Polytropy

Mathematics—Symbolic Logic

Certain phases of the total problem of education have received greater attention in some periods than in others.

Probably now, as never before, students of education are engrossed with the problems of the philosophy of education. A variety of factors have produced this result. Preeminent among them, however, must be the muddled times in which we live. Nothing throws one on the resources of philosophy more certainly than perplexity and doubt as to what ends and means should control inside and outside the school.

Confusion here has been chiefly compounded of the influence of science on education, the impact of industrialism on the schools, current emphasis on the accepted political ideal of education. Philosophy is not concerned with policy or practice but only with principles. It is concerned not with the single case, nor the general case, but with the universal. Thus it is the business of the students of education to determine a philosophy, i.e. a set of principles out of which may be evolved the policy relative to the general case and in turn the practice relative to the particular case.

Moreover the current literature of the field of mathematics is teeming with studies of the philosophy of mathematics. I shall refer particularly to what has been (2) justly called, “one of the greatest scientific discoveries of our age”. Viz. the enunciation of the thesis, Mathematics is Symbolic Logic, Symbolic Logic is Mathematics.

In order to appreciate the significance of such a result, it is important to note that the manner in which it was derived presents one of the most significant scientific phenomenon [sic] of the last century.

For many years after the invention of analytic geometry and Calculus, mathematicians reveled in the applications of these instruments to physical and geometrical problems. But with the coming of Euler, Gauss and LaGrange a more critical attitude developed. Under the leadership of Cauchy and Weierstrass in the field of analysis this critical movement took definite form and moved forward with increased momentum. We come finally to the work of Lobatschewsky and Bolyai, Grassmann and Riemann, Cayley and Klein, Hilbert and Lie on the foundations of geometry.

This critical study of the foundations of the several divisions of mathematics revealed a great ensemble of theories each built up by logical processes on its own appropriate hypotheses, or assumptions of postulates. As this study advanced it became more apparent that there was some single foundation upon which all these several divisions rest. (3)

Concurrently with this mathematical activity a critical study of logic inspired by the work of George Boole—“Investigation of the Laws of Thought” and carried forward by Peirce, Schroeder, and Peano and their successors revealed that the class-logic of Aristotle was inadequate to rigorous thinking. As a result we have today not only the Logic of Classes but the Logic of Relations and the Logic of Propositions. An attempt to determine the base or foundation upon which these three phases of logic must rest revealed a set of undefinable notions called logical constants.

Thus originating in apparently distinct domains and following separate parallel paths these two scientific movements led to the same conclusion, viz. that the basis of mathematics is the basis of logic also. Symbolic logic is Mathematics and Mathematics is Symbolic Logic.

The “Principia Mathematica”, a work in four [sic] volumes by Whitehead and Russell in England, and the “Formulaire de Mathematiques” of Peano in Italy bear evidence that symbolic logic has today achieved a high degree of precision.

It is not my purpose however to discuss sumbolic logic this afternoon. On the other hand I wish to consider with you the significance of such an important contribution of science to us as teachers of mathematics. It points out precisely that just as the world of sense is the universe of the physicist, or worker in natural science, so is the universe of logic the sphere of activity of the mathematician. As the natural scientist works with rocks and materials of sense so the (4) mathematician works with facts, classes, ideas, relationships, and implications. These are his materials. His processes and methods are similar to those of his colleagues in natural science. The importance of observation, experimentation, testing of hypotheses by the laboratory method, and incomplete induction cannot be too strongly emphasized. His Infinitesimal Analysis he uses, as the natural scientist uses his microscope.

It was by observation of the fact that the squares of certain numbers are each the sum of two other squares; the collection of these sets of numbers by the method of trial; the observation that apparently these and only these triplets are the measures of the side of a triangle. That is by observation, experimentation, incomplete induction processes, common to the experimental sciences, led to the discovery of the Pythagorean Theorem.

Through observation of the definitely lawful manner in which the coefficients of a system of equations enter their solution, a suggestion came to Leibniz. On the basis of Leibniz’ general idea grew up an algebra built on algebra; I refer to the Theory of Determinants.

We have said that the activity of the natural scientist is the activity of the mathematician. Let us consider for a moment a teacher of physics or chemistry. In order to examine the validity of an hypothesis, he does not rush through the proof, i.e. the experiment, concentrating on the findings, but spends [sic; “and spending”?] the major portion of his time trying to discover whether (5) or not his pupil has retained the results or findings. No, he sets up an experimental situation, gives time for observations and reflection on his observations. When a decision is reached as to the hypothesis, it is held up for reexamination. Are there any factors which might influence it which have not been considered? Only after time for reflection is the student ready to accept his finding as a truth. He understands it, he will retain it for it is his belief. It has not been imposed. Too many teachers of mathematics distribute their time in exactly the reverse order. Very little time is spent on establishing the truth, but there is great effort to fix it by repetition or use over and over.

If we as teachers of mathematics do not break down at least in the academic world the traditional fallacies regarding the nature of mathematics and the delusions regarding the kind of activity in which the life of the science consists we are neglecting our obligations. Mathematics is no more the art of reckoning and computation than architecture is the art of making bricks, no more than painting is the art of mixing colors.

Although a mathematical system is syllogistic and postulational in style and form, the assumption that syllogistic reasoning is the very foundation of all mathematical activity is another inherited fallacy which is the result of the reign of methodology.

However the observation of the mathematician transcends (6) that of the natural scientist in that it is not confined to observations of the physical eye. He is curious about pure relationships, about logically possible ordering, about implications, about functional relationships. He is not at all concerned as to whether they have any extension in our world of sense. The aggregate of things thinkable logically is his universe. He enjoys the construction of concepts, the contemplation of systems that are inconsistent with one another though each is entirely coherent.

The mathematician builds a hierarchy of hyperspaces, orders sets of worlds, worlds that are possible logically. Whether they are possible in our world of sense is of no concern to him. For example Euclidean geometry assumes the parallel postulate of Euclid. Several forms of Non-Euclidean geometry have resulted from the denial of that postulate; yet they are incompatible as regards any actual space. Euclidean geometry is based on the parallel postulate of Euclid while Non-Euclidean forms result from the denial of that postulate. Here we have an example of the meaning of truth—i.e. consistency with assumptions.—e. g.

Another example:

Are both true? In the world of mathematics there is no lack of harmony here, there is no inconsistency. (7)

Again what is the mathematician doing? He is building notions or ideas, he is constructing, inventing, adding to his body of science. With what is he working? Ideas, relationships, implications, etc. What are his methods? Observations, experimentation, incomplete induction. He is deliberately provides time for reflection and comtemplation.

Now I would like to consider with you some of the contributions of this abstract science to the development of citizenship. Not unmindful of its many and valued applications, students of mathematics realize that, though most abstract in form, mathematics is a very real part of living. The problems of mathematics are in reality mere mathematizations of the problems of life. It is precisely the job of the teacher of mathematics to assist the pupil in, shall we say, mathematizing life’s problems.

To better appreciate this point, it is worthwhile to consider for a moment some of the great concepts of the science.

The notions of constant and variable obviously have as their counterparts in life permanence and change. The notion of class, relation, order need no interpretation; they exist per se in human living. Interdependence the correspondent of the concept of functionality permeates our entire world of experience. It is very evident that contemplation of the concept affords opportunity for the development of an appreciation of the need of each individual for the other, and of the dependence of the ensemble upon the cooperation of all its individuals. (8)

Consider the concept of dimensionality. Its correlate in human experience is the notion of degrees of freedom.

Closely associated with this is the notion of natural law. Its mathematical counterpart is the equation or system of equations. The concept of limit is merely an expression [in] mathematical form of an ever receding goal of perfection for which man yearns and for which he strives, yet never attains. With each new approximation, he is merely closer.

In the concept of invariance we have the concept of that which remains stable in midst of change. Here is the notion of value or relation that survives in an ever changing world. Reflection upon this doctrine of invariance shows why mathematics is listed with philosophy and theology among the spiritual enterprises of man. The human spirit craves invariant reality, it craves freedom, it craves peace, a sense of being in harmony with a divine Being infinite and eternal.

Whether it is possible by the modern doctrine of Infinity as developed by Cantor and Dedekind to demonstrate the existence of the Infinite is not important here. Certain it is that it contributes much to the formation of our concept of Deity. Ths in this world of thought man realizes his aspirations for an unchanging reality, for freedom, for harmony.

Finally I wish to point out that the great mathematical concepts are not found only in the higher developments of pure mathematics as is sometimes argued. On the other hand they are present in the elements as well. (9)

That their significance which I have tried to portray be made clear, awaits only the skillful handling of a resourceful teacher.

The notion of degrees of freedom cannot be obtained if we confine our discussion of dimension to length, width, and thickness. All colors are three dimensional, because each is defined in terms of three primary colors. All musical notes are three dimensional, for each has three coordinates, pitch, length, and loudness. The line is a one dimensional, two dimensional, three dimensional, or n dimensional space according as its elements are pts, pt pairs, triads or sets of n pts each.

The continuum of all real numbers is 2-fold, 3-fold, or n-fold according as we view it as an aggregate of number pairs, triads, or ordered system of n numbers each.

Concept of limit—perimeter of regular polygon and circumference of circle. Meaning of solution of eqtu. [sic]—throwing [a] movie in reverse always operating under the laws of eqtu. solves the eqtu. Concepts of system of eqtu.—pts common to 2 lines.

To be resourceful our teacher must have knowledge gained by serious study, the insight and breadth of view which may be acquired only by prolonged contemplation of and reflection on the nature of the “mother of the sciences”. To be resourceful our teacher must appreciate the right of mathematics to rank among the forms of spiritual activity.

I have been concerned with fundamental principles, not with techniques, for I an convinced, given this philosophy, a conviction with respect to it, an inquisitive scholarship, a certain resourcefulness, we will not only work out techniques, we will (10) accomplish our goals. We will teach, we will do more than that, we will inspire.