Why do the constants of nature have the values they have?

The constants have their values because the world reads itself from inside, and only one setting lets the readings agree. Read one cell of the screen that bounds a region from outside as an area and from inside as the finest detail light can resolve, and requiring agreement gives a self-referring equation with exactly one solution. That solution is the fine-structure constant, at 1/137.035660 against the measured 1/137.035999, and closing that 2.5 parts per million is work in progress. Read the cosmic horizon the same two ways and its capacity fixes the cosmological constant. There is no panel of dials. A cell set to a different fine-structure constant would never agree with itself, so within this world no other value is on offer. If the self-reference equation for the whole universe has further solutions, those are other universes, and no signal could reach them. Nobody chose 137 any more than anybody chose √2.

Also in Chapter 36: Why These Laws and Not Others?, Chapter 33: Why Was Every Open Problem the Same Problem?.