addition (and by addition of parts a line must be constructed) of nothing to nothing cannot result in something. That is to say, beginning with the extensionally nothing, i.e. the point, subsequent concatenation of the extensionally nothing (adding points one at a time) cannot result in an extensum, the line. So points are not parts of a line. And from this follows : How far whatsoever a line is divided (not only by dichotomy but also by any other type of repeated division), it is not divisible into points, but only into smaller lines, which may again be divided, and so on, and so on. A consequence of this, which we will mention in passing, is the following : A real extensum may consist of a number of contigua, but however large this number is, ultimately these components must be continua, otherwise the extensum would be made up by points, which is just shown to be impossible.
In the original doctrine of sets [mathematical set theory], proposed by Cantor, a line indeed is considered to be a set of points, in the, by us, by the wole of classical mathematics, and by the philosophy of all schools, rejected sense. With these and the like assumptions are connected the infamous paradoxes of this doctrine, forcing a correction of these assumptions. So in this original sense of Cantor the line cannot be a set of points.
To arrive at this conclusion, the consideration of the method of dichotomy (and the like) is not sufficient. Therefore this second consideration is needed. That's why the result was called "sharper". Dichotomy only results in a so-called "countable set", that is to say, we can arrange (order) the members of such a set in such a way that in indexing each one of them with a natural number (1, 2, 3, 4, ...), we will not, on the way, leave out any such member, so long as we are still continuing the indexing. For instance we may arrange all fractions (including 1/1, 2/1, 3/1, etc.) in such a way that we obtain a truly countable set of them, proving the fact that there are as many fractions as there are natural numbers.
The continuum, on the other hand, has, taken as a set, a higher "power" than has the countable set, meaning that in trying to index the members of such a higher-power set we do not have enough natural numbers, not enough indexes, despite the fact that there are an infinity of them. As to the set of real numbers (supposed to be a continuum), this is proven by the famous "diagonal argument" of Cantor.
In a different sense, in which the line is not an aggregate of points, but, to speak with Brouwer, a "matrix of points" the line is, of course, surely a set of points, like in the older mathematics the "locus of points" is such a set (for example the perpendicular running through the centre of a given line segment is the locus of all points having equal distances to begin and end point of the line segment, where "all points" here should mean : whatever point you choose). The simple truth, that multiplication of a nothing or addition of a nothing to a nothing cannot result in a something, might have prevented this error.
As the parts of a continuum are not there actually, but only potentially, so are the points being nothing else than boundaries of these parts. That's why Brouwer correctly says : "But the continuum as a whole was given to us intuitively. Its synthesis [construction], which would create "all" its points individualized by mathematical intuition, is unthinkable and impossible."
It appears to us not unuseful to formulate, in passing, a general principle of which the above case was a particular instance. We would like to formulate it as follows : If a thing has a property as a result of which it may be subjected to an operation, producing one or more things having the same property, then this operation may be continued ad infinitum (there is nothing that would naturally bring it, i.e. the repeating, to an end). In this it is not necessary that then, without end, new things are produced. But in our case new things are indeed produced. A line may, as a result of its extensionality, be split up into parts always having themselves extensionality. And because of that the process can be indefinitely continued. In our case the process always results in new, smaller lines. The number of resulting individuals therefore grows without limit. ( This general principle also has a specification in the construction of the series of natural numbers, called, therefore, inductive numbers. [we start with 1, and then we repeat the following operation : arithmetically add the number 1. Every round produces a new natural number)
The mathematical consideration, accordingly, gives us the following result : The continuum cannot be constructed -- by addition or concatenation -- from indivisible, i.e. non-extended, elements. And the other way around : It is undefinitely divisible. (Physically, the continued division of matter may find a limit, but the resulting ultimate elements are then necessarily continua, and are still extensional, meaning that they cannot be "infinitely small", and especially meaning that we, as we did in the repeated division of the line, end up with continua with their inherent problems, unsolvable without the concept of potency.). The last formula -- undefinitely divisible -- will be made more precise in due course.
Above we had said : a metaphysics not possessing the concept of potency, potential being, cannot justify the continuum. One might perhaps have thought that unity and multitude can go together when the unity is accidental, a unity of contact of contigua. Then the antinomy vanishes. Accidental unity and actual multitude (i.e. in essence still a multitude, and thus per se a multitude) can perfectly go together. But the solution is worthless. When these contigua are each for themselves a continuum (and ultimately they must be continua), the solution (accidental unity, per se multitude) is valid, it is true, for the whole , i.e. the series of these contigua, but the antinomy returns for every ultimate element which necessarily is a continuum. If one wishes to avoid the return of the antinomy, then these elements must be non-extensional, be points, and from these no extensum can be produced. This leads us to the historical antinomies.
Antinomies of the continuum
These problems came from the fact that metaphysics seemed to arrive at an opposite conclusion : The continuum cannot be divisible ad infinitum, it ultimately must consist of indivisible elements. Leibniz and Kant have seen this problem very acutely. Seen also, and explicitly written, is that there was (seemed to be) an opposition between two sciences a priori, viz., mathematics and metaphysics
(I [JB] would like -- in anticipating things -- to remark the follwing : Mathematically there is no end of the process of repeated division, the process closes in on a point without ever reaching it. Metaphysically, on the other hand, i.e. the continuum seen as an ens, i.e. as a being, the process of repeated division must ultimately come to an end, because otherwise this process would, without actually resulting in elements that are not-beings, nevertheless close in on non-being as a result.) -- making clear the philosophical tragedy of the conflict -- but they were not able to elaborate their own metaphysics in order to create space for its solution. All problems, also of others than Kant or Leibniz, originate from the tacit or explicit assumption that the parts of the continuum in the whole (that is, before division) exist actually. They neglect or lack the Aristotelian concept of potency and its application, having given our first result in the metaphysics of the continuum : the thesis that the parts in the whole are merely potentially existing. Already the denial of only this truth makes the continuum collapsing under internal contradiction (namely the continuum as an ens : actually a one, and actually a many). Its unlimited divisibility only more clearly uncovers this consequence following from a false presupposition (namely parts, and that means here : infinitely many parts, actually existing in the continuum). Nevertheless it is instructive to consider these problems.
Infinity in the continuum
In some scholastics (schoolmen) in and after the Middle Ages especially this difficulty came to the fore : If the continuum is indefinitely divisible it must consist of an infinite number of parts, because a finite number would, in continued division, eventually become exhausted. But they held an infinite number of individuals to be metaphysically impossible, and, accordingly, concluded that the continuum is not indefinitely divisible, because they insist on the reality of the continuum, in contrast to many subsequent thinkers, who, as a result of the antinomies, decided to abandon the reality of the continuum. In what way these scholastics tried to escape from the mathematical conclusion that the continuum is indefinitely divisible, is, in this context, not important, because their problem is very easy to solve.
Let us, to begin with, distinguish : first, a multitude which is in the process of originating and may grow without limit, and, second, a multitude which is actual (i.e. already existing, not in the process of generation) and infinitely large (i.e. which cannot be exhausted, not be passed through). The first multitude is the potential infinite, the second the actual infinite. The first surely is not impossible. The series of natural numbers is an example : how large whatsoever a given number may be, we can always add unity (i.e. add the number 1), ad infinitum. Evidently, potential infinity is possible. But it is also evident that such a successive process cannot produce the actual infinite, because infinity can, by definition, never be exhausted. ( By continually adding 1 the resulting series of natural numbers remains finite as to the number of its members). May the actual infinite be produced in some other way? We don't think so, we fully admit to the philosophers that the actual infinite in quantity is a metaphysical impossibility.
(As to the mathematical continuum, we may draw, and thus produce, a line-segment, but we've not produced with this an actual infinity : we did not, in drawing, successively produce an actual infinity of points, because, as we saw, a concatenation of points does not produce a line. So we've only produced successive line-segments, and there are only finitely many of them making up our line-segment. In the ensuing intermezzo we will elaborate further on this.).
We here, for a short while interrupt our course, a course following HOENEN's text (1947), with saying something more about infinity as it is found in numbers. In natural numbers (1, 2, 3, ...) we have to do with a countable infinite set. The nature of this infinity is, apart from being countable, only potential. The set when produced by continually adding unity will never be concluded, there is no last number. And no matter how far we have gone in producing new natural numbers, the resulting set is always (i.e. in every instance) finite. No actual infinity is ever produced. Now it is proved that the infinity of the number of so-called real numbers (all integers, fractions, and all so-called irrational numbers), i.e. the number of elements in the set of real numbers, is of an even higher infinity than that associated with the set of natural numbers. The famous diagonal argument of Cantor proves this, by showing that the set of real numbers (in which it is already sufficient when we take all numbers between zero and one) cannot be counted, and that means that it is impossible to index every real number, i.e. label it with a natural number. There are not enough natural numbers to label all real numbers!. It is said that the real numbers form a true (arithmetic) continuum. Let us give and analyze this proof.
Cantor's diagonal argument
Let us take, because this is already enough, the set of all real numbers between zero and one (zero and one themselves excluded). It is clear that each real number of this set begins with " 0. -- " , so we can, in indicating these real numbers, leave this out and only write down the digits coming after the decimal point. Each such a real number, has, in addition, a countable, potentially infinite set of digits (which may be the same, for instance in [0.] 2222222 . . . , or partly or completely different).
Now we assume that the following list represents all real numbers between zero and one. And of course it is already clear that we cannot use the expression "all" to indicate the members of this set (of real numbers between zero and one) in the list mentioned. We simply cannot list them all. So by listing only a few of them we express the fact that we have to do with potential, not actual infinity. And the same goes for the infinity of digits after the decimal point in each such a real number. We cannot list them all, so we give only a few of them.
Here, then, the list of supposedly "all" real numbers between zero and one :
1 3 2 2 8 8 5 0 0 2 . . .
2 2 9 9 9 7 3 1 5 0 . . .
0 0 0 0 1 9 7 7 3 2 . . .
9 0 9 7 6 9 1 0 4 4 . . .
2 1 5 5 4 0 9 0 1 1 . . .
2 2 2 2 3 3 2 2 5 9 . . .
9 7 8 5 5 1 0 0 3 8 . . .
1 1 1 5 7 4 4 2 7 3 . . .
5 7 8 1 6 0 0 2 9 1 . . .
2 3 7 1 3 3 4 9 8 8 . . .
The digits that fall, in this list, on the diagonal are highlighted (but being not in any way special).
As has been said, it is assumed that the above list doesn't miss any real number whatsoever between 0 and 1. And because this list is sequentional (the next item comes after the previous one), the members may be indexed by natural numbers. So each member (not excluding anyone of them) of the set of real numbers between 0 and 1 is now, supposedly, accompanied with a natural number. And this means that, under the supposition of the completeness of the list, the set of real numbers between 0 and 1 is countably infinite, meaning further that the number of elements in the set of natural numbers is equal to the number of real numbers between 0 and 1. There is, under this assumption, a one-to-one correspondence between the members of the two sets.
But now, if we take the digits forming the diagonal of the above supposedly complete list, and add to each digit unity (whereby 0 becomes 1, 1 becomes 2, 3 becomes 4, . . . 8 becomes 9, and 9 becomes 0) we obtain the following real number between 0 and 1 :
2 3 1 8 5 4 1 3 0 9
But this real number, although being a number between 0 and 1, does not belong to the above assumed complete list, because its first digit differs from that of the first number in the list, its second digit differs from that of the second number in the list, its third digit differs from that of the third number of the list, and so on, and so on. And because we are supposed to have already used every natural number to index the members of the list, we have no natural number left to index this newly formed real number between 0 and 1. So it is now proven that the real numbers between 0 and 1, let alone all real numbers, cannot be exhaustively indexed by natural numbers. We simply do not have enough of them to do so. So there are more real numbers between 0 and 1 (and by implication more real numbers anyway) than there are natural numbers : The set of real numbers, the arithmetic continuum as it is called in mathematics, contains uncountably many elements. Or, said differently, their number is uncountably infinite (whereas the set of natural numbers is countably infinite).
In all this, that is to say, in the case of the countable infinite as well as in the case of the uncountable infinite, we cannot produce the corresponding complete sets. In both cases we operate, not with the actual, but with the potential infinite. And a set with a potentially infinite number of elements is itself always finite, but at the same time always still incomplete. And that is, what infinity really means : there is no end. So infinity, as a number, i.e. as a so-called transfinite number, does not exist. It cannot be actually produced by some specified procedure. And if such a procedure contains a guarantee that it will naturally come to an end it is then not a procedure producing infinity anymore. And intuitionalistic mathematics, as proposed by the school of the Dutch mathematician Brouwer, does not accept the reality of some mathematical entity (like infinity), or mathematical state of things, to be proven as long as it is not actually mathematically produced, i.e. constructed. So already in mathematics there is no actual infinity, only potential infinity. There are also no transfinite numbers as long as they express actual infinity. Only if they express potential infinity they do really mathematically exist, and may express the difference between countable and uncountable infinity.
(end of intermezzo)
Let us now continue to follow HOENEN again, and return to the (other) philosophers' problem with the continuum. Is it necessary that in order for the continuum to be divisible ad infinitum (because of it being an extensum) there must reside in it an actual multitude which is infinitely large? The first reply is already very simple : on the basis of our earlier results there is no actual multitude in the continuum [because a continuum as an ens cannot be at the same time and in the same respect one and many], and thus surely not an infinitely large one. Now the philosophers dealing with the continuum-problem supposed that the parts in the continuum were actual, and then indeed the conclusion follows that the number of these parts must be infinitely large. But their supposition is false, already beforehand destroying the unity, and thus the essence of the continuum. No wonder that from this supposition a false conclusion follows. And thus we can even from the falsity of this conclusion (that the continuum cannot be indefinitely divisible) find an affirmation of our first thesis saying that the parts in the whole are only potentially present.
The transition, in the use of "all" from the first sense ("all" as every (each)) to the second ("all" as totality), also in other cases than the one above, is called by Brouwer the "comprehension-axiom", and rightfully indicated as the main error of the classical Cantorial doctrine of sets : "dass das Komprehensionsaxioma, auf Grund dessen alle Dinge, welche eine bestimmte Eigenschaft besitzen, zu einer Menge vereinigt werden ... zur Begründung der Mengenlehre unzulässig bzw. unbrauchbar sei". [Each considered thing having a property A, now interpreted as a l l things with property A. So the expression, if referring to an infinite set, "the set of all elements being, as to what they are, of this particular type, or having this particular property, . . . ", so often encountered in mathematical (set) theory, does not, in fact, make any sense! Indeed, we often see the phrase "the set N of all natural numbers", or "the set R of all real numbers", or "the set of all points in line-segment L", etc.]
Various conclusions from the difficulty of the infinite.
The problems produced by the notion of infinity which (notion) is inherent in the continuum, were especially living among scholastic philosophers, earlier and later, up into our (1947) time. In 'modern' philosophers they remain in the background -- Leibniz saw so little objection in them that he even demanded infinite multitudes to be present in a perfect Nature -- except for some.
So in Renouvier, the leader of the neo-criticists in France, taking the infinite to be impossible, because it would be "un nombre sans nombre". Nevertheless there is a big difference between the scholastics and Renouvier. They wanted to save the real [i.e. not only the mentally conceived] continuum (some of them even at the cost of theorems of geometry) and tried to resolve the difficulties. Renouvier, on the other hand, sacrified (in order to deal with the contradiction of the infinite) the reality of the continuum : In reality there is no space, no time (i.e. nothing spatial and temporal), no matter, no motion. But isn't then this very same contradiction present in our mental images of all these continua, i.e. in the mere phenomena? No, replies Renouvier, because there the actual infinite is avoided, because there the parts are only potentially present, not actually. [indeed, although parts can, as a result of being thought, be present in the mind, they cannot materially be so present]. And he fully acknowledges that Aristotle, with his notion of potency, solves the problems completely, but holds that potency is only to be found in the mind, not in objective reality. And so he maintains that the ideal, phenomenal, continuum in our mind is free from contradiction, but not so the real continuum (the continuum as taken to be something real), resulting in the fact that continua have merely a phenomenal existence (i.e. as mental images, or mathematical entities). He holds that the parts of real continua must be actually existing things. He makes the mistake against which we already warned above : He confuses a row of contigua [parts not confluent into each other but merely touching each other], which as parts of an accidental whole do indeed actually exist, with the parts of the intrinsically one continuum. He, eventually must take the frog-embryo, about which we spoke above, as two individuals lying adjacent to each other. We will meet the same explanation also in other idealists or semi-idealists, denying the reality of extensa, with the same concession (the continuum with its potential parts is present in the mind) that the notion of potency saves the continuum.
Priority of parts before the whole? (reductionism or holism?)
Leibniz, as we said earlier, did not object against an actually infinite multitude. But from his metaphysics he obtained another proof against the reality of the continuum, a proof which was repeated later by Kant (18th century), and by Lachelier (19th century), in their own ways. It boils down to the following. Metaphysics demands, as they hold, that in a composed thing, a compositum, -- and the continuum is composed of the parts into which it is divisible -- the parts, as it is in the nature of things, have priority over the whole [i.e. the parts or elements are more fundamental than the whole they together form]. A composed thing that is real, would have its reality only from the reality of its parts. But then there must be parts that can be called 'first parts', because a going-back into infinity to [find] the [ultimate] conditions that must precede in order that the result be realized, is impossible [the going-back trajectory must be finite, in order that the corresponding going-up trajectory of successive conditions can be traversed to realize the effect]. So there must be first parts. However, in the case of the continuum, these parts cannot have extensionality anymore, otherwise they would still contain parts, and not be first parts [or last parts, if one prefers]. Accordingly, these ultimate parts must be points. But because mathematics rightfully denies that a concatenation of points results in an extensum, we have a contradiction between the demands of metaphysics and those of mathematics. [ (reductionistic) metaphysics demanding first parts guaranteeing the reality of the whole -- mathematics saying that these first parts are necessarily points, and points cannot form a whole with extensionality]
As we saw above, a line is, in the original Cantorian doctrine of sets, a "set of points" in this to be rejected sense. If a line could indeed be so [taken], the antinomy would, of course, disappear. This is the solution of B.Russell (1911) working out his logic as based on this doctrine of sets. But this solution cannot be accepted.
So the continuum cannot be real. Only a phenomenal (as in mathematics) continuum is possible.
These philosophers themselves lead us to the solution. They ask : "but, the continua of our imagination, the phenomenal continua, do we construct them from first, and consequently non-extensional parts, i.e. points? No, they reply, in our imagination the whole is given. There the whole has priority over the parts, so there that reasoning doesn't hold. Why the order is reversed there? Because there, i.e. in the mind, the parts are potential (parts), only having originated as a result of partition of the whole, the latter has priority over and above the parts. Let us listen to Leibniz in a letter to Remond :
"Dans l'idéal ou le continu le tout est antérieur aux parties, les parties ne sont que potentielles. Mais dans le réel le simple est antérieur aux assemblages, les parties sont actuelles, sont avant le tout".
[This rings a bell in our [JB] noëtic theory : Here we are going to assume that all truly holistic events and structures do involve the Implicate Order, which itself is noëtic and thus immaterial, whereas all truly reductionistic events and structures wholly belong to the province of the Explicate Order. It might be that a given structure, insofar as manifesting itself in the Explicate Order, is reductionistic, while in manifesting itself such that (also) the Implicate Order is involved, this same structure is holistic.]
Kant and Lachelier reason similarly. And again we arrive, as we did above, at the same result : 10, where the parts are potential they do not need to have priority, there are no first parts (possible division never stops), and the continuum is possible [When the whole is actual and the parts are potential, these parts may subsequently become actual, therefore potential parts are not prior to the whole. And as long repeated division is going on, the resulting parts are no first, i.e. ultimate, parts, no points]. And 20, the real compositum is called an "assemblage", indicating a same confusion of a continuum with a series of contigua. And always the same (opinion) recurs : A continuum is not taken to be an intrinsic unity, but naively taken as a collection of parts, existing next to each other. So the Aristotelian notion of real potency [as contrasted with potency in imagined structures] is again capable to free the real continuum completely from contradiction. All antinomies between mathematics and metaphysics disappear in Aristotle's metaphysics. Here the real continuum is a totality. [ The whole is earlier than the parts, because the whole is actual, whereas the parts are only potential].
For the same reason Kant is severely criticized by Schopenhauer. He [Kant] commits himself to "eine gar nicht feine petitio principii" [circular reasoning] because he takes the compositum only in the sense "dasz die Theile vor dem Ganzen da waren und zusammengetragen wurden, wodurch das Ganze entstanden sei", while in fact opposite from the "simple" (the indivisible) there is, not the "compositum" in that sense [i.e. in the sense of a reductionistic compositum], but "the divisible". "Die Theilbarkeit behauptet blosz die Theile a parte post [. . . only parts being present after division]. Das Zusammengesetztsein behaupted sie a parte ante" [parts already present]
Schopenhauer surely detects Kant's error precisely.
Finally, we [HOENEN] want to point to one more difficulty (having not detected it in the literature). A difficulty, that is, at first sight, looking to be unbreakable. We saw, that if the parts must be prior to the whole, the reasoning of Leibniz and followers is sound, implying that then first, and thus non-extensional, parts must be assumed, but from which no continuum can be constructed. As to the static continuum the difficulty has undoubtedly disappeared. But doesn't it return in the case of the dynamic flowing continuum, of motion and time? Aren't there necessarily the parts, at least the first parts, prior to the whole? We shall face this difficulty later when discussing te nature of motion and time. Then it will turn out that also there [i.e. in motion and time] the whole is prior to the parts.
Atoms and minima
We came to know the continuum as divisible (not divided!) ad infinitum, by considering it as extensum alone. After all('s said and done) its analysis as ens does not generate difficulty if we possess the Aristotelian notion of real potency. But the continuum is not only ens, not only extensum, it is also something physical [i.e. something having a specific content]. That such a continuum must also for us remain indefinitely divisible does not yet follow from its divisibility as extensum. After all, our tools may be in such a degree imperfect that there is a limit in the practically executed continued division, likewise our senses are in such a degree imperfect that they cannot continue to be able to perceive the increasingly smaller, while the intellect may well think the increasingly smaller, even ad infinitum so. Accordingly, there exist minima to our senses, but not to our intellect.
But these limits of divisibility are extrinsic. Other limits may be thinkable which come from the nature of the physical continuum itself. Because this is not a pure ens extensum, it is an extensum that has a specific nature. In this way we arrive at the philosophy of these "physically indivisibles", and historically we find two of them : atoms and minima. The first in Democritus [greek philosopher, 460-371 B.C.], the second in Aristotle [384/3-322/1 B.C.] and the peripatetici [Aristotle's followers]. This (historical) exposition is most important to the philosophy of Nature.
Democritus and Descartes [ Descartes : 1596-1650]
Both philosophers view the essence of matter in its extensionality. In Descartes this is clear enough, in Democritus we must -- as we believe -- also assume this. For him, ens is nothing but the "full" [no nothingness separating individual entia, beings]. Now to Descartes matter is nevertheless indefinitely divisible (sometimes, as we saw, divided all the way into infinity), but to Democritus this is impossible. Ens, the "atom", is, despite its extension, not divisible even into two parts : " It is impossible that from one originate two, [and] that from two originates one". This, not as a result of external circumstances, about which we spoke above, but as a result of an intrinsic impossibility. Who is right? Mathematically taken, of course Descartes, except as to an assumed actual unlimited dividedness into points. After all, as pure extensionality matter, and every given part of it, must be indefinitely divisible. Every given continuum, however small it may be, must surely be divisible into two parts. Therefore Descartes denied the existence of atoms [i.e. of indivisible, but still extensional, parts or particles].
But, on the other hand, while one may rightfully take matter as pure extensionality, if one refers to real matter, the "full" of Democritus, then one has to do with an ens, and this has to obey the laws of ens, and thus obey the laws of metaphysics. Now, to Democritus, as well as to Descartes, the ens was only the 'eleatic' ens, viz., unchangeable Being. In essence their metaphysics was that of Parmenides [greek philosopher, 540-480 B.C.]. And in applying this metaphysics to the extensum, Democritus, not Descartes, was consequent. For, given this position, even a division of one into two is already out of the question. After all, if from one, two would originate (or vice versa), then at least one of these two would be a new ens (respectively, at least one of them would have been "perished"), and then we would have genuine "generation and corruption" which in this metaphysics (i.e. the eleatic metaphisics) was held to be impossible. Ens must be "apathes" (it cannot undergo anything), unchangeable, and therefore also indivisible, and impossible to construct from parts. It is necessarily : a-tom [i.e. indivisible]. So Democritus has dug deeper than Descartes by his attention to ens and to the theory of ens.
The fact that Descartes has not penetrated so deeply is connected with his whole attitude as regards these problems. A continuum is for him already there where two bodies are at rest next to each other or performing the same motion. They are contigua when they are moving along each other.
Yet, also in Democritus one problem is left unsolved. After all, while Descartes had, in considering the "ens extensum", neglected the ens, and thus metaphysics, and only focussed on the extensum, Democritus almost entirely had focussed on the ens and did not sufficiently account for the extensum. Yet this [extensum as extensum] would necessarily contain parts, implying divisibility. Especially so, because all his extensa, his atoms, have the same nature, only being "the full", and only differ in size and figure. This difference in size is evidently a sign of the fact that this "full" was not confined to determined sizes, and consequently that it had to be divisible into smaller extensa. In this way, atoms, being absolutely indivisible things, cannot be justified.
Democritus, it is true, had put forward an argument to justify this indivisibility of the "full", an argument that we also find in Zeno the Elean, pupil of Parmenides. If the extensum (as we may reproduce his reasoning) is divisible in one point (i.e. in one plane), then it must also be divisible in every point (i.e. in every plane), because in the purely extensional every point is equivalent to any other. But if it is divisible in every point, then it is divisible in all points. [ This is the comprehension-axiom about which we have spoken earlier]. To be divisible means : can be divided. And now realize this possibility, and the result will be : a multitude of inextensa, i.e. of points. But from these a continuum cannot be built [i.e. the continuum cannot consist solely of points]. So the extensum cannot be divisible in even one point only. In the eleatic (and Democritic?) argument the last part read : the inextensum is nothing, and from nothing the ens, the extensum, cannot result. Also here we cannot deny the acuteness of eleatic reasoning. Nevertheless, the reasoning is false. The first part of it we have already analyzed earlier : the transition from every to all, i.e. from whatever (point) to the collection of all (points), as takes place in the argument, is not [in the case of an indefinite number (as contrasted with a finite number) of points] legitimate and thus false.
So in the conception of Democritus the unsoved problem remains, the problem in what way the "full", occurring in larger and smaller individuals, can yet be indivisible. Atoms cannot be justify in this theory. And in every atomic theory one should be alert whether in it there may be hidden a similar error.
So we come to the following result : Geometry, considering in "ens extensum" the "extensum", demands that this must be always divisible, however small it is. Metaphysics, as long as it sticks to the Parmenidean conception of Being (one, unchangeable, and indivisible), demands, in its consideration of the "ens" in the "ens extensum", that it must be always indivisible, how large it may be. Nevertheless both sciences must be applicable to the same object : the ens extensum. So here we already have a seeming antinomy even before considering the infinite in the extensum. The solution lies in the extension that Aristoteles had given to the eleatic metaphysics. The same extension, that can guarantee the changeableness of ens in general, also explains the "generation and corruption" that is implied in "being divided and united", namely the introduction of the notion of real potency, being-in-potency.
But such a consideration does not result in [the notion of] atoms or something like it. If we view physical bodies as " ens extensum" only, then they are indefinitely divisible, then no minima do exist. Yet Aristotle, and with him the peripatetici, have also justified [the existence of] minima. But for this the physical body must not only be viewed as ens extensum. Which is possible because natural bodies are indeed not only that [i.e. they can have, over and above merely being an ens, and having extensionality, a specific content, and so they can be ens physicum].
After all, the bodies in Nature are not merely ens, having them be subjected to the laws of metaphysics. They moreover are not merely extensum, in virtue of which geometry does apply to them. They also have a specific nature, principle of activity and passivity. They are not exclusively the "full". The fact that they have different qualities already witnesses to the fact that they do have such a nature. That they have a nature was already in the beginning so evident that this physis took part in constituting the name of a science [physical science, natuurkunde]. But then, also the laws of this nature -- not all of them we do, pitty enough, know a priori (most of them we do not so know) -- must hold for the physical body, which is not merely "ens extensum", but also "physicum". And these laws might modify the above derived results concerning indefinite divisibility. This was already noticed and applied by Aristotle, and further worked out by the peripatetici.
Aristotle and the minima
Among the predecesors of Aristotle there was a philosopher who, after and influenced by Parmenides, had constructed a (not consequent) mechanicism : Anaxogoras. Of the theses characterizing his system, one is interesting to us : Natural bodies are indefinitely divisible. Against this thesis Aristotle objects in the fourth chapter of the first book of his Physics. The stagirite (as Aristotle is called because he came from there), who, in the same Physics so strongly defends the indefinite divisibility of the continuum as such, and solving the seemingly implied antinomies in so ingenious a way, nevertheless opposes to the idea of indefinite divisibility of physical continua, by the following reason : the physical nature of these bodies resists it. It admits division only up until a certain finite limit. Remarkably, he not only demands a lower limit, a limit to continued division, but also wants to derive an upper limit, a limit to increase, growth. Not totally a priori, because as to the specific nature we lack [a priori] insight, but concludes it from experience. Indeed, he had observed that the size of individuals of a same animal species varies only within certain limits. Below a certain minimum, and above a certain maximum, -- of course co-depending on conditions -- no individuals of a given species exist. And these limits differ in different species, depending upon their specific nature, their "own proper nature" ("oikeia physis"). So every species has its own maximal and minimal size, like it indeed also possesses its own qualities. Then, Aristotle concludes against Anaxagoras that also the materials, building up the animal body, "flesh and bones", have their own minimum. Were these [materials] possible in every size, without upper and lower boundary, then from them also an animal could be built up having any size whatsoever, what militates against experience. So the components of these living beings are not indefinitely divisible. Absolute divisibility of the smallest possible particles of these substances [in the chemical sense] is, evidently, not excluded by this reasoning, but it must be true that division, if it succeeds, results in fragments having a different nature, being of a different kind, not being "flesh and bones" anymore.
Who is here not recalling modern chemistry, the molecule of a chemical compound, being divisible, but not into parts of the same physical kind as was the whole?
Succinctly formulated we find this thesis in THOMAS AQUINAS (13th century) : "ideo est invenire minimam aquam et minimam carnem, ut dicitur in I Phys., quae si dividantur, non erit ulterius aqua et caro". (= "therefore there must be found a minimum of water and a minimum of flesh, as is stated in the first book of Physics, which, when further divided, do not remain water and flesh".)
And so Aristotle concludes as to the existence of minima, indivisible particles, not by themselves absolutely indivisible, but only relatively so, i.e. such that subsequent division must result in disintegration into other materials of a different kind. And this thesis he wants to be applied to all materials forming "natural" bodies.
Yet one important remark : these minima of a same specific material must be equal in size among one another ("isa peperasmena"), which is not surprising, because the size of a minimum is determined by the specific nature.
Development of the theory in the peripatetici.
Aristotle does definitely present this theory, but yet in austere sketches only, as we [HOENEN] here have described it, following the 4th chapter of the first book of Aristotle's Physics. A short allusion can be found in some other places in the works of the stagirite. The theory of minima is fully worked out by the later peripatetici (followers of Aristotle). Simplicius considers it in his commentaries on the Physics and on About the Soul. He explains that the limited divisibility of the [living] body as "physis" does not contradict its indefinite divisibility as ens extensum. Further that the theory can be applied to all "physical" [living] bodies, including the inorganic ones, that the specific nature of the materials is the cause of this limitation [of divisibility], and, finally, that the minima of the same species [the same kind of chemical material] must be of the same size.
Also St Thomas, being, it is true, dependent on Simplicius, repeatedly returns to the theory. He accepts it without reservation, expounds its logical coherence, and its consistency with the demands of geometry. Let the size of the minima be determined by the specific nature, from which then automatically follows that the minima of a same substance (in the chemical sense) are equal among each other. And this conclusion is explicitly emphasized by St Thomas, following Aristotle and Simplicius.
After St Thomas, the theory was generally accepted by the scholastics, with the one restriction that the followers of Duns Scotus assumed the bounded divisibility (to be a fact) in heterogeneous substances [chem. sense] only, while they assumed indefinite divisibility, also physically, in all homogeneous substances. In these latter only extrinsic circumstances, not their intrinsic nature, limit their divisibility. And perhaps this reservation should not a priori be discarded.
Principle of Toledo
That one, being familiar with the theory of minima, without hesitation applied it in explaining the origin of chemical compounds (the mixta), may be shown from the words of a philosopher, Fr. Toledo S.J. (1579), who, directly after the Middle Ages, presented the communis opinio of the medievals as follows : "as to this point there is consensus that the substances [chem. sense] to be compounded [first] become divided into their natural minima, that the individual minima [of the reactant substances] come to lie next to one another, that they [then] interact until a third form of being results, namely that of the compound itself ". Later we will return to these words.
The wish of Pereira
It could not last long before the wish would come up to know also the size of the minima of elements and compounds, especially because these sizes are among the expressions of the specific nature of these substances [chem. sense]. Not only there was the wish, but it was also expressed, albeit in a mood of despondency, a despondency taking the execution of it [i.e. to determine those sizes], because of the foreseen difficulties, to be impossible. So, in the same century B. Pereira S.J. (1585) writes " To precisely uncover what precisely are the upper and lower boundaries of size [of the minima] in each species of natural bodies, is very difficult, if not impossible". How familiar one was with the theory of natural minima is clear from the words just cited. What was wished was, after all, no less than the determination of what nowadays is called : the atomic weight and the molecular weight of chemical substances. That here the problem was clearly recognized, the solution of which became the foundation of scientific chemistry, nobody will deny. Neither [anybody will deny] that it was posed as a natural consequence of the Aristotelian-scholastic theory of natural minima. The atomism of Democritus [Greek philosopher, 460-371] on the other hand, did not naturally result in the formulation of this problem : " how to determine the size of the atoms". It couldn't result in it, because in Democritean atomism all sizes, indefinite in number, of atoms were assumed, and there was no question of equal atoms of a same element [because in that atomism no specific natures were presupposed]. The considerations of the scholastici naturally had to lead to the question, and indeed have done so.
Must we give Pereira and his contempories the blame for not finding the methods to solve the problem practically, and thereby falling into despondency? Let us answer with a counter-question. Why is it that since the re-introduction of atomism after the Middle Ages it had to last two and a half century before by John Dalton the first attempts were made to anwer the question of Pereira by means of experiments, summarized in the stoechiometric laws?
In this, we must realize that there is a little shift of problem to be seen. One did not directly speak of "minima of extension", but of "minima of weight" (better : mass). The shift is not essential. The first coincide with the second, are measured by them in homogeneous and periodic-micro-heterogeneous substances.
To find a method in order to answer Pereira's question, one needed : application of mathematical methods to physical problems. These methods began to develop only in the century after Pereira. And they rather soon had success in physical problems. Why not in chemical problems? That it could not succeed in Cartesian circles is evident : there [physical] matter is indefinitely divisible. But why then not in atomistic theories, which, at least in this respect, quickly ousted the cartesian views? Also here the answer can easily be found. As we already said, the pure theory of Democritus does not result in posing this question, because there equality of size in atoms (if found at all [i.e. in cases where it actually was found and had to be explained] ) is something accidental. Atomism being the point of view, chemistry was only able to arrive at scientific results, when, as a result of Dalton's hypothesis of the equality of atoms of a same element, something essential was changed in the theory of Democritus, to which we will return later on. Success was achieved, when Dalton returned to the position of Aristotle and scholasticism [probably as seen so after the fact, but nevertheless significant]. In fact, Dalton's theory was opposed to the atoms of pure atomism. As to the natural minima of Aristotle it was already worked out in the Middle Ages as a naturally to be expected consequence, a natural specification of the theory. [i.e. the outline of Dalton's theory was, in the form of a natural specification of Aristotle's notion of natural minima, already worked out in the Middle Ages.]
Significance of the theory of minima
That this theory was by Duhem rightfully called "cette puissante théorie" is, after what has been said above, evident. It has, starting from philosophical considerations, out of still poor data of experience, led to a result that had to be taken as one of the foundations of scientific atomic theory [i.e. the theory of atoms and molecules, that is, chemical elements and compounds], as it was only developed in the 19th century. And the original theory, as already developed in the Middle Ages, can (as we will see later) serve as the foundation of an atomic theory, surviving the classical theory of the 19th century, i.e surviving the crisis of the latter theory, a crisis taking shape in the first half of the 20th century and resulting in quantum-mechanics. It is also a powerful theory in having so rightfully combined the demands of sciences of different degrees of abstraction. Let us clarify.
Indeed, in order to obtain the above result, three sciences had cooperated : natural science, mathematics, and metaphysics, where all three of them must be applicable to the same object : the intrinsically one, physical body. These three sciences remain, according to Aristotle and St Thomas Aquinas, at different levels of abstraction : respectively of the first, the second, and third degree of abstraction. Above we sufficiently recognized that both last two sciences (mathematics and metaphysics) are applicable to the physical body as ens extensum. The application, when a still imperfect metaphysics -- that of Parmenides -- was so applied, resulted in a conflict, later fully solved by a metaphysics possessing the notion of potency, potential being. While, according to Parmenides-Democritus, the ens extensum is not divisible [for Parmenides the whole of Being, for Democritus the atoms], even not into mere two parts, potential being, as predisposition to either oneness or multitude, explains the possibility of such a division, but it doesn't demand minima. If the (natural) body was only ens extensum, then minima would not exist.
At this point the science at the first level of abstraction (natural science), which must be applicable to the same object, interferes. Applicable to the same object, precisely because the physical body is not exclusively ens extensum but in addition having a qualitative nature, even such that, being revealed at least in long-term daily experience, there appears to be a specific difference between the (physical) bodies. Also in this case, in the meeting of two sciences with different degree of abstraction a conflict could arise. In Democritus and Descartes it was there, because they could not account in their metaphysics for qualitative and specific differences. In Aristotle the conflict failed to take place, because his metaphysics was by itself appropriate to take care of the new problems. A difference with the aforementioned meeting consists in the fact that this physics [chemistry] had to rely in a high degree on experience precisely because of its lower level of abstraction. Aristotle had the relevant experience at his disposal only with respect to living beings, and we saw how ingeniously he knew to use it also for the components of these beings, and how one did extend these findings to inorganic substances (materials). For this (latter), 19th century science had gathered an immensely rich experience precisely in those matters that served as foundation and further elaboration of the stoechiometric laws, for an undestanding of which [the assumption of] minima [is] are absolutely necessary. As appears from experience there is a limit of divisibility.
Is there then not a conflict with the result obtained by mathematics and metaphysics demanding and justifying divisibility for every extensum? No, because already Aristotle had a metaphysics that also explains this. Potency is not only a predisposition to unity and multitude in one and the same species (only already this is needed and sufficient to explain divisibility), but is also a predisposition to various kinds (species) of materials. It thus can be actualized by specifically different (substantial) forms. To explain this, the theory was constructed in the first place. Accordingly, the division of a given extensum may not only result in two smaller individuals of the same species, but may also generate two (or more) individuals specifically differing from each other and from the (original) whole. And then we have the minima of Aristotle, which, above, we called only "relatively indivisible", i.e. indivisible into parts of the same species. And these relatively indivisible parts are sufficient to explain the rich data of experience, summed up in stoechiometry. So here all conflict is avoided already in advance thanks to the intrinsic capabilities of the theory.
Having carried through this analysis this far, the reader might ask : may we then not expect in continued division (for which we might perhaps not have the physical means, but which intrinsically should be possible) successively new minima, specifically new materials? As far as we know, the problem is nowhere as such posed. Only St Thomas Aquinas asks a corresponding question with respect to an analogous case, and the answer that he gives according to his general principles would, for our case, result in an affirmation : There are, ad infinitum, new species possible insofar smaller minima are possible. We will not further go into this matter.
How it is, that Aristotle and the peripatetici, who did not yet have to their disposal the rich experience in the inorganic world, nevertheless manged to set up so acutely their theory of minima also with respect to the inorganic bodies? We already saw how Aristotle reasoned from data of the organic nature. But there is more. Also the inorganic materials are not solely extensa, but show qualitative properties rightly be preserved (i.e. not reduced to quantity or to imaginary properties) in the Aristotelian theory. Now there are in the inorganic world also constant qualitative differences, pointing to the specific differences. Therefore there is in Aristotle constant emphasis on the specific "nature" of things, in which "nature" means : the intrinsic principle of activity and passivity. And then it is obvious to suppose -- St Thomas repeatedly refers to it -- that also the limits of quantity are own proper, i.e. specific, manifestations of these specific differences. Insight into the necessity of specific quantitative limits we do not have, but the assumption of such a necessity of nature, as St Thomas calls for, is yet plausible. In modern science it is clear that a chemical compound or crystal can only exist within a certain range of temperature. The internal heat energy has a maximum and a minimum. Something like it may be expected with respect to quantity, as St Thomas argues : in the way that materials have their own specific qualities, they will also have their own, specific, quantity. So in this we do not have -- as is the case in mathematical and metaphyscal laws -- insight into the necessity, insight based on simple general observation, but the assumption nevertheless is very plausible. And then the rich modern experience beautifully confirms this assumption, in which we want to put forward the following cognitively important remark : As long as we do not yet have sufficient insight, we may doubt the value of this thesis of the existence of minima. So we still have uncertainty and contingency. But if it is true, then the relation it expresses is not merely a contingent, accidental connection. It is a per se connection in the nature of things. The contingency is not objective, but only in our poor insight or also in the relative insufficiency of observation. When modern experience thus gave us a wonderful confirmation of this insight of the medievals, it is a confirmation of an invaluable truth, invaluable, because it expresses a real and necessary connection.
So rightfully Duhem could -- although not having analyzed all this -- call the scholastic theory of natural minima a "powerful theory". It will yet be more evident when, as we will see later, it, with the help of other modern discoveries, lets itself to be further developed.
The theory of minima as a specification of general principles.
And thus we find out for this problem, what we at every turn also find elsewhere : The general principles of natural philosophy, clarifying the possibility of generation and corruption and of change in general, and determining the conditions for it, do need, in order to be processed into specific explanations, further determination, further specification. The latter cannot, as Descartes held, be found in an exclusively a priori way, but only partly so. Indeed, also other [empirical] sciences may participate in such specification of metaphysical principles, and [so] experience will [and must] play a role in it. The specification which the theory of minima (we heard it from Toledo) contributes to the problem of "generation and corruption" was, in essential terms, already conceived from experience by Aristotle, and was further worked out in scholasticism. This specification can and must, as we will see later, be united with other theories also from the Middle Ages into a modern atomic theory specified by the most recent discoveries.
We only spoke of the lower boundary which Aristotle had derived, and this has turned out to be of the utmost importance. We did not spoke about the maximum, also assumed by him because observation in the organic nature seems to demand it. Is there enough reason to assume this limit also in inorganic materials? We shall devote only a few words to this question, which surely is not as significant as is the first problem, and will probably never be so.
Of course the question of upper boundary in size is not about that in aggregates of bodies, because, as a result of continuous addition, they can grow in size indefinitely. It is about bodies which are intrinsically one, one substance [in the metaphysical sense], one individual. Do these inorganic individuals have an upper boundary in their size, dependent upon their specific nature, and so being equal in one and the same species and different in different species? If somebody holds only chemical molecules, or chemical atoms, or even only the constituents of atoms, to be true individuals, and larger complexes as mere aggregates, he will immediately say "yes". In that case upper and lower boundary coincide. As to larger complexes : gasses and liquids certainly are mere aggregates. But how about crystals [as such a larger complex)? Later we will meet important reasons why we must conclude that any given single crystal is a continuum, and consequently an intrinsic unity, a single substance [in the metaphysical sense], one totality. [see about crystals in this respect Fourth Part of Website ]. With this assumption our question (as to a maximum in size] is not yet immediately solved. Can a crystal grow indefinitely, or is there some upper boundary to it? Retgers, the well-known Dutch crystallographer, was, at the time, of the opinion that in the world of crystals there do exist specific upper limits. And even according to mere superficial knowledge of crystallography it is conspicuous that some sorts of crystals so readily do grow into larger individuals -- rhinestone may become several meters long -- while crystals of other chemical substances hardly reach microscopical size. And all intermediate gradations, all of them apparently specific, do occur. There are, however, opponents to Retgers' opinion. In my [HOENEN's] student years there was in the Leiden inorganic laboratory an amanuensis, who, to refute Retgers' opinion, grew crystals of a [chemical] substance which very easily yields large crystals, I think it was chrome-alum. Well, the growth of these crystals was only limited by the limits of the vessels having to contain the mother liquor. Does this prove that they can grow ad infinitum? Of course not. This is all about the subject of maxima.
In Nature we encounter extensa [natural bodies] which at least appear, that is, in macroscopic and even microscopic observation, not to show any difference in properties in their different parts, and [we encounter] other extensa that do sow such differences. The first bodies are homogeneous, the other are heterogeneous. So heterogeneity means diversity of properties in a given extensum, in the sense that there is in one part or region [of that body] a quality, which, in some other part or region of it is replaced by another (of the same genus) quality, or simply is absent, or present with a weaker [or stronger] intensity.
Temporary and permanent heterogeneity
Heterogeneity sometimes can be of a mere transient nature, varying in time, arising and disappearing again. Thus a body may show in its different regions differences in temperature soon disappearing again.
But we also encounter permanent heterogeneity in Nature, a permanent difference in properties in parts or regions of an extensum. This is even characteristic in organisms, without being limited to them. Also in inorganic Nature it occurs. There, there may be tensions in certain parts of a given body, lacking or being weaker in other of its parts. Detailed investigation has found -- we will meet it later -- that even crystals carry a permanent heterogeneity according to a determined structure. And even in atoms the one part has different properties than another. But this, of course, escapes from direct observation, even through a microscope. So these entities (crystals and atoms) appear to be homogeneous, but are only seemingly so. Of course permanent heterogeneity is to natural philosophy much more important than temporary heterogeneity. To be "permanent", namely, could imply "to be constant in the species", i.e. not being different in different individuals of the same species of inorganic body (atoms, molecules and crystals), or, said differently, a permanent heterogeneity may be a direct result of the individual's specific nature. The great importance of it [i.e. of specific heterogeneity] will become evident later when we derive the "principle of heterogeneity of chemical compounds".
Abrupt and continuously going off heterogeneity
Another way of classifying heterogeneity is also evident. Let us, for example, consider the density of a given extensum (i.e. a given physical body). If it is, with respect to this property, heterogeneous and also permanently so, then it has different density in different parts. If we now draw a line through this extensum, then, following this line, the density in and around different points [of the body, indicated by those of the line] will accordingly be variable. Now, evidently, two cases are possible. The change in density may be continuous [flowing], but we may also encounter points at which the density suddenly decreases or increases. Graphically displayed, where the line is the abscissa and where the densities are measured by the ordinate axis [perpendicular to the abscissa], the density curve will in these points ascend or descend parallel to the ordinate axis. Of course there exist cases in which the density varies in a continuous fashion, but in a small interval so rapidly that it cannot experimentally be distinguished from discontinous change. We shall call the first form of heterogeneity "continuously going off " [or continuously distributed], and the second "abrupt". It is clear that these two forms may also appear, in an extensum, in a periodic fashion, in which along the above mentioned line we see successive maxima and minima of density appearing. We will see this realized in crystals. It is also clear that both forms of heterogeneity do also exist with respect to other properties.
Above we deliberately spoke only of heterogeneous extensa. Now the question could be asked, which question is, as a matter of fact, easy to answer, : can these extensa at the same time be continua? For one might think that, as a result of these different properties, the parts of the continuum become actual things, and thus constituting an actual (finite) multitude. And actual multitude neutralizes the actual unity, and thus destroys the continuum. Fortunately the answer is easy enough : heterogeneous continua are possible.
A continuum -- we here do consider only a body, a substance [in the metaphyscal sense] [and not, for example the continuum of a given motion, its trajectory] -- is an extensum that is intrinsically one, so, in our case, one single substance [and a substance is intrinsically one in a different way as does a geometric continuum). Now it is immediately clear that one single substance [for example a crystal or an organism], without losing its substantial unity, can be a carrier or substrate of an actual multitude of accidents.
REMARK ON ARISTOTLE'S "CATEGORIES"
"Substance and accidents" are distinguished in Aristotle's classification of the highest genera of things (whether concrete things or properties). So it is a classification of the main type of things. They are Aristotle's "categories". They are forms. They represent the different ways in which things in the world are determined, namely determined as to be substances [in the metaphysical sense], or as to be quantities, or as qualities, or as to their place, activity, passivity, or their relations to other things. In knowledge the "accidents" are predicated of a given substance, while to be a substance is predicated of some not yet specified 'thing' (a 'this'). In fact the "categories" is a first classification of beings. They together form a so-called "pros hen unity" of instances of Being, in which "substance" is the primary instance of Being, because it possesses the true nature of Being, while the accidents (which are necessary or accidental determinations of a given substance) do not possess this true nature of Being. They are secondary instances of Being. They only possess it insofar as they, as to what they are, refer (by imitation one might say) to substance, i.e. to the primary instance, namely by always (ultimately) inhering in substance, and only in substance, as their ontological substrate or carrier. A substance in the metaphysical sense (and carrying accidents) is a unity throughout, a per se unity, i.e. a holistic unity, such as a crystal or an organism. Its form, its substantial form, is its matter-actualized (and its matter is its form actually, and other forms potentially), and is the substance's intrinsic content or "essence". Accidents are not part of but only inhere in substance. Some of them immediately result from the essence of that particular substance and thus inhere in that substance necessarily, and are called "propria" (singular : proprium), while other [accidents] only happen to inhere in some particular substance. They are the true accidents [Of course a same accident may be in one substance a true accident, while in another a proprium]. No accident has independent being. For actually to be, they need a substrate, and this is substance, which itself can independently be, thay is, not needing an ontologial substrate.
To repeat, substance, without losing its actual intrinsic unity, may be determined by an actual multitude of accidents. After all, actual unity and actual multitude do only exclude one another if they have to be taken in the same respect, here both with respect to the substance itself. So a true being, i.e. its primary instance, is actually one in substance and actually many in its accidents. In an extensional substance, a substantial continuum, this actual multitude of accidents evidently can be expressed differently in different parts. These parts then become actual, but only accidentally so, that is, not as concrete parts but as (additional) properties or features of the substance. [they are, afterall, mere (further) determinations of the substance, inhering in it, and consequently no parts or particles in it]. So there will be actual multitude in accidental respect, while unity in substantial respect remains. So with this we have a heterogeneous continuum. That this holds for abrupt as well as for continuous heterogeneity, and for permanent as well as for temporary heterogeneity, is clear. Our consideration was general.
What we have deduced here, we even can experience in introspective observation. In the expression "cogito ergo sum" ( I think, therefore I exist) it is said that we, in our selves experience the accidental "being" of the thought as inhering in our substantial "being". Well then, we not only experience in our selves the presence of thoughts, but at the same time, for instance, expressions of our will. We then experience a multitude of further (temporary) determinations in our one single substance. And even more : we, at the same time, experience the multitude of sensible perceptions and feelings in different organs of the body. So here we have a direct observation of heterogeneity in the unity of one and the same substance.
Later in our exposition (when we treat of inorganic substances) heterogeneous continua will have important applications. Here we cursorily present some other applications.
From the possibility of heterogeneous continua immediately follows : When we observe a heterogeneous extensum, also when in observation it appears to be abruptly heterogeneous -- we saw that it cannot be decided upon purely experimentally -- then we cannot immediately conclude that it is a multitude of bodies. It might very well be a true continuum, one single body. The abrupt heterogeneity is no obstacle. Let us directly apply this to living beings. If we see there a difference in properties between various tissues or cells, properties that abruptly give way to other properties, then these different cells and tissues are not different substances (in the metaphysical sense) : in such a body there exclusively are different properties, different qualities, of regions or 'parts' of one single substance. Also when parts (regions) of such a body move relative to one another they are not separate substances, contigua, i.e. then also they, as implied by the intrinsic substantial unity, remain parts of the one single substance, which parts only have a different accidental "being", meaning that they, as further determinations, (ontologically) inhere in that particular substance (today, in "parts merely qualities of the whole thing" we speak of its "holistic constitution") [meaning further that the parts themselves are not, as to their being, this substance, but, as to what they are, merely refer to it : As "beings", accidents and substance together form a pros hen unity of beings].
Of course all this applies only to those parts of the body that truly belong to the one substance. At first sight these conclusions may seem unusual and therefore weird. For those who have understood the heterogeneous continuum it is clear.
[As to this latter statement of HOENEN (p.116) it is certainly not immediately clear : In an organismic body the various organs (bones, heart, blood-vessels, etc.) certainly do not look like mere "qualities of the one body", they, although connected with each other, look like concrete separate things, i.e. things with form and matter. The connection with each other of these various organs (of a given organismic body) may indeed be a continuous connection instead of a mere touching, but, as has been said, that is hard to establish empirically. HOENEN has correctly demonstrated that in general the notion of "heterogeneous physical continuum" (in addition to the more self-evident homogeneous continua) is not in itself contradictory. Having said this, and having, in addition to that, defined a substance (in the metaphysical sense) to be an intrinsic unity, and not a mere aggregate of different substances, and having, finally, presupposed that any given organismic individual is such a substance, then indeed (but only then) the conclusion follows that the organs and parts of such an organismic individual are not substances themselves, but are different properties of the one single substance that is the organismic body. Their connection with each other and with the whole body then must be a continuous connection. Later HOENEN will call these organs, parts and particles of and in the organismic body "virtual" (in the sense of not-actual), here virtual, not with respect to their qualitative content or quantitative extension (size and shape), but with respect to being substances themselves, meaning that the 'concreteness' of them is virtual, while their qualitative content is actual.
And of course HOENEN's presupposition that organismic bodies are true substances in the metaphysical sense is reasonable, were it only because of the fact that in Aristotle's metaphysics the organismic body is the 'primary instance' of substance.]
We spoke about movement of parts of a substance with respect to one another. This movement surely is real. We, for example, move our limbs. But not everything that to superficial observation appears as movement of parts of a substance needs to be so. The theory of the heterogeneous continuum opens up prospects of other possibilities. Such a "movement of parts" could be -- and this possibility necessarily follows from insight into the heterogeneous continuum -- a pure alternation of the distribution of the heterogeneity across the substance, without movement of its parts. Also this looks weird at first sight, but reflection on the heterogeneous continuum will take away this weirdness. Here we cannot dwell on it any further. Later, when we discuss qualities and where we will attempt to determine the essence of "mass", we will take yet another step further in considering these conclusions.
It is perhaps instructive to insert here a short section of a discussion, written down earlier, already in Fourth Part of Website , part XXIX (sequel 4), on the nature of a heterogeneous continuum as compared with a tight contiguum, where the latter may be viewed as a "formerly heterogeneous continuum" in which the regions, formerly representing different qualities of the one body, have now (conceptually) turned into concrete parts, bodies, i.e. discontinuous parts (contigua), merely touching one another, and together making up an aggregate, and consequently having turned the heterogeneous continuum into a tight contiguum (a mozaic of contigua). The discussion was (in Fourth Part of Website) already in terms of the Explicate / Implicate Order duality of Reality, but not in completely the same way as we've done in the present (sixth) Part of Website.
Continuum or Tight Contiguum?
While, according to our earlier considerations, free atoms, free molecules and single solid crystals seem to be (heterogeneous) continua, and snow crystals (and then with them, all solid crystals) seem to point to, we could say, an even stronger sense of continuum, namely where the parts and elements are non-locally connected [connected with each other through the Implicate Order], organisms seem to be different, although they can reasonably be interpreted as continua, even in the just mentioned stronger sense.
Because to see all these entities (atoms, molecules, crystals and organisms) as continua of some sort is intuitively difficult, but nevertheless very intriguing, it is perhaps instructive to summarize and further elaborate on the notion of "holon as (heterogeneous) continuum".
In his interesting book "Philosophie der Anorganische Natuur (1947), referred to earlier, HOENEN shows that the results of classical physics do not decide whether any given intrinsic being (i.e. a being which has the specific causes of its generation all within itself) is a continuum or a contiguum, i.e. whether it is (1) a strict continuum , and thus a holon in its strongest, but one, sense ( The stongest sense is that of a non-local continuum), in which its elements exist only virtually -- the continuum is, as continuum, not divided, but only divisible -- and are -- in a physical continuum -- just a spatial distribution of qualities of the holon, or (2) (whether such a being is) at most a tight contiguum, i.e. an entity consisting of well-defined tightly packed material parts and elements that exist actually in it, and are not qualities of something, but beings (i.e. things) themselves, and so resulting in a unity, it is true (because it is organized and patterned), but not a unity in the strict sense.
All results of classical physics and chemistry, about molecules and crystals (which are the supra-atomic intrinsic beings of the Inorganic World) are, according to HOENEN -- for which he advances strong arguments -- not in fact aimed to answer this question, and consequently do not decide on it. However, as we have seen above, quantum mechanics seems to point to continua and not to contigua. Further, as we saw above, dendritic snow crystals point in the same direction.
With organisms, however, things are a bit different, because they show -- it seems -- actually existing parts (like organs, bones, cells, etc.). Nevertheless organisms, and especially organisms, are considered to be true holons by many authors. We ourselves also consider it very likely that organisms, it is true, have been naturally evolved from inorganic precursors, but that they are nevertheless not derivable from these precursors, which means that during the emergence of organisms from the mentioned inorganic precursors a NOVUM has appeared. And this NOVUM we explained by means of holistic simplification within Implicate Order. So when we accept that an organism is a true holon, it seems to be so in a different way as it is in inorganic entities. And, as has been already found out, it is this way by which an organism is a continuum, that forces us to interpret the continuous nature of all intrinsic beings, be they organic or inorganic, in a special way : Their continuity (i.e. each one of them being a (heterogeneous) continuum) is so (i.e. is what it is), always in virtue of their parts being non-locally connected. But for this we need the Implicate Order : The parts of a holon are non-locally connected via (and in) the Implicate Order ( In an 'ordinary' continuum -- a local continuum -- there are still distances present between given areas or points of it).
In fact the two metaphysical interpretations of the unity status of any given intrinsic being -- absolute continuum [local or non-local] or (tight) contiguum -- left open by natural science (more or less decided for inorganic beings in favor of continua by quantum mechanics, left open for organisms), are, in a sense, both realized :
- Intrinsic beings appear as contigua in the Explicate Order (as is most evident in the case of organisms). However, they must be interpreted as local heterogeneous continua (in the sense of HOENEN).
- Intrinsic beings reside in the Implicate Order as absolute non-local continua.
And because the Explicate Order is inextricably connected with the Implicate Order (by the incessant projections, injections, re-projections and re-injections), we must interpret all intrinsic beings (inorganic and organic) as absolute non-local continua, and thus as holons in the strongest sense. This is because the local continua and contigua have a derived nature.
The fact that these intrinsic beings are, in the Implicate Order at least, non-local continua, can be seen to be expressed in the Explicate Order as follows : Intrinsic beings appear in the Explicate Order ultimately as h e t e r o g e n e o u s l o c a l c o n t i n u a , i.e. continua with a spatial distribution of qualities of the given intrinsic being.This "spatial distribution of qualities of the given intrinsic being" is, first of all a pattern of different qualities across that being. Secondly, it can be a pattern of different intensities of a same quality across that being. The extension of the mentioned qualities is not something of these qualities themselves, but derives from the extension of that given being. So while a quality intrinsically has intensity (i.e. it has intensity per se), it has extension only per accidens. Where a given quality is prominent in a certain area or site of the given intrinsic being, we are persuaded to say that it is the quality of the corresponding part of that intrinsic being. However, if we consider the latter as a holon, then it is the quality of that holon. It is then heterogeneously distributed across that holon with a maximum of intensity at the mentioned site of that holon.
If we interpret, then, also organisms as heterogeneous local continua, then there seems to be an additional problem : In organisms we see movements of parts, which (parts) then seem to be ontologically independent entities all by themselves (instead of just qualities), destroying the continuum. Although many of these movements are real, e.g. the movements of our arms or legs, or the contractions of our heart, other 'movements' need not to be genuine movements at all. The a priori given possibility of heterogeneous (local) continua makes clear that many such 'movements' allow to be interpreted as just a change in the distribution of heterogeneity across the body (i.e. across the local continuum), without any movement at all. This interpretation sounds strange and unfamiliar, but metaphysically it is sound, and has no implications for biology or medicine ( The latter are neutral with respect to both interpretations).
One should realize that sense observation and the result of processing it theoretically, namely natural science, can never discriminate between (1) a given stable being that is a continuum with a heterogeneous and variable distribution of qualities-and-their-intensities of that being , and (2) that same being , equally stable, but now consisting of concrete parts -- which are themselves beings, and which cause that being not to be a continuum anymore, but a contiguum -- representing those qualities (i.e. parts that, with all their features exactly correspond to the mentioned qualities).
This implies that aggregates and totalities (intrinsic beings) cannot be distinguished by their appearances alone. They must be distinguished by the way they are generated. And of course the way of generation leaves its visible trails in the resulting beings.
A continuum is an extensum (i.e. something which is spatially extended) that is intrinsically one. And this means that a true continuum has no actual parts, that constitute it by their addition, because if a continuum as continuum were to have actual parts, these parts would then be points. But points can never lead to a continuum (which is an extensum) by addition. So a true continuum has, as continuum, no actual parts, it is not divided. It has, however, virtual parts, because it is divisible.
Of course there do exist also processes that directly and mechanistically lead to certain patterns, which are then patterns that are wholly reducible to initially present elements, and thus are not holons. Indeed they figure in our scheme of two main types of ontologically independent beings : aggregates and totalities.
True aggregates come into being by a purely mechanical process, which here means that the entities that now figure as elements of that aggregate are actually existing (as beings themselves) in that aggregate. They are in every respect separated from each other ( They at most touch each other, i.e. are immediately adjacent to each other). That these parts are beings themselves is, however, just a metaphysical interpretation. It means that every part is ontologically independent (as contrasted with accidents, say, qualities, which are ontologically dependent on a carrier, which ultimately is that being of which they are qualities). And we consider this to be so when the causes of the aggregate are extrinsic with respect to this aggregate. And of course in the process of aggregation the initial elements could be transformed by their interaction resulting in a collection of holons, while these themselves are driven together, resulting in the aggregate.
( End of insertion from Fourth Part of Website)
In the above insert we see that in the question of the metaphysical interpretation of material continua, organisms force us to involve the Implicate Order as actually realizing true material continua (inorganic as well as organic), by non-locally connecting their material parts. The connection is indeed non-local, because the Implicate Order is supposed to lack all extensionality (it is folded up), while in the Explicate Order the connection appears as a local connection. So through the mediation of the Implicate Order material entities (inorganisms and organisms) can be true continua, despite the fact that the Implicate Order itself is not a continuum because it lacks extensionality. So when we speak of a continuum (homogeneous or heterogeneous) we always mean a material continuum. But, as will now be clear, the Explicate Order alone cannot account for true continua, it needs the mediation of the Implicate Order.
Impenetrability and penetrability
Meaning of "impenetrability"
The extensa, discovered in Nature in elementary experience, the (physical) bodies, do have a property which, apparently, is closely connected with extensionality : they are impenetrable to one another. To the modern philosopher, more so than in earlier days, it is very important to further enquire into this property. For, more detailed than elementary experience and its theoretical interpretation have led to the question whether this property does indeed necessarily connect with extensionality, as often was held earlier, that is, whether it indeed is so general a property as it seemed to be according to simple experience. For his theory of electrons Lorentz constructed the "aether of Lorentz", a real extensum, an imponderable material (i.e. a material without mass), which is penetrable to ordinary ponderable bodies (i.e. into which ponderable bodies can penetrate). We will often meet this aether later on. But, see also Fourth Part of Website , part XXIX (sequel 5), Section The Ether of Lorentz.
There are also other facts in modern  physics that raise doubts as to the absolute impenetrability of ponderable materials to each other.
With "penetrability" is not, of course, meant the property that a body -- a knife -- can divide another body, i.e. can penetrate between its parts. Also not meant is that a body, as is the case in a sponge, can be porous, and allowing other materials to pass through the pores. It is meant that two extensa, that are and remain perfect continua, can coincide with each other, part with part, point with point, which we might call : these continua touching each other intrinsically along three dimensions. And this precisely is what one wants to exclude if one speaks of bodies to be impenetrable.
Impenetrability is a datum of experience.
There are properties, resulting from extensionality, of which we see the necessary, and thus general, "resulting" a priori. See here one that will be very useful later on : It is immediately evident that extensionality immediately implies that two bodies can touch each other externally. Two cubes with two of their sides. This is a trivial truth. We have immediate intellective insight into this. It is no proposition that merely states some fact of experience.
Not so with impenetrability. Rather its opposite will appear to be true, and indeed turn out to be true. This property is a pure datum of experience. In this, two such different minds as St Thomas Aquinas and Newton agree. The first says : "Sensible experience teaches us that if a body arrives at a place, then another will be expelled from it. Therefore, it is qua experience evident that two such bodies cannot reside at the same place". The second says : " That all bodies are impenetrable is not revealed by reason, but by sensible experience. The bodies with which we are dealing turn out to be impenetrable, and on the basis of this we conclude that impenetrability is a property of all bodies."
But with solely the simple fact of experience we cannot be satisfied. We must try to find the cause, to understand the "how" of it, like it has always been attempted. The fact that impenetrability is closely connected with extensionality is immediately clear and is presupposed in all explanatory attempts. But what prcesely is that connection? This is a question put foreward by many philosophers. But there are only a few of them who have arrived at a clarity of conception. And this holds for philosophers of all schools.
Extensionality is necessary as a condition.
That impenetrability is connected with extensionality is clear, namely in this sense that the latter is a necesary condition to the former. Whether it is a sufficient condition is not so clear.
Extension is a necessary condition. After all, physical "impenetrability" only makes sense in a world of extensa. It then expresses the fact that two extensa are necessarily "outside one another", and [being physically] "outside" [something] presuposes an extended [extensional] world. " Impenetrability" accordingly means that two extensa cannot approach each other any further than up to immediate (external) contact. Movement of two extensa toward each other necessarily stops when they touch each other, necessarily so, as a result of impenetrability. Also its opposite, physical "penetrability", necessarily presupposes extensionality of that which is penetrable. So it is clear that extensionality is a necessary condition for something to be impenetrable or penetrable.
Extensionality also sufficient?
But is extensionality alone also sufficient for impenetrability? Here is the big problem. Aristotle thought it was. But he seemed not to have investigated the problem any further. As did Newton, he appears to have just generalized elementary experience. And so he accounted for impenetrability, also as such, in terms of the property commonly possessed by all physical bodies, extensionality, and only extensionality. For him the penetrable aether of Lorentz would be an impossibility. St Thomas first proves that extensionality is a necessary condition, but holds that extensionality, as a mathematical datum, is not sufficient, but in addition demands influence of a physical principle. Something qualitative, residing in the extensum, in order to provide sufficient ground for impenetrability, he cannot, however, indicate. Therefore he decides physical matter, as the root of extensionality, to be the ground for impenetrability. Yet he, at some other place in his writings, seems to demand a certain amount of activity in order to account for impenetrability. It is certain that he does not assume absolute impenetrability. Also by him, the whole problem was not further investigated, and no clarity was reached. We cannot accept his incomplete solution. For complete clarity sharply defined mechanical conceps are, apparently, necessary, or at least very useful.
But if activity is needed to understand impenetrability, activity working outwardly, exercising influence on other extensa, then impenetrability cannot be understood on the basis of extensionality alone. Extensionality as such is not active.
There have been philosophers who viewed extensionality as a result of active principles or factors, that is, of forces, and therefore thought to find in extensionality a sufficient ground for impenetrability [namely as a result of these forces]. So Kant did. But this attempt cannot lead to success, because it supposes force to be prior with respect to extensionality.
Kant assumes that in every physical body there is an expansive force dispersing its parts. Were this force the only one, the body would inflate indefinitely resulting in the disappearance of all matter. Therefore also an attracting force is postulated, and, from the equilibrium that may result, if one attributes a certain dependence on distance to both of these forces, the finitude of extension would be explained. So extensionality is explained as a resultant of a play of forces. The same expansive force expresses itself as repulsive force with respect to other bodies : two bodies can never penetrate into each other. The aether of Lorentz would be anathema to Kant.
The impossibility of this construction is evident, when one notes that the forces, if they are to produce their effect, must seize upon parts of a given extensum. A single point may be seized by these forces and being moved, but not such that its parts (if it has any) are being dispersed into different directions. Here, also the concept of force (whether attractive or repulsive), the magnitude of which depends on distances of parts, loses its sense. All this, it is true, would make sense in the case of a very small extensum, but in the case of a single point it is totally without meaning. And so this explanation of extensionality boils down this : In order to understand an extensum, there must be forces that scatter the extensum as to its various parts into different directions, respectively holding it together. To explain the extensum, extensionality must already be presupposed. It is clear that we here end up in an infinite regress, something absurd. Extensionality cannot be viewed as a resultant of forces. It is something primary, that necessarily precedes forces. So if, accordingly, the impenetrability of two bodies presupposes a mutual activity, it cannot be identified with extensionality. Surely not with forces from which extensionality itself supposedly originates. So after [i.e. posterior to] extensionality a force must be assumed, meaning that extensionality alone is not sufficient to explain impenetrability.
Remark concerning the electron.
The latter consideration has an application to a problem present for some physicists regarding the electron. The electron of Lorentz, carrier of the elementary negative electrical charge, is sometimes considered as something that is in itself ununderstandable. It has extensionality and a, even changeable, shape. Now charges of the same sign are repulsive to one another, and this force can, in the case of short distances, take on high values. Why, one asks, doesn't the electron explode as a result of the repulsive forces of its parts? Here one considers the electron precisely as an extensum of Kant, but then without its attractive forces, of which no trace is present. It would suffer from the impossibility of Kant's extensum [because, apparently attractive forces, demanded by Kant, are evidently not needed to stabilize the electron]. But these problems can only originate if one takes the continuum, here the electron, to be an agglomerate of its parts, i.e. to be an "assemblage". Were this the case, yes then the repulsive forces [inside the elecron] would be there, scattering the electron when no hypothetical attractive forces are introduced to prevent this. And also this play of forces turned out for us to be impossible, if one does not introduce an extensum preceding the play of these forces. But also the supposition that a continuum is an "assemblage" of its parts was already absurd to us. It would lead to the impossible construction of a continuum out of point-like contigua. One clearly sees the connection of both impossible suppositions [no actual parts of the electron, because these would repulse one another, - a continuum cannot consist exclusively of points, these are mere boundaries of potential non-pointlike parts].
The extensum, being a continuum, is a given intrinsic unity, it is only potentially many. Also because of this its extensionality does not result from forces [because these would result in an actual multitude]. It is prior to forces. If then the electron is -- as it is for Lorentz -- a continuum, then it is a given unity, a totality, and not an aggregate. And then also its charge is a given extensional unity. Then there are no repulsive forces between its parts [because there are no parts]. Then the problem vanishes. Outwardly the electron exercises a repulsive Coulomb force onto its fellow electrons. That also its parts, which are just potentialities in a totality, must do this to each other, is a supposition which only can be supported by a false view of the continuum. And so it will also be as regards "impenetrability" (next section).
Active nature of impenetrability.
The fact that proper impenetrability, i.e. the property that two bodies -- (not the parts of one and the same body [where "body" is here taken to be a substance in the metaphysical sense, and in which, therefore, its parts are not concrete material parts, but regions of the substance with different qualities] ) -- must remain "outside" each other, presupposes activity, certainly follows from experience from which we have learnt this property.
A given body flies in and meets a second body. If they were penetrable to each other, the first would happily continue its path through the second. But it doesn't do so, as a result of impenetrability. The second body undergoes acceleration, while the first undergoes deceleration. Therefore it, and also the first body, undergoes a force. The primary cause is the impenetrability, so this must bring with it activity. The impenetrability does not need to be itself the repulsing force. One may apply the classical view of "collision" : first there is deformation in the colliding bodies, this generates elastic forces, and these act upon -- again via impenetrability -- the other body. The impenetrability must be active, first in generating deformation, then by transmission of the influence of the elastic forces.
A second example. On a table lie several objects, being attracted by the earth's gravity. If the table were penetrable, they would fall with accelerated motion. This is prevented by the impenetrability, they stay where they are. Then, according to mechanics, in addition to the force of gravity, yet a second force must strike the bodies, equal to the weight but in the opposite direction. Again impenetrability expresses itself as something active. In what way all this takes place, is again clear in general outline only : The first effect is, again, the deformation of the table. The resulting elastic forces make -- again through impenetrability -- equilibrium with the gravitational force. The impenetrability surely is -- insofar as it should not be identified with the forces acting upon the other body (see further down) -- an effective cause of the deformation and of the transmission of the forces.
"Impenetrability" does not need to be of the same nature in all cases. When for example we are dealing with electrons, the repulsive forces are of an electrical nature, which (forces), being transmissed by the electric field, already act at a distance [i.e. without a collision]. If we, on the other hand, are dealing with neutrons, then we must have to do with true elastic forces and all that is entailed by them.
All this deserves a deeper analysis, still not yet having, as far as our (HOENEN's) knowledge goes, been carried out. But the above considerations are sufficient to understand that "impenetrability" presupposes activity. And this could have been foreseen : "impenetrability" has an effect upon the position and motion of other bodies. And this is impossible without effective cause. This was already noted by the acute Duns Scotus [end 13th, beginning 14th century].
Now, extensionality (as a property of something) is in itself not active. So extensionality surely is, as we saw, a necessary condition of impenetrability (as it is also of penetrability), but it is not sufficient. In addition to extensionality there must be something else, something active, being responsible for "impenetrablility".
Possibility of penetrable continua.
From this we can draw important conclusions. If something already having extensionality demands yet some other feature, an activity, in order to render that something to be impenetrable, then surely the possibility of extensionality [i.e. of a thing having extensionality] lacking this other feature, is automatically given. So if more sophisticated experience demands [in some given case] penetrability, then philosophically there can be no objection.
Let us go in to it still deeper. Penetrability and impenetrability are symmetrically-relative concepts. Always we must add : of two bodies with respect to each other. One cannot simply speak of bodies that are penetrable and bodies that are impenetrable. Two bodies that are impenetrable to each other, both may be penetrable to a third. The latter then is also penetrable to the first two bodies, but not necessarily to a fourth body of its own species. And so it is with the aether of Lorentz. We cannot contrast it with the ponderable bodies [bodies having mass] as a penetrable extensum against impenetrable ones. The aether of Lorentz is penetrable to ponderable matter, but ponderable matter is penetrable to it too. Ponderable materials are only impenetrable to each other. And if one would posit two "aethers" next to each other, then they might be impenetrable to each other too. So if an extensum is not, solely because it is an extensum, impenetrable to everything else, i.e. if for something to have extensionality does not necessarily imply it to be impenetrable to everything else -- and we saw that this is the case : extensionality is not sufficient for impenetrability -- then in this respect no objection can be raised against the [reality of the] aether of Lorentz (which is supposed to be a penetrable extensum]. A precious result.
A second result. If impenetrability is an activity, being a force or being able to generate it, then it is possible that the force has a limit, such that it cannot make equilibrium with just any other force. And so it can be that a continuum, at least in some of its parts -- and we have already acknowledged the possibility of heterogeneous continua -- can only develop impenetrability forces that do not guarantee absolute impenetrability. A small particle with high speed colliding against such a place in the continuum, may pass through it. Where impenetrability results from an electric field this is quite clear. So, fast electrons, alpha-radiation (Helium nuclei), protons, and neutrons penetrate through thin, or also through thick, metal plates. Surely, in classical physics this is explained such that those metals are discontinuous and these particles pass through the pores. That such an explanation is not necessary is evident from our consideration. So if one would have reason, taken from elsewhere, to assume a continuity of these metal plates [and thus without pores] -- and then of course constituting a heterogeneous continuum -- then it would not at all oppose penetrability. If the particles do not encounter places in the continuum where there is a strong electric field [effective on flying-through-electrons and alpha particles] or where large masses are concentrated [effective on all kinds of flying-through-particles], the continuity of the metal plates will not be an obstacle. Extensionality alone is not sufficient ground for impenetrability. Activity must be added, and this may be too small. A second precious result.
A possible third result could be the following : Now the hypothesis cannot be ruled out anymore that in a collision always a degree of penetration takes place, a small-depth-penetration of the colliding bodies. In that case no deformation to take place is needed as a condition to generate [repulsive] forces. The penetration itself could -- in virtue of the activity of the bodies -- generate these forces, and the latter could initially be proportional to that penetration. Then proper impenetrability could simply consist of these forces. Above we'd said that all this deserves deeper analysis. Of this the last consideration might become an element.
With all this we have concluded our exposition about Extensionality. In the next document we continue our exposition (still largely following HOENEN, 1947) of the organic world's inorganic context as seen by the Aristotelian-Thomistic metaphysics as the latter is integrated into our Explicate / Implicate Order ontology. In that document we will, to begin with, discuss Place and Space.
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