Thoughts about a priori space:
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Space is a concept that is critical to understanding science and those areas of mathematics that use some form of space.
Space, in mathematics, can be considered the infinite set of all points.
In science, we often think of space as having three dimensions with a fourth dimension that is thought of as a dimension of time. These four dimensions constitute space-time, largely derived from Einstein's theories about the nature of empirical reality.
In mental space, or the pure space of mathematics, space is infinite and three-dimensional and, though we can think of non-Euclidean space, mental space is constituted however we choose to think of it. It is hard for us, if not impossible, to think of space as bent without something being bent in it because it is only our focus on the bent thing that gives us the understanding of curved space.
Thoughts about empirical space or space in scientific observation of the universe:
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The following portion of this page concerns the nature of empirical space or space in scientific understandings of reality rather than pure mental space.
That we are theorizing that a graviton or gravitational waves exist is interesting to me. Why is it not possible that the force of gravity could simply be accounted for given the warping of space-time itself? Because I can think of nothing else, I will call this the CSTAG conjecture, or the
"Curved Space-Time Accounts for Gravity" conjecture which I made on about the 4th of January, 2010.
Here is a pure summary of CSTAG:
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(i) gravity is a concept that need not be thought of as a force but only as the warping of space and time in the presence of solid-state energy (mass), and (ii) the warping of space-time may be inverted in the absence of mass on cosmological scales related to galactic superclusters or higher and (iii) the warping of space-time may oscillate between appearing as an attractive force and a repellant force given various cosmological-scale quantities of mass.
It must be true, that logically, gravity as an ontological and/or forceful entity is not necessary because we could imagine otherwise. It is my conjecture that gravity is not a force but a feature of curved space-time only and that no forceful substantial particle or wave can be found to support the "law" of gravity, which, in all science, should only be a theory of gravity (laws of science hinder fundamental revolutions in science and should be abolished by any reasonable philosopher of science).
This is possible because (a) no substantial carrier of gravity has ever been directly observed and (b) space need not be considered acted on by a force in the same way other substantial particles and waves are.
Saying gravity does not exist will get you laughed at but this is precisely why institutional thinking is so anathema to change and progress because of the human psychological desire to not be ridiculed and not take risks where there will certainly be resistance and derision. For each of the greatest revolutions in science there was an entire scientific community scoffing at the idea that they could be so wrong. But, as it appears, all scientific theories will always be wrong based on the very empirical method science uses.
If all theories of reality's consistency in terms of parts of it interacting with other parts, have been wrong and undermined completely, in the past, then why should this trend not continue indefinitely or until some point at which there are no more anomalies or misfit forces or things that are not fully understood? If gravity was no longer a force but a misconception, then would it not be easy to see the other forces as mere aspects of the same basic or fundamental force? Gravity is the oddball.
Space is not a particle or a wave it is the fabric in which particles and waves can be acted on by force. I am only saying here that the relativistic curvature of space-time can possibly account for the appearance of a force like gravity. I am not saying that some mysterious unknown force holds us to Earth but only that curvature of space and time around massive objects can internally to relativistic theory, be said to account for the accelleration of substantial objects in what we call "graviational fields".
A gravitational field then, would be revised to be understood as only curved space time and then we would stop, principally, looking for a substantial carrier of gravitational force, such as the graviton or gravitational wave, because we know that gravity is then, only a feature of curved space time, but not, as it has bee considered since Newton's paradigm, a substantial force in itself.
This, then makes gravity unnecesary for our ontological scheme and it becomes a force that is, I think, more properly thought of as a feature of space in the presence of mass or energy, not necessarily acted on by that mass or energy by waves or particles.
If I am wrong, then gravitational waves or particles can be or may be empirically observed. If these particles of waves that are carriers of gravity are ever found, then I am completey and totally wrong. But, I claim that this is only a conjecture, not strong enough to be theoretical, until such time as gravity need be rethought because a project to observe it, in the future, has failed in all tests and methodologies for such tests.
There need not be any particulate or wave-like intermediate to convey gravity if it is just a built-in feature of the curvature of space and time. I could very well be wrong about this, but to my knowledge, no empirical evidence has been found for the graviton or gravitational waves. I think Occam's razor applies here in that many scientists are simply confusing the situation by holding on to an unnecessary ontological entity in their paradigm.
It is my conjecture that gravity is the curving of space-time and it need not be instantiated in any substantial thing like a wave or particle. Gravity, then, does not exist. Only the curvature of space-time exists and this alone is why things tend to fall into other things.
Thinking of space-time as a thing that can be bent by a force could be wrong. Space-time may not be a substantial thing, it may be the substrate of reality in which substantial things exist and are bent. Nothing may be able to bend space-time through force but only presence.
There is, then, possibly, no gravity but only curved space-time. Curved space-time gives the appearance of force where there is none. It is more like a built-in feature and not so much a forceful thing in scientific reality. Space and time themselves are not things but features of reality.
And, perhaps, it could simply be a built in feature of our universe that extreme distances of vacuum space create repulsion because space is so constituted. If it curves into mass, perhaps it curves out, against itself, where there is no mass in some symettrical and elegant way, the way in which nature typically reveals itself to us. This is just plain simpler and more elegant and not really too different from what we have already come to believe with a tweak of simplification.

These figures were created in the open-source freeware 3D imaging program, Blender.

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It is impossible to completely mentally represent these conjectures, I think, because they would look too complex. We can only imagine this conjecture in isolation, as in the pictures above. Space-time is not a plane but we can not visually represent what it would actually look like and make sense of it.
If this is right, then nothing "forces" space-time to curve, it just does in the presence of solid-state energy (mass). Saying that force curves space-time is like positing that something "forces" strings to vibrate in string philosophy/theory.
The strings, if they exist, simply do. Gravity in a black hole can simply be thought of as tightly warped space-time, so warped, that even light can not escape. But there are no gravitational strings acting on the light, it is simply that space-time is so wound up into itself that light can find no way to escape.
Gravity, then, could very well be thought of as a conceptual vestigial organ in science, heldover from the Newtonian paradigm and kept in relativistic theory only because it was so ingrained in the minds of scientific theorizers, theories, and educators. Like Pluto banished from planethood we should banish gravity from our understanding of space-time.
I do not mean to imply that the idea of gravity is not useful but only to correct our deepest level of understanding about it. We say, then, not that the apple fell on his head because gravity pulled on it but that the apple fell on his head because space and time were so curved toward the greatest nearby mass. It should be noted that Newton's laws can still and should be used in Earth applications where the relative sizes of objects basically negates the need for deeper theory. Although, nanotechnology may be able to use this conjecture.
If I am right, then no substantial carrier of gravity can ever or will ever be found.
Again, I could be completely wrong about these conjectures and I make no claim to certainty about any of this as no scientific theory should ever be so certain as to reject criticism.
Under this conjecture, the Big Bang never needed to have happened because the red shift we observe could simply be caused by the movement of super-massive bodies away from each other. This conjecture says nothing about the origins of the universe as no empirical theory, in at least the deepest sense, can.
This conjecture, would, if proven valid, fundamentally undermine contemporary cosmology, which is currently in a state of limbo anyhow. The universe could well be infinitely old if space-time allows for further super-super massive (super clusters of galactic super clusters), curving of space back inwards. But, this possibility and others can only be determined by empirical data.
This is a potential semantic and ontological clarification that has obfuscated a, perhaps, more elegant theory for some time.
The mathematics for this conjecture would look exactly like current mathematics for non-Euclidian relativistic space but would include new interpretations of space as it is curved in instances where mass is not present or in instances where mass has reached some threshold.
This is a conceptual reformulation not a mathematical reformulation but only a potential mathematical reformulation based on new empirical necessity for some mathematical reformulation of what has been thought to be forceful gravity but which I say could be accounted for by the accelleration caused by the curvature of space and time near massive bodies. What causes then, the curvature? What, then causes the warping of space and time?
Perhaps, I posit, it is only the presence of mass because it may be possible that the way matter affects matter (through force) and the way that matter affects space-time (through presence) are different. It is not necessary but it is possible and it fits our difficulty in finding substantial gravitational force in carrier particles and waves.
That gravity does not exist just makes things more clear because the term gravity refers to an ontological entity that I do not think helps, but rather, confuses, the situation we understand by our cosmological theories.
This is nothing more than clarified and expanded relativity theory.
I also posit that a refutation of my conjecture is possible by other possible conjectures such that Dark Energy is, simply the case in which space itself is a substiantial carrier of energy and, therefore is acted on gravity forcefully but in which case my latter conjecture that space is curved inwardly and outwardly is only true whereas the rethinking of gravity need not be.
Also, I accept that these conjectures are all possibly false but they will only be false empirically not necessarily a priori.
If there is a flaw in this conjecture it is that matter affects space differently than matter affects matter. But we say that matter exists in space, not that they are substantially identical but not that they are substantially different because space, is, I believe, we think, the absence of substance not some different kind of substance. How does matter affect what is not there, but empty? It may be different and that is all I have said.
All of this being said, I would never throw out the theory of gravity as an extremely useful concept in practical applications on Earth. I can't begin to imagine the complexity of equations that would have to exist in order to actually use this theory, in that it would require knowledge of the interaction of three-dimensional and dense topographical map interactions.
When would my conjecture qualify as a theory? If it works and can be found to work in a way that is verifiable, consistent, and falsifiable. Then it could be a theory but never a law because laws in science should never be thought of as so impervious to revision.
Maybe the reason we have not been able to squeeze gravity into our GUT(Grand Unified Theory)s or TOE(Theory of Everything)s is because it wasn't ever really there.