The Challenge of Plato's Cave
"Life's but a walking shadow, a poor player, That struts and frets his hour upon the stage, And then is heard no more. It is a tale Told by an idiot, full of sound and fury, Signifying nothing." - Macbeth by Shakespeare ( 1564 - 1616 )
"The creative principle of science resides in mathematics" - Albert Einstein ( 1879 - 1955 )
What is time?
This is not an easy question to answer. We all intuitively "understand time" by the clocks and calendars we invent to "measure it". In a sense, therefore, the "concept of time" is given an "operational definition" by these invented instruments which ultimately relate back to earth's axis of rotation as well as its orbital transit about the solar system's sun.
Perhaps a more elegant definition of time would involve the relative motion of objects [ or event - objects ] such as the statement "this event - object occurred before that event - object" in a volume of human - perceived 3-d space.
And the "units of time" are suggested to the human mind by the regularity of recurring event - objects such as the periodicity of the sun's rising and falling. Other units of time were also suggested to the early Greek astronomer - philosophers as they studied the celestial movements of stars and planets.
( We use 12 months )
June solstice sunrise viewed from the western observing point
at the astronomical observatory at Chankillo ( Chanquillo ), Perú
© Professor Iván Ghezzi,
Pontificia Universidad Católica del Perú
Dictionary definition of time according to Webster's College Dictionary, 2001:
" ... the system of those sequential relations that any event has to any other, as past, present, or future; indefinite and continuous duration regarded as that in which events succeed one another."
A more precise yet simpler definition of time given by this author:
time is an accounting of the relative motions in space of bodies without which neither time nor space can be understood. Or, time is a system of accounting for the relative motion of bodies in space. And how we keep "score" is by means of human - invented clocks and calendars.
Einstein's definition of time, from "Space - Time", 1926:
"The physical time - concept answers to the time - concept of the extra - scientific mind. Now, the latter has its roots in the time - order of the experiences of the individual, and this order we must accept as something primarily given."
The "parts of time" that the mind creates, like that of points and lines, are obtained directly from those subdivided parts of space for which the relative positions of bodies the mind identifies from the underlying noumenon.
That is, the mind identifies the "grand sweep" of the relative motion of bodies, such as earth and sun, but then subtly begins to subdivide these relative positions of space which the relative motion of these bodies provides. Thereby these subdivided relative positions ( regions ) of space produce the subdivided periodic "parts of time" such as day, hour, minute and second with which our clocks are normally associated.
In other words, the "parts of time" consist of "parts of space" which the relative motion of bodies provides.
Let's all accept that there exists both an external physical world and an internal world of the human mind. Also, let's examine all too briefly what does or does not exist in the external physical world: essentially only two salient things exist in the external world and these are 1). physical bodies and 2). the perceived relative motion of these same physical bodies. In other words, think to yourself as to what appears salient to the mind of your doggie or pussycat. Yes, I know ... love and food. But seriously, only physical bodies and the perceived relative motion of these bodies are what essentially matters to the mind. Beyond that, all else is an invention of the human mind. For example, music. Musical notes and bars do not exist in the external physical world, do they? Rather these are invented by the human mind and then translated into a mathematical system of notes and bars which in turn are given further expression by human - invented musical instruments. Straight lines in nature? Do straight lines actually exist in the external physical world? How about points? Or circles and triangles? How about squares and rectangles? Have you ever seen or touched a straight line? A pure circle? In nature? In the real physical world? Yes, you have seen straight lines and pure circles drawn on pieces of paper but these have been expressions of human imagination. In truth, none of these "things" exist in the real world. They are ideas in the human mind which in turn are given expression by means of mathematical notational symbols. That is, try yourself to describe in words only what is a point or a straight line. You can do it, but it definitely lacks the clarity, precision and power of further using mathematical notational symbols.
In fact, "straight" light beams are not purely straight but rather are bent in the vicinity of gravitational mass; the surveyor's theodolite gives only a physical approximation to the mathematical ideal of point and line.
All of the aforementioned applies to the concept or idea of time - essential time in and of itself does not exist in the external world. That is, for example, when was the last time you actually saw time? You saw time passing, yes, but this was because of the relative motion of bodies, not because you actually saw "disembodied essential time". Or, you felt time as something having happened or occurred "before" or even "now". Precisely the point being made here! In this instance of your mental activity, time calculation is something internal to your mind and not anything external except that there had occurred the relative motion of bodies which in turn had affected your memory and perhaps your emotional system. Time rather is an invention of the human mind in order to make an accounting of the relative motion of bodies in the external world. Or, time is a mental system of accounting for the relative motion of bodies in the external world. And, operationally, we humans invent clocks and calendars in order to better keep score of this accounting. But time is still an invention of the human mind. And the power of mathematical notational symbols and its related calculus makes the concept of time [ always shown as ' t ' ] an extremely powerful intellectual idea.
There will be one further intellectual experiment in order to elucidate the concept of time being propounded here: suppose that everything stops. Everything! Your heart stops. Your eyeballs stop all movement. All other bodies in and around you stop. Everything stops completely and totally! Now, what about time? Does time exist? Is there such a thing as time? Is there even a need for time?? That is, a mental system of accounting for the relative motion of bodies in this completely stopped external world? Of course what next will drop away will be the concepts of velocity and distance. Why distance? Because there will be no need to travel by definition. The human mind will adapt to its circumstances as is needed in other words. All of what has been described here is what is theoretically the reality beyond the event horizon of a black hole. Inside a cosmic black hole, in other words.
Let us continue in this discussion ... so we live in an external world whose arrangement and ordering in our minds is made possible by all sorts of concepts and ideas: time, points, straight lines, circles, ellipses, squares and rectangles, and, of course, musical notes and bars. Now, in order to survive successfully in this external world, we humans need a sort of feedback. That is, if some preconceived idea or concept applied to the ordering and arrangement of perceived ephemera of the external world does not produce "consistency" with other known ideas and concepts of the external world and further does not possess a sort of "predictive quality" as to what "should come next", then we humans discard that preconceived idea or concept as being useless. Therefore, the human mind is more than fully capable of conjuring ideas and concepts which are wacko, if not only untestable, to use some vernacular. Hence, we arrive at the mind's concepts of empiricism, 'cause and effect', and the scientific method of knowing about the external world. Look, if we humans were at the level of, say, a crocodile, then our "perceived world" would be a solipsistic series of disconnected snap - shot images of perceived ephemera, no? Nothing about time, points, straight lines, musical notes and so forth. And certainly nothing about the application of these ideas in the ordering and arrangement of perceived ephemera of the external world in order to more successfully survive in this chaotic external world.
So you say now that the "external world" is also an invention of the human mind, that its "reality" is also unproven? Well, yes. In fact, it is unprovable. It just is. It just exists in the human mind. This is where psychology and philosophy meet. Furthermore, ideas such as 'cause and effect' and the scientific method are also unprovable besides being inventions of the human mind. We cannot "prove" the idea of 'cause and effect' any more than we can "prove" the efficacy of the scientific method and empiricism. These concepts just work for us better than anything else that we can invent in order to better make sense of our invented idea of an external world. We just accept it. It works for us. It is solely based upon a faith that this best works for our survival. But it is certainly not "a faith" like that to which we can go to a church or synagogue and listen to some priest or rabbi and pray. No, no, no! In fact there is absolutely nothing to pray to! You going to pray to the idea of an external world or 'cause and effect'? Or pray to the efficacy of scientific empiricism? Ha, ha! It is just a common sense sort of faith that now and into the future that the ideas of an external world, 'cause and effect' and scientific empiricism and methods will better make sense to us humans regarding our understanding of the external world and thereby enhance our survivability in this crazy mixed up world.
Nevertheless, mathematics is the ultimate language of the human mind to translate abstract thoughts into some sort of comprehensible ontology. And, the scientific method is the human mind's best practical method for testing its mathematics against perceived physical reality.
According to Book VII of Plato's "The Republic" ( 514A - 520A ), human understanding of the external world is tightly chained to only the perception of moving shadows of light reflected off of the interior walls of a cave to which human minds and limbs are physically bound by immobilizing chains. We as humans are existentially limited to only "seeing" shadows of light's reality. In other words, what is the difference as between "reality and appearances"? Shadows appear, but is that also reality? Or are the shadows themselves the best that can be achieved for understanding perceived reality?
These are unanswerable questions.
But these are nevertheless the constant challenges given to the human mind. The human condition can neither escape the question's challenge nor can it escape the reality of Plato's eternal cave of light's shadows.
Plato's Cave and its Challenge:
So here we are, human minds existentially trapped between perceived appearances and Plato's constant question of light's reality. But the human leitmotiv does not admit to any sort of limitations in our understandings of the external world, does it? And therefore we humans with our minds invent ideas and associated mathematical notational symbols including musical notes and bars in order to escape our human boundedness. And hence we arrive back to where we started: the mathematics of music, time and space are all ultimately creations of the human mind in order to humbly answer the challenge of Plato's Cave. And what follows in Relativity Science Calculator [ both the web site mathematical essay and future related software ] is just another small attempt to penetrate Plato's Cave and its Challenge.
Author Relativity Science Calculator:
Throughout my entire thinking life I have been constantly amazed at the power of mathematics. I mean, why do mathematical notational symbols with their related rules of logical connections give the sort of intimate noumenal insights into nature's reality to which we all have become so accepting? And, if one body of mathematics does not work so well, well then we move on to another body of mathematics which does better for us. Sometimes much, much better. This is truly amazing when you think about it! A bunch of scratches on paper or blackboard and the entire universe comes better to human understanding! Doesn't this amaze you? Have I forgotten that even the chemistry of human biology is essentially mathematical? Why this is true is also one of those unknowables. And unprovables. It will never, ever be proven with complete and total logic [ mathematical logic, that is - ironic isn't it! ] that mathematics is the ultimate means by which the secrets of the cosmos and sub - atomic realities will be obtained.
So, my further hypostatizing question is this: is ultimate reality what mathematics describes or is ultimate reality the mathematics itself? This then returns me always to Plato's Cave and its Challenges. One possible solution to this conundrum is "Model - dependent realism", by Stephen Hawking ( see: "The Theory of Everything", Scientific American, October, 2010 ). I only can say that mathematics is both a shadow of light's reality outside the Cave, and that mathematics is itself the ultimate reality, if only partially.
Einstein's Special and General Relativity
So as astronomers peered into the macro of nature and other scientists of the microscope peered into the mysteries of the micro, questions arose. And one of the myriad of questions which arose was about the very nature of light itself, that quintessential part of nature which brings messages and information to the human eye and brain. What is light? From where does it come? Of what is it made? How does it behave under glass prism? In the sky? How fast does it travel, if at all? Are light's messages instantaneous? Some Greek philosopher - thinkers actually thought that light was corpuscular and therefore travelled with a finite speed - this was during the Classical Period of Greek Thought, way, way earlier than Isaac Newton who believed that light's messages were instantaneous. Some of the answers were given by Galileo Galilei, father of modern astronomy, and Danish Olaf Roëmer who produced one of the first significantly important and accurate calculations of the speed of light using a Galilean telescope; and still other answers were given more recently by the experiments of Michelson and Morley. Along the way the equations of Lorentz ( Arnheim, Holland ) and Dublin - born FitzGerald today stand at the very heart of the mathematics of Special Relativity. Of supreme importance were the equations of electromagnetism by Scottish mathematician and theoretical physicist, James Clerk Maxwell, unifying magnetism and electricity and in telling us what is the essential wave - like nature of light. Throughout this mathematical essay several, perhaps many, of these giants of human thought have been overlooked. The point is that Einstein stood on the shoulders of philosophical and mathematical giants preceding him by several hundreds if not millennium of years. And he would be the very first and most humble of persons to admit this obvious fact.
Oh, what knowledge was lost at the burning of the Library of Alexandria∗, founded during the Greek Hellenistic reign of Ptolemy II of Egypt, 3rd century BC!
So what in a nutshell is Special Relativity and what does it tell us about nature's reality? Well, because light possesses finite speed all sorts of other realities become apparent. Among these other realities are that nothing in nature travels faster than the speed of light but that at speeds approaching the speed of light, time slows down appreciably; physical bodies actually contract and get smaller - truly that mass itself is a variable quantity; light itself acts like matter and therefore reacts to the gravitational forces of other bodies of mass; black holes exist and either completely trap light or bend nearby traveling light; and that huge amounts of energy are contained in bodies of all masses. In fact, energy and mass are interchangeable quantities of nature such as what The Big Bang itself demonstrates ... it all depends upon the speed of light! Other little tidbits of consequence arising from Special and General Relativity are that gravity and acceleration are one and the same phenomenon. And because of Special and General Relativity much is understood - and much is yet to be understood - of the cosmos.
source: Hubble Space Telescope
But you knew all this before, didn't you? But you don't know why you knew all of this, do you? Except that someone told you or that you had cursorily read about it in some magazine or watched some science program on commercial television with 5 - minute interspersed commercials way back when. But because of Relativity Science Calculator software and its mathematically related web site essay, it will be shown to you in simple but oftentimes extended algebra and simple diagrams and drawings why all these things of Special Relativity are true in addition to their related historical contexts. And beyond Special Relativity it will be attempted in subsequent web pages the same for General Relativity.
"Astronomy compels the soul to look upwards and leads us from this world to another" - Plato, The Republic
∗click here for a thumbnail history of the conflict between humanist science and the historic religious persecution of it.
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Cathedral III music score © Dmitri Tymoczko, Princeton University
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