But I don’t know which bizarre analytic properties are implied by an exotic structure over R^4. Surely the differential forms must have isomorphic structures, since they’re a topological invariant.

If this model is not correct (even if producing accurate results), how would we know and what might give us clues to where to look for a different model?

I will suggest that the clues come in the form of physicists predicting, desiring, new mathematical tools in order to further elucidate the current theories. This would suggest that the problems are actually mathematical in nature, rather than ‘physics-al’ at this point in time.

I think physicists are considering more advanced mathematical tools - extensions of current tools. What if there was a more basic change in mathematics needed?

Are there areas of mathematics that are having problems today? Not problems like an unsolved conjecture, but problems where multiple, non-complimentary, interpretations are being made by different mathematical scholars. Areas where mathematicians disagree.

So far as I can tell, such areas appear to be in the foundations and philosophy of mathematics. Infinities, liar’s paradox, others?

In terms of analytic ideas, the geometry of an infinite dimensional cube confuses me. Yeah, I have heard of and tried to grok the Hilbert cube. But I am trying for myself to understand even the most elementary functions, e, sine, cosine, in terms of volumes of prismatic simplicies.

On a related note and in relation to the initial question here, the terms in Pascal’s triangle correspond to the linear slices of the n-cube by the hyperplanes \sum x_j = k . The recursion relation is manifest as joins of the lower dimensional sets.

There is higher dimensionality all around us if we have the eyes to see it.

There are physics theories published in journals and the arXiv that presume 2 or 3 dimensions of time, as well as at least 3 dimensions of space. One of these was written up in the past few months by New Scientist (albeit they do a shoddier job on Physics than on Biology, as John Baez and Greg Egan have actively pointed out).

My wife (a Physics professor and science fiction author) has a bunch of linked short stories, someday a fix-up novel, about submicroscopic intelligent beings in the first seconds after Big Bang, able to travel in two orthogonal time axes. The anisotropies that became galaxies are distorted remnants of the infrastructure that they built.

Theory, and science fiction. No experimental data.

Oh, and as to linear, there’s a saying I coined at Caltech circa 1968 that caught on for a while:

“The road to hell is paved with linear approximations!”

Walt:
You are correct in presenting what I mean - that “we model time as a single real number, and assume that the properties of physical objects vary continuously with that real number.”
However, I am not overly concerned with the continuousness of the variable (that involves a much deeper mathematical issue) and am more concerned that we assume a single-dimensional variable is capable of adequately representing time.

To return to my last email, how do we define time? If we define it as representable by a single variable, then it becomes a tautological definition, also known as an assumption. I believe this is rather typical in physics today - from McGraw-Hill Dictionary of Scientific and Technical Terms 5th Ed.: “1. The dimension of the physical universe which, at a given place, orders the sequence of events.”

The definition implies a single dimensional variable. Are we justified in assuming this? If so, why? Can we define time without a tautological reference to a single dimension? What would that be?

I am suggesting that we may find a number of the strange aspects being put forward in physics models today might be solved differently if we do not limit time to a single dimensional variable. And I also suggest that we might then find better models.

What I think you’re saying is: we model time as a single real number, and assume that the properties of physical objects vary continuously with that real number. What if we need to go beyond that assumption in physics? That’s certainly possible, and physicists have entertained theories along those lines. I know that in loop quantum gravity they consider something more general called causal sets, but I don’t know much about it. There are lots of mathematical theories that have a single discrete or continuous time parameter in them, and do sometimes wonder what would happen if you wrote down some axioms for “causality”, and worked in a more general framework.

In the specific example you give of using logarithms to transfer power functions into linear functions, then mathematically no information is lost (you can always undo the transformation). In that sense, the important thing is whether the function really is a continuous function of our time parameter; then, we don’t lose any information by modeling time as a single real number. If there are additional parameters that we’re implicitly throwing away, then we are losing information.