Quote:
Originally Posted by tdvance
Mathematics is amazing in that you can take some axioms that appear to hold in practice, do a lot of stuff on paper involving manipulating symbols without actually touching what you are modelling, derive some theorems, go out and measure things, and find out they agree with the theorem to many decimal places. Some author called it "the unreasonable effectiveness of mathematics". An example, take a rectangular patio. measure the two sides, square the lengths and sum them, then take the square root. A mathematical theorem says that this will be the diagonal length. Measure it, and find that it is right. In fact, if it doesn't come out right, it means you screwed up measuring!!!
Yes, if you use a poor-fitting model, it won't give the right results. But the fact is, good-fitting models have been found for so many things--Newton's "incorrect" mechanics is still good enough to put a man on the moon. The mathematical consequences of Einstein's general relativity give precise enough answers that GPS works--and we know this is not a complete model either because it contradicts Quantum Mechanics which also gives very accurate answers in its domain.
Roger Penrose suggested that mathematics is "unreasonably effective" because there is a "platonic world of mathematics" that exists independently of human mathematicians (i.e. if aliens did math, they'd come up with the same stuff, though with different names of course!), and that the universe's fundamental laws, whatever they are, are mathematically consistent. ... [snip]...
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From
Torsten’s
referenced paper:
Quote:
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Originally Posted by Wigner
The principal point which will have to be recalled later is that the mathematician could formulate only a handful of interesting theorems without defining concepts beyond those contained in the axioms and that the concepts outside those contained in the axioms are defined with a view of permitting ingenious logical operations which appeal to our aesthetic sense both as operations and also in their results of great generality and simplicity.
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More to the point, these are axioms chosen in response to conditions demanded to solve a particular problem, whether in the natural laws of physics, or probabilistic conditions observed. Once these axioms are correctly chosen, the rest of the math, which is nothing more than an ‘interrelationship’ of how these axioms interact, is useful in defining a solution, which when tested against real events gives predictability of what the solution should be. If the predictability fails, then the math may be inadequate, which may either be because axioms chosen are wrong, or the interrelationships defined are wrong. But when they are right, and predictability is achieved, then the ‘magic’ of math becomes evident. Though the math may be perfectly self consistent per axioms chosen and how they subsequently interact, it is not a language of the natural world unless it passes the test of predictability.
IMHO, where math and reality may become confused is when we use a limiting factor to our axioms, such as the 'light speed limit' of our observations (a reasonable axiom), to qualify the interrelationships that occur from this limiting factor. This is what Relativity is all about. But if the limiting factor applies only to our observations (we see with light), axiomatically, but not to how the observed interactions interact in and of themselves (which may be faster than light), then our axiom merely limits our ability to observe things in and of themselves, but not necessarily from their own intrinsic point of view: Viz., an event at time zero, and zero distance, will be different from the same event seen at a distance over time, since it is not limited by light c.
This, in effect, highlights an axiom of Relativity, that there are ‘no preferred reference frames’, which then limits our ability to understand the interactions without such light speed limitation. So from our local point of view, we are limited observationally, but this does not mean that (at a distance) the events under observation are constrained by our light speed limit. The end result is that we then are left with a light distorted view of the world, where natural laws interacting within themselves may not be constrained by the light speed limit; but we per our axiom of light speed limit c are constrained from seeing it, so we do not see it as it really is, merely as we can
observe it to be. If so, the math may be ‘beautiful and elegant’, but it may miss the point of what is actually happening, because of the axiom chosen. And if this is so, then that axiom of light speed limit c is not the right tool mathematically to fully understand nature on its own terms, if it interacts faster than light. (Of course, it is totally the right axiom if the universe interacts at light speed c only.) And if this is so, we have the wrong tool to understand the interrelationships of physics, except as limited by our observational limit; though what we observe is correct as an observational artifact, since we must use light or electromagnetic energy to observe phenomena at a distance, which is therefore corrected by the Relativity effect of light speed c. Taken outside of its ‘domain of applicability’ the result of observation may be different from the facts involved, if they interact at above light speed!
The only way to know if our observation is true or not is then to test it for predictability: Observationally, it will prove correct within its ‘domain of applicability’ and line of sight; but realistically, it may prove wrong locally (at time and distance zero), though we cannot know it! Why? Because our axioms chosen would not let us know we are wrong, since we chose axiomatically that there are ‘no preferred reference frames’ within the ‘domain of applicability’ for observations using light. But this may be wrong, since it
assumes that what we observe at a distance is what is being observed locally. This is true, but only if one assumes that there
is a ‘preferred reference frame’, that of the
observer, since the universe may work above the light speed limiting factor (and not the other way around). Circular reasoning then takes effect, where we are proven right within our domain of applicability, but
only observationally, and not necessarily true for the reality observed, which may break the axiom. Using the light speed limit c then of necessity gives us a limited understanding of what nature is really doing, if so. To break out of this circular reasoning can be done only one way, and that is with independent observation at a distance, where time and distance are both zero. But in astronomy this is impossible! So there is the conundrum for using mathematics as a tool of astronomy, which may prove correct observationally, but we do not know it to be true in fact. What we think we see may not be what is there. In fact, our ‘act of observing’ skews the result, just like in the Quantum world, because we are limited by the speed of light in all our observations.
And if distant aliens were to do the same mathematical observations they would come up with the same results, from their point of view within their ‘domain of applicability’, but with different units of measure, perhaps?
Unless... they do not use the light speed c limit in their mathematical modeling… different mathematical tools... they'd come up with a different 'Arelativity' model.
