Divine Neutrality

Living cells multiply. Their number grows exponentially. The more there are the faster they increase. An exponentially growing population has a doubling time: the time it takes for the population to double. Having a doubling time is a characteristic of exponential growth. In the time it takes each individual cell to divide into two cells, the whole population of cells doubles. The population doubles because each of its members doubles. Anything that is growing will eventually double in size and then double again and then again if one waits long enough. But only exponential growth has a single characteristic period of time that is the same for every doubling.

Things that do not grow exponentially are the distance the train carries you away from the train station or the amount of coffee in the cup you are pouring. These increase only linearly with time. The increase has a rate of growth but no characteristic doubling time. The time for the second doubling is not the same as that for the first doubling. Rather it is twice as long. The third doubling takes four times as long. So, in linear growth, no single period of time characterizes a doubling.

An amount of money invested at a compounded interest rate of, say, 7%/year, grows exponentially. It has a doubling time. Ten years. After 10 years the return on the investment will be as much as the original investment itself. The original investment will have doubled in value. Each of the dollars in it will have doubled. In the next ten years the money will have doubled again - to four times the original investment. The next doubling - to eight times the original amount - again takes ten years. The rate of growth, r, is related to the doubing time, T, by the simple formula: rT = Ln 2 = 0.7 (approx). At a growth rate of 10%/yr the doubling time is 7 years.

The motion diagram shows exponential growth through four doubling times at the rate of 7% per second. The exponentially growing brown bar doubles in size every 10 seconds. The green bar increases in size linearly at 7% per second. Clicking on GO starts the growth. Clicking on STOP freezes time. You can assess the doublings by stopping at 10 seconds (first doubling), 20 seconds (two doublings) and 30 seconds (three doublings) etc.

No matter what mathematics governs its increase, any physical quantity must eventually stop increasing. Nothing goes on increasing forever. Eventually the mathematics of growth fails to describe the phenomenon. Growth is never sustainable.

See Can growth be sustainable?

Jennifer says, with enthusiasm, “I love equations.”

What’s to love?

An equation is is a statement that says ‘this equals that’. It’s hard to imagine, from that raw and basic idea, that something called an equation could be of any use. Whether ‘this equals that’ or not seems a matter of little consequence!

But, in fact, we know that to state what things are equal can have powerful consequences.

Newton’s law - that the net force on something causes it to accelerate - is a matter of things being equal. Force = mass multiplied by acceleration. This law of nature governs an extraordinary panoply of phenomena: the solar system, the entire NASA program of space exploration, the working of engines, the nature of energy and thus of pressure and of temperature and thus our understanding of weather. It underlies thinking in engineering, geology, chemistry, biology, and, of course, physics. All from ‘this = that’.

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I am reading some wonderful papers on the Measurement Problem.

What strikes me is how measurement is visualized. It is visualized as taking place in a laboratory. The system - an isolated entangled state - encounters a macroscopic measuring device. In doing so the Hilbert Spaces of the two become entangled. Somehow a pointer state of the device must result. This is the scheme set down by von Neumann in 1932 and lucidly explained and expanded upon by Schlosshauer in a well written review article. Another gripping article is by Geoffrey Sewell, who says, effectively, that there is no measurement problem.

But is measurement about laboratories?

In the laboratory a photodetector signals the arrival of each photon and a counter accumulates the counts. It works because the photon is absorbed, ejecting an electron. (A current of electrons moves pointers.) The reaction

(photon + electron) yields (electron*)

is what marks the measurement.

But is not any green leaf a photodector? The photon gets absorbed via photosynthesis. The leaf’s vitality is a photon count accumulator. The reaction

(photon + water + carbon dioxide) yields (sugar + oxygen)

marks the ‘measurement’.

Surely every chemical reaction that goes to completion is a measurement event; the reactants disappear and the products appear. Isn’t every inelastic scattering a measurement event? In every such event the original quantum system is destroyed and something new emerges. It is just the property of any chemical reaction.

What, then, constitutes a measurement?

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