In calculus, infinitesimal quantities are an idealization, and the concept of limit, provided to support the results obtained, is a rationalization. This dynamics going from idealization to rationalization is inherent to the liberal-materialism or material liberalism of modern science. Idealization is necessary for conquest and expansion; rationalization, to colonize and consolidate all that conquered. The first reduces in the name of the subject, which is always more than any object x, and the second reduces in the name of the object, which becomes nothing more than x.
But going to the extremes does not grant at all that we have captured what is in between, which in the case of calculus is the constant differential 1. To perceive what does not change in the midst of change, that is the great merit of Mathis’ argument; that argument recognizes at the core of the concept of function that which is beyond functionalism, since physics has assumed to such an extent that it is based on the analysis of change, that it does not even seem to consider what this refers to.
Think about the problem of knowing where to run to catch fly balls—evaluating a three-dimensional parabola in real time. It is an ordinary skill that even recreational baseball players perform without knowing how they do it, but its imitation by machines triggers the whole usual arsenal of calculus, representations, and algorithms. However, McBeath et al. more than convincingly demonstrated in 1995 that what outfielders do is to move in such a way that the ball remains in a constant visual relation —at a constant relative angle of motion- instead of making complicated time estimates of acceleration as the heuristic model based on calculus intended . Can there be any doubt about this? If the runner makes the correct move, it is precisely because he does not even consider anything like the graph of a parabola. Mathis’ method is equivalent to put this in numbers.
How much can we synthesize knowledge? If there is no way to prove that an algorithm is the shorter or the faster, there can be no explicit limit to its comprehensibility either, and the same applies to all formal knowledge. However, there is an informal but effective guide both to synthesize knowledge and to improve its quality: always pay attention to the invariable mean, to what does not change in the midst of change. And although it is an informal principle, it is always possible to recognize its lineaments: the example above is clear enough both in the real world and in formal analysis.
In fact, it should be said that it gives us an incomparably straighter and simpler path than the «alternative models», in this case the established methods of standard calculus and the analysis of algorithms for tasks in artificial intelligence. Over time we can find an infinite number of instances with a potential for convergence at least as great as the potential for divergence of the usual analysis in terms of change.
Examples of idealizations are the inertia principle, infinitesimal quantities, point particles and the existence of single individual systems in quantum mechanics, reversibility at the fundamental level, the global synchronizer in both Newtonian and relativistic physics, universal physical constants with dimensions that are isolated from the environment, or space as a differentiable manifold. These are force-ideas born out of an intimate need of a subject and a culture, expressing the will to expand till the maximum extent. This idealization required an intense faith in the program that today, when even the rationalization phase seems unnecessary, we can only underestimate.
It is for the present moment to see through idealizations and rationalizations, not to try to move beyond them, since they are already the expression of extremes and of an extreme exercise of abstraction and experimentation. That being reified is only a thought, but the thinker is a thought as well, and the real thinking process, the Logos pervading the world, can only reverberate in a distorted way between both.
The idea of an invariable mean and the idea of dynamic equilibrium, principles so far removed from the scene by long overdue idealizations, are the two sides of the coin and define a new spectrum of relations between formal and non-formal knowledge, between duality and non-duality.
To think that today we are closer to a «theory of everything» in physics than we were in Newton’s time is just a mirage. We could increase the amount of knowledge by a trillion without getting any closer to any ultimate truth. In practice, exhaustion comes much sooner than true knowledge; but just as in aging, claudication results from the inability to eliminate and renew oneself.
Nor are we any closer to «unraveling the mystery of consciousness» than in the time of Leibniz and Newton. In fact we are closer to that time than to «the final solution»… among other things, and to begin with, for failing to recognize things like why the calculus works. We haven’t even suspected that both things may be connected.
Formal knowledge can increase indefinitely without ever reaching what we are experiencing right now, and the advance in informal knowledge only takes place by the direct recognition and appreciation of something always present but unnoticed. Just as power and knowledge limit each other, so do formal and formless knowledge, but there can be no law specifying their relationship.
So there is no «transcendental horizon» of gradual approach to truth for accumulated social knowledge, but neither for knowledge prior to formalization. For the same reason, in formalized knowledge the possibility exists of «transcending» forms, at least in the sense of leaving behind idealizations and rationalizations.
Much more could be learned today by gently modulating the states of particles than crashing them in accelerators, just as more could be learned by developing a proper theory of the extended particle than speculating about the origin of the universe expanding from another point. But to do this we would have to remove the extreme weights of our bets, that is, the unnecessary restrictions of our present theories. To say that in present day science «theory is superseded by correlation» would be the clearest example of extreme propaganda for the prevailing theory and practice. In fact, if many laboratory experiments do not have a greater significance today, it is because a standard interpretation of facts is immediately enforced even if the theory has nothing to do with it.
It is known how in 1956 Bohr and von Neumann came to Columbia to tell Charles Townes that his idea of a laser, which required the perfect phase alignment of a great number of light waves, was impossible because it violated Heisenberg’s inviolable uncertainty principle. The rest is history, but Bohr’s and von Neumann’s words are not recorded. This is no exception, but what happens constantly and routinely. It is clear that a theory that subtracts infinity from infinity every time is needed can predict anything, especially after the events. They call it «powerful methods».
So, there is no observation that quantum electrodynamics can not rationalize. And the same goes for the two theories of relativity, with statistical mechanics, classical mechanics or calculus. This is what means to speak of «powerful methods», what puts the search for truth in a desperate situation. Ptolemy’s epicycles were also undoubtedly a powerful method, as they could cope with all kinds of celestial observations with an unbeatable predictive power for the time.
The problem with formalized knowledge is that once we accept a standard and push all sorts of things in, it is terribly difficult to get out of it. In the case of relativity, a certain reform of classical mechanics was accepted because, in addition to the urgency of solving blatant contradictions, the unification with electrodynamics promised an even greater expansion. Another issue is that since Gauss and Weber everything could have been done differently, sacrificing other things and obtaining different advantages in return.
Even if the amount of accumulated knowledge does not increase one iota, the formalization of knowledge goes through mathematics, and mathematics is always capable of expressing any concept and relation of concepts in a new and even unrecognizable way. We have already seen this: the same phenomena can be described by saying that the movement of bodies is determined by external forces, as by saying that the bodies themselves determine their movement. If this does not surprise us, nothing will do.
It would be much more attentive to reality to try to recombine to some extent our ideas than to try to recombine the whole world to its smallest parts only for the sake of manipulating them. It would also be more attentive to the quality of our knowledge, too. In formal knowledge anything can become anything else in the end with due transformations, but here we are not talking about arbitrary transformations, but about historically possible transformations, transformations of meaning and sense.
In other words, one cannot change everything at once, far from it. But there are clear lines of action to turn a glove inside out without breaking it, and here we have been talking about such lines. The same quantitative increase in knowledge has nothing to do with improving its quality, and now is rather the symptom of a great imbalance.
Conversely, the improvement of quality is intimately but not expressly related to the balance between formal knowledge, based on change, and the awareness of the invariable and undivided, which does not seek infinity because it knows it has it within. This balance also depends on the harmony between principles, means and ends; on how the circle of interpretation closes on the principle using the straightest means.
Even if mathematics does not care about what reality is, it still depends on the form, which thus becomes a reality on its own. It is through mathematical physics that it has come out of her splendid isolation, but physics has used and abused mathematics to conquer the world rather than to see it. Of course, the opposite is always equally possible: math can use physics to investigate its own relation with reality, in a way totally different from the one used up to now.
Since reality is first and foremost what has no form and what is the support of forms, but mathematics can begin to glimpse another relationship with it through the recognition of unity in the invariance of change. It is in this sense, as an interpreter of physics in its most basic sense, that it can transcend itself and access the transcendental plane.
In the Platonic dialogue Meno, Socrates poses questions to a young slave with no more culture than his knowledge of Greek by drawing a square on the floor and then another. After a skillful interrogation, asking him about the length that the side of the second square must have in order to double the area of the first, and after intermittent phases of stupefaction, he manages to awake in the young man the idea of irrational numbers, the first «great revolution» of ancient mathematics. Socrates prides himself on not having teaching the slave, but of having helped him collect the knowledge just questioning his previous answers.
As always, various readings can be made of this famous pedagogical moment, and Gómez Pin, in a magnificent book whose subtitle is precisely The Knowledge of the Slave, takes the reasoning further until the slave finds out himself the idea underlying infinitesimal calculus . Certainly, for us it seems that one can pass in two hours from irrational and real numbers to the inception of calculus. But why then did it take more than two thousand years since Plato’s time? Many great minds devoted to this problem not only two hours, but a great part of their lives, without approaching the crux of the matter.
The culture and knowledge we treasure as a society serves both to inscribe new things in our minds and to erase them. But high-level knowledge, very specialized wisdom, is a very delicate plant. Mathematicians naturally tend to think that theirs, not that of philosophers, is the true factory of ideas; to this it could be argued that pure mathematicians handle far more rarefied concepts, and that they only have delivered true ideas when they are philosophers to begin with, as in the case of Pythagoras, Descartes, or Leibniz, or at any rate «natural philosophers» and physicists like Newton.
The advanced concepts of mathematics only serve at the cutting edge of instrumentalization. What can a modern-day physicist or mathematician convey even to an educated public about his research? Gómez Pin says that the categorical knowledge inscribed in language is more basic than mathematical knowledge: differences such as quantity and quality, or the category of measure, which is precisely a mediation or synthesis of both.
No doubt there is quite a lot of truth in this; there are ideas more basic than mathematics, which are not exclusive to any discipline, and which channel the drift of mathematical concepts. These ideas are at the basis of a cultural syntheses of an entire epoch or civilization. And finally, there are symbols, which do not even pass through the antinomies of ideas and concepts but can adopt them for mere external convenience. In another time, these symbols were supports of various traditions that sought to transmit a knowledge beyond the reach of forms.
There is a formless knowledge that is truly our common root and soil; the ideas are like the sap that ascends through the trunk of the diverse cultures, and the mathematical concepts, would be rather at the level of the branches, the leaves and the fruits. If fresh sap ceases to ascend, what we have is the autumn of a culture, the season of dry leaves. And instinct is also a part of the implicit knowledge, but shaped and conditioned by the social environment. No one tells us how to catch fly balls, but whether one does it playing baseball or practicing different skills depends on the culture and environment.
The point is that instinct is presented to us as opposed to reason when it is obviously reason that is opposed to instinct, to nature trapped within us. Or perhaps it should be said that it is not even reason, but a series of rationalizations. It always seemed to me that Newton’s explanation of the ellipse contradicted not only reason, nor reason and intuition, but reason, intuition and instinct; and it also seemed to me that if people accepted this sort of explanation, it was certainly not because of instinct or reason, but because of a certain interest. The same could be said of wanting to enclose the universe in the laws of mechanics.
Let us see what reason and instinct tell us about the next question. Following the same logic of his amendment of standard calculus, Miles Mathis states that the true value of the constant π in any situation involving motion is not 3.14… but 4. This is something that he has argued in detail showing, to begin with, that he has read Newton and other classics much more carefully than his critics .
Mathis is simply saying that length is not the same as distance and that to advance around a curve like a circle one has to move in two directions at once. It is just adding up the vectors, not taking the graph literally. Of course pi gives us 3.14… in relation to the diameter in a straight line as if we were measuring it with a string; but the point is that moving around a curve involves simultaneously a velocity and an acceleration.
If anyone is in doubt, he can ask himself whether he will use the same fuel driving in a car for 3.14 kilometres in a straight line than driving around a circle 1 kilometre in diameter. And if the difference is attributed to friction, he may wonder if an object in space with an external impressed speed, which according to the principle of inertia should continue with the same direction and speed indefinitely, can go round and round indefinitely. As far as we know no law of inertia with perpetual motion has been enunciated for circular motion.
And yet that is what Newton’s fundamental lemmas at the beginning of the Principia imply. According to Newton and all celestial mechanics since him, a planet in a perfectly circular orbit could orbit around a central body forever, exactly like a perpetual motion machine. But I think it is clear that friction has nothing to do with this here, since anyone understands immediately that a contribution of force is needed not only to change speed, but also to change direction.
The question, then, is not how it is possible for Mathis to say what he says; the question is how it is possible for Newton and Euler and Lagrange and Laplace and all the others to have been able to accept this for over 300 years without a blink, and how we can continue to accept it without giving it the slightest attention. Mathis himself asks this question repeatedly, and hesitates between the hypothesis that they did not perceive it and the hypothesis that they did perceive it but hid it.
Surely they were aware of it. But we underestimate the ascendancy that over them had certain ideal of science and nature alike, that of a clockwork governed by a clockmaker. Then, simultaneously with the perfectly utilitarian idea of expanding the realm of calculus at any rate, they felt morally justified that this contributed to bringing us closer to an ideal of nature perfectly passive in relation to its creator. This intimate mixture of utilitarianism and disguised theology, in which a separate personal creator and an equally separate cognitive subject melt together, still has a great ascendance even today; but the important thing for the instinct is not to know itself. On the other hand, while eager to play the game, surely they believed they were taking «the shortest path» to their goal —which was also their ideal of efficiency. Besides, no one could imagine that calculus understood as reverse engineering —truth could not escape- was amenable to such kind of mistakes.
Another way to visualize this issue, if necessary, is by means of the curve known as cycloid, drawn by the rim of a wheel rolling in a straight line without slipping. The length of this curve is exactly 4 diameters or 8 radii, and the motion it describes is extraordinarily elusive only if we are thinking of the ordinary circular graph.
It is ironic that the first diligent study of this curve was made by Galileo, the father of the principle of inertia, —a principle so elusive for the time, that it took fifty years to consolidate and only after the contributions of Descartes and Newton-, in order to calculate quadratures. As he made his laborious calculations, he was pushing with his own hands time and again the best possible counterexample to that idea of the rolling ball that had already taken possession of his mind.
As is well known, Galileo opposed Kepler’s elliptical orbits and and continued to adhere the ideal of circular orbits. Huygens made an even more thorough study of the cycloid to improve the accuracy of the pendulum clock, whose very idea was due to Galileo. Later Newton had the matter at hand again while dealing with the celebrated problem of the brachistochrone curve, whose solution is precisely the cycloid.
I don’t think that the subject π = 4 deserves more explanations; and whoever needs more information already knows where to find it. If we want to express it in terms of limits, it can be said that bodies travel in the curves to the limit of the shorter legs of the triangle, instead of the limit of the longer leg. It is called the «Manhattan metric», but it casts by the way a deep shadows on the basis of the vast majority of metrics. However the purely «technical» question pales in comparison with the effect which it should have on our minds to find such holes going back to three or four centuries. This alone should radically change our perception and relationship with science.
What the derivations of Huygens and Newton precisely show, round about the crystallization of calculus, is the delicate pass from talking about proportionality in forces to talking about their equality. As in the case of the definition of the central forces, Newton is extremely cautious and avoid expressly to speak of equal forces; but once calculus paved the way, equality will be taken for granted. With the emergence of the «vanishing quantities» the constant proportions of geometrical figures left the scene forever.
It is possible that a great deal of the mismatch between the calculus and the continuous proportion is due to the portion of motion vanished in the representation of graphs, of which the kinematic circumference is the best example. It has been said that the continuous proportion belongs to the realm of statics, in contrast to the changing world of calculus —but in reality the opposite is true, it is calculus, the tool designed to describe change, that adheres to the static figures. This could have implications for proportionality analysis, a possibility that was uprooted and wiped off the map by the same operation of standard analysis.
Calculus, as it has been understood, really involves a inversion in our natural view of the physical world. As Krishna Vijaya observes, instead of determining geometry from physical considerations, deriving from it the differential equation, since Leibniz and Newton the differential equation is set up first and then one tries to solve it to get the physical answers. Both methods are far from equivalent, but the same belief in the reality of the differentials follows from the procedure adopted. It is still necessary to reverse this method in order to open the eyes and recover the right perspective .
Ideally, description and prediction should be balanced, as should memory and anticipation, with which we constantly and reflectively create our perception of time. Inevitably, if there is a perception of an «unreasonable effectiveness of mathematics» in predicting natural occurrences, it is because a great part of the description has been obscured or concealed. This creates enormous cognitive dissonance in both our perception of nature and our idea of science and knowledge. Modern science is always urged to be more and more «creative» in order to be less and less aware of the nature of its manipulations. But a way exists in which the liberation of Nature and the liberation of the knower coincide.
One can understand Nature without calculus, as one can predict things of Nature without the least understanding of them. Venis’ morphology is an instance of the first, and modern physics is certainly the best possible example of the second; but that does not mean that one kind of knowledge excludes the other; everything depends on how the connections are elaborated and derived. The crux of the issue is that modern technoscience is far more interested in manipulation than in understanding, and to such an extent that understanding becomes inconvenient. In turn this limits the degree of disorder that we humans can create.
Philosophers have repeatedly complained that calculus involves a geometrization of motion, but they have never been able to substantiate their claims. Now that they have it, they can seize the opportunity to delve further into the subject.
The cycloid and the wheel could serve as an indication of a different quadrature than the one Galileo intended; rather the opposite, though still with an important point of contact. It is well known that throughout history, and for very different cultures, the circle and the square were the symbol of Heaven and Earth, the active and the passive, the dynamic and the static.
However, the understanding of what is «dynamic» and what is «static» was definitely reversed in the short time span from Galileo to Descartes. Before this reversal, motion and changes in extension could reflect change, but rather accidentally; and so potentiality and actuality in Aristotle had an incomparably broader meaning than that now attributed in physics to potential and kinetic energy —a pair already defined in terms of motion.
Physics takes off when it starts to define everything in terms of extension and motion, although in a very unclear and indistinct form, one is aware that not all physical reality can be reduced to motion and extension.
«What moves does not change and what changes does not move.» The closer we get to our time, the more important motion becomes, and the further we move away, the more secondary and related to appearances it is considered. But in this continuous transformation there is permanently a double movement; for it is evident that calculus has also frozen motion and not only reduced it to geometric forms but has assumed its static quantitative relations.
This double movement of ascent and descent, of condensation and volatilization is really a natural and spontaneous process that can take place at different levels —individual, collective, physical and non-physical. The history and cycles of different civilizations can also respond to this double pattern, and the same history of science shows intermittent evidence of it.
Rivers of ink have flowed about the Cartesian dualism that separates mind and extension, but much less has been said about the duality inherent in physical quantities, in which an extensive part always coexists with a non-extensive one, being now mediated by motion itself. From this stems pairs such as space and time, force and mass, vector and scalar, intensive and extensive quantities, until the extremely complex measurement units of today.
This duality of physical magnitudes, antinomic or not, is nothing but the resolution in the plane of motion of that first duality; and there is no reason why this endless dialectic should have an end, since it is the very deployment of reason. Only ignoring our role in all this could we believe that the cause of consciousness can be found at some explicit level of causation. The thinking substance is already threaded into the description of any level of physical causation since the times of Galileo and Descartes and even much earlier; so in vain do we wait for the paradoxes of quantum mechanics to solve these enigmas for us.
There is a big difference between an idea, and even a mathematical idea, and the manipulation of mathematical symbols. Profound ideas arise when concepts try to return to interpretation or representation, and when representation pays due tribute to the unrepresented, to the implicit in knowledge. But the current drift of science prevents the full circle of this motion that should always be expectant towards the most basic, the really fundamental.
The world would only be an illusion if it could be reduced to extension and movement, yet science fails to tell us what it can be beyond these attributes. Here is its limit, but it would have to be a fertile limit. And the limit between that which cannot represented and that which is representable passes precisely through this double movement: this is the best exponent of true activity, in Nature and in Spirit alike. We can find exponents of this double movement on many levels, from the electron to the constitutive relations of materials, from our own breathing to the movement of the vortices in the Venis sequence, from the process of individuation to the same self-consciousness.
The swastika offers us a good example of a symbol containing implicit knowledge. It is first of all a sign of the Pole and its action on all things in the world; but it can also be the clearest expression of squaring the circle through motion, of the double movement itself, of the dynamic equilibrium, of Dharma or the reciprocity intrinsic to the Law. Surely there was little concern with the issue of calculus in prehistoric times, and yet we always can actualize and give voice to any mute expression of reality.
Change itself is only the visible side of the transcendental plane, but this change has an infinite number of aspects that escape our limited ideas on motion. In this sense, all modern physics and science in general continue to have a huge Platonic deadweight, and we have gone from metaphysics to mathephysics without hardly noticing it.
The mere fact of thinking that there are fixed physical constants or identical particles is already a total lack of taste when it comes to describing Nature, an eloquent exponent of our incapacity. And yet, in a framework that aspires to «maximum predictive power», these assumptions are indispensable. Can identical electrons exist, as if they had just come out of a factory, independently of our complex web of assumptions, constants, and measurement conditions? And yet within this framework it is nearly impossible that it could be otherwise.
An electron, like ourselves or any other entity, can only be an ephemeral configuration that changes from moment to moment. It is not the individual what matters, but the process of individuation, which constantly connects what appears to us as the part and the whole. This connection is pure activity, balance or fight at present, not a frozen event that creeps through the universe for thirteen billion years.
The analysis seems to dissolve everything, yet the fluidity of the world escapes it completely. How can this be possible? That’s an excellent question that should be answered. Because, currently, it is not even approaching the real flow of things and the great unity of life, but on the contrary is moving further and further away from them.
In fact this is a fatal question for modern science. Because it is clear that the immutable cannot be the object of knowledge, and that if something can be known, it is a matter of change, and nothing else. How is it that analysis, which is the study of change, ignores almost everything about it? Is not it amazing that scientists do not ask themselves more questions about this? But there is not even a proper frame to pose the question.
What can be predicted is an expression of regularity, and in that sense, of law. However, to subordinate everything to prediction one is forced to separate and isolate aspects of the processes to cut out the forefront from the background and the context. But even if the ultimate background could be neutral, the context never is. Any context already flows in a definite direction, but now we have a different direction, because to be guided by predictions alone is like running with blinkers.
Modern science tries to compensate for the void of analysis and prediction with disciplines of a «synthetic» nature such as cosmology and the theory of evolution in an attempt to reconstruct the contexts and directions that analysis has destroyed —but in doing so completely relies on the assumptions established by the analytical criterion. And so the whole of cosmology depends implicitly on the principle of inertia, and the theory of the evolution of life on inertia plus the purely random nature of the processes. We have already seen that the first assumption is unnecessary and contradictory, and the second false.
Since the analytical-predictive part took precedence, and the synthetic part relies on its assumptions, the whole is completely unbalanced. The underlying idea is that we can dissolve everything, and then put it back together again and give it a direction and a narrative. But in this way it is impossible to respect the reality of things.
The «analytical part» already has a direction, and therefore cannot be separated from the synthetic part. In other words, nature already has its own narrative all the time, so it does not need any other one as a supplement.
Venis’ infinite sequence presents us the unity of the analytical and morphological aspects, of balance and direction in Nature. This unity is the real challenge of modern science and of science in any age; the contrast between its perspective and the perspective of modern analysis and synthesis will prove to be supremely instructive.
Heraclitus understood the world better than contemporary scientists; if he already disdained Pythagoras’ knowledge as polymathy, as a mere variety of learning, it is better not to imagine what he would think of our theories and specialities. The problem is not the quantity of knowledge, but its quality; and mathematics can be a master of deception in this regard. You end up having the knowledge you want, but the point is what kind of knowledge you want.
One thing is the spontaneous organization in Nature and another is the recurrent presence in it of a certain mathematical constant. It has yet to be demonstrated that there is any kind of functional link between both, and if there is, nothing can yet be said about its relevance. What admits few doubts is that fundamental physics can be interpreted most directly as a phenomenon of self-organization, through relational mechanics and the retarded potentials; as well as with the due recovery of the «thermomechanical» irreversibility, in fact the only real dynamics.
But theoretical understanding alone is incapable at this point of modifying in the slightest the highly dissolving and destructive drift of current science, so in tune with the rest of the social dynamics —we cannot change ideas without first changing what we do and what we want to do. Today man does not seem to be between Earth and Heaven, but between a Nature in perpetual setback that we only see as a resource pool and machines that materialize the human spirit to exploit both the external nature and our internal nature.
We say «human spirit» because not only our intelligence but also our will is materialized in the machine: and the substantial unity of both has become inconceivable to us precisely because of the growing separation made possible by machines. Yes, machines, like other human creations, are an obvious crystallization of a certain water and fire, a feminine desire and a masculine will: a small abbreviation of nature isolated from the rest that tries to perpetuate its movements at the expense of the environment. We pay dearly for this simulacrum of closure that is not closed at all nor can it be; in fact, perpetual motion and its telos never ceased to exert a fatal spell on us.
Throughout this writing we have been indicating a common axis that runs through man, nature and the machine, and must undoubtedly go beyond them, since we know of them nothing but momentary planes of manifestation. If current technoscience separates in order to recombine at will and unleash the universal confusion of planes, we may well do the opposite: see where different planes coincide in order to survey their vertical, their height and depth.
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