You asked for examples of predictions about the world that can be made based on understanding physics minus the math. It would be disingenuous in the extreme to then dismiss all examples of predictions about the world given because they are, in fact, mere predictions about the world and therefore could have been memorized without real understanding.
However, there is indeed a difference between rules memorized in isolation without any additional understanding and, on the other hand, real understanding of physical theories and generating predictions based on them. Just like there's a difference between understanding electromagnetic theory and knowing a few common-knowledge technical facts on how to deal with electrical devices, there is also a difference between understanding relativity and knowing a few facts that follow from it in isolation.
You are now employing a rhetorical tactic where you try to make this obvious and relevant point look like weaseling, but in reality it is a pertinent and adequate response to your example.
In the various responses you have been given your claim that you can't have any understanding of physics without math has been overwhelmingly refuted. All you are left with is "But that understanding isn't true understanding, true understanding means you remember the math!" To that I reply "No, not all Scotsmen like haggis. You can only tell a true Scotsman by the kilt they are wearing!"
This is sheer rhetoric. You latch onto one point I made, completely ignoring the context and making the most extreme uncharitable interpretation of it (one that is in fact bordering on caricature), all for the greatest rhetorical effect. Instead, a rational approach would be to see if there may be some validity behind my point even if its original statement was imprecise, especially since I readily admitted this imprecision on first objection. Not to mention that your own example is largely irrelevant in the original context, which was about lengthy pop-science works purporting to explain whole physical theories to lay audiences, not about isolated examples such as yours.
In any case, if you think the distinction I outlined above is invalid, or that I am applying it incorrectly, please go ahead and explain why you believe that. If you're going to latch onto a caricature of what I wrote while treating the discussion as a rhetorical context, I have no further interest in continuing this exchange.
I think it might be helpful for you to taboo the word "real understanding". It seems like a lot of the disagreement stems from luke and wedrifid being unable to understand what you mean when you use that phrase. To be honest, while I agree with many of your points, I also don't think I understand what "real understanding" is supposed to mean. Perhaps you could restate your original point without use of the word "understanding"?
Richard Dawkins
My private school taught biology from the infamous creationist textbook Biology for Christian Schools, so my early understanding of evolution was a bit... confused. Lacking the curiosity to, say, check Altavista for a biologist’s explanation (faith is a virtue, don’t ya know), I remained confused about evolution for years.
Eventually I stumbled across an eloquent explanation of the fact that natural selection follows necessarily from heritability, variation, and selection.
Click. I got it.
Explaining is hard. Explainers need to pierce shields of misinformation (creationism), bridge vast inferential distances (probability theory), and cause readers to feel the truth of foreign concepts (quantum entanglement) in their bones. That isn’t easy. Those who do it well are rare and valuable.
Textbook writers are often skilled at explaining complex fields. That’s why I called on my fellow Less Wrongers to name their favorite textbooks (if they had read at least two other textbooks on those subjects). The Best Textbooks on Every Subject now gives 22 textbook recommendations, for fields as diverse as scientific self-help and representation theory.
Now I want to jump down a few levels in granularity. Let’s pool our knowledge to find great explanations for each important idea (in math, science, philosophy, etc.), whether or not there is equal value in the rest of the book or article in which each explanation is found.
Great explanations, in my meaning, have four traits:
A great explanation does more than report facts; it uses analogy and rhetoric and other tools to make readers feel the target idea in their bones.
A great explanation is not a single analogy nor a giant book. It is, roughly, between 2 and 100 pages in length.
A great explanation is comprehensible at best to a young teenager, or at least to a 75th percentile college graduate. (There may be no way to seriously explain string theory to an average 13-year-old.)
A great explanation is exciting to read.
By sharing great explanations we can more often experience that magical click.
List of Great Explanations
I’ve barely begun to assemble the list below. Please comment with your own additions!
(The list below is exclusive to written explanations, but feel free to share your favorite explanations from other media. My favorite explanation of BASIC programming is a piece of software from Interplay called Learn to Program BASIC, and of course many people love Khan Academy’s videos and The Teaching Company’s audio courses.)
Epistemology
Aumann’s agreement theorem: Landsburg, The Big Questions, chapter 8.
Occam’s razor: Yudkowsky, Occam’s razor.
Math and Logic
Physics
Special relativity: Wolfson, Simply Einstein, chapters 2–12.
General relativity: Hawking, The Universe in a Nutshell, chapters 1–2.
Infinite, flat universe: Greene, The Hidden Reality, chapters 1–3.
Timeless reality / block universe: Greene, The Fabric of Reality, chapter 5.
Inflationary cosmology: Greene, The Hidden Reality, chapter 3.
Rainbows: Dawkins, The Magic of Reality, chapter 7.
Biology
Psychology
Anchoring: Kahneman, Thinking, Fast and Slow, chapter 11.
Availability heuristic: Kahneman, Thinking, Fast and Slow, chapters 12–13.
Prospect theory: Kahneman, Thinking, Fast and Slow, chapters 25–26.
Modularity of mind: Kurzban, Why Everyone (Else) is a Hypocrite, chapters 1–4.
Economics