Divine Neutrality, Blog. Science, Philosophy

Axioms

September 8th, 2015

Hypotheses I cannot prove; taken on faith

I’m thinking about my axioms of faith. On what unprovable hypotheses do I confront the world. My axioms are my prejudices; the fewer the better. The first one is this:

1. That reality exists. That a world exists in whose catalogue of entities I am one. And what I encounter of its other entities are my access to reality.

But illusions exist. They may be mistaken for reality but are not ‘real’. One such is that the sun sets! The sun’s apparent motion in the sky is an illusion. The effect is entirely due to the rotation of the earth on which we are fixed; that is the reality. Other illusions are not on such a grand scale; most are of a personal nature. That a snail speaks to you.

How are we to know that we know reality? How are we to distinguish between what exists in reality and our interpretation of reality? To distinguish objective from subjective? These words, ‘objective’ and ‘subjective,’ are the ones which are commonly used to partition the two; to differentiate reality from strictly personal experience.

Many philosophers contend that nothing is objective. You can’t ever be sure that you know reality. The moving-sun illusion epitomizes that idea. Until a few hundred years ago the whole world’s consensus was that the sun is an object that physically moves across the sky from east to west. Like a bird or clouds move across the sky. Mankind surely classed this as an objective fact. But, that the sun moves is a communally subjective experience. It is not an objective fact but an illusion shared by many.

Since our notions of what constitutes reality can change, how can we, in fact, trust that we know what reality is? The axiom of faith that reality exists is of no consequence without addressing how one can know reality. And there is no assurance that one can. So to proceed further I need another axiom of faith. It’s this:

2. That the ‘provisionally objective’ findings of science are genuine pictures of reality. So all we can know of this reality comes from studying the world via the methods of science. In exploring the world scientifically we are learning about reality; its parameters and properties, the laws governing it. That’s the axiom.

A problem arises in that science can change its mind. That ideas be revised in the light of new evidence is at the kernel of scientific inquiry. What is classified by science as objective reality may change. So how can we speak of an objective reality if what is considered to be objective turns out later not to be so. To cope with this we have to take ‘objective’ to mean the ‘provisionally objective’ idea of science.

The axiom, then, is that the current findings of science constitute what we know of reality. Science gives a valid picture of reality; pictures that contradict science are not valid. By this axiom what is classed as objective, changes with scientific understanding. So that the word, ‘objective,’ must be understood as ‘provisionally objective’. Objective is whatever current science says it is.

The methods of science include these two features:

a. To be classified as objective, an event must be verifiable. Events not verifiable by independent observers may not be ranked as objective.

No one knows the experience of another only the outward signs. The outward signs can be verified. The color called blue is consistently indicated by all parties. That this color is blue is thus verified. But whether, in fact, Smith’s experience of blue is the same as Snell’s is not verifiable. The experience is, therefore, subjective, not objective.

b. Theoretical deductions from verifiable experience about how the world works must bear the signature of refutability. Out of the theory must come a prediction for the outcome of an experiment. Such a deciding experiment is one that includes possible outcomes, which if observed, would refute the theory. This notion is charmingly portrayed in an essay by Karl Popper – as valid today as when written. (‘Science: Conjectures and Refutations’ in “Conjectures and Refutations: The Growth of Scientific Knowledge” by Karl Popper (1963) Routledge, N.Y.)

Using these axioms a remarkable finding emerges: that data collected via science can be represented by mathematical rules. We see that nature is governed by laws. These can be cast as mathematical equations. The universe is not capricious. There are underlying rules which govern its behavior. Order in the universe resides precisely in the laws that govern it. It is this that gives rise to the conviction that everything proceeds by natural process by which is meant, processes investigatable by science.

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What is Energy?

June 1st, 2015

An Abstract Reality

my wayWhat IS energy; that you buy it, use it, have it and notice it in others? (“The kids have such energy!”) You never see it, never touch it. Never smell it or hold it. What is the nature of such a substance? Energy is not coal. Nor oil. Nor sunshine. Nor boys playing soccer. These are said to possess energy. But what is this energy that they posses?

Here is the remarkable answer: “It is a numerical quantity which does not change when something happens.” R.P. Feynman 1962

That energy is a “quantity which does not change” means that if it grows less in one form then it must increase in another. So that the total amount doesn’t change. When I pay, say, $.15/kilowatt-hour for electrical energy, my purpose is to see it transformed into mechanical energy to run my refrigerator and into light energy at night for my reading convenience. Energy is something that can be transformed from one form into another. The totality of it persists (is conserved) as its form changes.

Electrical energy is made from the mechanical motion of wires in magnetic fields. The mechanical energy driving this motion comes from the thermal energy of heat which in turn comes from the chemical bond energy stored in coal or oil. There is energy of motion. There is energy of position – say of an apple feeling the force of earth’s gravitational pull. There is chemical energy stored in the binding of atoms to one another. There is nuclear energy stored in the binding of the nucleons inside the nucleus of an atom.

But what, then, is energy?

It is a mathematical attribute found to exist in nature. That there is a quantifiable something called energy emerged from the mathematics invented to describe the physical world.

The concept was born about 1807 when Thomas Young gave the first quantitative formula for the energy that is to be associated with motion. It was wrong by a factor of 2 but soon corrected. The energy to be associated with a mass, m, in motion moving at a speed, v, is given by the formula mv2/2. The energy of position (in the gravity field of the earth) to be associated with the same mass being at a height y above sea level is mgy, where g is a known numerical factor.

The genesis of the idea that such a thing as energy exists is exemplified in this joystick graphic. In it you see that a mathematical quantity is conserved throughout the turmoil of physical events.

You can throw or drop the apple by clicking on the appropriate button. The graphic numerically displays the apple’s attributes as they change with time; its speed (positive means ‘going upwards’, negative ‘going downwards’), its varying position and the time at which each is measured. (The actual time is displayed numerically but the motion graphic shows it slowed down for convenient viewing. A 1-second interval is viewed in 10 seconds.)

As the apple, in its motion, follows the laws of nature, its position (height above the ground) grows or shrinks drastically and its speed keeps changing all the time. But remarkably, an arithmetic function of its speed (the square) plus another function of its position is shown to be constant throughout the motion.

In the graphic, the sum shown is actually computed repeatedly from the two terms above it. This sum turns out to be the same number no matter what is going on in the motion. You may freeze the motion and check the computation at any stage of the trajectory. Then release the motion to freeze it again at another time – say 0.2 seconds later. You will see that although all the physically measurable quantities – the ones above the summation line – have changed markedly, the mathematical construction combining them is constant. This mathematical construction is conserved! It doesn’t change with time even though so much is happening.

It was from the reckoning illustrated in this joystick graphic that an idea was born. The idea is that there exists an ethereal mathematical something called energy that is conserved in all physical processes. Any such process can only change the form of the energy – not its magnitude. In the falling apple, position energy changes into motion energy. Mathematical expressions soon followed describing electrical energy, magnetic energy, chemical energy, heat energy and all the other forms. Over a hundred years ago Einstein taught us that mass is a form of energy. The equation for mass energy is E=mc2.

Afer a few centuries of hearing about it we have come to adopt, as something completely familiar, a mathematical relationship between measurable quantities. This mathematical insight brought a completely abstract and invisible property of nature into common acceptance. The word energy is popular with everyone. It has achieved recognition.

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Exponential Growth

June 8th, 2009

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 filling. 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?

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