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Comment by Todd Kelso
Mostly, I agree with you, but I will focus here only on what you said with which I disagree.

You don't mean "infinite dimensions" of space and time but infinite duration of time and infinite extent of space.

An expanding universe, if there were one, would not violate conservation laws. There isn't any law that says that density is conserved. As long as matter is not created or destroyed, pieces of matter are free to move away from each other.

Galaxies and atoms can move apart from each other and still retain their sizes, since the size of a galaxy is determined by the strength of forces with which the stars and other stuff in the galaxy interact with each other and the size of an atom is similarly determined by the internal forces of its parts. (I am here using the word 'atom', as I presume you are, in its Daltonian sense.)

I don't much care for the identification of distant galaxies as young.

It is not obvious to me that space and time cannot be subjects of scientific investigation since we are within them. We are inside the universe also, but we can still investigate it. There is of course no reason to think of space and time as physical objects.

I'd be interested in hearing why you consider plasma physics to be a good candidate for a source of explanation of the cosmological red shift.

You might profitably study non-Euclidean geometry. There is nothing sacrosanct about the Euclidean notions of 'distance' or 'straight' nor any known inconsistency in non-Euclidean geometries, or geometries that, from a Euclidean point of view, are 'curved'. There is no reason to suppose that the geometry of the real world is anything other than Euclidean, but that it is Euclidean cannot be argued on the basis of logic alone.

Your notion that one can not assign any properties at all to space or time is nonsense. One can, for example, assume that they are Euclidean.

Reply
Although I feel that my original arguments are sufficient to invalidate the concept of a Big Bang cosmology, let me address your major points:

The concept of 'infinity' is apparently even not fully understood by many mathematicians: ANY object in the universe has necessarily finite coordinates (both in space and time) or otherwise you couldn't localize it; it also has necessarily a finite size or otherwise different objects couldn't physically exist. 'Infinity' is not a coordinate value but only describes an asymptotic mathematical process which can not be assigned to any specific object or event. If you wanted to restrict the principle of cause and effect (which implicates an infinite recursive chain) to a finite life time of the universe, you would have to use a non-uniform time scale, a procedure which has already led antique philosophers to logically absurd claims regarding the possibility of motion (Zeno's paradoxes).
On the other hand, the hypothesis that time itself started at the 'time' of the Big Bang is a logical self-reference, i.e. a sentence of the logically paradoxical form 'This statement is false' and therefore nonsense.
Similar absurdities arise from the application of a non-linear spatial scale, i.e. the assumption that the distance between two objects is not given by a straight line (curved space).

Regards the matter density: there is of course no law of conservation of density; however there is a continuity equation, i.e. any decrease of the density in a given volume element can (in the absence of physical destruction processes) only be due to particles leaving this element, leading therefore to a corresponding density increase in neighboring volume elements in accordance with particle conservation. It is therefore logically impossible that the matter density decreases simultaneously everywhere in the universe and any assumption to this effect can certainly not claim to be based on physics.

As indicated, the redshift of galaxies (as well as the background radiation) could well be caused by the intergalactic plasma, but this is a separate issue here as the Big Bang theory is logically flawed for the above reasons. Nevertheless, let me address the related point of 'observational evidence':
Unless one knows exactly what one is dealing with (and cosmologists could hardly claim this), observational data can serve only as a guide for a theory but do not prove anything conclusively.
Furthermore, published observational data are often a) highly selective samples, b) processed in various ways and c) matched to a theoretical framework with almost arbitrarily scaleable parameters and which could therefore practically fit any data set.
In fact, I am already aware of some published data, which show differences for the redshift factor of the same object at different wavelengths of up to 10%, which clearly invalidates the Doppler - hypothesis and could well support my plasma theory as the redshift depends here to a certain extent on the wavelength and possibly also the coherence length of the radiation.
For related aspects (like Olbers' Paradox) see the main page .

Todd Kelso (2)
My point was that you said infinite DIMENSIONS of space and time rather than infinite EXTENT. Space has (apparently) three dimensions and time has one. It is the length of time and the volume of space that are infinite, not the dimensions of either.

What you said otherwise is true: even some mathematicians do not understand infinity, any specific physical object is some finite distance away and has some finite size, etc.

I don't understand how Zeno can be considered to have used a non-uniform time scale.

That time started at some time is obvious nonsense, but that time is finite in extent is not. Assuming that Euclidean geometry is itself consistent, so are other geometries in which all of space has a finite volume and each line has a finite length. In these geometries, the "lines" are sets of points such that for any two of those points, the shortest path from one point to the other is along the "line," but these lines are not straight in the Euclidean sense. Time could consistently have such a geometry also. In view of the fact that there is no evidence from the empirical world that even suggests that either space or time have non-Euclidean geometries, and in view of the fact that Euclidean geometries are much simpler than the alternatives, it is reasonable to adopt as a postulate of a theory of physics that the universe has a Euclidean geometry, both in space and in time. However, this is an assumption that has alternatives rather than something that is necessarily true.

Your supposition that any decrease of the density in a given volume element can (in the absence of physical destruction processes) be due only to particles' leaving this element, leading therefore to a corresponding density increase in neighboring volume elements, is a theorem in a Euclidean geometry but would not be true in various consistent alternatives. This is not, therefore, a matter of logic alone. It IS logically possible that the matter density decreases simultaneously everywhere in the universe and a theory of physics can certainly claim this and still be consistent.

I agree with you that the big bang theory is not only wrong but also ridiculous, but you go too far when you claim that anything that is not Euclidean is inconsistent.

I understand, with you, that published data are often highly selective samples, processed in various ways and forcibly matched to a theoretical framework. Worse things than this happen too.

Reply (2)
Your statements concerning the properties of non-Euclidean geometries may be correct in a mathematical sense, but - as indicated in my earlier reply - these do not correspond to a possible physical reality:

Even if you assume for instance that a perfect sphere physically exists (which of course is not possible due to the finite size of atoms), the 'space' defined by its surface exists only in connection with the physical body but has no independent reality (destroy the body and the surface disappears as well). It is obvious that in real (3-dim) space any two objects can be connected through a straight line (this is easy to visualize even if you can't connect them physically) and that therefore the Euclidean geometry is SUFFICIENT for describing the universe, as by definition everything in it is part of this real space. As the extension of a straight line always increases (proportionally) the distance between its end-points and you can extend it to an infinite degree, this also shows that Euclidean geometry is NECESSARY, because for instance in a curved spherical geometry the line eventually runs back into itself (unfortunately the useful and important logical qualities of 'sufficient' and 'necessary' seem to have become neglected in mathematics as well as physics).

I can't understand how you can on the hand assume that the universe IS Euclidean but still allow for the logical possibility that it could as well be non-Euclidean: it certainly can not be both at the same time and admitting the latter possibility logically rules out your first preposition.

The expression of 'infinite dimensions of space and time' in my original message should mean of course 'infinite in extent' and not 'infinite in number' (there are obviously, for whatever reasons, only 3 spatial and 1 time dimension).
You could argue that the use of the adjectives 'infinite' and 'Euclidean' in connection with the word 'space' contradicts one of my previous statements that space has no properties; however in view of the lack of actual alternatives, I feel that this formulation is still valid.

With regard to Zeno's paradox: it was Zeno's mistake to generally assume that an infinite number of steps equates to an infinite length in total. Apparently he was not aware that by successively reducing the length of each step by a factor 1/2 he was creating a geometrical series which obviously always sums up to a finite value (1 +1/2 +1/4+ 1/8 +...... =2 and not infinity). One could ascribe this mistake to the general ignorance at the time, but unfortunately today's science is full of similar flaws (most of which are probably caused by logical self-references, as indicated in my previous message).

Todd Kelso (3)
Obviously the geometry of the physical universe cannot be both Euclidean and non-Euclidean, but it could be either. If Euclidean geometry is consistent, then so are the various non-Euclidean geometries. We would need to observe the world in order to decide which geometry the world has. Unfortunately, no amount of observation would suffice, since the non-Euclidean geometries are not measurably different from Euclidean geometry in small regions. Thus there remains always the possibility that looking at larger regions of space will prove that the geometry of the physical world is non-Euclidean.

Various other things that you have said are just not so. For example, one can connect two objects by a straight line if and only if the geometry of the universe is Euclidean, but it does not follow from logic alone that this is so. If you wish to say that it is reasonable to hypothesize that space and time are Euclidean, or that there is no empirical evidence to the contrary, then I will not argue with you. But you cannot prove that space or time is Euclidean and in particular assuming it does not necessarily make it so, while it is worthless to argue in a circle by, for example saying that certain things that are true if and only if the universe is Euclidean are obvious. These things, such as the existence of lines that are straight in the Euclidean sense, are obvious only if you are already committed to a Euclidean point of view and such commitment not self-justifying.

I agree that Zeno did not understand that infinitely many numbers could add to a finite sum, but what has that to do with using a non-uniform concept of time?

Reply (3)
I fully understand your claim that each metric should be possible a priori. However one always has to distinguish between formal and real possibilities: if one said for instance 'All evidence points to the fact that the universe does exist, but theoretically it might as well not exist' then it is obvious that the latter option, although formally apparently possible, is not consistent with the circumstance that one has formulated the sentence in the first place.

The problem of self-consistency also has to be considered in connection with the metric of space:
it is for instance not in question that the shortest distance between two points on the surface of a sphere is given by a segment of a great circle. However, the shape of these and other 'minimum-curves' in non-Euclidean spaces can only be uniquely defined relative to a Cartesian (Euclidean) coordinate system: the circle appears as a circle because you view it relative to a straight line. Embedded into a higher dimensional non-Euclidean space it could take on any form dependent on your assumptions. The only invariant representation is obviously given by a Cartesian (Euclidean) coordinate system where in fact you can choose the coordinate base such that the lower dimensional space forms a direct sub-space of the higher dimensional space (a curve in a 2-dim Cartesian (x,y) coordinate system does not only have the same shape in any 3-dim Cartesian (x',y',z') coordinate system but even has identical coordinates if you let the x and x' as well as the y and y' axes coincide).
It is therefore not a question of 'experimental evidence' that the universe can be described by Euclidean geometry, but this is self-evident as it provides the only self-consistent representation.
Any observational evidence that is claimed to contradict Euclidean geometry (e.g. Olbers' Paradox) does therefore necessarily have a natural explanation in terms of physical processes of some kind.

Todd Kelso (4)
You are confused about all of this.
1) A set of axioms or the theory determined by that set of axioms is consistent if and only if there is no proposition P such that both P and ~P are deducible from those axioms and thus theorems of that theory. It is not known whether Euclidean geometry is consistent. It is known that if any of Euclidean, hyperbolic or elliptic geometry is consistent then so are the other two.

2) I'm not sure what you mean by "irreducible," but a set of axioms is independent if and only if none of them is deducible from the others. This is a matter of elegance only and has no bearing on the whether the theory is consistent.

3) Whether the axioms make sense is independent of whether they are consistent. Perhaps what you mean is that an inconsistent theory is worthless (since every statement that can be formulated within the theory is a theorem of the theory), so that it makes no sense to study it.

4) I don't know what you mean by "The 'problem' of finding the true metric of the universe is in fact not one of deductive logic but of self-consistency (identity)." The problem of finding the true metric of the universe is one of observation of physical objects. A proposed metric must be consistent with (concordant with) empirical data. This is different from (self-)consistency.

5) I don't know what you mean by 'identity' but anything that I would mean by that word is something different from self-consistency.

6) Euclidean, hyperbolic, elliptic and projective geometries all exist in any number of dimensions. Non-Euclidean 3-dimensional spaces do not depend on the existence of 4-dimensional or larger Euclidean spaces. One can have a three-space of some sort of geometry imbedded in a higher dimensional space with some other geometry (or of the same geometry, as Euclidean 3-space imbedded in Euclidean 4-space), but there is nothing inconsistent in the assumption that the physical universe has a three-dimensional space that is not embedded in any space of more than three dimensions. Such an assumption could consistently be made whether the 3-space is Euclidean or not. As a pedagogical device, it may be useful to think of a 3-dimensional elliptical space as imbedded in a 4-dimensional Euclidean space, but this conception merely helps in learning about a space that is curved relative to a Euclidean space provided that the student is already familiar with Euclidean spaces. It is not necessary to think this way any more than it is necessary to think of a Euclidean space as being imbedded in a space of more dimensions.

7) If you'll tell me what a tangent space is, I'll tell you whether Euclidean space is the only one that contains all its tangent spaces. I suspect that you're wrong, though, In all three spaces one can draw tangent lines to differentiable curves and tangent planes to differentiable surfaces and the tangent lines and planes are always subsets of the space that contains the original curve or surface.

8) If a theory of physics can consistently assume the existence of a three-dimensional Euclidean space as the home of physical objects and the arena of physical activity, then competing theories may consistently assume the existence of a three-dimensional space that is elliptic or hyperbolic.

9) The only way to determine which geometry, if any, appears in an accurate description of the real world would be to measure things in the real world and find the empirical formulas for the circumferences of circles and other such things, in which the geometries differ from each other. Even then, though, we would need sufficiently large circles, etc., since, given the finite limit of our ability to discern tiny differences or make accurate measurements, all the geometries look the same in small regions. Thus the situation is that if the Euclidean postulate is correct, then we will never know whether the Euclidean postulate is correct, since there would always remain the possibility that measuring larger things or measuring smaller things more accurately would disconfirm the hypothesis.

Reply (4)
You seem to forget that Non-Euclidean geometries are derived (albeit generalized) from curved surfaces in 3-dim space and have therefore inherited the corresponding characteristics of these 'pedagogical devices' (not the other way around). One may be able to ignore these connections for certain (formal) purposes, but this doesn't mean they don't exist (it would be bizarre to claim that languages in general are only formal constructions which exist independently of reality, with the latter being only a 'pedagogical device' for demonstrating the properties of the former). In the end, the formalism of spherical geometry for instance is in practice only good for one thing and that is describing the geometry on a sphere.
Fact is, as already indicated before, that the Euclidean metric always yields the shortest distance between two objects in the universe, and there is no reason whatsoever (not a physical one anyway) why the usual 'minimum principle' of mathematical physics should not hold here.

I do not want to address all the other points you mentioned as they are merely different aspects of the problem, and people interested should be able to work out the correct view by themselves from what has been said in the course of this discussion and by using their common sense.

Todd Kelso (5)
Your understanding of the relation to each other of Euclidean geometry and non-Euclidean geometry is faulty. Your views on this matter are simply mistaken. You ought to modify your web page to say that there is no empirical evidence that the geometry of physical space or the chronometry of time is anything other than Euclidean and cease insisting that the non-Euclidean geometries depend in some way on the prior existence of some Euclidean space that contains them or that the geometry of the physical universe must be Euclidean on logical grounds.
The facts of the matter, which you may check with any mathematician who has included this much geometry in his arena of expertise, is that the three geometries (elliptic, Euclidean and hyperbolic) have the same relationships each to each other. Each of them implies the existence of the other two. The lines and planes of each of them appear to be curved from the point of view of either of the other two. The inhabitants of an elliptic world, if they were as confused about this matter as you are, would say things such as "The concept of a 'curved space' is logically flawed because space can only be defined by the distance between two objects, which is however by definition always given by a straight line," but he would be talking about a line that is straight according the meaning of this word in elliptic geometry, not in Euclidean geometry. The same is the case for hyperbolic geometry. The inhabitants of a space of any of these three kinds have no right to say that his lines are really straight and his planes are really flat and his distances are really shortest, etc., and that the other two are using strange notions of distance that depend on artifices of curved n-dimensional surfaces in a space that is itself actually (n+1)-dimensionally flat.

Reply (5)
As I said above, the Euclidean geometry always yields the shortest distance between two points (that's why people build tunnels through hills etc.), and this singles it out the against all other geometries.

Note added later: I have learned with regret that Todd Kelso has died some time ago. Despite our differences over certain scientific and philosophical issues (as illustrated above), our discussions were always enlightening and have helped me to shape my own opinion more clearly. Todd Kelso's writings (mostly mathematical-philosophical essays about physics issues) have been posthumously published on his homepage Pythagorean Physics (again, I would probably not share all of his conclusions, but there are certainly a number of interesting and valid points being made).

Comment by Michel Guillor
My comment on Todd Kelso(4):

Mr Kelso wrote : "It is not known whether Euclidean geometry is consistent."
Euclidean geometry indeed IS consistent. It is complete, decidable and consistent, provided 1) no induction on arbitrary integers (using reasonment per recurrence) is performed, and 2) no use of general axioms of set theory is performed. These reservations are for Gödel's incompleteness sake.
In other words, elementary geometry based on Euclid's (Hilbert's, actually) axioms only is decidable and consistent ( http://mathworld.wolfram.com/EuclideanGeometry.html). And, of course, the same is true mutatis mutandis for Lobatchevsky's and Riemann's, under the same reservations.
As for the geometry of the physical universe, it's likely that it's Euclidean at a big scale, or at least that's the conclusion of three chains of observations on CMBR, namely, in the chronologic order, Boomerang, Maxima and Archeops. Of course these experiments were performed assuming the BB is true. But that doesn't invalidate the main conclusion.

Reply
As you indicate yourself, the mutual consistency of a set of axioms does not necessarily mean that they are true (as an everyday example: assume a person has committed a crime, but he denies this and in order to 'prove' his claim asks a friend to issue him with a false alibi; then the claim would be consistent with the alibi-statement, but would nevertheless be untrue).
What this means with regard to the true geometry of the universe is that it has to satisfy not only the basic mathematical axioms of the geometry, but also the wider logical requirements associated with applying the geometry to the corresponding physical situation. As indicated in my replies to Todd Kelso above (and also to Juan Casado on page 2), this requirement can only be satisfied by the Euclidean Geometry. This circumstance is not dependent on experimental results, but is a conceptual necessity.

Comment by Peter Rizo-Patrón
I am in full agreement with your basic views in Physics. I am bewildered at the attitude of the 'academic establishment' that has adopted such absurd stances as can be seen in current theories of relativity and cosmology.
However, I am at a loss at one statement you make: 'It is obvious that the assumption of a 'creation' is logically inconsistent with the scientific principle of cause and effect.'
Could you please explain what you mean by this? Obviously 'creation', or rather 'the universe' is impossible for us human beings to understand, at least at the moment. Many of the cosmologists' attempts are ridiculous, as you clearly explain. But there are compelling arguments made by brilliant scientists and philosophers that lead us to the conclusion that the marvelous order of the Universe must have been caused by 'someone' who designed this order. This 'someone' must be independent of matter, and able to form it at will. If we appreciate the extraordinary balance of elements required for the existence of life forms as we know them (that we reckless human beings are unfortunately contributing to destroy on this planet) then we cannot imagine that blind evolutionary processes have caused all of that beauty. Life is not only amazing due to the physical and chemical equilibria that exist, but is also a marvelous work of art, and seemingly this can only be appreciated by human beings. But it is possible that human beings must be trained for the appreciation of beauty, or to have been exposed to art since childhood. Maybe many physicists today have lost appreciation for 'quality' since they put too much emphasis on 'quantity'. Fortunately this seems not to have been the case for example for the famous Maxwell who clearly expressed his faith in an all-powerful and good God. Finally, do not think I am a bigot. I have good friends who are agnostics or atheists. The important thing is for people to be kind with each other and to be sincere. Religious or not we are all ignorant about the ultimate causes of life as we know it. At the same time we are amazed at the progress that scientists and inventors have made in the last centuries that has led to the possibility that life in this world may not be destroyed after all. Objectivity and straight thinking are necessary though, but we seem to be a long way from this.

Reply
I do not at all object against the notion of a 'creation' if this is meant in a metaphorical way, i.e. not referred to the dimensions of time and space. Most cosmologists appear to have fallen into a logical trap however and seriously try to limit the existence of the universe both in a temporal and spatial sense. This is plainly a conceptual mistake from a scientific point of view and, on the other hand, effectively blasphemy from a religious point of view (it is simply ridiculous, and revealing, if the Pope considers the 'findings' of Big-Bang cosmology as a support for the teachings of the Catholic Church; true faith should not need any scientific support, but then again the dogmas of the Catholic Church have probably little to do with true faith).
In my opinion, true spirituality can only grow on the basis of the factual truth. The latter can only be obtained however by adhering strictly to rational and logical principles. This leaves no room for a creation as a scientific object (because this would be self-contradictory), but only as a spiritual, metaphorical notion.
In this sense one can certainly use a creation as an 'explanation' why certain fundamental things are as they are and, more importantly, why they exist in the first place. These 'questions' however have no answers within logic and science but they can be seen as complementing the latter, like the reverse side of a coin complements the face side. Unfortunately, people often try to mix these intellectual and spiritual aspects of existence which is at best meaningless and at worst the source of unnecessary conflict.

Peter Rizo-Patrón (2)
Your use of the word 'metaphorical' is not clear to me. Matter and the incredible order and beauty of the Universe is something quite tangible to me. The fact that we do not understand how the wonders that we behold every day have come about does not belie their reality.

It seems to me, intuitively, that time and space have existed always, and that God created matter (atoms and all sorts of particles and their links and interactions) into that space at some point in time. I'm not sure if you are thinking in the same terms in this point. I am assuming the existence of a 'God' that 'created' the material universe (before creation I assume that space was void). This is impossible to prove, at least 'for the time being' (no joke by the way). Given the impossibility to prove this it is no wonder to me that either (a) some scientists would shrug away the existence of a 'creator' or 'God' (since they only accept what they can prove or confirm experimentally), or (b) that people that claim to be 'believers' would concoct some crazy theory about creation and the 'beginning of time and space' (viz the 'big bang') in order to try to accommodate their 'physics' views to their 'faith'. The fact that Monsieur l'Abbé Georges Lemaitre proposed the big-bang theory is not surprising in this context.

What is clear is that without SINCERITY there cannot be true spirituality. The important thing is to try to look for the Truth sincerely. Reality is a great Mystery, not only the physical aspects but also the spiritual and tragic aspects that include all sorts of apparent contradictions that are a simple reflection of our ignorance. Although we know a lot of things, increasingly so, it is folly if we jump to many conclusions, especially in the cosmological realms, where there are more unknowns than certitudes.

I agree that we cannot understand creation in its origin, so pretending to apply scientific method to discover the origins of the universe is at best a chimera.

Reply (2)
Many people (scientists and philosophers alike) forget that their intellectual investigations are restricted by the language they use (I have emphasized this already on the Relativity- Forum page). Apart from formal logic (which is an empty exercise) one can only 'prove' a statement if its elements are clearly defined and part of a self-consistent system (in this sense it is for instance easy to prove (by induction) that the universe has to be infinite in space and time). It is obvious however that statements which are not fully defined within this system can not strictly be proven (or disproven). This limitation of the cognitive process by the language effectively defines what can be 'known' and what on the other hand has to be 'believed' (we still try to speak about the latter in terms of words defined for the former and it is in this sense that the idea of a 'creation' has to be seen as metaphorical; this does not in any way belie its reality but only its objectivity (and the objective rendering of reality is what science is all about)).

Unless one is doing science, there is obviously no need to 'know' the world and one can prefer to 'believe' instead. Although one can still be 'sincere' in this case, this kind of spirituality is in my opinion on rather shaky grounds as it is based on loyalties or dogmas rather than the Truth (I think history has shown this often enough). Of course, nobody is perfect and people will therefore sometimes not recognize or misinterpret the true nature of things (although they may have tried sincerely), but this again only emphasizes my point because eventually this presumed 'truth' will come to nothing (one could be tempted to adopt a psychological point of view where it doesn't matter if one is dealing with the actual Truth or not, but such an indifferent attitude would really make little sense here and just lead to people fooling themselves (and others)).

This is not to say that science will ever be able to explain everything. As already mentioned, the scientific language (as well as the everyday language) is only a rather crude instrument through which we analyze and describe the world and there is consequently an infinite number of aspects of reality that cannot properly be described by it, and secondly, all theoretical models have to be based on certain fundamental postulates (axioms) which can not be further deduced in a strict way from other principles. The view that God is forced out of existence by science is therefore rather short-sighted and naïve. On the contrary, in my opinion science puts him actually on much firmer grounds as there is less room for falsehoods (people who don't realize this not only can be accused of having a lack of faith but also of being incompetent scientists).

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