Space and time are fundamental categories of philosophy - in this, they can not be "defined", rather they can be "understood" - reflected/interpreted in philosophical analysis. Space and time are attributes of Matter - there is no material object out of space/time - and thus they are fundamental forms of Being. Space and time are aspects of Motion - because in Motion only they are revealed. With the evolution of the scientifical analysis of various forms of Motion, philosophical understanding of space and time has been also evolving.
I am assuming you do not wish to go into Physics and Theory of Relativity here, so... From an Euclidean point of view, in our Universe space has 3 dimensions, x, y and z, while time measures of durations of events and the intervals between them. Of course you might want to consider time as a dimension that orders events from past to future. Yours is an opern question and one can answer as I did (no Physics, theory of relativity and other complications) or one can add quite a lot to it. Without you specifying what you want it is difficult to determine the depth of the answer!
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V.Vlasov
Hi Fernando
A very well thought of answer- great. But this answer suggests that these dimensions are orthogonal, and independent. Thus, time is a measure of events that are defined in x,y,z axisis, or, is a measure of movement? Am I on the correct path?
Time is not a measure of movement only. It is a dimension that allows measurements of changes of situations in the x,y,z frame. And yes, time would be orthogonal to x, y and z, if you define orthogonality as integral of (dimension 1 x dimension 2), as we normally do.
An interesting question, but is it possible to have an 'objective' definition of space and time in the first place ? So your first question would have to be 'Can there exist an objective definition of space and time'
In other words: Are Space and Time provably independent of any possible Observer Effect ?
Unless and until you convincingly answer that one, your question must be held in abeyance.
Space and time are fundamental categories of philosophy - in this, they can not be "defined", rather they can be "understood" - reflected/interpreted in philosophical analysis. Space and time are attributes of Matter - there is no material object out of space/time - and thus they are fundamental forms of Being. Space and time are aspects of Motion - because in Motion only they are revealed. With the evolution of the scientifical analysis of various forms of Motion, philosophical understanding of space and time has been also evolving.
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While I know that there seems to be a tendency to look at this question philosophically, that is not the intent.
My question is- if we are unable to define space and time effectively, which seems to be evident from the above, then, how can we define frequency? And thus, all that is signal processing.
I went to my school books and found that space was defined in geometry by the 'presence of distance'. So, my question is- what is distance? If I was to travel at infinite speed, will then there not be any space? I am aware that this is a hypothetical question, but then so is Fourier Transform.
There are things that are not "defined", only introduced. For example, in geometry the notions of a "point" or of a "straight line" are not defined, yet allow one to define other notions.
So, I think, the question with your question is whether you seek for a formal mathematical definition, or for a physical-sense definition like "what space really is?"
In the former case, like Fernando already suggested, it would be sufficient to define an N-dimensional (e.g. N=4) space (in the mathematical sense of the word). This given, there are no problems to further define distance, interval, frequency etc.
In the latter case, since mathematics may or may not reflect the reality of the objective world - and whether it does we can know only through experience - there is no "definition" of space and time, only understanding thereof as attributes of matter, the way in which material objects do exist.
Dear Anton and Fernando and others
Do you state that time is the 4th dimension, and that is orthogonal to the space dimensions. That is the concept I am not able to come to terms with, because if it is, convergence at infinity (speed) should not happen. To my head, based on the above discussions, it is appearing that these dimensions are not orthogonal, but dependent.
The reason I bring this up is because this has an impact on how we see the future of signal (and image) processing. It would suggest the convergence of time and frequency (and space).
It also helps me understand the philosophy of physics. Because, time cannot 'start', and this helps me understand the concept of infinity.
Dinesh
Dear Dinesh,
Yes, for example in the special theory of relativity (in which, I must confess, I am not an expert, but anyway), the model is that time (or, more formally, c*time) is the fourth dimension, and it is orthogonal to the "space dimensions". The "mathematical space" is the so-called Minkowski space.
Introductory paragraphs of the "Field Theory" by Landau & Lifschitz describe this model:
http://www.scribd.com/doc/3209787/Landau-L-D-Lifschitz-E-M-Vol-2-The-Classical-Theory-of-Fields
In the special relativity perspectives, the answer goes rather simple. The coordinates for space and that for the time intermix when the space/time coordinates of the moving frame and stationary frame are described. It gives a rather weird consequence that completely defies the common notions about time, which is linear. One famous consequence, as noted by Einstein himself at the funeral of his friend Michael Besso, is that " the distinction between the past, the present and the future is an illusion, a persistent illusion of mankind. "
In this context, we have to completely discard the conventional notion that the time and space coordinates are separate, mutually independent parameters; now we have to admit that they are no more than the different aspects of a single physical object, i.e. space-time, which is never to be confused with the " space and time". The space-time is a hybrid physical object invented to reconcile the aforementioned weird intermixing between time and space coordinates when describing the physical laws in reference frames with differing relative speeds; in this context, time and space are mutually complementary ( and yes, mutually essential, since you asked) in describing the remarkable world we live in. Such remarkable conclusion follows almost solely from the constancy of the speed of light, which is indeed remarkable again.
The second paragraph of this answer was written today, 28 Nov 2017, almost 4 years after the first paragraph, since I discovered that you gave another question to my first paragraph.
Hi Wook
Yes, that is weird, but interesting. Does that mean that time is an essential dimension for describing space?
On a mathematical point of view, spacetime is the mathematical model that combines space and time on a continuity or a topological multiplicity where events can be sufficiently described by coordinate system. Using space coordinates (x,y,z-Euclidean space) it is possible the determination of an object location. By adding the time coordinate as fourth coordinate, the coordinate system determines where and when an event takes place. Time coordinate in a spatial dimensionless system has no meaning due to the fact that both its definition and measure units are connected with space alterations
Hi Wei and Panagiotis
Does that mean there is no space and time if there is no object? What is an object? And, what is an event? Also, I am not sure what the word, 'dimensionless' means in this context, when we are defining the 4th coordinate.
Dinesh
Space – The unlimited or incalculably great three-dimensional realm or expanse in which all material object are located and all events occur.
Time – The system of those sequential relations that any event has to any other, as past, present, or future.
(Webster’s Encyclopedic Unabridged Dictionary of the English Language, Gramercy Books, New York, 1996)
To summarise, space and time cannot exisit without each other. By simple logical deduction, time cannot start, and thus, space cannot start. As our Universe is largely space, does that mean that the Universe cannot start? Does that not negate the Big bang theory? I am confused.
'space' and 'time' are jointly composing the only four dimensions that human can naturally comprehend. However giving an objective definition of these four dimensions is not possible because they are intrinsically associated to the observer perspective (as per the special relativity theory). And since the observer in this case is 'us' which negates the objectivity condition.
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