Our scientific models of reality, beginning with the ancients are simply an expression of a mixture of the remarkably narrow possibilities of our own neurological structures with fairly fantastic confabulations, extensions and interpolations and mathematical metphors derived from our direct perceptions.
The word "time" is overloaded with many different but interrelated concepts. Some psychological and physical types of time are parsed and related to each other. A most fundamental concept of time is sought through these physical and psychological understandings, from which the others may arise in a hierarchical tree of concepts. The necessary properties of such a fundamental are discussed, and reference is made to a specific mathematical model which allows these properties to be fulfilled. Necessary digressions to history, physics, logic, mathematics, geometry, linguistics, biology, physiology, chemistry, neurology and psychology are made along the way.
This is a companion to [Classical Geometry & Physics Redux]
In the presumed mists of time past we were as a species born, and in the mists of the future we will continue exist - so we believe. There is in human beings, a sense of timelessness or eternity of existence, that would seem to defy the experience and knowledge of science, if only because the eternity, an infinity, is as unobservable and unknowable as any reciprocally conceived infinitesimal: dividing by zero is not defined, nor is multiplying by "infinity". We have no real memory or experience of physical or personal beginnings, yet memory is an essential requirement for our development of temporal concepts, and the various concepts, mostly all of them, that are dependent on temporal concepts. Our memories are assassins of the past; a process difficult to teach to computers and history.
An assumed infinity in any model of essential existence is a metaphysical rather than physical assumption; it is an assumption for which we have little justification; there are indications from existing physical theory itself that the assumption is logically untenable - not to mention simply physically and scientifically wrong. The essential use of continua [Wikipedia] in physical models is then almost a classical example of religious faith: credo quia absurdum. Continua also seem to stem from illusions created by our own neural systems as Gestaltists showed long ago. Gestalt psychology [Wikipedia] Show a sufficient number of points equally distant from a given point, and it will be conceptualized as a circle, while intellectually and epistemologically it is obvious that it is not a circle, nor is it an n-gon: it is merely a collection of arranged points that are certainly not dimensionless. I feel a bit like an Oscar Wilde, who when offered a "white wine", preferred the yellow wine, merely exposing the rather obvious lingual silliness in a seeming paradox, by implication.
The same thing can be done with almost any geometric figure. Show four distinct points and the modern mind immediately conceives of a quadrilateral, or possibly a tetrahedron extended into three space. Three non collinear points become a triangle, and with two points, a line connecting them pops into the head, though it has not at all been presented. I have wondered whether this is more a consequence of language, culture, the execrable rote teaching of geometry and the use of metaphor than it is a genuine perception or neurological construction; regardless, both effect and leap are quite real.
The very sense of timelessness with which we anthropomorphize our gods, however, is rooted in a deep sense of time against which timelessness and its implied infinities are played out. That deep sense is an extrapolation of normal human experience, and perhaps those short isolated periods where it seems that time has stopped, and within which the infinite can be conjured.
It is presumably the manifold of cyclicities, and "almost cyclicities" of the physical universe that gives rise to the locally relatively stable human view of the universe, and which suggests the concept of local cyclicities played out against a "long time" that is later idealized and enshrined in the modern culturally intuitive, but rather mystical underpinnings of Newtonian mechanics [Wikipedia]. Do we want "cyclicities" or periodicities" here?
Early man seems to have developed both a sense of local cylicities and extrapolated a sense of a linear "long time" on the basis of many solar or lunar cycles. The classical philosophical question of whether physical time is linear or cyclic is mooted with the understanding that there does not exist one single physical time; the concept can be revived only when a specific well defined physical time is chosen. A number of distinct physical times are given below, and a longer list of such physical concepts is contained in [Appendix K] related to another work that we return to below. It might be remarked that the Mayans had a collection of calendrics that were used for different periods or cycles of time. (I would really like to see their specific computations.) For the Mayans, there were different times (cyclicities) for different kinds of events or happenings. They relied on a collection of celestial clocks.
We have become, in a real way, an infestation of the only planet we experience directly, inhabiting the now metaphorical four corners of the earth. Lazily, though possibly having come to grips with our own individual mortalities, we barely notice nature's clocks of the seasons that divide and measure the years and the lunar cycles that once defined the months, cycles of light and darkness that define the days. Species die, in time. Some of us are sensitive to this reality, but few of us seriously contemplate the death of our own species, though all our hominid ancestor species are now extinct. There is this deeply set illusion of our infinite existence that motivates the untenable assumption of an infinitely extended linear time as some magical fourth dimension conforming to our equally illusory pictorial concepts of Euclidean space that even defy its own defining mathematics. [Classical Geometry & Physics Redux]
Will a cousin subspecies arise from the inevitable death of the current deranged "species" that will be smarter, wiser and more human? Perhaps. Maybe this turn in evolution was such a grand mistake that it is utterly lethal; the informal probabilities look pretty good for that ignominious end, even if one only considers the Peter Principle applied to the obvious global condition.
While we can, let us speak of time. In so doing, we speak *in* our deeply constructed psychological time, for, seemingly, we cannot escape time, unless time travel and vacation miraculously become possible for us. For such reason of being immersed in our own constructed time concept, we have had great difficulty understanding this time's nature, i.e., what it is and perhaps more importantly, where it comes from, how it arises. Any thoughtful physicist knows that "why" is not a reasonable scientific question, but that "how" is. A "why" question can be fraught with teleological ambiguities, but it is also a matter of "metacausality" that is outside of the arena of science.
Time appears for us, an inextricable aspect of all that we call "experience": experience is always over a time interval, no matter how short it might be. We do not experience or measure instants of time, and idealizing these preconstructed neuromodels of "instants" into our mathematical models of fundamental physics is not only ultimately futile, but generally misleading and even structurally (epistemologically) erroneous, creating paradoxes, incompatabilities and incomputabilities arising from singularities, where such things as infinitesimals and infinities are physically unreasonable.
A particular kind of time is the time of our experience, often called psychological time, which a priori has multiple contextual meanings. There are many other concepts of time in theoretical physics, that determine many senses of causality, and they, together with psychological times are the subjects of this essay.
I do not, however, want to focus on the numerous sociologically determined or conceived concepts of time that are associated with cultural processes (like language) since they also derive from the psychological, i.e., neurological invariants through which psychological time is subconsciously constructed.
I am interested here in theoretical concepts of "physical time" as construed in Western science, that are related to psychological times, unreasonably and thoughtlessly.
In particular, I am interested in how, deeply rooted, and subconscious constructions of time (and space), which are largely unexamined, intrude themselves into physical theory as givens - and how this is a fundamental error of unexamined metatheory.
Well known among physiologists, psychologists and others is the "Weber-Fechner law" [Wikipedia] (One might better call it an arcane principle.) which holds that perception is generally related in measurements to a logarithm of a stimulus. It is not a law, of course, merely a phenomenological rule of thumb that seems to work; it is not satisfactorally "derived" from fundamental theory or even reasonably understood. Much of science is the attempted expansion of such understandings.
Weber-Fechner gives rise, with further experimentation, to a general "logarithmic hypothesis" of perceived time intervals as a function of age which can be described by saying something like, the passage of 1 year to a 10 year old appears as the passage of n years to a (n x 10) year old.
with 'A' understood as a constant, not necessarily universal. But many very simple experiences tell us that the phenomenon of psychological time is not near so simply put. Given the way nature so frequently provides for both redundancies and specializations, the idea that somehow an animal with time sense should have one nice neatly placed simple clock in its biology seems fairly silly, dimwitted almost.
There is no simple little cuckoo clock ticking away in each of us, any more than there is a little person "the self, or homunculus" sitting inside our heads at some control panel, making all our decisions and controlling our actions, however much that immediate illusion may seem naïvely to be true.
If the logarithmic hypothesis holds for time, the realities of experience should be explained generally by assuming that there are different clocks for different time scales, exactly as there are different mechanisms for memories of different time length. Perceiving a length of time does, after all, depend on memory spanning that length of time, so the idea that clock mechanisms are associated with memory mechanisms should not seem very strange.
Perception of lengths of time relative to some relatively long clock time is also decidedly nonlinear. If we listen to a musical composition, at any point of the listening, the time that has elapsed will seem to be shorter than it really (measured by any objective and local metronomical clock) is.
Most composers of music are aware of this, if only intuitively, and so they will arrange for the climax of the piece to be not at an objective temporal midpoint, but put it rather significantly displaced toward the end of the piece, artistically creating a subjective midpoint for listeners, because what has come before will be psychologically foreshortened or compressed in a nonlinear way.
The perception of pitch is also nonlinear, in a logarithmic way, as the organ of Corti which lines the cochlea of the inner ear is a nonlinear transducer of pressure stimuli to neurological impulses.
There are clocks within clocks, subclocks and superclocks like subsets and supersets, and they are all around us, and within us, and containing us. The rotating galaxy in which we live is a an enormous clock with a very long period. The galaxy contains subclocks that are rotating stars, like our Sun. The solar system of our Sun is a complex clock of rotating planets, which are complicated subclocks of the solar system with physical rotating bodies that are in orbits about the planets. The orbits themselves are complex clocks in their cyclic eccentricities and their orientations relative to both the Sun and the galactic plane.
Even one of these physical bodies is a complex clock with three independent faces since it requires 3 rotational degrees of freedom to specify a rotation relative to any fixed frame of reference in a 3 dimensional space. If the space were n dimensional, it would require (n 2), a binomial coefficient degrees of freedom. This, not coincidentally is the dimension of the Lie group SO(n) of rotations in a real n dimensional Euclidean space, E^n. This is true for any classical, massive, spatially extended particle or solid model in E^n, not involved with spin degrees of freedom.
The suggestion is that every "almost closed" physical system is in fact, a clock in principle. I say "almost closed" because, in truth, there are no truly closed systems, even though their conjuring is part of all theoretical physics, via an "adiabatic assumption" that in the end needs to be trashed. It is after all simply a convenient calculational feint.
A major conflict should arise conceptually for any student of physics who is asked to engage in a willing suspension of disbelief that involves a single particle in the alleged infinite universe having a wave function that extends throughout the universe. That is not a physical assumption: it is something magical out of a theater of mathematics - out of the Theatre of the Absurd, on a par with Newton's universal time. In fact, it depends conceptually on Newton's universal time, which Relativistic physics says quite clearly is impossible, or at very least, undefined. See, e.g. [Friedman 1983] on the matter of spatially separated simultaneities. Special relativity does not provide any definition for such things, and while one can make additional definitions to fill in the gap, any such definition is completely arbitrary - and can never be a relativistically invariant concept.
Different kinds of cells in our bodies have different lifetimes. Hepatic cells have a fairly rapid turnover rate, neurons that build the very center of our beings do not. The purpose of all these various physical systems seems to be, through understanding of how they work, the support of our neural system; the neural system is the core of our being. The neural system has first dibs on all incoming resources, and the body will, through its cooperative biochemistry, gladly sacrifice musculoskelatal calcium to provide for the functioning calcium gates that provide neural conduction and tie everything together with interfaces to the various physical systems (hematologic, cardiopulmonary, lymphatic, hormonal, ....) that we conveniently think of as isolated. Now, there is such an emerging area of study called Psychoneuroimmunology; it is not difficult to figure out the kinds of connections that are being made: neurological interfaces between psychological process and immunological process. "Worrying yourself sick" is not just a metaphor.
There are deep biological clocks that determine our breathing rate, our sleep cycles, hormonal cycles, eating cycles, waste elimination cycles, peristaltic process, even mating cycles, and finally, a grand clock that measures our approach to inevitable death, a clock that is most likely programmed in our genetic material, as one understands in every other thing, living or not. Biological species have a large range of life times and cycles, as do even stars - which even have different kinds of deathlike transformations, as also, seemingly, black holes. Birth - Maturity - Death. As the old saying goes, "soon ripe, soon rotten." An entropic relationship of some sort springs easily to mind.
That deep clock of life and death has not yet been found explicitly, but there are very reasonable understandings to say that it exists. Material existence is merely an entropic blip of an open system, that cannot be thermodynamically separated from its surroundings. Theoretically, we merely pretend that such separations are possible, and that the local thing dies, when, actually, the larger thing merely undergoes a minor transformation.
Even stars and galaxies, also open systems, are born and die, and there are clocks there too that order the progression of the march to death. Birth and death are perfectly normal, natural and ubiquitous, and that it should be also for what we call the universe, should come as no particular surprise. Our universe will also die. Everything that is born, dies. That seems as ineluctable and inexorable as the second law of thermodynamics, as may be derived as a theorem from quantum statistical mechanics. See any good text on the subject. We are, after all, agglutinated star dust. Without the death of stars, we could not exist.
If you take the viewpoint of a photon in the context of relativity theory, its birth is coincident with its death. Contrary to various terrified arguments to the contrary, it is not an illegal operation to take the limit of the velocity approaching c, the supposed invariant and precise (uncertainty 0) speed of light in a vacuum (but, vacua do not really exist, and the speed of light changes in the presence of matter), if that limit exists mathematically in the sense of a removable singularity [Wikipedia].
There are ways to look at c as an emergent parameter rather than a given constant of the universe, as we discuss. As an emergent parameter, a statistical parameter, c will not have some "god given" precise value, but a mean value with a highly peaked statistical distribution, i.e., a relatively small variance. The idea of c being a cosmological constant seems to be bound up with the assumption of both time and space being modeled correctly by continua, plus a few more assumptions depending on those assumptions.
Our physicomathematical modeling is guided by our sensory perceptions under the illusion that what we experience is somehow an exact model of reality. Most of the modeling is based on the sense of vision. Though there is vision, hearing, touch, smell and taste, our most sophisticated and dominant senses are vision and hearing. It is no wonder then that these dominate our fundamental sciences, where we are concerned with measurements, observers and observables that are all rooted in concepts of sight and sound. Little science, indeed little language derives from our senses of touch, smell and taste. These are considered too subjective, while somehow sight and hearing are considered remarkably objective.
For all the marveling we can do about the genius of its intricate design, it is a pitiably inferior sensory organ that just barely serves its purpose. If it has been designed by some intelligence, that intelligence needs to go back to school, or trade in his manifestly deficient intellect for something more clever. There remains, of course, the problem that intellect, mind and even neurological process of a Central Nervous System (CNS) are not "things", but barely cohesive, and incoherent processes, much like the very processes that give rise to the evolutions themselves of the CNS that take place on structural levels descending to the cellular, then to the molecular, and then to the atomic where quantum theory is unavoidable. Descending further into the quantum theory of the minute, we reach particles constructed from particles in theory, discovering the key idea that some particles, at least, are resonances (processes) that behave like particles.
Ontology turns more and more away from notions of "things" as fundamental and toward the notion of "processes" as fundamental. The fundamental kinematical quantum postulate wraps together the impossibly precise point with an equally impossibly precise process (momentum), the precision being within time since the fundamental commutation relation is understood at an "instant of time". Something is not quite right there, and I propose that an erroneous contimuum is being presupposed, and that an illegitimate limit is being taken, if only conceptually, in order to understand the Canonical Commutation Relation (CCR) that happens to lie at the foundations of the theory of Nilpotent Lie algebras. It is a pity that this elegant foundation of quantum theory should prove so physically, logically and mathematically problematical as a foundation for quantum theory. It creates more problems than it solves.
While some of the engineering solutions of mother nature seem to us as ingenious, because we would likely not think of them, e.g., the calcium gate structure of neural conduction, they can also look quite kludgey and absurd, though from an evolutionary viewpoint they do not look quite so ridiculous. Examination of the neurology of retinal rods and cones shows the afferent nerves coming unreasonably from the front end, not exiting behind the retina as any kindergarten child would design them. Hence the necessity for the central hole in the retina that provides us with a not exactly useful (except for amusement at Gestaltist cocktail parties) blind spot. Ooops! Not designed by the brightest light on the Christmas tree; but, at some point, it got locked in biologically, and became a given of an evolutionary branch, and here we are.
On the front of intelligence and intellect, things actually look as chaotic as they do also in any evolutionary process. As a scientist, I have become quite accustomed to being wrong in off the cuff thought better than 99% of the time, and this is rather apparently not an uncommon experience; it is the essential chaos of thought, as it gropes through the personally accessible parts of an ill defined solution space of possibilities attached to an always rather ill defined problem.
As Linus Pauling (1901-1994) [Wikipedia] once quipped smilingly when asked how he had so many good ideas, "The way to have a good a idea is to have *lots* of ideas." Similar words apparently came from Thomas Edison on inventing the light bulb; he discovered a thousand ways how *not* to do it.
The composing of music goes the same way: at any given point in the process, most of the time you are deciding what *not* to write. Our creative thinking is based on selection from a limited chaos according to some certain fitness for a prior intellectual hierarchy.
We have a very narrow band of the EM spectrum to which we are sensitive, and our world, constructed from that narrow spectrum, is equally narrow, as a viper sensing infrared, or a horseshoe crab sensing ultraviolet might tell us.
We have a very narrow band of the pressure wave frequency spectrum in air to which we are sensitive, and our world of sound from this narrow spectrum is equally narrow - as any dog or mouse might tell us. Mice also apparently have a sense of humor, and laugh in high frequency chirps. Douglas Adams (1952-2001) [Wikipedia] may have been right about white mice.
Not only are our senses restricted in their range of perception, and also restricted in their powers of discrimination, they are also filtered and modulated by our neural systems to create a body of neurological signals that can be "comprehended", i.e., turned into conceptual (even if illusory) models. Unfortunately, we mostly take these perceptions and the concepts derived from them as somehow physically and ontologically "real", as if that were some sort of monolithic concept with absolute Platonic validity. Even philosophers and scientists take their self perceived being entirely too seriously as somehow isolated from the rest of physical existence, which, in a sense it is, but then confusing that perception with their physical existence. The cut between those two notions, and the resulting problems, of being, defines what is philosophically call "psychophysical parallelism". This equates the level of physical ontology with the level of psychological ontology. That the two are related is reasonably necessary to assume, but to assume they are in some way equivalent is different. One often gets to such an equivalence by being burdensome about the word "reality", as if there can be only one level and concept of ontology, derived from a necessary perception.
While a physical brain has a given ontological level, a thought as process taking place in the context of that physical brain has a different level of ontology. The difference is not unlike the ontologies of source and field in physics; these are two related kinds of things which behave quite differently, and "appear" quite different, as well as differently.
Perhaps the major epistemological lesson to be learned from the 20th century developments of quantum theory and relativity theory is that physical reality has little to do with our garden variety perceptions and concepts that we attempt to impose on it. Ultimately, we are indeed part of the very system of physical reality that we seek to understand.
Sometimes it can be useful to try to consider things that are beyond our experience. This is especially so in scientific situations because in a very real sense science is an abstraction and formalization of experience that is rooted in our direct experience. Making sense of the extremely abstract and mathematical language in which physics is cast frequently means finding pictures, or organizing relationships, directly abstracted from common experience - even though perhaps only experience common to physicists who routinely think about such things - that describes the mathematics, or that can be described by the mathematics. This translation from abstract mathematics to pictures is both easier and more difficult than might first appear. It can also be dangerously inaccurate.
Nonphysicists will speak of "the second law of thermodynamics" with no idea whatsoever of its various statements and interpretations in various theoretical contexts: in classical thermodynamics it is an axiom (from a "principle"), while in statistical mechanics it is a theorem that also involves and allows fluctuations. In both cases, it is not a "physical law"; the language itself is decieving, and only an intelligent physicist would be able to understand the various parsings and applications of the phrase. The collection of concepts and their interrelationships being obvious to him, it might be tediously time consuming (and it is) to make all of the careful conceptual and mathematical distictions, except to a student of mathematical physics.
Popularization of science has its dangers - especially when reptilian, corporate politicians are ruling what used to be the process of science, only to use paid for purportedly scientific pronouncements (slogans) as political and emotionally charged footballs.
"Global Warming" comes to mind as just another terrorizing slogan; "oil peak" comes as another: there is no scientific basis for the fairytale of the biotic (cretaceous) theory of the origins of petroleum, while there is dated petroleum of precretaceous origin.
There is science, and then there is a descriptive language that describes science that is not science at all; the populus has been taught by the corporate government educational system that there is somehow no difference between them, when there is all the difference in the world, on the levels of linguistic, conceptual and scientific abstractions.
For example, consider what kind of simple mathematics some unknown extraterrestrial species might come up with. In an immediate sense, this is ridiculous to consider, but then physicists should be in the habit of considering ridiculous things seriously. It is ridiculous only because no seemingly directly pertinent empirical evidence might be readily available.
It ceases to be completely ridiculous, however, if one understands the principle of abstraction that leads to the compound question that is outside of the realm of mathematics: what is mathematics all about and what is necessary for its existence and development? If physics is an abstraction of an exactly unapproachable reality, mathematics is a kind of abstraction of all possible kinds of physics that is understandable by us.
It is a systematics of relationships between arbitrarily defined objects. The obvious systematics depends for its obviousness on the overall structure of the beings who define and relate such objects.
I am, of course ignoring what it true (reasonable to assume) and what is false (unreasonable to assume) has become entirely a political matter.
Asked about the nature of mathematics, one mathematician might say that mathematics is "really" about number, and that it is the "science of quantification". Another mathematician might claim that mathematics is "really" about relations between mathematical objects, the mathematical objects being experiential abstractions of physical perceptions. Yet another mathematician might claim that they are both wrong and that mathematics is "really" about scribbling symbols in a formal way and making up rules about how to manipulate them in some sort of game that creates the distinction between correct scribbling from incorrect scribbling while also giving rules to make the corresponding categorization. These are different metamathematical viewpoints.
Mathematical Logic is the rule language of algebra and its correct manipulations of postulates and axioms; Mathematical Logic *is* also an algebra (in the loose sense, and in the technical sense) of strings of symbols. What I have just said is not a mathematical statement, but a metametamathematical statement; neither is metametamathematics vacuous nor meaningless.
You are already assimilated in the regression or talking about talking about ...; resistance is futile: we have difficulty avoiding the obviousness that our mathematics always involves a formal manipulation of strings of symbols, about which one must speak; indeed, that is its very strength and purpose, that of formal calculation by algorithms - of course, within our mathematical models, and even within mathematical models of mathematics. How consoling it is, at first blush, that the language in which we make mathematical models of mathematics is already mathematical - or is it?
Suspicions arise when it is obvious that the "doing" of mathematics in terms of computations is deductive, while the creation of those rules of deduction is inductive: they did not drop from the sky, are not built into our neural systems, any more than our human languages are built into our neural systems - though Chomsky may have thought so. The capacity for language is almost certainly built in, as it is even for insects, but familiarity with various human languages tells that their various grammars are so disparate that for some languages, their grammars have even yet to be figured out; still, there is no unique expression of any given language's grammar.
Each of these afore conjured mathematicians, in his own way, says something true about mathematics, its development and its practice, but also, each is deficient in that he fails to account for the equally valid viewpoints of the others, and even most often excludes them. Even scientists and mathematicians can become, and most often do become stuck in their own metaphors, mistaking those metaphors for reality. We are largely a product of our genetics (80%-95%), and our genetics dictates the structure of our Again, CNS at the core of our physical being, and so also dictates what is possible for us to experience and so purportedly think. Thought is not dependent on language, but its transmission and expression is.
[Recently found, June 12, 2006] For a more personal, very similar and interesting discussion of the nature of mathematics that is more detailed, see Mathematics Itself: On the Origin, Nature, Fabrication of Logic and Mathematics See also, regarding concepts of backwardness and forwardness of time, Backs to the Future regarding the linguistical studies of the Aymara people and their language.
It is always very difficult to say what something "is" without becoming circular, ultimately defining something in terms of itself. This is illustrated nicely, though perhaps not so forcibly as might be appropriate to the current discussion, with the "undefined objects" of standard Euclidean geometry presented as an axiomatic system as it may have been understood in High School, Gymnasium, etc.
By assuming and recognizing these undefined objects, in the case of geometry, points and lines, we break into the circularity of the logic, and so destroy the logical circularity. On first thought, this looks to be devious and artificial because we have simply avoided saying what these undefined objects are; but, the method turns out to be rather more logically clever and subtle than it is artificial. The key to that subtlety is understanding what we mean by understanding what something *is*. There are two basic ways to look at this matter of defined existence.
One way is to think of a "thing" (weither or not a Ding an sich) together with an additional list of properties. The other way is to understand the thing's relationships with other things. The difficulty of the first way only becomes clear when you try to carry the program out, as Bertrand Russell, and Alfred North Whitehead discovered in the limited formal scope of Principia Mathematica.
First, it must be understood that this *is* a formal and linguistic business; it is "symbolic modeling", in desperate need of interpretation, nothing more. It has little to do with an intrinsically unapproachable physical reality, or unapproachable thought.
Formally, you run into either circularity, or an infinite regression of definitions, or some imperfect finite truncation of the list of properties that approximates the thing itself. Often we are stuck with this last possibility in a theoretics of phenomenology as "the best we can do right now"; this is at, or close to an embryonic level of theory that is simple taxonomy.
This is a matter of organizing nouns of a language. Some linguists take the view that somehow verbs are the essential aspects of language; I disagree, and would claim that nouns and their hierarchies are the essential epistemological aspects of language. They are also the most pernicious critters about which there is the most wrangling and arguing, not that verbs are excluded from the arena.
Simply put, nouns are almost always about immediate visual concepts. Verbs involve a sense of time that is a more neurologically complicated and sophisticated (unconscious) construct - or so I understand right now: time arises from changes of visual patterns and concepts that demand some mechanism of memory, together with a concept of memory ("I remember"), derived from the neurological mechanism. [I ignore here the distinctions between various neurochemical mechanisms of different memories from short term to long term. This is not insignificant.]
Biology, e.g., begins with ancient taxonomy based on the "looks like" principle, with no knowledge of any underlying chemistry, and absolutely no knowledge of underlying genetics and molecular evolution. Knowing no such underlying basics, ancient biologists still managed (quite wonderously) to make a reasonable taxonomic scheme that largely works today. A taxonomy based on a "looks like" principle is then not necessarily too primitive to be useful.
The point is that while taxonomy may appear simplistic, it is an essential part of epistemological and theoretical beginnings, and done with thought and knowledge, that it is most often very helpful in developing theory further. But, it is still possible in this way to become locked into a fundamentally erroneous path of development, as seems to have happened with certain life forms.
How is it, in all these millions of millennia, that cockroaches have not developed mathematics? On the other hand, perhaps, in the end they are the "intelligent" ones; certainly, they adapt and are efficient, hence a more perfect lifeform. Value judgments are just that.
A Second way avoids these difficulties, and is indeed more subtle and abstract, but it is the way of mathematics. The points and lines of geometry acquire their defining properties by the postulated relationships among them; that is what they are, and that is all that is needed to define them.
This is a completely nontrivial move: it seems to complete the logical system (except as Gödel has shown us that it may not), but it also cuts the logical system off from any development of physical understanding. The physical science then ceases to be physical science, and becomes an abstract mathematical model of a still ambiguously percieved reality.
What is needed is a common unassailable understanding, and that is exactly what is most often missing among humans. This, despite the common neurological structures that organize what we call perceptions. Apparently, that commonality is not near so great as one might be led to believe. So then, how much different is it to consider the possible thought processes of an extraterrestrial from considering the thought processes of another human being? In cognate form, we should have plenty of experience considering the former in terms of the latter. Psychologists have considered the latter, formally, for about a century and a half, and in that time a good amount of the substructure of the neural system on which this depends has been parsed at least analytically.
This is a little like trying to understand what a radio or television or computer really does by pulling it apart into its smallest constituent pieces. The relationships of the parts are lost, and these have everything to do with the final machine and its functionings. I had this exact experience as a very young boy with a big old radio filled with vacuum tubes, and fingerable and coded resistors, capacitors and inductors of a few types, so this is not at all much of an abstraction to me.
Is it not strange that it has been so such a short period of time that formal psychology has existed?
Actually, the beginning of such considerations can be seen in the writings of Aristotle, e.g., in his "De Anima", and before that, also in the communicated thoughts of Anaxagoras (ca. 500 BCE - 428 BCE) Anaxagoras [Internet Encyclopedia of Philosophy] [Anaxagoras] Anaxagoras [Wikipedia]
Given such early beginnings in considerations of the psychology of how we "think", the long gap in the doing of it seems even stranger.
The idea of psychology is not all that new; it simply took a long time to develop into something solid, i.e., something with a physical basis such as a neurology rather than a speculative "metaphysics" of presupposed existences. [I assume that the translations of Aristotle from Greek into Arabic and then into Latin and subsequently into English have not suffered *too* much; anything close to the original was deliberately denied us a millennium and a half ago.]
There is another very long and slightly scary story about why psychology did not develop, and again, in the end, why it did finally develop and shine briefly from a culture of thought that is now mostly dominated by a culture of manipulation and nonthought, and what might be called the frightening concept of "psychological engineering". Frightening, because there is little real science in back of it, and it is exercised with the grand stupidity that its consequences will be only and exactly those desired. For another time, perhaps.
At any rate, the preceding should make the abstract consideration of the mathematical intelligence of extraterrestrials appear slightly less bizarre. At the same time, it should reinforce understandings of the dependencies of physical and mathematical concepts on both large scale and small scale human neurology.
Our illusory notions of time, by which we are conceptually impeded, depend on the physics of our neurology, which in turn depends on the realities of fundamental physical existence which we seek to understand. The circularity of this unavoidable situation (that we are physical beings in the physical world that we seek to understand) indicates that all of our grand edifice called "theoretical physics" while it may be ultimately and optimally refinable in finite time, is not necessarily unique, and therefore is also not necessarily optimal, i.e., it (we) may not have taken an optimal path in its (our) development, any more than any biological evolution can be claimed to have taken an optimal course.
We do the best we can, because that is all we can do; what we can do does have limitations, and if the metatheoretical concepts of quantum theory and relativity theory have taught us anything, it should be precisely that.
While spatial extensions can be "seen" (and it would seem therefore more easily "understood"), laid out and attacked with rulers, it usually being physically possible to make such human seen lengths stand still long enough to be measured with a ruler, time extension cannot be seen or be made to stand still. It is not directly available to our customarily defined physical "senses", but is instead a matter of sequences of differences by transformations that are remembered.
For those of us intimately involved with these considerations, the discovery of Descartes that the Greek model of axiomatic geometry can be combined with the Arabic models of number through algebra is taken for granted in almost the same way that most people take for granted the arithmetic of decimal numbers; neither are remotely given or obvious, and much of the mathematics and the algorithms depend on the very notation that we use, and that have been invented. Again, try doing what you consider simple arithmetic, to be perfectly risible, try long division, using some extension of Roman numerals as your arithmetic (symbolic) language.
Descartes has given us, indirectly, our standard method for "coordinatizing" space and time by mapping copies of the real numbers onto these geometrical spaces. This Cartesian method finally seeps into Descartes notions of history and ontology itself because it does not contain within itself any notion of its essential limitations. Everything can thus, according to Descartes, be understood in terms of functions of certain primitive variables. We are treated to a delightfully fanciful (and more realistic) extension of the idea in the concept of "psychohistory" as described by Isaac Asimov (1920-1992) [Wikipedia] in his Foundation Trilogy. Asimov makes it clear, however, that the equations of evolution themselves change and evolve, though they refer to a single isolatable, yet open system. His concept is already of a synthetic physics and not merely an analytical physics.
So far, notice that without saying it, I have already assumed a specific kind of time in saying this, namely a time which is immediately present and immediately local, for this is the sort of time that we experience as "psychological time". The sensory experience has a subjective elasticity to it regarding duration (measure). As Einstein is reputed to have quipped about the "relativity" of time, sitting on a hot stove seems like an eternity while being with a pretty girl (substitute boy alternatively for the metaphorical generality) seems only a moment.
We have sayings in English like, "A watched pot never boils", and "Time flies when you're having fun." The second probably has more truth than the first, but they both illustrate not only that psychological, perceived time durations are subjectively flexible, but that we are also aware of that flexibility.
To some degree, clocks can often be reset (Think seriously about actually doing this), or their cycles can be entrained to different cycle lengths, but always, even in saying such a thing, there is this elusive idea that somehow "a time", as we conceive of it, exists in some sort of background, independently of any clock. For some people, melatonin and tryptophan can help reset their "sleep cycle clocks in a situation of jet lag.
This practically unconscious idea beomes a deep and fundamental assumption beneath Newtonian physics, but also beneath both the Special and the General theories of relativity with their assumption of a given four dimensional spacetime manifold of "events" within which "everything takes place", according to those theories as models.
The postulated manifold of relativity theories is also the space of "partially ordered local lightcones" [Zeeman 1967] of all physical "events" of all existence. All of anything is always a dangerous concept, as is the concept of "always", meaning for all times, with some infinity implicit. Let us recall that the spacetime of SR is a "vacuum solution" for the Einstein equations of the general theory of relativity, but again, there is no real vacuum spacetime.
So, say these theoretical models of a physical spacetime. We are also not at all relieved of such deep philosophicomathematical assumptions in standard quantum theory. It is Zeeman who finally proves that the group of automorphisms (topological homeomorphisms) of the special relativistic spacetime (Minkowski space) defined by lightcone ordering necessarily acts *linearly* on Minkowski space. This result is not exactly trivial.
The time concept of quantum mechanics is, in fact, *exactly* the time concept of Newton (1643-1727). Time, in quantum mechnaics (QM), does not even pretend to the magical quantum status of position by appearing as an operator. That is also, exactly, one of the formal aspects of quantum theory that is at odds with relativistic theories, i.e., space and time are not on the same conceptual footing in QM, and advancing to the relativistic Klein-Gordon (K-G) equation is not exactly a help - nor is reinterpreting the K-G equation in terms of second quantization; it is, overall, an unhealthy situation, where, even yet, the only models of a relativistic quantum field theory that can be solved and made sense of are completely trivial.
An important thing to notice is that our psychological sense of time is the fundamental notion from which physicists have derived their various more formal and mathematical ideas of time more suitable to scientific objectivity and to being treated logically and mathematically.
The fundamental aspect of any of the cognitive constructions of psychological times, is that of the ordering of events where we make the outside of time construction, inside of time:
filling in between events like all good Gestaltist subjects to create
the idealistic model of a "continuous" line, the present being
the cut between past and future. We may understand that in
a cosmic sense this line may have a beginning and an end, and so the model
will then be a line segment, rather than a line in the technical sense
of Euclidean geometry.
Again see, Backs to the Future
We cling to this linear model of an ordered continuum rooted in illusions created by automatic and subconscious neurology, while the evidence of physics from our environment, our interactions with it, as well as our own constructed theories suggest rather strongly that:
1. Time is discrete in a fundamental Q sense (linear combinationa of discrete operator eigenvalues); not in the simple sense of classical mathematical discreta, i.e., discretizing an artificial classical parameter divorced from any quantum mechanical operator. A clock must be finite, but it cannot be arbitrarily small. 2. Clocks are the only way we have of defining time, and that such times are necessarily meaninglful only "locally" (within some neighborhood) to the clock. 3. The times told by such finite, local clocks are necessarily circular (toroidal), finite and very specifically not an a priori linear time, which is merely an apparently convenient metaphysical fiction.
The second, (after ordering) and disturbingly variable aspect of psychological times is measure. While we have sought to transfer ordering of psychological time to the abstract time of physics, we seem to need to create a standardization, or universal measure for physical times. It is clearly this that Newton had in mind when he spoke of time "progressing uniformly". A funny part, is that the equations of physics are mostly utterly indifferent to such changes of measure - except in relativity theories, where even there, there is significant "wiggle room".
Early physicists, Galileo and Newton made the first two contributions to formalizing a concept of "physical time". Galileo intuitively understood that one must factor out human subjectivity and assume that some objective physical time exists and is passing in the immediate vicinity of any human experiencing the passage of time, and that a clock could reasonably make up for a human's subjectivity. Clocks, of course, vary among themselves as to their rate, and any given clock can usually have its time telling rate adjusted by some physical adjustment of its physical design characteristics. These would be mechanical clocks; the length of its period, and its precision generally depend on its mass/energy and size; the earliest manufactured clocks were sundials which seem to have been invented in Egypt around 1500 BCE. Their rates were not adjustable, and depended on the speed of revolution of the earth about its axis, and to a much lesser degree on the varying speed and position of the Earth in its orbit.
These ancient clocks were not especially good for very small time intervals. That a sense of such time solar time was actually much earlier is contained in the structure and precise orientations of many earlier architectural works in various parts of the world.
Needless to say, during the hours of darkness sundials were not exactly useful, and other means such as hourglasses filled with sand, marked candles and water clocks using water instead of sand were later developed. Astronomical observations and even standardized oil lamps served a similar purpose.
Also, needless to say, the cyclicity of the seasons provided a natural clock suitable for much longer periods of time (the cycle of a year), imprecisely measured in units of days that became codified by many ancient civilizations in calendrics, the most remarkable being those of the Maya in central America that comprise a calendric system that is more accurate and precise than the Gregorian calendar used now by most of the allegedly civilized world as the standard solar calendar - speaking of erroneous paths of scientific and social evolution.
A solar calendar uses as its reference (as we now understand it) the cyclicity of the earth in its orbit about the sun. Lunar calendars obviously reference the cyclicity of the Moon's orbit about the earth, and may have developed well before the agricultural revolution (ca. 30,000 BCE) where the seasons of the year became critically important. The earlier lunar calendar was experienced, of course, more directly in terms of the grossly observable phases of the moon.
It is no great secret that the human female menstrual cycle is lunar, and there are good anthropological reasons to believe that this measurable connection was understood long before the agricultural revolution. It proves almost always an embarrassment to underestimate the intelligence of long dead hominids as it is equally embarrassing when we underestimate the intelligence of other animals. We are not the exclusive "toolmakers" (an idea already of the slightly sloppy Descartes): chimps and the great apes make tools too. We are not the arbiters of all language: whales have language, and even honeybees have language. We may, however, be the true inventors of stupidity, cruelty and arrogance - in all of which which we seem alone to excel above all other species.
Our expectations of ourselves are high; overall performance is close to nonexistent. It is a very sick species that probably deserves its impending and inexorable death by the thousand cuts of stupidity. The evolutionary process and the Hegelian dialectic are are not tools of engineering, and certainly not tools the most cretinous among us have the ability to wield without utter ruin. Certain politically powerful groups seem to be of a singularly contrary and psychotic opinion.
In any case, these earliest clocks, and indeed all clocks are dependent for their basis on some reliably cyclic physical process; the process may not be absolutely cyclic, but it would have to seem so "for a while", i.e. through many counted cycles. The processes that are used now depend, directly or indirectly, on cycles of molecular vibrations, where it is assumed that, e.g., an ammonia molecule (NH3) is always "the same as" any other ammonia molecule, and so one can have as many clocks, exactly duplicated, as are needed. If there are differences in the behaviors of such clocks, they can presumably be attributed to local physical conditions in which they exist, and conditions which we might somehow discern, but any ammonia molecule clock is assumed as good as any other. For atomic clocks (Cesium) that are even more refined, the mechanism is not so easily visualized, yet they still basically depend on a harmonic oscillator.
While in principle, a clock can be conceptualized from any "persistent" apparently cyclic process, the favorites among physicists are those that are examples, to greater and lesser degree, of what they call "simple harmonic oscillators" (SHOs). [Simple Harmonic Oscillator] For very small excursions, a simple pendulum (if it were frictionless) is approximately describable as an SHO. In a practical pendulum clock, we make up for the existence of friction by providing a spring mechanism that provides a compensatory trickle of energy to the pendulum that matches the average amount of energy lost through friction. That is of course why one winds such pendulum clocks up. Reality, of course, is never quite so simple, and the compensatory force almost always winds up being a nonlinear function of time. Reality is very complicated, while simplistic theory may appear fairly elegant, useful, but unreal, and even surreal.
Practically speaking, time is always measured with respect to some other time, and always will be measured so - unless there is a genuine fundamental fiducial time lurking in Newton's centuries old hypothesis. If there is no fiducial time, then the whole idea of time interval is rather arbitrary, and is merely derived from what might be called human unconscious cognitive constructs. The question is how it comes to be that there is such conceptual uniformity among us, despite the great varieties of experience. A partial answer is the natural theoretical adoption of Newton's model that agrees so well with our more thoughtful perceptions. Has this thoughtfulness gone far enough?
It appears, also, that humans have no unique claim to this kind of neural construct of time since other animals, such as dogs and cats clearly are possessed of a sense of time much as we are, a reality consistent with recent scientific results that the sense of short range times originates in the basal ganglia. The clocks of fundamental life functions serve as the references for clocks of higher order cognition, obviously in a rather free floating relationship. Is it unreasonable that the thigmatropisms of amaoebae require a structural level of clock? What is the reaction time? It is certainly not instantaneous.
We have "biological clocks" that provide circadian rhythms, and even the seemingly lunar female menses cycles. These, apparently molecular clocks are not the immediate and total referents for our psychological time which seems most likely to be a rather complicated and sophisticated unconscious cognitive construct upon which we rely constantly.
Yet, in a hierarchically linked collection of clocks, ultimately our time senses must depend somehow on the functioning of molecular clocks. Molecular clocks must depend on atomic clocks, and atomic clocks must depend on the quantum theory that describes their behavior - and probably on the particle clocks underlying atoms. There must be a bottom to this reductionism; that reduction cannot be based on concepts of "instants" that are points in some continuum.
Somehow, a correct fundamental quantum theory must show us where an ultimate concept of "time" comes from, and at least suggest how its various manifestations arise. The kinematics of standard quantum mechanics, embodied in what is called the Canonical Commutation Relations (CCR), and any quantum theory derived from it, does not and cannot give such an answer or explanation because it automatically assumes the Newtonian model of time that is ultimately derived as an abstraction from human perceptions.
The idea of time then becomes a logically circular one, the circle being broken into only by the ad hoc and otherwise inexplicable descriptive assumptions of Newton.
The general idea of quantum theory, however, gives us a clue by its indications that a Planck Regime of physical existence is quite real, and that therefore quantum theory (which does not acknowledge a priori the necessary realities of the Planck regime) needs an essential correction.
That essential correction must be a formal mathematical structure that does, in its fundamentals, acknowledge the Planck regime as fundamental and primary, and in particular acknowledges the physical existence and significance of the Planck time as an indivisible and fundamental unit of time.
Again, In saying what things "are", we always wind up defining those things in terms of other things, which we presumably "understand". Practically speaking, this works, until we start asking "too many questions", by continuing to ask what the terms of the definitions are.
As before, we have two options in this extended logical game:
1. We admit that there is no end to the game and that the definition extends in infinite regression of definitions. (An infinite linear structure of metalogic) 2. We interrupt the aforementioned infinite regression in one of two ways: i. Place boundaries on the regression, saying essentially, we stop here: there are fundamental things by definition, and we do not know what they are but we accept that what they are will be sufficiently explained by their relationships. This is the standard "axiomatic method" of mathematics. (A finite structure of metalogic) ii. Allow logical circularity, so that the regression of definitions (explanations) close on themselves as a circle rather than an infinitely extended, and unbounded line. This is a more accurate representation of the axiomatic method in mathematics and mathematical logic. [To a musician of mathematical inclination I would suggest thinking about how "well tempering" creates the closed "circle of fifths" - and how closure can be created by different means.] (A circular - and finite - structure of metalogic) The "real" explanation of the full question of tempering involes the structure of the Riemann zeta function.
This is, of course, mathematics, and while physics is cast in a mathematical language, it is a bit different from mathematics, especially in matters of logic and of time, where all the history of physics (and mathematics) up to this time, tells us that there is, indeed, a fundamental quantum of time, and that if we want to make a good fundamental theory (if one even exists) that we must take that fundamental time quantum into account in the structure of our mathematical models. Also quoting from Einstein in FCCR, section I.
More generally, of course, mathematics concerns itself with *provable* hypotheses, while science concerns itself with *disprovable* hypotheses. In this distinction, mathematics is more like theology: the assumptions are largely arbitrary.
Time is, as Einstein also said, "what you measure with a clock." This is one reasonable and practical way of looking at it, but of course, it depends on your knowing what a clock is. The implied concept in back of this is somewhere between the concept of psychological time and the idea of a classical, local (in time and space in the sense of physics) fiduciary time.
Actually, Einstein said a number of interesting things about time, showing how deeply he thought about its essential problem; another is that it is a parameter invented to make the dynamics simple - which as it turns out, gives a delightfully teasing circularity when we ask what dynamics is. In this case, he is rather specifically talking about a formal (it being invented) time concept created by physicists, using as a model the formal ideas of Newton.
Here are a few of the interesting and probing things said by Einstein et al. concerning space, time and related concepts.
"Experiment alone can decide on truth ... But the axiomatic basis of physics cannot be extracted from experiment."
It cannot, of course, because theory is necessarily a matter of guess, intuition and thoughtful induction - not deduction.
"More careful reflection teaches us, however, that the special theory of relativity does not compel us to deny Ether. We may assume the existence of an ether; only we must give up describing a definite state of motion to it, i.e., we must by abstraction take from it the last mechanical characteristic which Lorentz had still left it .... [There] is a weighty argument to be adduced in favor of the ether hypothesis. To deny ether is ultimately to assume that empty space has no physical qualities whatever. The fundamental facts of mechanics do not harmonize with this view. ... According to the general theory of relativity space without ether is unthinkable; for in such space there would not only be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and -clocks), nor therefore any space-time intervals in the physical sense."
"For us believing physicists the distinction between past, present and future is only an illusion, even if a stubborn one."
"... every reference body has its own particular time; unless we are told the reference body to which the statement refers, there is no meaning in a statement of the time of an event."
"Time and space are modes by which we think and not conditions in which we live."
"One can give good reasons why reality cannot as earlier be represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers). This does not seem to be in accordance with a continuum theory and must lead to an attempt to find a purely algebraic theory for the description of reality. But nobody knows how to obtain the basis of such a theory."
"Space-time does not claim existence in its own right, but only as a structural quality of the [gravitational] field".
As long as I seem to be on a roll with pithy and penetrating quotes, here are three more relevant ones from other people:
"Space and time coordinates are just four out of many degrees of freedom we need, to specify a self-consistent theory. What we are going to have [in any future Theory of Everything] is not so much a new view of space and time, but a de-emphasis of space and time", -- Steven Weinberg
"In the theory of gravity, you cannot really separate the structure of space and time from the particles which are associated with the force of gravity [ such as putative gravitons]. The notion of a string is inseparable from the space and time in which it moves.". -- Michael Greene
"Space is a form abstracted from matter and exists only in consciousness". -- Ikhwan al-Sufa, Arabic physicist, ca. 900 CE
That notions of space and time as continua, and the notion of causality are basic and primitive subconscious constructs of mind, and that they should be viewed with considerable suspicion is nothing new. The view of physical theory, however, has been historically to take these constructions as essences of the physical reality in which we are embedded, and make formal mathematical models of them assuming their intrinsic ontology extended to this essentially unapproachable physical reality which we can only model. The concepts have been modified in relativity theory, but in ways that smack only of Ptolemaic epicycles. The essential truth of their illusory nature has yet to be truly grappled with, though the complaints and understandings regarding difficulties of the situation are both numerous and ancient.
Newton said that time (whatever it is) progressed at a uniform rate at every point of space. This is axiomatic for his Principia. But truly, it is not so much axiomatic as it is merely descriptive, and as axiom it is a bit embarrassing since it gives no reference for the assertion of uniformity. Again, Newton had little choice in the matter, and all things considered, he made an optimal choice and was honest about his dissatisfaction with the necessity of the choice forced on him.
Notice that there are two senses of "uniformity" here that are necessary to convey the idea of Newtonian time, a distinction, that Newton seems not to have been very careful about. Part of the uniformity of progression is contained in Galileo's idea that depends on the smooth unchanging cyclicity of some ideal and universal clock.
Galileo is reputed to have observed the isochronous nature (period independent of amplitude) in the swaying of chandeliers in a cathedral. Exactly how he did this (legend says using his own pulse), and how accurate his method could have been escapes me, and history - not to mention more than a few historians.
In Newtonia, one must assume that such a "universal ideal clock" can exist, at least in principle; this is dubious in the sense that it is what we think, Newtonianly, it is. (Is that too obtuse?) On the other hand, if I might claim that all physical systems really *are* clocks telling their own time, then with the universe itself as a system, there must also be an associated sense of clock. If so, a reducible system as the universe has a clock which is somehow determined by the collection of clocks which it contains, otherwise note that this ideal reference clock is necessarily (Newton) outside of space in the sense that it is independent of space altogether, and so is applicable uniformly over the preexisting Euclidean space of Newtonian dynamics; but, there is yet the second sense of uniformity to be considered.
That ideal Newtonian and extra universal clock is naturally to be considered to be a "classical" clock, which is to say a mechanical system which can be accurately described by classical (Newtonian) mechanics; either that, or it is pure metaphysical magic, a notion to which Newton himself may not have been averse, alchemist that he was, in fact. Suggestions of logical circularity arise again since Newtonian time is defined by a system which obeys the laws of Newtonian mechanics which necessarily involves the time which is being defined. There is a fine distinction between such circularity and the matter of consistency of the definition with the axiomatic system upon which the definition depends.
In the context of Newtonian mechanics, however, the distinction between the local Galilean concept of time and the (really unnecessary) Newtonian concept of time remains. But, the distinction really only appears in the alchemical context of "As above, so below", only in the operational converse: "As below, so above", the "universal law of alchemical correspondence" which is quite obviously Newton's progression of alchemical thought in going from the laws of earthly mechanics to the gravitational theory between celestial bodies, with which, "owing to some merciful providence, we have not yet" been able to play experimental games in this area - paraphrasing Hermann Weyl in matters of exhorbitant energies implied by "E = mc²" before fission bombs. [Weyl 1950]
[See however, NASA's recent attack on the comet Temple I - on July 4 no less - how jerkily quaint. Next, they will be blowing up Jupiter, just to see what happens. It is called "government science"; think of MkUltra, its methods and purposes, the HAARP project, etc. - a governmental world gone mad in its fundamental stupidity.]
We may now see even many more of these concordances of above and below in the understanding that both the photon (mediator of the electromagnetic force) and the putative graviton (mediator of the gravitational force) are massless, giving a "1/r²" force law for both, in three spatial dimensions, and "1/r(n-1)" in n spatial dimensions. Further connections of that with Laplacian harmonicity (harmonicity of the solutions to Laplace's equation) and holomorphy of functions of complex variables may yet turn out to be more than just curiously mysterious.
Related to this question of the dimension of physical space is the profound flippancy that were physical space not 3 dimensional, we would not be able to tie our shoe laces: knots only exist, nontrivially in three dimensions. Below three dimensions they cannot exist at all, and above three dimensions, they are all trivial "nonknots". So, there really is a sense in which our existence, in some sense, takes place in 3 dimensions, otherwise, the knottings of polymers which give physical proprties would not exist. It is not then an amazing idea that in dimensions other than 3, for our existences, the molecular conformations of our knoted existence would also not be possible. This, of course, does not exclude "existences" in other than 3 dimensions, but it does restrict the dimension of our existence, and what we might be able to perceive on a molecular level.
Physics accepts and attacks questions of "how", attempting to describe the suitable relationships mathematically, i.e., determinately to the degree that seems possible; teleological questions of "why" are not scientifically answerable and are left to the babblings of religionists who have the freedom of absurd concoctions demanded to be real, and various other lethal psychopathologies. Notable in the middle ground is that the principle of least action Maupertuis (1698-1759), a student of Johann Bernoulli (1667-1748), was originally a theological assertion at a time when "God" was being thought of by the Deists [Wikipedia] alternatively as the supreme mathematician or supreme engineer - or perhaps the supreme Platonist of maximal efficiency. Connections with Masonic concepts, dispensing with surrounding abracadabra, are obvious.
In making Newton's implied (below → above) syllogism, it becomes necessary to extend the Galilean concept of a local uniform time to a unified global concept of time that obtains throughout all space, and this is the second source of the assumed uniformity; it might more properly be called "temporal homogeneity", as it is an expression of a homogeneous distribution through points of space - interestingly, this is a homogeneous distribution of *process*, without actually saying so.
A mathematician and differential geometer will understand immediately that this "local to global extension" of the time concept immediately involves what he would call an "affine connection" and that by Newton's standards, that the affine connection is "flat", which is to say that spatial translations of a temporal duration (or clock) leaves that expressed temporal duration invariant.
The technical way of stating the flatness is that the curvature [Wikipedia] tensor derived from an affine connection vanishes identically. The idea of spreading out fiducia is not unique and could also be thought of in terms of Lie derivatives [Wikipedia].
The spatial translations, according to Newton, take place in a Euclidean space which is automatically flat and isotropic (invariance under translations and rotations, respectively - described now with a simple linear velocity addition as "Galilean invariance"). From the isotropy, one deduces the invariance under rotations of the clock which tells the time, and then the invariance of the "Newtonian time" is, mathematically, a constant scalar (a constant function in the sense of tensor analysis) associated with the Euclidean space that models physical space. That means that Newtonian time is axiomatically independent of space, and that the S x T topological product may, as a model, be formed freely, as can a trivial bundle with T as base space and S as "fibre" (mathematical meaning), with which to discuss the dynamics of particles.
Note that when we speak of cosmological time, as in the "age of the universe", yet another, and very different abstraction from psychological time through pictures of certain general relativistic cosmologies has been made. This is a transferred thought and concept from the mundane notions of time, to Newtonian theory, to the thoughtpictures of a general relativist.
This constructed theoretical reference is based in the mathematics of Einstein's General Theory of Relativity, and depends on what is called the "Ricci scalar" that is a scalar function (full contraction) of the spacetime fourth order Riemann-Christoffel curvature tensor, from which the Einstein tensor is constructed. Visually, the idea of the radius of the universe is the radius of an expanding three dimensional hypersphere which is the generally curved physical 3-space in which we live.
This cosmological time is not a global homogeneous time, and it is not the time coordinate of some putative four dimensional continuum, but an additional derived parameter. Instead, it is a measure derived from the "Ricci Scalar" function obtained through two contractions of the Riemann-Christoffel curvature tensor of the postulated four dimensional continuum, where the affine connection is the Christoffel symbol in terms of which the curvature tensor is defined and derived from the symmetric second rank gravimetric tensor that describes the "gravitational potential field", and is also rather specifically attached to a very specific set of solutions to the Einstein equations possessed of spherical spatial symmetry. (Friedman, Robinson, Walker)
Note that there is a mathematical singularity in a 4 dim spacetime for both a black hole and and a "Big Bang" creation story that can be avoided in higher dimensions; hence, an interest in higher dimensional Kaluza-Klein theories [Wikipedia].
An argument for the Ricci scalar's global interpretation is by conceiving of the universe as a vast, but somehow coherent and causally connected "thing" that also with regard to its spatial aspect is possessed of spatial spherical symmetry, so that the Ricci scalar R = R(t) is only a function of the time coordinate, in what is technically called a (Friedmann)-Robertson-Walker Universe. The topological product, or its generalization, "fibre bundle" comes to the fore again.
The relatively subtle second law of thermodynamics [Wikipedia] (which is strictly applicable only to closed systems) tells that the entropy of a closed system does not decrease in time. As Ilya Prigogine [Wikipedia] pointed out many years ago, this conceptually provides a direction to the arrow of time. Something to determine this direction since we observe one: as a good rule of thumb, broken plates do not jump up onto tables and reassemble themselves. The world appears to us to be time asymmetric; we remember the past, not the future, and so have a habit of imputing to the past a physical ontology, but become personally reserved on the matter of future ontology, unless if one accepts the idea "prophecy", a fabrication in denial of physical reality designed to assuage the horror of the unknown future in the minds of mental midgets.
Yet, all the fundamental laws of physics are invariant under time reversal t → -t, and so fail to distinguish past from future. Yes, equations for mechanically dissipative systems do distinguish past from future, but these are then not closed system: energy leaks from them in the work done against the dissipative forces.
This kind of time of reversal is what allowed Feynman to consider a positron to be equivalent to an electron running backward in time, that, and the CPT theorem. [Streater 1964] See also Spin, Statistics, CPT and All That Jazz by John Baez. Or, CPT Theorem, or do your own Internet search.
Thermodynamics [Wikipedia] is a slight misnomer since it is not a dynamics in the sense of Newtonian mechanics that uses a time parameter to talk about time rates of change of states, but talks rather simply of initial and final states. Modern physics tends to understand the origins of thermodynamics through a combination of atomic theory and statistical mechanics applied to it, with thermodynamics emerging from how calculated mean values relate to one another: the pressure in a balloon filled with an ideal gas is the result of the average momentum transfer from the energetic constituent molecules of the gas, and so is a matter of statistics of a physical model.
Entropy is a strange concept that first arose in the context of thermodynamics, and had to to with the irreversibity of thermodaynamic transformations in an ideal Carnot Engine without work being applied. In one sense, we now think of it as a measure of "disorder" (Farts increase entropy: the gaseous expansion is not a free expansion; think of the vast earthly population of flatulent animals!), but we measure it at a level of the disorder of a statistical distrubution, basing that on a validity of thermodynamics emerging from statistical mechanics of simply specified "particles".
In the statistical mechanics due to Boltzmann (1844-1906), entropy is calculated from
where Si is the entropy of the state i, pi is its probability, and k is a physically dimensioned constant. This looks very much like the negative of the Shannon entropy in information theory, and many physicists now accept the Entropy-Information Hypothesis that these two concepts of entropy and information are intimately related so that an increase in entropy is associated with a loss of information of the system. The future is then about local losses of information.
On a cosmological scale, this can prove to be problematic when the cosmological time related to an entropic time. The concept of "local", though understood, is not exactly well defined in the context of the 2d law of thermodynamics.
Please note also at this point that even Aristotle understood that time cannot exist conceptually without physical space, and that similarly space cannot exist without time, by primitive yet valid argument. There is something rather wrong with the Newtonian argument separating time and space into independent concepts, for ultimately, Aristotle was right, even basically in his reasoning, even if you do not believe that the relativistic models are exactly right, or that they are not being correctly interpreted. [Aristotle 1941]
Interestingly, the "time" that Aristotle had in mind, as I read him, was essentially the global time of Newtonian physics: a universal extension of an idealized (i.e., externalized, objectified and formally uniformized) personal time; he was less precise about it than Newton.
If you have gotten this far, I have found a longer, historical explication of the history of physical time by Bradley Dowden: Time [Internet Encyclopedia of Philosophy] I have a few minor quibbles of wording with this, but nothing substantive; it is a brief history of thought (right or wrong) on the subject. It would be worth reading this now. Also, see the article Time [Wikipedia].
One point, however, that should be emphasized about SR that is not made in Bowden and that is that the calculated results of SR depend completely on observations being made with lightlike signals of vacuum velocity c. Apposite to this point is yet another point that "signals" are not the same as "waves", and that this distinction is what informs the difference between Poincaré invariance and the obviously generally covariant nature of the Maxwell equations in 4-tensor form. Highly recommended are the lecture notes by R. M. Kiehn that enlightened me on the details of this. In particular, start there with What are Signals and what are Waves? An important point explained in the lecture notes is that the mathematical language of differential forms is not simply an elegant recasting of tensor calculus, but is more general, and in such a way as to encompass an irreversible dynamics that is not restricted to thermodynamical argument and expression. The subject is then irreversible flows on manifolds [Wikipedia], under the aegis of the Lie derivative; this transcends the notions of Hamiltonian flows [Hamiltonian vector field - Wikipedia] on symplectic manifolds [Wikipedia] as these are all reversible in time flows.
In the standard interpretation of the formal mathematics of SR, the information obtained about anything is obtained through these signals with the specific finite velocity c, and it is assumed that these observations imply physical ontology. This is assumed to be operative, whether or not the Minkowski "block form" of spacetime is assumed.
Up to this point, most has been recorded history and about various kinds of understandings about concepts of time and their relationships.
A few lessons to be learned concerning concepts of time from the history of philosophy, physics, psychology and neurobiology:
1. The fundamental human concept of time is derived from a flexible "sense" of time that has unconscious origins as a cognitive construct which depends either on some biological clock for longer periods, or depends on a closely related sense of "simultaneous pattern" that is also a cognitive construct for short periods of time. [The distinction between short and long term memory mechanisms should be remarked on here as being a likely structural parallel.] Regardless of the regime of memory span, our notions, derived from perceptions, of time are very much dependent on the function of memory. Without memory, we could have no concept of past, present and future; that is not to say that memory is sufficient for the existence of a concept of future, but, it is surely necessary, and a biological contraption that exists in lower life forms more than we might imagine in our customary hubris. This does, at least suggest that while a time, as we understand it does not exist fundamentally, that it arises as an emergent phenomenon, and sufficiently so that it is available and perceivable aspect of reality by most all life forms. 2. This neurologically based sense of time is what gave rise, by formalization, to the formal physical notions of time. 3. The formal notions of time within any given physical theory are inequivalent. All these are are distinct from psychological time, and so there exists no one single concept of time. 4. Present (local) time must be distinguished from global time, the usual assumed Newtonian parameter of physics. 5. Psychological time must be distinguished from the molecular biological times told by "biological clocks". 6. Psychological time is a cognitive construct created unconsciously by the cerebrum, and unavailable to the mesencephalon, which probably has its own more primitive temporal function. 7. Since Newton, the notion of physical time has been global, and this may be (undoubtedly, I believe) a great conceptual mistake. 8. Relativity theory specifically exploits the globality of time. At the same time is denies the possibility of meaningful spatially separated simultaneities, as well as distant concepts of "before" and "after". This is made painfully clear in one of the classic texts on Special Relativity [Bergmann 1942]. SR's invariant (proper) time is only a concept "at a point", whether or not you believe GR. [Mathematically, SR is GR "at a point", or as is more usually said "GR reduces to SR, "at a point", and in a tangent space, which is a bit different than "in the manifold".] Contrary to some interpretations of SR, a "proper time" [Internet Encyclopedia of Philosophy] is the only real, most directly definable typus of time: coordinate time is a theoretical fiction with no especial experimental foundation. Again, see [Friedman 1983]. We assume the reality of coordinate time because we have this neurological function called "memory", which we then assume is a perfect model of physical reality - which, of course, it is not. To "parallel transport" time to another point means engaging in the fairly silly ideal notion of adiabatic parallel transport of clocks that many relativists still accept as reasonable "in principle". Though the mathematical model of SR does indeed imply that such a transport is logically possible, there is more to consider than simply the simplistic model of SR, most especially if it is to be understood as being consistent with a quantum theory that understands "clock" aright. 9. Quantum theory obviates the precision with which time can be measured (think energy of pair production and the problem of distinguishing particles by position), and so obviates modeling it by a measurable continuum: it cannot be done, most especially in principle. If you believe QT beyond QM, then you must disbelieve in the measurability of time with arbitrary precision. There are no options, and there is no weaseling sophistry to avoid this. This means that the essential assumption of Newtonian time for standard QM is an illogical and self contradictory hoax. QM, as it stands, is physically and logically unacceptable. 10. Putting together an ensemble of interacting clocks can create, as cooperative phenomena, a hierarchy of larger clocks that dynamically coordinate themselves, but whose clarity of physical existence is less than obvious, not fundamental, and clearly statistical in nature in a way that goes beyond the fundamental statistics of quantum theory. One can think of these Q clocks, prototypically, oscillators as being related either coherently or incoherently by their overall phases. Properly, they should probably also be thought of as open systems, even if only a little open. A major problem here is coming to grips with the elemental concept of "interaction". We think this is not even any real problem only because we have been taught a collection of mathematical incantations that purportedly describe interactions. Knowing the incantations is a, but ignoring utter lack of understanding them is not. The incantations in the context of QM and QFT are particularly noisome. 11. The very language (both human and mathematical) that we use to babble to each other as physicists is intimately related to its specializations, and to the conceptual apparatus in back of it. We do seek to transcend our language with thought, and yet our thoughts, and specifically those that we can communicate, are restricted by the language we use. The current language and the current conceptual set is clearly insufficient to the task. Within human languages, I speak in terms of most European languages that possess tenses, and not of the many Amerind languages that have no tense structure, and where there is no past, present and future. Kolmogorov (1903-1987) was surely right in saying that what one can say depends on the language in which one chooses (or has available) to say it. However, Kolmogorov missed the converse, that where the language does not constrain, the thought can be more free; Kolmogorov was not a Zen Buddhist. 12. Interestingly, the primary idea of "causality" as it is formulated in physics is, in what ever way, merely a partial ordering of events, and is far more generally that any philosophical or "common sense" notion of causality. That latter have the idea of ontological causation, while the causality of physics is instead "the possibility or impossibility of a causal relationship between events". The emergence of this notion of causality appears to be due to Hans Reichenbach around 1924. One speaks of event B being within or without the lightcone of event A, not whether A actually caused B. In this context, we cannot affirm causality (A→B), we can only rule it out if we believe the causality of partial orderings. CORRELATION DOES NOT IMPLY CAUSALITY. Such a general structuring of physical causality is not inconsistent with the fundamental notions of quantum theory, and so one might wonder what the problem is concerning the consistency of QT and SR. One problem is obviously the language itself: the physicist's definition of causality (rooted in the time reversibility of fundamental equations of physics) is something quite different from what is commonly called causality. A physicist's mathematical models of causality is more akin to the "synchronicity" as discussed long ago by Jung and Pauli. Pauli was then already concerned with the "entangled states" that Schrödinger had earlier pointed out, and how their existence messed with simplistic notions of causality. Einstein was, apparently, not conversant in this theoretical subtlety and so we have still the consideration of the EPR paradox, as it is periodically revisited, most significantly perhaps by Bell, et al. [Bell 1975] Many things that should not be lost, are lost. The concept of causality is one like time and space that is constructed upon those concepts in human minds at a subconscious, post neuroperceptual level that humans, because they have not thought to construct causality consciously, automatically assume that is an aspect of physical reality. Yet, ultimate physical reality is already utterly beyond human perception, and accessible only through the ever refined process of scientific modeling. To infer (and it is inference, not deduction) that physical reality must also contain the causality that is a structure of our models is a madness of some significant magnitude that impedes the progress of that intellectual modeling process that we call science. 13. It might be a good thing if SR, which is historically and conceptually still hinged on Maxwellian EMT, were reformulated as a restriction on the propagation of information: to save SR from EM anomalies (Sommerfeld suggested how to do this. Cf. [Stratton 1941]) in propagation velocities, the subtle switches from phase velocity to group velocity to propagation velocity of information in the EM wave is assumed. Although Shannon's definition of information, SUM p log p, is not the only definition of information; there are general problems with defining such things in QM (but, not in FCCR). 14. In SR, time dilation in the direction of motion is different from the time dilation in a direction orthogonal to the direction of motion, so, in a more primitive and fundamental context, it may not be silly to have a concept of time associated with each position operator. 15. There is a big difference between the quest to understand and the quest to control; perhaps, that enormous gulf is being ignored while wisdom would indicate that ignoring it is more than perilous. What is the probability that an infant playing with a scalpel will cut himself or others?
Something that becomes obvious relative to this accumulating understanding of the various multiple concepts of time, and what is required of any fundamental concept is that the Newtonian concept does not, and cannot fill the requirements of a fundamental concept. At very least, in the small, it breaks down and becomes meaningless and inapplicable. In the large, it also breaks down and becomes fundamentally inapplicable. In both cases we are breaking assumed symmetries. That size/number matters in physics should be cleae from the velopments nanophysics.
The question then is really whether there exists a mathematical formulation of quantum theory that diverges not far from what we have known, returns it as a limit or special case, allows for the emergence of relativistic structure and provides all these requirements for a quantum theoretic concept of time which is democratically on the same footing as space, where both space and time are "quantized" in conformity with the logic and understandings of the Planck regime.
The answer to that question is affirmative, as demonstrated in the work referred to in footnotes  and . While the mathematical existence of such a theoretical framework is clear, its physical validity is another matter.
It one takes the derivation of a local Newtonian time in the work cited in footnote  seriously, then it becomes clear that it arises rather quickly at a level beginning with 4 points in what amounts to a finite noncommutative geometry of the class normally called "0 dimensional". That may seem trivial by virtue of language and nomenclature, but it is not.
If one takes the FCCR undepinnings seriously, the quantum theory is written locally and boundedly, and so describes physical reality in terms of open systems with various degrees of "in the manifold" locality measured by a positive integer n > 1, that are necessarily nonunitary, non Hamiltonian and possessed of a stochastic, dynamical evolution. [Jadczyk 1993]
In discussing the concept of a classical relativistic simple harmonic oscillator it is shown that in a relativistic context, the formal Hamiltonian function loses its most important property of having invariant value for conservative systems under dynamical transformations, i.e., under translations in proper time for the observer's frame that is stationary with respect to the oscillator's center of force. For the classical relativistic oscillator, the Hamiltonian function value varies periodically with respect to the displacement parameter, as a fairly complicated modulated cosine function.
Curiously, and interestingly, while the ostensibly nonrelativistic FCCR oscillator energy expectation value, computed at usual, in time operator "eigenstates" is the same for all such eigenstates, the transition amplitudes are cotangents periodic in (Ek - Ej). The transition amplitudes for the energy in q eigenstates is reasonably complicated. See [Theorem 8.3]; the quantities q(n,j) that you see there are the roots, indexed by j, of the n-th Hermite polynomial. Perhaps, looking at Mittag-Leffler expansions of sinusoidal functions, .... I do not know yet.
Finally, if we want to understand the time of a clock associated with a large reducible system, it is composed of the clocks of all of its reducible and irreducible interacting sybsystems, all running incoherently, and at bottom, erratically in and out of phase. With considerations of randomness in the implied statistics, one can think of a kind of Riemann-Lesbegue lemma [Wikipedia] That cancels out the collection of phases in an averaging over an ensemble, giving an illusion of a coherent time belonging to the original reducible system. Time is a many splendored thing.
"Time", as we perceive it through our neurology, has no ontological basis in fact, logic or reason, except as an emergent concept that is the result of putting many fundamental clocks (systems) together. For fundamental physics to enforce this neurologically determined concept on the structure of fundamental physical theory, particularly quantum theory, is a fundamental epistemological error with erroneous consequences, and a block to understanding how it is that this macroscopic variable arises.
In standard classical and quantum mathematical physics, time is always defined as a parameter through which a translation preserves/conserves energy; the idea fails, of course, in dissipative (open) systems. In relativistic physics, the idea becomes kinky, in that space is inextricably mixed with time, and energy is inextricably mixed with momentum. The seemingly "simple" concepts of nonrelativistic physics then become more complicated and murky.
Alas, most fundamental physical theory concerns itself with conservative (nondissipative) systems, which are to a large extent treated with much "hand waving" (sometimes shouting) as to why the dissipation of energy is somehow not important in whatever case is under consideration, when, in fact, it is important, but happens to be only difficult and unruly. Easy problems are more satisfying to solve, probably because they provide the comfortable illusion that you actually understand something about reality. While the process of science seems to be a good and convergent one, the people who are engaged in it seem only rarely to match its efficacy.
A two state clock can only distinguish between 0 and 1. If a second two state clock exists fortuitously structured and coupled to the first, it can capture two cycles of the first clock. If a third clock exists, again fortuitously, it can capture the cycles of the second clock, and so extend the notion of at least a quantum linear time that distinguishes 8 points of "time". The n-th clock in such a sequence enables the distinguishing of 2n points of time, the clocks acting as memory buffers. Is this time linear or circular? Linearization of time from a circular one is one idealization in the limit of infinite clocks; another idealization is the precise continuum of the unit circle as states of the clock, and this never really exists, physically. Truly, all realistic concepts of time are of a discrete circular time. This is physically unavoidable. All legitimately parsed, physically trustworthy temporal concepts are necessarily circular, and always told by physically bounded clocks, i.e., physically bounded systems, which are the only physical systems that can be properly defined as known.
Reality, of course, is always more complicated than we would suppose, or like it to be: there are irreducible clocks with n > 2 states, and a quantum statistical mechanical arena within which to treat them mathematically.
An alternative interpretation, and mode of thinking of the 2n result above is thinking of clocks and state/processes, and how states/processes may be distributed over elementary clocks with two states, as points are distributed over subsets of a given set.
My thanks to Prof. Michael Leyton for prior discussion of the cognitive construction of psychological concepts of our world and how they can be understood in terms of group theory, to Mitch Smith for discussions on Model theory and continua, to Martina Schubert for very helpful comments and suggestions of form, to Prof. R. M. Kiehn for subtle and complex reasons involving the structure of thought that I cannot quite explain, undoubtedly to Elihu Lubkin for conversations of which I cannot even now remember details, and always to Alan Bellamente for more things than merely my continued existence than I can count.
1. This was written as a nonmathematical, and fairly indulgent prequel and background to an earlier, yet ongoing essay On the Quantum Theoretic Origins of Newtonian Time which is largely mathematical in its content. A small amount of mathematical reference has still managed to creep into the material here, even though the particulars have been avoided. 2. Finite Canonical Commutation Relations (FCCR) is the algebraic system which connects CCR with the Canonical Anticommutation Relations (CAR) of quantum theory, has all the properties required for a more general quantum theory, and is the foundation for the derivation of local Newtonian time presented in the paper linked to in footnote #1. 3. Credo quia absurdum (I believe *because* it is absurd.) Religion *is* the antithesis of science (σοφια) and all spirituality, and this is the crux of that irreconciliation. See also, Tertullian, "Certo est, quia impossibile est." Religion is an essential denial of reality and reason, and insistence on the absolute word of fabricated language, not to mention denial of humanity, and is also a fundamental annihilating psychosis. There is little more to say about its diseased and delusional mythologies pretending to some sort of absolute reality exuded in psychologically poisonous language. While mythologies are about understanding; religions are, and always have been about social, political and economic control, and that is all they have ever been about. "God is conscience. He is even the atheism of the atheist." -- Mohandas K. Gandhi (1869-1948)
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