Yes, we expect the laws of mathematics to still apply.
The laws of physics also apply, but we don't know all of them. For example Newtonian physical vs. relativity. Both are correct, but one only comes into play at high velocity or energy. Same with the big bang, there may be a more complete description of what happens that only comes into play at high energy.
This doesn't mean the current laws don't apply, but rather they are inaccurate, but we can't tell since the inaccuracy is too small at our energy level.
A much better mental model of this stuff is Newtonian mechanics is always wrong at every scale. At human masses, velocity's and time scales you can't detect the error but it's still there.
As such the physical laws are unchanged at the big bang, but the simpler aproximations in current use may be noticeably off at these scales. If they are, then they would also be wrong at all other timescales and energy levels etc. The trick is looking for edge cases we can detect that seperate our aproximations from the actual laws running reality.
Given all that; simpler aproximations continue to be useful as long as the errors are hard to detect.
Math is independent. If physics works different, it may not support humans, so there's no one around to think those thoughts, but if you accept the axioms of math everything else follows.
Granted, in some bizzaro physics 2+2 may actually equal 3, (along these lines http://lesswrong.com/lw/jr/how_to_convince_me_that_2_2_3/) but you should still be able to construct a successor function and mechanically evaluate 2+2 = 4 according to our rules, even though the answers would seem weird.
We may indeed be assuming stuff we don't know we're assuming. Still the other people would see, oh they're assuming X which is wrong. but the other people take X to be true, it all works out.
It's always been my understanding that the normal laws of physics don't work at big bang time as well. One thing I've wondered about is how people are so adamantly confident in the theory of the Big Bang in light of this. Is it simply because they don't like the alternatives, because they've never been heard an alternative, or is there a better reason?
The big bang hypothesis could be stated (in a simplified way) by stating that the universe was once a lot more hot and dense than it is today. Using General Relativity, we can model the expansion of the universe. When we do the math, we find that out come correctly the relative abundance of light elements in the early universe as well as the spectrum for the cosmic microwave background, and many other quantities. It holds very well together.
If we extrapolate far enough back, we reach a point where the predicted density would be infinite. We call that point the beginning of time (or the big bang). This "prediction" of an infinite density is taken as a sign that our model breaks down at (or before) this point (by "before" I mean when we extrapolate backwards in time towards the infinite density time...).
So, it works extremely well for a time when the density was extremely large (but not infinite) until today. We just don't have enough information to know what would happen when densities larger than a certain value so as to make meaningful predictions.
To make a completely silly analogy: it would be like predicting the motion of a rocket aimed for a comet. Our knowledge of celestial mechanics is good enough to predict the trajectory up until the rocket "touches" the comet. We just do not have enough information about the comet composition to know if the rocket will just blow it apart or if it is the rocket itself that will splatter on the comet, or whatever.
NASA uses Newtonian mechanics to send spacecraft to the outer planets and beyond. Everybody knows that Newtonian mechanics stop being accurate in certain conditions, but that does not prevent it being an incredibly useful model outside those conditions.
The best experimentally-verified cosmological models we have point to the fact that the Universe was in a hot and dense state a few microseconds after a specific point of time about 13.7 billion years ago. Extrapolating those few microseconds back in time we get an unphysical result from our current models, but that simply means those models are inaccurate if applied to the beginning of time.
Contrary to the popular belief, The Big Bang model simply does not concern itself with what happened at the exact moment of Big Bang; what it does is describe the evolution of the Universe after that, and does it well.
>Contrary to the popular belief, The Big Bang model simply does not concern itself with what happened at the exact moment of Big Bang; what it does is describe the evolution of the Universe after that, and does it well.
That sounds as much of a cop out as the answers religious people give.
> It's always been my understanding that the normal laws of physics don't work at big bang time as well.
That's wrong. As soon as the universe's mass/energy expanded sufficiently for there to be a time dimension (IOW as soon as the universe was something other than a singularity), the present physical laws applied. Some of the physical constants with which we are familiar may have had different values, but the same set of rules have been in effect since there were four distinct dimensions. This is borne out by the existence and properties of the CMB, as well as other observations.
Your use of the term "big bang time" implies that there was, in fact, time, which means four dimensions existed, which means our present physical rules were in effect.
One idea has it that the natural forces we evaluate separately now, were unified by the conditions of the early universe, but this idea doesn't contradict the notion that basic physical principles were in any way different then.
One of the more interesting cosmological ideas is that the positive mass/energy of the universe is exactly balanced by negative gravitational potential energy, as the universe expands and continuing to the present, but only at one specific expansion velocity: what we call escape velocity. If this idea is true, it means the universe could have sprung into existence spontaneously, like a virtual particle pair, without violating the law of conservation of energy. Stephen Hawking explains this idea in his book "The Grand Design": "Because there is a law such as gravity, the universe can and will create itself from nothing. Spontaneous creation is the reason there is something rather than nothing, why the universe exists, why we exist. It is not necessary to invoke God to light the blue touch paper and set the universe going."
It's amusing how confidently you state "wrong" and then make the same logical error in using a time-dependent conjunction ("As soon as...") in your own description.
The OP specified a time other than zero, so there are four dimensions. This means the usual physical rules apply. My use of "as soon as" is completely consistent with the era specified by the OP, i.e. when four dimensions are available and physics is "normal". And I made this perfectly clear in my reply.
It depends on what you mean by "big bang time". Standard physics started a few microseconds after the big bang, followed by what appears to be a rapid expansion of the early universe.
The Big Bang is widely adopted in light of this. Preceding the expansion there has to be a state of greater compression, but there is no explanation on how this state could be possible.
Only by ignorant people..
The more knowledgeable one will tell you that the big bang is the description of the state of the "young" universe but that we don't have the tools to describe/know what was before/what created it (yet).
The big bang, unlike theology, tells us why distant objects are more red than close objects, why we can't see anything further than ~13bil light years away, why there is microwave noise everywhere, etc. The mathematics that describe all those things imply certain things about earlier cosmological history.
Yet. As we gather evidence ever closer to the actual moment of the big bang (such as the possible recent discovery of a particular kind of polarization in the CMB), we might figure out ways to prove or disprove theories about what might have preceded the big bang.
As I understand the process, theoreticians will invent mathematical constructs that explain everything we have measured already, plus make predictions about something else we can test. If all of the testable predictions are verified, then we are somewhat safer in accepting any untestable predictions of the same model. Then, an expanded mathematical theory will be devised with new physical tests, etc. ad infinitum.
The laws of physics also apply, but we don't know all of them. For example Newtonian physical vs. relativity. Both are correct, but one only comes into play at high velocity or energy. Same with the big bang, there may be a more complete description of what happens that only comes into play at high energy.
This doesn't mean the current laws don't apply, but rather they are inaccurate, but we can't tell since the inaccuracy is too small at our energy level.