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When is explanation important?

1. When is explanation important?

These are Informal notes on the importance of explanation in the context of scientific theory (rather than pedagogy, where it is also important). These are heuristic notes, trying to lay ground work for clarifying a concrete, practical problem in science. Comments about any errors of argument, or history or philosophy of science are warmly welcome.

Explanations are important when there are unsolved problems with a theory. With what theories are there no such unsolved problems?

2. When explanation is less important

When there is only one theory, and its scope of application is universal, and it is clear how to apply it in new cases, and the theory is relatively easily to teach and use and understand, it makes predictions which are never contradicted by credible competent repeatable experiment and it isn’t generally thought of as problematic.

Partial examples of situation where explanation less important

• Newton's theories of motion and gravitation before the electromagnetic storm clouds formed. (Though people worried with some justification about action at a distance, and Mach's Principle, and a few other things. Various attempted naive "explanations" failed to account for the phenomena.)
• Euclid before non-Euclidean geometries. Lobachevsky etc. (though lack of good explanation for parallel postulate bothered some theorists)

3. When explanation is important

• When there are no good theories.
• When there are two or more competing theories.
• When dealing with 'vexatious litigant' theories.
• When the scope of validity or application of a theory is unclear.
• When it is unclear, or there is no consensus, how to apply a theory in a substantial range of cases.
• When the theory cannot be explained satisfactorily to proverbial 12 years olds or undergrads.
• When no transferable supplementary non mathematical intuitions about the theory have yet been developed.
• When predictions are met only some of the time.

4. Examples of key, decisive explanatory moves

Note – paradoxes are sometimes negative, but valuable explanations - explanatory moves that show that something in the current theory is currently unexplained and may be wrong.
0/ Newtons "hunt the force" idea
1/ Einstein catching up with a light beam
2/ Einstein falling freely
3/ de Broglie and electrons
4/ Realizing that i is instantiated by rotation (or is always about rotation)
5/ Feynman "Strange case of light and matter" sum of the histories
6/ Deutch argument for reality many worlds simply of basis of youngs slits (no QM needed)
7/ Gailileo explanations for what he saw on moon
8/ Gallileo explanation for satellites of Jupiter
9/ Gallilean relativity argument
10 /Hestenes realising that geometric algebra subsumes large and obscure areas of mathematics in physics
11/ Robinson and non-standard analysis
12/ Feynman explanation of conservation of momentum (for beginners)
13/ Avogadro
14/ Arguments collated by Lucretius for atomism
15/ Zenos paradox
16/ Cantors arguments
17/ Ways forward from Bohr atom - esp stopgap move by Heisenberg.
18/ Crick and Watson
19/ Newton & Leibniz doing something that looked suspiciously like dividing by zero but inventing calculus (this is a hugely illuminating example with big implications for pedagogy)
20/ The evolution of the concept of complex numbers

5. Examples where lack of taking explanation seriously caused (or compounded) failure or missed opportunity

• Planck's constant – while still uninterpreted by Planck
• Copenhagen interpretation
• Lorenz before Einstein
• Dirac half-hearted prediction before empirical discovery of positron
• Astrology and precession of zodiac (ha ha only serious as instructive example)
• Famous comment by American academic on mystery of Euler's equation
• Problems lurking in Maxwell

6. Places where mathematics won big while mostly ignoring explanations

Happens all the time. Does not contradict role of explanation. At one level exemplifies it. But see below on best practice.
  • Dirac
  • Gell-mann
  • Maxwell
  • Heisenberg and Schrodinger (Deliberately listed for for more than one strategy - since explanatory strategies can be applied at different levels)
  • Most mathematics as conventionally taught

7. Best practice - Combined Strategies

Best practice is for explanation and mathematical models and experiment to be in constant intricate dialogue.
Mismatched but but highly successful example of this using two incompatible theories, no less, was Bohr interpretation constrained by correspondence principle. This kind of thing happens much more than one might expect - cf physical explanation for i to the power i.

To Do

  • Dig out detailed case studies where explanation was essential to the heavy lifting, and where long or critical volleys of interplay between the math & the explanation & experiment proved vital.
  • Be explicit about why conceptual metaphor ect is more than just impedance matching of arguments to our animal brains - or why this can be of decisive importance in such episodes.

Great question from Alistair Willis

Why Eulers equation?
Fact about universe or human minds?
My take - fact relating two ways to do same physical task - so fact about physics -
but if you like, via Tegmark a fact about math underlying physics,
and via evolution, fact about our cognition