Karen,
A quick addendum to my comments above, which is a hypothesis:
In the absence of perturbative opportunities, the computational costs of fully formal methods for complete, end-to-end solutions trends towards infinity.
The informal proof is that full formalization implies fully parallel combinatorial interaction of all components of a path (functional) in some space, that being XYZ space in the case of approaching an electron. The computational cost of this fully parallel optimization then increases both with decreasing granularity of the path segment sizes used, and with path length. The granularity is the most important parameter, with the cost rapidly escalating towards infinity as the precision (inverse of segment length) decreases towards the limit of representing the path as an infinitely precise continuum of infinitely precise points.
Conversely, the ability to use larger segments instead of infinitesimals depends on the scale structure of the problem. If that scale structure enables multiscale renormalization, then the total computational cost remain at least roughly proportional to the level of precision desired. If no such scale structure is available, the cost instead escalates towards infinity.
But isn't the whole point of closed formal solutions is that they remain (roughly) linear in computational cost versus the desired level of precision?
Yes... but what if the mathematical entities we call "formal solutions" are actually nothing more than the highest-impact granularities of what are really just perturbative solutions made possible by the pre-existing structure of our universe?
Look for example at gravity equations, which treat stars and planets as pointlike masses. However, that approximation completely falls apart at the scale of a planet surface, and so is only the first and highest-level step in what is really a perturbative solution. It's just that our universe is pre-structured in a way that makes many such first steps so powerful and so broadly applicable that it allows us to pretend they are complete, stand-alone formal solutions.
So, I'll end for now with an even more radical hypothesis:
All formal solutions in physics are just the highest, most abstract stages of perturbative solutions that are made possible by the pre-existing "lumpy" structure of our universe.
So... ah... hmm! No, I'm not done. The above hypothesis is not radical enough. One more issue needs to be addressed.
Human cognition must rely on bio-circuitry that has very limited speed, capacity, and accuracy. It therefore relies very heavily in the mathematical domain on using Kolmogorov programs to represent useful patterns that we see in the physical world, since a Kolmogorov program only needs to be executed to the level of precision actually needed.
Furthermore, it is easier and more compact to process suites of such human-brain-resident Kolmogorov programs as the primary data components for reasoning about complexity, as opposed to using their full elaborations into voluminous data sets that are more often than not beyond neural capacities. In addition to shrinking data set sizes, reasoning at the Kolmogorov program level has the huge advantage that such program capture in direct form at least many of the regularities in such data sets, which in turn allows much more insightful comparisons across programs.
We call this "mathematics."
The danger in not recognizing mathematics as a form of Kolmogorov program creation, manipulation, and execution is that as biological intelligences, we are by design inclined to accept such programs as representing the full, to-the-limit forms of the represented data sets. Thus the Greeks assumed the Platonic reality of perfect planes, when in fact the physical world is composed of atoms that make such planes flatly impossible. The world of realizable planes is instead emphatically and decisively perturbative, allowing the full concept of "a plane" to exist only as unobtainable limit of the isolated, highest-level initial calculations. The reality of such planes falls apart completely when the complete, perturbative, multi-step model is renormalized down to the atomic level.
That is to say, exactly as with physics, the perfect abstractions of mathematics are nothing more than top-level stages of perturbative programs made possible by the pre-existing structure of our universe.
The proof of this is that whenever you try to compute such a formal solution, you are forced to deal with issues such as scale or precision. This in turn means that the abstract Kolmogorov representations of such concept never really represent their end limits, but instead translate into huge spectra of precision levels that approach the infinite limit to whatever degree is desired, but only at a cost that increases with the level of precision. The perfection of mathematics is just an illusion, one engendered by the survival-focused priorities of how our limited biological brains deal with complexity.
The bottom line is this even broader hypothesis:
All formal solutions in both physics and mathematics are just the highest, most abstract stages of perturbative solutions that are made possible by the pre-existing "lumpy" structure of our universe.
And looking at what I just wrote... yes, will be so bold as to assert with a high level of certainty that the above hypothesis is correct.*
In physics, even equations such as E=mc2 that are absolutely conserved at large scales cannot be interpreted "as is" at the quantum level, where virtual particle pairs distort the very definition of where mass is located. E=mc2 thus more accurately understood as a high-level subset of a multi-scale perturbative process, rather than as a complete, stand-alone solution.
In mathematics, the very concept of an infinitesimal is a limit that can never be reached by calculation or by physical example. That makes the very foundations of real mathematics into a calculus not of real values, but of sets of Kolmogorov programs for which the limits of execution are being intentionally ignored.
Given the indifference and often lack even of awareness of the implementation spectra that are necessarily associated with all such formalisms, is it really that much of a surprise how often unexpected infinities plague problems in both physics and math?
Cheers,
Terry
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* This real-time, on-the-fly restructuring of all of physics and mathematics has been brought to you courtesy of The FQXi Essay Program, 2017, which has encouraged just this kind of re-examination of fundamentals by folks like yours truly. How's that for blame shifting?... :)