A Passage. An Illustration. Of Imagination. Of Mathematics. Of Science.

To illustrate what Pascal and Leibniz knew:

The first process . . . in the effectual study of the science, must be one of simplification and reduction of the results of previous investigation to a form in which the mind can grasp them. The results of this simplification may take the form of a purely mathematical formula or of a physical hypothesis. In the first case we entirely lose sight of the phenomena to be explained; and though we may trace out the consequences of given laws, we can never obtain more extended views of the connexions of the subject. If, on the other hand, we adopt a physical hypothesis, we see the phenomena only through a medium, and are liable to that blindness to facts and rashness in assumption which a partial explanation encourages. We must therefore discover some method of investigation which allows the mind at every step to lay hold of a clear physical conception, without being committed to any theory founded on that physical science from which that conception is borrowed, so that it is neither drawn aside from the subject in pursuit of analytical subtleties, nor carried beyond the truth by a favourite hypothesis.

In order to obtain physical ideas without adopting a physical theory we must make ourselves familiar with the existence of physical analogies. By a physical analogy I mean that partial similarity between the laws of one science and those of another which makes each of them illustrate the other. Thus all the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers.

. . .

It is by the use of analogies of this kind that I have attempted to bring before the mind, in a convenient and manageable form, those mathematical ideas which are necessary to the study of the phenomena of electricity. The methods are generally those suggested by the processes of reasoning, which are found in the researches of Faraday, and which . . . are very generally supposed to be of an indefinite and unmathematical character, when compared with those employed by the professed mathematicians. By the method which I adopt, I hope to render it evident that I am not attempting to establish any physical theory of a science in which I have hardly made a single experiment and that the limit of my design is to shew how, by a strict application of the ideas and methods of Faraday, the connexion of the very different orders of phenomena which he has discovered may be clearly placed before the mathematical mind.

. . .

It has been usual to commence the investigation of the laws of these forces by at once assuming that the [electrical and magnetic] phenomena are due to attractive or repulsive forces acting between certain points. We may however obtain a different view of the subject, and one more suited to our more difficult inquiries, by adopting for the definition of the forces of which we treat, that they may be represented in magnitude and direction by the uniform motion of an incompressible fluid.

. . .

The substance here treated of must not be assumed to possess any of the properties of ordinary fluids except those of freedom of motion and resistance to compression. It is not even a hypothetical fluid which is introduced to explain actual phenomena. It is merely a collection of imaginary properties which may be employed for establishing certain theorems in pure mathematics in a way more intelligible to many minds and more applicable to physical problems than that in which algebraic symbols alone are used. The use of the word “Fluid” will not lead us into error, if we remember that it denotes a purely imaginative substance. . . .

—James Clerk Maxwell, “On Faraday’s Lines of Force”

A Passage. Some Thoughts. Of Language. Of Mathematical Physics.

The opening paragraph of Klein’s Greek Mathematical Thought and the Origin of Algebra (tr. Evan Brann):

The creation of a formal mathematical language was of decisive significance for the constitution of modern mathematical physics. If the mathematical presentation is regarded as a mere device, preferred only because the insights of natural science can be expressed by “symbols” in the simplest and most exactly manner possible, the meaning of the symbolism as well as of the special methods of the physical disciplines will be misunderstood. True, in the seventeenth and eighteenth century it was still possible to express and communicate discoveries concerning the “natural” relations of objects in nonmathematical terms, yet even then—or, rather, particularly then—it was precisely the mathematical form, the mos geometricus, which secured their dependability and trustworthiness. After three centuries of intensive development, it has finally become impossible to separate the content of mathematical physics from its form. The fact that elementary presentations of physical science which are to a certain degree nonmathematical and appear quite free of presuppositions in their derivations of fundamental concepts . . . are still in vogue should not deceive about the fact that it is impossible, and has always been impossible, to grasp the meaning of what we nowadays call physics independently of its mathematical form. Thence arise the insurmountable difficulties in which discussions of modern physical theories become entangled as soon as physicist or nonphysicist attempt to disregard the mathematical apparatus and to present the results of scientific research in popular form. The intimate connection of the formal mathematical language with the content of mathematical physics stems from the special kind of conceptualization which is concomitant of modern science and which was of fundamental importance in its formation.

Not all languages are equal. We sometimes cannot teach in one language what has been learned in another. In order to teach what we have learned, should we not begin with the language behind it?

A Musing. Of Research. Of the Old and the New.

Often I joke to myself (and sometimes aloud) that the “research” we study has all been said by Plato and Aristotle already. I am tickled whenever I find researchers who are willing to agree that, yes, most of our new conclusions are all very old. I just wish they would go one step further to say that our research does not contribute much new to our understanding of humans and human essence; “research” is all about convincing ourselves again of the things we had been made to forget.