"One-Half of Wisdom"

Far too frequently than I care to admit, I hit myself on the head wondering, “How come I never even thought of the question (let alone finding the answer)?”. Turns out Francis Bacon knew the importance of asking the right question:

“A prudent question is one-half of wisdom.”

Take the number ‘e’. Those with the maths background to know of it would probably have been introduced to ‘e’ via this equation:

In case you wondered what the term with the exclamation means: it is math-speak for “multiply all the numbers from 1 till that number” (Thus, 2! = 1 x 2; 4! = 1 x 2 x 3 x 4 and so on). Given we instinctively think of the number ‘e’ via the above definition, no wonder it seems like a number defined as an arbitrary sequence of calculations. And yet, this arbitrarily defined number pops up across science equations across topics as diverse as the charging of a capacitor or the rate of radioactive decay. The question I read on the Net was: Is the occurrence of ‘e’ in such diverse topics a coincidence? Or is there a deeper reason?

As that article explains, the right way to think of ‘e’ is via a different definition that make the continuous growth/de-growth aspect of the number obvious. And since a capacitor charges continuously, a radioactive material decays continuously… it should actually be a surprise if something that involves a continuous phenomenon does not involve the number ‘e’!

Another instance was about this statement: a photon of light always moves at light speed. And yet it never struck me: Whoa! Don’t they also say light moves at different speeds through different materials (glass etc)? So which is right?

The answer, as I learnt in Tales from the Quantum by Art Hobson, is subtle:

“Light moves more slowly through water or other materials, but this is because the material’s atoms occasionally absorb (destroy) and then reemit (create) each photon so that a photon’s average speed is less than light speed. Every photon, whenever it actually exists, moves at light speed.”

The explanation hit me with the force of a Richard Feynman explanation: it’s so obvious once it is explained like that.

And the last example is from an Infinite Monkey Cage podcast. While talking about how gravitational waves were discovered, the host asked the panel: if gravitational waves cause space to bend/warp/distort, won’t the length being measured change by the same amount as the scale used to make the measurement? In which case, won’t they just “cancel each other out” leaving you with no way to know that a gravitational wave had passed?

The answer requires a knowledge of the experimental setup:

•           It involved sending a beam of light and reflecting it back;
•           Measuring the time the round trip took;
•           Then calculating the length travelled using the known speed of light and the time it took.

See the answer? If you’re like me, probably not! So let’s spell it out. The length wasn’t being measured directly; it was being calculated via a time to travel measurement. And gravitational waves distort not just space but space-time, and nobody said space and time were being distorted by the same amount, ergo the calculated value would be different with/without a gravitational wave!

A prudent question followed by the answer is the best way to wisdom indeed.