Mathematics and Reality - Introduction to Mathematics and Reality

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INTRODUCTION TO MATHEMATICS & REALITY

Arthur M. Young

"The most abstract of all the sciences is mathematics. . . . Mathematics is only busied by purely hypothetical questions. As for what the truth of existence may be, the mathematician does not care a straw."

- Charles Sanders Peirce

To state at the outset my reason for writing this book, I am a lover of theory, and because I love theory, I love mathematics. But theories may not be fully developed, or they may be incorrect. Even if the physical world with its recalcitrant facts, its objects that get in the way of theory, had no other function, it would be invaluable because it exposes the flaws in theory. It forces theory to correct itself and invariably reveals that there is an even better theory to be discovered.

Theory, even if it has to be revised in the face of new facts, is always the ultimate victor. Theory grows and builds on itself, achieving a noumenal status, whereas facts cease to be important after their work of correcting theory is accomplished. They remain phenomenal.

The physical sciences began with Natural Philosophy, the recognition of the importance of this interchange between fact and theory. But as mathematics developed, it became increasingly independent of the physical applications so essential to physics. Mathematics differs from the physical sciences in that there is a gift peculiar to mathematicians that has enabled giants like Euler, Newton, Gauss and others to create whole worlds of relationships out of thin air, in which they sustain themselves like birds on the wing with no need to touch the earth. It is this gift that justifies Peirce's statement, for it is incontrovertible evidence of a "higher world" that has its own laws and needs no validation from the world of facts.

But the very perfection of mathematics can so captivate the minds of physical scientists that they neglect their prime mandate, to respect facts and revise theories.

This brings me to the subject of the present work. Often it is not that a theory is incorrect, but that it has been misapplied. Rather than try to argue an issue that might require volumes and would still leave questions unsettled, I will mention one example illustrating the unfortunate consequences that follow when a valid mathematical formalism is erroneously interpreted. (The subject is discussed in greater depth in Part Two.)

Newton's calculus is a theory that makes it possible to deal with change and thus to extend measurement to include motion and permit prediction. This was the birth of western science as distinct from geometry, which does not deal with motion. Thus Newton's fluxions, now called derivatives, were rates of change. His first derivative was the rate of change of position with respect to time, or velocity; his second derivative was the rate of change of velocity with respect to time, or acceleration.

But there is a third derivative, the neglect of which in physics has led to the widely-held assumption that determinism negates free will.

In deriving the laws of motion it was necessarily assumed that energy is not admitted or subtracted from the system (the closed system), and for this purpose the third derivative is not required because it is dependent on the others. But the third derivative, or change of acceleration, is control. Because of the third derivative, we can drive a car. It is our option to control the car's speed or direction, and thus reach a destination.

The existence of the third derivative, with the same formal status as the first and second, because it enables us to take advantage of the determinate laws governing inert matter, removes all conflict between free will and determinism; indeed, the reliability or determinism of the world of physical objects, including our bodies, makes freedom of choice effective.

This example shows that the laws of matter (Newton's calculus), far from denying free will, not only sanction free will but show how its scope and power are extended into the world.

It is often argued that science deals with only one aspect of the world and is not concerned with freedom and values. But this is not the point I am making. I propose to show that an examination of the ideal world of mathematics will indicate not that it is not ideal, but that mathematics, in spite of itself, contains in addition to its formal elements, other aspects such as value and purpose not recognized by current science; and it owes it to itself to develop or at least acknowledge these other functions.

I said that I was a lover of theory. Perhaps I should rather say, a searcher into first principles. I believe that in addition to its other duties, mathematics could, if properly interpreted, become a science of first principles and hence truly queen of the sciences.

Francis Bacon, often credited with setting modern science on its course (and also blamed for doing so), prescribed the role of science as investigation into secondary causes. First causes, he said, were the province of philosophy and religion.

But religion has increasingly concerned itself with ethics, and in its division into sects it has become more concerned with dogma than with first principles. Philosophy, outdistanced by the advances in science, has been pushed into the background. In fact, the prestige of science has so surpassed that of philosophy that philosophy no longer has the courage to challenge science even on the matter of first principles.

In any event science itself has outgrown its limitation to secondary causes. The major problems with which science is now concerned-the origin of the universe (currently referred to as the Big Bang), the nature of ultimate particles, the mass discrepancy of galaxies, the experimental evidence of phenomena that transcend space and time as in the EPR experiment-are problems that involve primary causation. They are problems that will require changes in the philosophy of science, but because they will necessarily require experimental evidence, they cannot be solved by philosophy alone.

There can no longer be a division between mathematics and physics, between science and philosophy, perhaps even between science and religion.

Ken Wilber, a contemporary philosopher who was himself at one time a scientist, takes the position in his anthology, Quantum Questions (1984), that there can be no wedding of science and religion. This view he supports by quoting a number of prominent scientists from the first part of the century. But science and religion had the same beginning, man's interest in his origins, and both deal with the same universe. So to say they can never be reunited is to make their division, originally a concession to convenience and expediency, into a fundamental principle.

We could say that oil and water won't mix, but this is at the molecular level. At a deeper or more fundamental level both are made of atoms; if you burn oil, you get water.

As Ilya Prigogine writes in Order Out of Chaos (1984):

For the ancients, nature was a source of wisdom. Medieval nature spoke of God. In modern times nature has become so silent that Kant considered that science and wisdom, science and truth, ought to be completely separated. We have been living with this dichotomy for the past two centuries. It is time for it to come to an end. (pp.88-89)

What has this to do with mathematics? Simply this that, as queen of the sciences, mathematics deals with foundations, with first principles. It is for this reason that physics draws on mathematics for its certainty. As Hempel says in "Geometry and Empirical Science:"

The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which accounts for its fame as the queen of the sciences, is no doubt the peculiar certainty and necessary of its results. (quoted in The World of Mathematics, James R. Newman, ed., [1956], p. 1635)

Of course, all sciences, insofar as they do not merely catalogue facts but go further to deduce laws and draw conclusions, invoke this "peculiar certainty and necessity." In fact it is because of such theoretical implications, which lead to empirical experimentation, that science can correct its initial assumptions, and has grown from the armchair suppositions of Aristotle ("Bodies fall with speeds proportional to their weight") to its present status.

But there is a difference between the certainty and necessity of a mathematical proof and the certainty and necessity of a law that interprets the physical universe. Mathematicians do not concern themselves with facts; such is not their province. The statement of Charles Peirce, given at the start of this introduction, is evidence of this, it was written in 1856. More familiar is Bertrand Russell's often quoted statement, "mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."

What we have, then, is that mathematics disclaims any necessary connection with the real world. But this doesn't stop physicists from using the formalisms of mathematics to support their predisposition toward causal explanations, symmetry, determinism, etc. And therein lies the problem, because mathematics, which is invoked to provide this sanction, is itself improperly understood. There are many mathematical tools available, and the choices as to which are appropriate and how they are to be interpreted are crucial. I will try to show that buried within mathematics itself are the clues to how this judgment as to application can be achieved.

To start simply, three important contributions of mathematics will be examined in this essay: the notion of dimension, the notion of the degree of an equation, and the concept of derivatives. In all three areas, mathematics tends to overlook critical distinctions, qualitative differences that have an important bearing on cosmology. More specifically, mathematics does not concern itself with the intrinsic character of one-dimensionality versus two-dimensionality, with the qualitative difference between equations of different degree, or with the different meanings of the time derivatives. I will consider these differences in the hope of showing that they provide the potential for a science of epistemology and cosmology. These distinctions are the seeds whose growth and unfolding produce physics and the other sciences. Their neglect, especially by mathematics, has led to confusion and error, and the time is ripe for their recognition.

In the later sections, I take up the multiple infinities of Cantor and the related question of the number of transcendentals, with a suggestion as to how a different definition of number could help resolve this issue.

Finally, I take a mathematical notion, the concept of modulo and residue, and apply it to the problem of consciousness itself. This may seem inappropriate in a book about mathematics, but I intend to show the profound contribution mathematical thinking can make to such "unscientific" questions as evolution and the purpose of life.