In fact each of these examples is counterintuitive. Imagine yourself five hundred years ago, before science. What did people back then use numbers for? Counting things of commercial import: money, seed, goods, lengths, people. Numbers were also used in symbolism and religion: the twelve tribes of Israel, the four winds. They had nothing to do with images, communication, weather, understanding your body, or the like. In daily life there were no calorie counts, no 98.6 degrees for the normal human temperature, no speed limits, no wind speed measurements, no measuring your pulse with a stopwatch. Numbers had their niche in daily life and they stayed there. Certain mystics and accountants might devote their lives to numbers, but that was their specialty.
No sensible person would have suggested that numbers are at the center of everything from understanding your own health and sickness to facilitating dialogue between cultures to agriculture. To get an idea of how incongruous that statement would have seemed, pick some other important part of daily life (fire, postage stamps, clouds, etc.) and substitute it for "numbers" in the preceding sentence. Sure, you can find connections between fire and practically everything else, but saying that fire is at the center of communication would be more than a little far fetched. Someone who is not simply holistic or poetic about fire, but really believes that it is the answer to everything, would be considered at best a dreamer or enthusiast or mystic, and at worst just crazy. That's the same reception that any visionary of today's numbers would have received before the success of numbers.
Even our own language betrays the unusual importance we give to numbers. We now speak of "quantum leaps", and the word "quantum" refers to counting by ones; a quantum simply means a single indivisible unit, the difference between one and two. Imagine encountering a culture where people spoke of "fire leaps" as the apotheosis of progress. What would you think? And what were the chances that someone before the victory of numbers would have thought our present attitudes as anything but crazy?
The worst of it was that there were no hints that numbers might be so useful for so many things. Throughout the ages, our visual experiences never gave us any hint of the billions of numbers that could be used to describe them. No doctor putting his hand on a fevered patient's forehead ever had the temperature suddenly revealed as a number. Even at the height of concentration no musician began hearing numbers emanating from her instrument. Our universe appeared then as it ever does, without numbers.
I don't think that science does anything to take away the mystery of the link between numbers and nature; nor does it give warning of the limits beyond which numbers provide no insight. Instead natural philosophers simply began with the intuition that numbers are the most important way of describing their experimental observations, and then began inventing and refining machines which attach numbers to the world around us. Every number we use to describe our actual physical world was mined from the world using special machinery: thermometers, speedometers, cameras, etc. (Footnote: There are two important numbers, pi and e, whose values can be derived independently of machinery, but neither are observed directly in the natural world - this is related to the issue of physical units, which we may discuss later in this blog.)