*

NEWTON'S DERIVATIVES

Arthur M. Young

Modern science had its origin in Isaac Newton's extension of geometry to include motion. Until then geometry had been a science of position. The first order of motion was velocity, the rate of change of position with respect to time, or, as Newton called it, a fluxion. The second order of motion was the ratio of change of velocity to time, or acceleration. Gottfried Leibniz made the same discovery, and his name for these ratios, derivatives, is the term now used. Through the use of these derivatives Newton defined force as mass times acceleration, and momentum as mass times velocity. Energy, or work, was later found to be distance times force - or feet (distance) times pounds (force). Power was the rate of doing work, the derivative of energy.

These quantities, most of which are derivatives with respect to time constituting the measure formulae of physics, have become the basic vocabulary of the science of motion. They make it possible to describe and predict the motion not only of the planets but of any inert body. This led to the philosophy of determinism, the theory that an all-knowing mathematician, the LaPlace mathematician, knowing the velocity and position of all the particles in the universe, could predict their future.

Note that these measure formulae, made possible by the concept of derivatives, with the exception of power, do not go beyond the second derivative. Energy is (the famous ), and power is , the third derivative of moment of inertia. Are there other third and higher derivatives? While in theory they would exist, such derivatives are not used, and have been ignored by theoretical science. To see why, we must remember that the laws of motion are considered to apply only when energy is not added to or subtracted from the system. Thus the laws of motion prescribe that a pendulum will swing indefinitely provided there is no friction. Science thus deals with a hands-off or ideal case. Newton thought the orderly motion of the planets was evidence of God, but Pierre LaPlace told Napoleon that their orderly motion made the hypothesis of God unnecessary. There began to be a split between science and free will, with science holding to the view that the laws of motion, which correctly predicted the behavior of most bodies, could also account for living organisms. As Albert Szent- Gyorgyi put it, "As scientists we cannot believe that the laws of nature lose their validity at the surface of the skin." Szent-Gyorgyi didn't leave it at that, but went on to show that something else, some drive, was needed.

This split becomes apparent in the difference between science and engineering. The scientist tends to think of the laws of nature as inviolate; the engineer thinks of the laws of nature as something to be used to make machines that work. It does not occur to either of them that when they control a mechanical device - by adding or subtracting energy from the system - that this interference does not involve any violation of nature's laws.  

Thus it is possible to control nature and make it do what you want it to do. While it would not be practical to cause Mars to change its orbit, it has been possible to control an orbiting satellite to fly past Mars, to visit Jupiter, and by guiding the satellite to take advantage of Jupiter's gravitational field, to get the extra impetus to carry it to Saturn and beyond.

But how, if the laws of nature are inviolate, can they be taken advantage of? How do we square this opportunism with Newton? How can creatures, themselves the product of laws, produce results that could not occur in nature as interpreted by science?

To answer this question consider the derivatives beyond the first and second. What would the third derivative be? The first, or velocity, is rate of change of position (governs position). The second, acceleration, is rate of change of velocity. It follows that the third is the rate of change of acceleration. Now change of acceleration is what we do when we drive a car, by pushing more or less on the accelerator pedal, pushing the brake pedal, or steering. It is our control of the car, and is effected either by adding energy to the system or by withdrawing it. Control is a free option, to be used by the driver.

So the laws of nature, so often invoked to support determinism, do nothing of the kind. The third derivative, or control, has the same right to status as velocity and acceleration. It is not so much one of the laws of nature as it is an implicit permission to use nature's laws.

But why is control ignored by theoretical science? It is true that since it is an option, it cannot be measured as can velocity and acceleration. It may also be true that it does not contribute to the edifice of exact laws so respected by science. This does not justify the neglect of control in cosmology in the old sense, one that includes life, where the belief in determinism would make self-maintenance, or control, an illusion. Surely plants grow and store energy against entropy by controlling their metabolism; and animals, while subject to instinctive drives, must use control in pursuit of prey or to avoid capture.

Here we might take time to answer the claims of behaviorism, whose prestige is based on the assertion that living creatures are subject to "drives" just as inert bodies are subject to laws, and that therefore consciousness is a superfluous or erroneous notion. But hold on a minute. Let us admit that when, say, a seal migrates northward in summer for breeding purposes, it does so in response to a drive triggered by the seasons. Even if this were interpreted to mean the seal has no free will, note that the seal is an organization of many trillions of cells, and each cell an organization of trillions of molecules. This enormously complex association of molecules behaves in unitary fashion, and not according to the Newtonian determinism that would apply to the individual molecules if they were not so organized. How does the seal control all those molecules in a way that Newton's laws would not? Even if we say the seal has no free will, it does have control of its own metabolism, of its musculature, of its growth and self-reproduction. Instinct is not due to laws of gravitation and electromagnetism.

While we cannot release the behaviorist from some responsibility for his interpretation of instinct as equivalent to Newton's laws, the real blame falls on the theoretical physicist who draws his credo, his dogma, from a partial reading of the derivatives.

Of course the physicist is entitled to define his own discipline, and if he wants to base this discipline on the first two derivatives only, he is at liberty to do so. By the same token, he cannot claim to know the workings of a universe that includes life.

What about the derivatives beyond the third? It might be thought that since the third derivative is an option, there is no point in going further. We cannot even measure free option, much less find its derivative, but a closer scrutiny shows that there is the equivalent of a fourth derivative. What is it that changes (governs) control? While control is an option, or at least not mechanically determined, it is also not completely free. If a child runs in front of the car, we steer to the side or put on the brakes. If the road curves, we steer accordingly. Our control of the car is continually governed by its position relative to other cars and objects. In fact, our destination is the ultimate governing factor.

Our destination is a position - not the position we started with, but the same kind of measure. It is something we observe - it is not velocity, which we can compute, nor acceleration, which we feel; nor is it control, which we exercise. If the fourth derivative takes us back to position, which was the zeroth derivative, we have what is called a four-operator. After four 90° steps, we get back to the starting point, position.

 

This concept has important implications for science. Not only does it limit the time derivatives to four, but it permits an important step in finding a definition of dimension in terms of angle. Since dividing by time four times brings us back to the start, we can equate division by time to a rotation of 90 degrees. (Elsewhere I have shown that mass has the angular dimension of 120°, and length 30°, thus providing a quantifiable relation between M, L, and T.)

The correlation of time to 90° is confirmed by the fact that square root of -1, or i, is used as a coefficient of time in the quadratic formula for interval in relativity, as well as in equations for oscillation, such as the Schroedinger equation. When multiplied by itself four times, i returns to itself:

1.  i

2. i x i =-1

3. –1 x I = -i

4. –i x i = +1

The fact that physics has so far stopped midway through the cycle at (in both the second derivative, acceleration, and in the quadratic formula for space-time) is an omission of crucial importance. It means that physics - and the rest of science following its lead - has tended to regard time as fundamentally linear, rather than cyclic.

 

Mathematics, Physics & Reality

 

Mindfire