One may ask:
How could a mistake in the arithmetic of physical representation possibly
have continued for three centuries?
The idea may seem unthinkable.
The following is my sense of what has happened. It is based on some
extensive reading, but I can't prove it step by step, and don't feel I have
to right now. Many people on *historia*Matematica* know more about this
ground than I do! Still, this is a scenario, that I believe may be mostly
true, or in significant resemblance to a story that is true. The scenario
makes an error in the arithmetic of representation "thinkable." The word
"guess" is to the left of every line in it, for clarity. I'm an engineer,
not a historian of mathematics. Professor S.J. Kline was also an engineer
(he did a sabbatical at Harvard's math department, and was considered among
the most mathematical and theoretical of computational fluid mechanicians.)
I'm hoping the scenario below suggests interesting things, and practical
things, to historians of mathematics. I consider it a reasonable "guess."
and would be interested in comments.
I sent a scenario essentially like this to George Johnson of THE NEW
YORK TIMES two years ago, and have been permitted some extensive space on
his forums since that time. Counterexamples to the Showalter-Kline position
have not been suggested there.
I hope this "guessed scenario" makes question A below more "thinkable,"
more plausible, and more potentially promising. Here is question A again:
A. Could the ARITHMETICAL rules we use in physical modeling have
been incomplete?
If question A is "thinkable" and right, something like the following may have
happened.
Robert Showalter
***************************************************
"Guessed Scenario"
Guess Once upon a time, at the end of the middle ages,
Guess people began to see clearer relations between
Guess mathematical description and nature than anyone since
Guess the ancients. Over time people came to surpass the
Guess ancients. A big break came at the end of the 17th
Guess century, when Isaac Newton, "standing on the shoulders
Guess of giants" devised the differential and integral
Guess calculus, and showed that this mathematical tool could
Guess be made to describe many things, terrestrial and
Guess heavenly. Although there were many difficulties,
Guess people throughout the next century were able to refine
Guess and apply analysis (applied calculus) to very many
Guess problems. Major difficulties occurred in the study of
Guess the motion of planets and moons, celestial mechanics.
Guess There was no direct way of relating the motion of three
Guess or more orbiting bodies together (a "naive" formulation
Guess "showed" that the interactions between three and more
Guess bodies always amounted to 0.) This was not then a
Guess "rigorous" concern, because at that time the notion of
Guess "rigor" did not exist. The thing was to solve
Guess problems, as cleanly and neatly as possible, but to get
Guess on with the work, neatly or not. Before the American
Guess Revolution, Pierre LaPlace and others worked out
Guess "perturbation" methods that approximated planetary
Guess motion very closely- closely enough to be the
Guess foundation of Nathaniel Bowditch's PRACTICAL NAVIGATOR,
Guess the standard work on celestial navigation for more than
Guess a century thereafter.
**************************************
Guess A "MISTAKE" had been made, and patched over by
Guess perturbation techniques. The notion of
Guess "dimensional parameters" or "dimensional numbers in
Guess the measurement domain" was not then thought
Guess of or defined. It was not then known that
Guess multiplication (division) of dimensional
Guess parameters and increments in extensive form was
Guess only defined when it exactly matched the
Guess definition of the dimensional parameters, but
Guess not otherwise. Arithmetic was applied to undefined
Guess entities, and trusted. Products of dimensional
Guess parameters and increments in an undefined form could
Guess yield false infinitesimals (so that interactions
Guess between multiple planets looked like they were 0).
Guess Sometimes undefined entities produced false infinities.
Guess The mistake was particularly hard to see because,
Guess most often, the false infinitesimals really were
Guess negligibly small. In many cases, the problematic
Guess infinities could simply be ignored.
**************************************
Guess No one noticed that this mistake had been made, and
Guess this mistake has only been identified by Showalter and
Guess Kline in the 1990's. For a long time, this mistake
Guess caused little difficulty, because perturbation theory
Guess usually provided good patches, and because people chose
Guess to work on problems that were not problematic.
Guess As the 19th century progressed, there were problems
Guess traceable to the mistake that are related to the
Guess separation of physics and mathematics as disciplines.
Guess Then, in the 20th century, the mistake caused a series
Guess of major difficulties in neural science and medicine,
Guess and may be suspected to have caused Einstein and others
Guess significant difficulties in physics, at the interface
Guess of Maxwell's electromagnetic equations and electron
Guess orbit behavior and elsewhere.
Guess Basic difficulties in celestial mechanics were
Guess central to a crisis in mathematics that continues to
Guess this day. That crisis of the 1880's bears many
Guess resemblances to the crisis of physics in the first part
Guess of this century. Until about the 1880's, mathematics
Guess and physics were considered to be the same field, or
Guess were not clearly separated. Perturbation theory
Guess became embedded in notions of differential equations
Guess tightly connected to celestial mechanics. Differential
Guess equations were defined by series that included
Guess perturbation approximations (patches on top of patches
Guess on top of patches without end.) Some significant
Guess calculations went wrong. Henri Poincare was able to
Guess show that the series that defined planetary motion
Guess (later essentially all differential equations defined
Guess by series) did not converge. The basic foundations of
Guess calculus were undermined, and remain undermined to this
Guess day. In the 1880's and thereafter, the "crisis of
Guess analysis" was felt to be so serious that people
Guess attempted to reform mathematics of radically other
Guess grounds, and have done so with significant but still
Guess incomplete success. The glorification of "axiomatics"
Guess and stark disconnection from the measurable world in
Guess math came after Poincare and others had shown that, for
Guess mathematical purposes, the mathematical world had lost
Guess contact with the measurable world. Many important
Guess mathematicians, including John Von Neumann, worked hard
Guess trying to find the error at the foundations of
Guess analysis, without success.
Guess At this time there was a schism between academic
Guess mathematicians, who distrust analysis, and engineers
Guess who have continued to use analysis as the foundation of
Guess their work, essentially as if the "crisis" never
Guess occurred. Kline and I are both of the engineering
Guess school - we use analysis because it works, and because,
Guess to do our problems, we have no alternative.
Guess I believe, but have not proved, that the
Guess problems in celestial mechanics can all be traced
Guess to misuse of the dimensional parameters in
Guess crossterms, and can be fixed now that the
Guess arithmetical limitations of the dimensional
Guess parameters are known. I believe that the same can
Guess be said for all the other sources of the "crisis of
Guess analysis." But to prove these things, on a large
Guess subject matter, will require work that is not yet
Guess done.
Guess In the 1860's and later, James Clerk Maxwell was
Guess setting out his electromagnetic equations. (During
Guess this same period and later, Maxwell devoted great
Guess effort, without success, to making clear connections
Guess between the measurable and the abstract, largely
Guess because he wanted sensible justifications for including
Guess or deleting terms in series. This work occupied much
Guess of his attention before he died unexpectedly. Maxwell
Guess set out but did not solve problems that have occupied
Guess George Hart and me in the 1990's.) In Maxwell's
Guess electromagnetic equation, an endless series of
Guess crossterms were truncated on the basis of a false
Guess "infinitesimal in the limit" argument. The truncated
Guess terms were far too small to detect with anything
Guess Maxwell had or could well imagine, but these crossterms
Guess become interesting at very high frequencies and very
Guess tight radii of curvature. To me, they appear to
Guess generate eigenvalue (quantal) behavior with
Guess nonradiating states for electrons orbiting about
Guess nuclei. At yet higher frequencies, there should be
Guess very many other (more or less stable) eigen-states,
Guess that seem to me like they ought to behave as
Guess "resonances" or "particles" (at higher and higher
Guess energy levels, without end.) I can only speculate as
Guess to whether the particle eigenvalues would group into an
Guess "eight-fold way" or not.
Guess In the late 19th century, and into the 20th,
Guess physics had to be redefined because of quantum effects.
Guess The domain of particle physics came to be, for reasons
Guess no one has clearly explained, RADICALLY separated from
Guess the ordinary, sharp world of classical physics.
Guess Quantum equations predicted some experiments very well,
Guess but did so in ways even the experts found counter-
Guess intuitive and counter-logical.
Guess The most famous of physicists spent much of his
Guess life trying to bridge the gap between quantum and
Guess classical physics, spending much time trying somehow to
Guess get quantal (eigenvalue) type effects out of Maxwell's
Guess equations. Insofar as I have read him, Albert Einstein
Guess seems to have used all the mathematical tools he had at
Guess hand, in many combinations, again and again. When
Guess Einstein said
Guess "Do not worry about your difficulties in
Guess Mathematics. I can assure you that mine are
Guess still greater."
Guess he was speaking the plain and painful truth. Could
Guess Einstein have bridged the gap between classical and
Guess quantum mechanics, and so unified much of physics? He
Guess would at least have had eigenvalue behavior out of an
Guess expanded set of Maxwell's equations, and that would
Guess have been a big step. Knowing what I know, I speculate
Guess that a unification of quantum and classical physics
Guess should be possible. Although some explanations would
Guess require modification, I'm not sure that many equations
Guess now in use that work well would have to be modified.
Guess In one way, quantum analysis WOULD have to be
Guess modified. Much of quantum calculation connects to
Guess problems with "infinities" that do not exist when the
Guess dimensional parameters are properly used. This point
Guess can be clearly shown by measuring conduction velocity
Guess in nearly pure water - the velocity is about 10,000
Guess times slower than the speed of light, and is dominated
Guess by a finite term that, in current usage, would be an
Guess infinity. There would be some analytical carnage
Guess related to this.
Guess A correct understanding of the use of the
Guess dimensional parameters has some other implications. A
Guess technique for numerically integrating coupled equations
Guess by computer, that is now much used, is inexact. The
Guess error is often negligible, but can be explosively
Guess large. The error appears to be important in controlled
Guess fusion calculations (that have been expensive,
Guess prolonged, and largely unsuccessful), and in nuclear
Guess weapon simulations (that do not work well, despite much
Guess effort.)
END OF "Guessed Scenario"
I placed the word "guess" beside every line of the speculation above
for a reason. I'm not a historian of mathematics, and cannot vouch for
any details. But the speculation above does seem reasonable to me, and
does illustrates how a buried mistake in the arithmetic of physical
representation might be consistent with the history of mathematics and
physics. Identification of that mistake, and fixing of it, might make
mathematical physics more powerful, more interesting, and more connected
with other fields, both academic and practical.
The main implication of this work, for me, is in neurophysiology,
where a modified neural conduction equation based on the work is consistent
with a great deal, and seems plausible to many. The issue there, set out
in "A Modified Equation for Neural Conduction and Resonance"
( http://xxx.lanl.gov/html/math-ph/9807015 14 July 1998)
is a plain matter of life, death, and very many dollars. To motivate the
direct, specialized tests on neural conduction that will need to be done
to prove or disprove my equation, tests that I cannot do myself, I need to
make question A "thinkable."
A. Could the ARITHMETICAL rules we now use in physical modeling be
incomplete?
People in the HM group may be the people best qualified in the world to
address that question. I am asking respectfully, fearfully, and carefully
for your comments.
M. Robert Showalter