by George Anastaplo**
The role of algebra in modern mathematics is extremely important, and there is a trend towards further “algebraization” of mathematics. A typical way of studying many mathematical objects that are sometimes far removed from algebra is to construct algebraic systems which adequately represent the behaviour of these objects. . . .
It would appear at first sight that the translation of problems into the language of algebra, solving them in this language and translating them back, is merely a superfluous complication. In fact, such a method turns out to be highly convenient, and occasionally the only possible one. This is because by algebraization one solves problems not only by purely verbal considerations, but also by using the powerful apparatus of formal algebraic calculation, so that one may occasionally overcome highly involved complications. This role of algebra in mathematics may be compared with the role of modern computers in the solution of practical problems.
–“Algebra” entry, Encyclopedia of Mathematics (1988), I, 75
Isaac Newton’s Principia Mathematica is probably the most difficult text on the Basic Program reading list for the Staff as well as for the students to work with. And yet it may well be the text on our list that the Staff, at least, can learn most from.
We, of course, are all respectful of the authority of modern science. Even so, someone as learned in these matters as Curtis Wilson of St. John’s College (a distinguished historian of science) can speak of “that strange kind of ‘knowing’ that is mathematical physics.” (See Samuel S. Kutler, ed., Essays in Honor of Jacob Klein [St. John’s College Press, 1976], p. 189.)
Strange, also, is what can be said by some observers nowadays about these matters. Thus a feminist has spoken of Newton’s Principia Mathematica as a “rape manual” because “science is a male rape of female nature.”
On the other hand there was the respect generated in Subrahmanyan Chandrasekhar when he devoted the closing years of his life to a study of the Principia. That respect was dramatized by him in a comment he made to me, as to others, when asked about his impressions of Newton, whom he had never really studied before: “I am like a small boy going to the zoo for the first time and seeing a lion.” (See Anastaplo, “Thursday Afternoons,” in Kameshwar C. Wali, ed., S. Chandrasekhar: The Man Behind the Legend [Imperial College Press, 1997], p. 125.) Mr. Chandrasekhar also observed that he could conceive of himself discovering what Einstein and others of note had discovered in the Twentieth Century, but he did not believe he could ever have done what Newton did.
Perhaps the most useful way for us to proceed on this occasion is to notice a few questions prompted by Newton’s work.
What, for example, does it mean that Physics, in Newton’s day (and for some time thereafter), could still be called Natural Philosophy? What does this suggest not only about what the emerging Physics was taken to be, but also about how traditional Philosophy was regarded? Natural philosophy, we are told by a 1705 lexicon, is that “Science which contemplates the Powers of Nature, the Properties of Natural Bodies, and their Mutual Action one upon another.” (Principia Mathematica, [University of California Press edition, 1999], p. 793n)
Three great predecessors, among others, are drawn on by Newton in the Principia: Copernicus, Kepler, and Galileo. Copernicus, with his heliocentric “hypothesis,” can be thought of as clearing the ground upon which Kepler could develop laws about the orbits of planets. Newton can be understood to have combined this work by Kepler with that by Galileo about the law of falling bodies on earth, developing thereby his revolutionary system of universal gravitation.
That system, described in the Principia, was at once widely acclaimed as a magnificent achievement. Indeed, the reception for this work suggests that the scientific world may have been ready for such a synthesis. It may even suggest (despite the Chandrasekhar assessment) that if it had not been Newton, then others (albeit perhaps in a less magisterial form) would have developed the critical elements of the Newtonian synthesis.
The organization of the Principia has been described as this fashion (I. Bernard Cohen, University California Press edition of the Principia, 1999, p. 128):
The Principia, in its final form, consists of four prefaces, a set of “Definitions,” “Axioms, or the Laws of Motion,” books 1 and 2 on “The Motion of Bodies,” book 3 on “The System of the World,” and a concluding “General Scholium.” The subject of book 1 is motion in free spaces, that is, in spaces devoid of resistance, while book 2 deals with motion under several different kinds of resistance. Here Newton’s subject is extended to include many other topics of natural philosophy, such as the principles of wave motion, along with some aspects of the general theory of fluids.
More can be said by Mr. Cohen about the organization of Book 3 of the Principia (ibid., p. 195):
Book 3 is composed of six distinct parts. The first contains a set of rules (“regulae”) for proceeding in natural philosophy; the second, the “phenomena” on which the exposition of the system of the world is to be based. Next comes the application of mathematical principles (primarily as developed in book 1) to explain the motion of planets and their satellites by the action of universal gravity. The fourth part sets forth Newton’s gravitational theory of the tides. The fifth part of book 3 . . . contains an analysis of the motion of the moon . . . The sixth and last part . . . is devoted to the motion of comets . . . [Book 3,] unlike books 1 and 2, is not formally divided into sections.
Thereafter is “the celebrated General Scholium” (p. 23) previously referred to, about which I will have much more to say further on. (See, also, ibid., pp. 59-60, 230-31, 274f.)
A distinction should be drawn here between what Basic Program students at large can be usefully engaged by and what we as the Basic Program Staff might be equipped to investigate on our own. Consider, for example, one line of inquiry that I have been developing, an inquiry that is never hinted at in the literature I have seen but which nevertheless seems to me, as very much an amateur in these matters, to be instructive (however unproductive it is likely to be for direct classroom use).
This inquiry proceeds from my identification of the central proposition in the final version of the Principia. That turns out to be Proposition 97 in Book One. It also turns out that this proposition, along with Proposition 98, was inserted by Newton after the first edition of the Principia. I have the impression, but again as an amateur, that these two propositions could have been placed elsewhere.
I find it intriguing that Newton, upon revising the Principia after having had time to reflect on what he had done, somewhat hurriedly in the first edition, should have added at the very heart of his masterpiece a proposition (No. 97) which includes (in his discussion of it) the observation that he had undertaken “to reveal . . . by this proposition” something that Descartes had concealed in his treatises on optics and geometry.
That is, I find intriguing this quiet suggestion of a fundamental difference (not explicitly developed here) between Newton and perhaps the greatest of those modern predecessors with whom he differed, a predecessor of his who too was remarkably gifted in mathematics. The Cartesians, mostly on the Continent, would see the Newtonian approach as far too mechanical for their taste. Newton, on the other hand, might have seen them as far more open to “the occult” than he believed either necessary or useful for a productive account of universal gravitation, however open Newton might personally have been (as he evidently was) to an intense study of the prophetic books of the Bible.
I presume to suggest that fundamental to Newton’s own understanding of his work is his recognition of his own ability and willingness to expose to view matters that Descartes had concealed. (Descartes had died in 1650, eight years after Newton was born.) The importance of the Cartesian alternative, which I am not competent to be sure about, is further indicated both by what is added after the first edition in the General Scholium (at the end of the Principia) about Descartes (along with Aristotle) and by what is said about Descartes by the editor (Roger Cotes) relied upon by Newton in his second edition.
All this suggests that it would be useful, if not even necessary, for a study of Newton’s own understanding of his principal work, to determine not only what he learned from Descartes but also what he believed was fundamentally flawed in Cartesianism and its antecedents (which antecedents might have included Medieval Aristotelianism).
I am reinforced in this suggestion by an observation in an 1855 book which I happened to discover after I had worked out what I have about the centrality of Proposition 97:
The Fourteenth Section [of Book 1] concludes [in Proposition 97, reinforced by Proposition 98?] with an elegant solution of a local problem in Descartes’s Geometry, for finding that form of refracting glasses which will make the rays converge to a given focus, a problem, the demonstration of which Descartes had not given. The brilliant discoveries made by Sir Isaac Newton upon the refrangibility and colours of light, not belonging to dynamics, he pursues the subject no further in this place, having reserved the history of those inquiries for his other great work, the Optics, perhaps the only monument of human genius that merits a place by the side of the Principia.
(Henry Lord Brougham and E. J. Routh, Analytical View of Sir Isaac Newton’s Principia. [London: Longman, Brown, Green, and Longmans, 1855], p. 152.) If I were far more competent I would myself “pursue the subject . . . further in this place.”
I hope that I have not been simply incompetent in singling out Proposition 97 as I have for others, who do understand the technical matters involved here, to develop further. It can be seen, in any event, that Newton does bring together here his pioneering work in both physics (dynamics) and optics. Proposition 97 (reinforced by Proposition 98) seems concerned with the development of lenses, perhaps suggesting thereby that a study of light may be critical to the study of matter and its motions. (In addition, we have heard, in recent decades, of the significance of interstellar sounds for astrophysicists. See, for example, Kenneth Chang, “Vestiges of Big Bang Waves Are Reported,” New York Times, January 12, 2005, p. A15.) Thus, it can be wondered whether Proposition 97 reflects, for the natural philosopher, the importance of the appearances of things. (The number 97 itself, the twenty-fifth prime, does not seem to be in itself a particularly interesting number–but, about this, too, I welcome correction, even as I venture further in my speculations: Did Newton, in combining Proposition 97 and Proposition 98 as he did, intend to echo what happens in Euclid’s Elements, where Proposition 48 is the converse of Proposition 47 [the Pythagorean Theorem], which is the culminating proposition in that work?)
I have already indicated that the speculations I have just sketched are not likely to be directly useful in guiding the discussion in a typical Basic Program class. But they are perhaps instructive for us as we consider what to discuss, and how, in the massive Principia text that is available for our classes.
Various parts of that text virtually recommend themselves for classroom discussion, in addition to the attempt we might make to suggest the instructive organization of the whole. Those parts include the introductory material, the definitions, and the axioms, or the laws of motion.
And then, of course, there is the General Scholium appended to the Principia after its first edition. The typical student can be encouraged to believe that there are things in that text which can be studied with profit. Something is to be said as well for their being encouraged to have on their shelves a massive and mysterious text which obviously challenges them.
The reader is left with two great problems by the concluding General Scholium. One is with respect to the Action at a Distance that seems to be essential to the Newtonian system of universal gravitation. Dependence upon such action had long been suspected and hence shunned as a reliance on occult properties. (See, for example, the discussion of this issue in the New Catholic Encyclopedia. See, also, Mary B. Hesse, “Action at a Distance,” in Ernan McMullin, ed., Concept of Matter [University of Notre Dame Press, 1963], p. 372. “[For Descartes, as] with Aristotle, there are no pulls but only pushes in nature.” Ibid., p. 375.)
It is in this context that Newton issues his famous declaration that he frames (feigns?) no hypothesis. ([H]ypotheses non fingo.) This should not be taken to mean, as some have done, that Newton always considered the use of hypotheses improper–for he does use them again and again both in the Principia and elsewhere. Rather, he seems to say that he himself does not have available any useful hypothesis in this situation.
The critical concern here is suggested by Galileo’s having evidently dismissed as unpersuasive, if not even as absurd, the suggestion that had been made by Kepler about our moon affecting the tides of the Earth. Descartes had had problems as well with Kepler’s planetary theory. (See Curtis Wilson, pp. 190-91.) Does anything pass from one body to another at a distance? We hear talk these days about gravitons, particle-like things which are said to move even through a vacuum, and evidently without the “emitting” body being in any way depleted thereby.
The astrophysicists I have consulted about this matter seem to be persuaded that the attraction exercised by one body upon another at a distance “moves” no faster than the speed of light. This can mean, for example, that by the time our tides respond to any particular position of the moon, the moon itself will have moved. (One could, if only as a diversion, calculate the extent of that movement thus (on average): first, determine the distance the moon moves per second in its orbit; then, determine the distance between the moon and the earth, which can then be put in terms of the seconds it takes light to travel that distance [at 186,000 miles/second]; finally, apply that number of seconds to the distance that the moon moves, on average, per second in its orbit.)
Mr. Chandrasekhar, in his 1995 treatise on the Principia (p. 535), observes, “In the penultimate paragraph of the General Scholium which concludes Book III of the Principia, we have Newton’s testament on gravity written with a sentiment and a style that are unsurpassed.” Newton, in this paragraph, after confessing, “I have not yet been able to deduce from phenomena the reason for these properties of gravity,” can yet conclude with the observation, “It is enough that gravity really exists and acts according to the laws that we have set forth and is sufficient to explain all the motions of the heavenly bodies and of our sea.” (Emphasis added.)
We have an authoritative “update” with respect to these matters in the Chandrasekhar postscript to the Newton paragraph that he had extolled: “There is only one additional statement one can make, at this date, three hundred years later [this was in 1995]: the quest for the ‘cause of gravity’ still continues.”
This, then, is one of the two great problems left by the General Scholium at the end of the Principia, problems that our students can usefully be encouraged to contemplate.
The other great problem there has to do with the cause of the arrangement of the orbits of the bodies in our solar system. Newton is much taken by the evident arrangement: “The six primary planets [that is, the earth and the five planets visible to the naked eye] revolve about the sun in circles concentric with the sun, with the same direction of motion, and very nearly in the same plane. Ten moons revolve about the earth, Jupiter, and Saturn in concentric circles, with the same direction of motion, very nearly in the planes of the orbits of the planets.” Newton then explains why he does not believe that “all these regular motions . . . have their origins in mechanical causes.”
He is moved to offer here, in the Principia, an explanation for what he considers a remarkable uniformity (p. 940):
This most elegant system of the sun, planets, and comets could not have arisen without the design and dominion of an intelligent and powerful being. And if the fixed stars are the centers of similar systems, they will all be constructed according to a similar design and subject to the dominion of One, especially since the light of the fixed stars is of the same nature as the light of the sun, and all the systems send light into all the others. And so that the systems of the fixed stars will not fall upon one another as a result of their gravity, he has placed them at immense distances from one another.
He rules all things, not as the world soul but as the lord of all. And because of his dominion he is called Lord God Pantokrator.
Further on, he says of the Lord God (p. 941):
And from true lordship it follows that the true God is living, intelligent, and powerful; from the other perfections, that he is supreme, or supremely perfect. He is eternal and infinite, omnipotent and omniscient, that is, he endures from eternity to eternity, and he is present from infinity to infinity; he rules all things, and he knows all things that happen or can happen. He is not eternity and infinity, but eternal and infinite; he is not duration and space, but he endures and is present. He endures always and is present everywhere, and by existing always and everywhere he constitutes duration and space. Since each and every particle of space is always, and each and every indivisible moment of duration is everywhere, certainly the maker and lord of all things will not be never or nowhere.
Of course, Newton’s academic successors today would rarely consider themselves either entitled or obliged to rely upon the suggestion (the hypothesis?) that Newton confidently offers here. Rather, they would much prefer to speak of the law of the conservation of angular momentum which “kicks in” when a massive body so spins off parts of itself that planetary relations are established, etc.
Can what Newton says here, if not also elsewhere, about the Divine, as well as how he says it, help us sense how he grasps the physical world? Or, put another way, does Newton’s conclusion (if not his assumption from the outset of his speculations) that he needs the Divine to fill out his system–does all this suggest that we may not understand his account as he does or even somewhat as it is?
It seems to be indicated, again and again, that Newton’s sense of these matters, including with respect to the role of the Divine in the grand cosmic drama–it seems to be indicated that his understanding of things is fundamentally different from Descartes’. And, we are told, William Blake’s is significantly different from that of both men. Thus, an inquiry into any one of these visionaries might help us see better the other two. (See, for example, Donald Ault, Visionary Physics: Blake’s Response to Newton [University of Chicago Press, 1974], esp. frontispiece and pp. 3-4.)
However all this may seem, we do have perhaps the greatest physicist in modernity unable to be content with his account of the cosmos without an invocation of the Divine. One may even be reminded here of the opening speech in Sophocles’ Antigone, where all of the dreadful things that have happened to the family of Oedipus seem to be attributed by the heroine to Zeus.
How ambitious and magisterial Newton may have aspired to be is suggested in a grand assessment of his work by Thomas Simpson, another St. John’s College historian of science (“Science as Mystery,” The Great Ideas Today, Encyclopedia Britannica, vol. 1992, pp. 115-16 [final emphasis added]):
The Principia is organized in three books. The first two contain the body of basic propositions which constitute what Newton calls the “mathematical principles of natural philosophy”–strictly, perhaps, these two books are the principia, the “principles.” The third book, called “The System of the World,” is then appended to the work as an illustration, or one might say, an initial realization. Where the first two books, as the books of the new mathesis, only await their application in philosophy, the third book is a brilliant example of that new philosophy itself. The first two books are thus books of mathematics in the new mode; the third book is an exemplary book of philosophy. As such, the third book is the first step on a way which is ultimately, I think, intended by Newton to yield a total replacement for Aristotle’s Physics in its full range. Newton aims to be forging here a mathesis adequate to the one truth of a created world.
The Simpson assessment continues in this fashion (ibid., p. 116):
The sample in the third book deals with only one of those natural forces Newton has alluded to, namely gravity, and addresses the cosmos only in its aspect as a gravitational system. It conspicuously does not, for example, deal with the optical or chemical systems of the world, nor with the causes of the vital motions of nature, those which alchemy calls “vegetable” and “animal” and is primarily interested in mastering. We recall that “physics” no more denotes to Newton than it does to Aristotle that “inorganic” realm which we now call physics and distinguish from biology and psychology. Newton clearly intends a mathematical physics of all natural things, including most especially all those which live, grow, move, and feel. The gravitational system unfolded in the third book is a brilliant but relatively easy initial step into the new philosophy. The cosmos is only the beginning. The real third book, which it was not possible for Newton to compose, would be the Book of Alchemy. There is evidence that Newton was trying hard to bring his alchemy into shape in time for inclusion in a more complete picture of The System of the World, balancing the macrocosm of the planets with the microcosm of a mathematical alchemy. Newton was not able to write the Principia he intended. One clue to his ambitions is found in the great “Queries” appended to his Optics, in which he did his best to pass his vision on to future generations. It is a vision of alchemy realized.
It might even be wondered whether Newton’s greatest predecessor here was Plato.
We can be more mundane in our inquiry by wondering whether Newton’s recourse to the Divine, at least as an explanation of the substantial uniformity in planetary orbits in our solar system, reflects something like the stellar parallax problem for the advocates of a heliocentric theory of our system. That is, it will be remembered that it evidently took a long time to recognize that interstellar space was so vast that stellar parallax could not possibly be detected with the equipment available to human beings. (It is said that it was not until 1838 that such parallax could first be perceived by observers on Earth.)
Would Newton have considered himself entitled, if not even obliged, to invoke the Divine as he did in his General Scholium if he had been aware of the scope of the universe now routinely available to our astrophysicists? True, he does refer to distant stars which may be “the centers of similar systems,” and as such they would be “constructed according to a similar design and subject to the dominion of One.” But such an extension of the universe that Newton explicitly contemplates and provides for can seem almost trivial compared to what astrophysicists talk about these days, with their hundreds of billions of galaxies and with hundreds of billions of suns in an average-sized galaxy.
When the universe is believed to be of such a scope, it becomes highly unlikely that a physicist, nurtured and evidently sustained as Newton was by the Bible, would speak of the Divine superintending the movements of innumerable galaxies with the interest and attention evident in Genesis and elsewhere with respect to the Earth and its attendant celestial bodies. Of course, “the laws of nature” could simply be a modern way of talking about Divine regulation of the material world–but that does not seem to be in the spirit of the piety exhibited by Newton when he devoted years to the study of Biblical prophecies, especially those in Daniel and in the Book of Revelation.
However all this may be, it can be startling to hear Newton’s successors today speak confidently not only of billions upon billions of distant systems but even of the number of orbits that our own galaxy has made around its center since it was formed (some billions of years after the Big Bang). One orbit by our galaxy, I have been told, takes about two hundred and fifty million years, which means that our galaxy has had thus far forty or so such orbits during its career.
When the universe contemplated is of this scope, it can be more easily supposed that there will emerge, here and there, modes of planetary organization such as that in our solar system, one consequence of which could be the origination and development of the kind of life with which we are familiar. (Fitting the orbits of comets into such vast systems becomes far less of a challenge than it had once seemed.)
In the circumstances described by modern astrophysics it can become easier for the more sophisticated to talk about an eternity of time and matter–or, at least, to develop calculations and an overall system which assumes such an eternity, even if “all” of what may be observed by us (history, fossils, and everything else) was created, say, only ten thousand years ago with the appearance of vast space and time.
The professional physicist today tends to forego philosophical speculations. He (and it is, almost always, a “he”) has more than enough to do without worrying about what one of them (when questioned by me) would dismiss, genially enough, as “theology.” Again and again I have heard the observation from top-notch physicists, in response to my questions, “I don’t think about that.”
However all this may be, we see recourse by Newton to an overall account of things (both physical and theological) that various of his peers (such as Galileo and perhaps Kepler) were not inclined to rely as much, if at all, upon. Even so, much of what Newton accomplished depended on the work done by others–on the data collected and on the explanations offered by others.
It seems that Newton could be stimulated to venture forth with his Principia only because of the deference paid him by distinguished scientists such as Edmond Halley (after whom the comet is named). He could also be stimulated, in a less attractive way, by the challenges he perceived in the work of Robert Hooke and of Leibniz.
Newton, in his competitive mode, could be unduly “sensitive,” sometimes being moved to carry on a doctrinal feud for years. Even his greatest admirers among the historians of science (such as Richard S. Westfall) can be troubled by the ferocity with which Newton conducted himself in such contests. This aspect of his temperament evidently found particularly ugly expression in the merciless fashion that he, as Master of the Royal Mint, could insist upon the execution of counterfeiters–and this even after he was “secure” as an international celebrity. (Newton is to be contrasted in this respect to, for example, Charles Darwin, who evidently got along easily with everybody that he personally encountered. Newton’s bouts of indignation can remind one of Homer’s [and Sophocles’] Ajax: neither man could bear to be slighted. Did it ever occur to Newton that the alchemy he devoted himself to would-be counterfeiting on a grand scale?)
Newton’s temperament may perhaps be traced back, at least in part, to how he happened to have been brought up by his family. His father died before Isaac was born; his mother, when she remarried, evidently did not want her three-year-old son around.
But however intriguing and whatever the causes of the Newtonian temperament, we can wonder how it bears on the scientific work that Newton did. Warnings about the effects of character on one’s effectiveness as a scientist are given in Helmut Fritzsche’s talk, “Of Things That Are Not” (John A. Murley, Robert L. Stone, and William T. Braithwaite, eds., Law and Philosophy [Ohio University Press, 1992], vol. I, p. 3). Particularly to be guarded against, this former chairman of the University of Chicago Physics Department warns, is that self-delusion which leads to the unconscious distortion either of the conduct or of the interpretation of experiments.
Even more critical than this may be the limitations that one’s character may place upon one’s ability to grasp properly the highest things. Particularly suspect here could be the apparent Newtonian conviction that the Divine is needed in an account of the ordering of the world.
How reliably can a deeply disturbed, or at least disturbable, soul see the most important things? On the other hand, does the modern mathematization of science tend to moderate or at least to mask the consequences of an observer’s disturbed character? Even so, is not levelheadedness useful, if not necessary, for a reliable grasp of the Divine? Still, may there not be some assurance of levelheadedness, reinforced perhaps by a sense of humor, in the recourse that Newton had to Proposition 97 of Book 1 of the Principia?
In short, we can be reminded here of the twin injunctions for human beings laid down (it is said, by Apollo) at Delphi: one should both know oneself and be moderate.
Newton is spoken of as discerning in the fall of an apple that “law” of universal gravitation which explains (or, at least, describes) the grand movements of heavenly bodies. The Principia brings together, in an authoritative manner, the movements both of the trivial and of the celestial.
At the same time, and for decades after the publication of the Principia, Newton applied his “methods” to studies of Biblical prophecies. It almost seemed, at times, that he was confident he could do for “theology” what he had done for “natural philosophy.” The Divine can be invoked by him, as we have seen, to account for the reliable ordering of planets in our solar system. (Related to this approach may be what we have heard, in recent decades, about “the anthropic principle,” something to which I was personally introduced by Mr. Chandrasekhar, who was about this account of things somewhat skeptical, or so it seemed to me.)
Newton himself yearns to be godlike, in that he yearns for total mastery of whatever he undertakes. This may be seen not only in the Principia but also, perhaps even more, in the considerable efforts that we have noticed that Newton evidently devoted for years at a time, both to alchemy and to the study of Biblical prophets. Thus, he can take quite seriously, as a scientist, the guidance provided by Moses about the existence and workings of heavenly bodies. (See, e.g., H. S. Thayer, ed., Newton’s Philosophy of Nature: Selections From His Writings [Hafner Publishing Company, 1953]. Moses Maimonides, on the other hand, seems to have relied on the astronomy of his day, not on the Bible, for critical heavenly calculations.)
How seriously can all this be taken? Some scientists today tend to believe that Newton was in a somewhat deteriorated mental condition when he devoted himself in the intense way he did, in the last decades of his life, to Biblical and related studies. Nor are they apt to take seriously – and in this they may be somewhat naive – the sort of thing that Newton says (about the Divine) in the concluding Scholium of the Principia.
We have recalled the story of how the fall of an apple can lead to an understanding of the greatest movements in the universe. But to put this development thus is to distinguish, in a way perhaps inappropriate for modern physics, between the High and the Low.
The collapse of the distinction between High and Low was nicely illustrated, two days ago, by the most entertaining talk this quarter in the Weekly Colloquia of the Physics Department of this University. That lecture, by a lively Russian, Andrei Varlamov, who works in both Rome and Moscow, was anticipated by this departmental announcement, “Physics in the Kitchen”:
The lecture is devoted to discussion of some physical aspects of cooking, eating and drinking. Among them are problems appearing in the process of cooking in a microwave oven, secrets of Italian espresso, boiling of an egg, and physics of wine and Italian pasta. Using mainly the method of dimensional analysis, the author will estimate the penetration depth of electromagnetic field into Thanksgiving turkey, the time of coalescence of two droplets of fat on the surface of bouillon, cooking times for different types of spaghetti, and will give recommendations on how to choose the right viscosity of the sauce for a perfect pasta of given size. The results of recent experimental studies of the wine-tears formation (Marangoni effect) and Bordeaux paradox will be reported and analyzed.
Although the lecture did not cover all of the culinary processes advertised, it dealt with enough of them (especially with respect to coffee, spaghetti, and wine/vodka) to be both instructive and highly entertaining. (See, for more on these and related matters, the book this lecturer published, with L. G. Aslamazov, The Wonders of Physics [World Scientific, 2004].) The various terms, equations and graphs employed, whether or not altogether serious, could remind one of what is routinely seen and heard in such Colloquia. (An out-and-out spoof of much of this could be witnessed, the preceding Friday night, at the Physics Department’s Holiday Party at Ida Noyes Hall, a party which a senior member of the Department insisted that we should attend.)
When the “language” of physics is used, it does not seem to matter whether one is dealing with the incredibly tiny or with the astronomically vast. And so, the names and formulae of august physicists could plausibly be used in explaining why the coffee at an Italian bar (at 65¢ a cup) is so much better than what one can make with the same ingredients in one’s home (at 5¢ a cup). One can be reminded here of the medieval alchemists. One can also be reminded of the concerns expressed by Jonathan Swift in his accounts of the struggles between ancients and moderns: the high and the low, he seems to have complained, can become interchangeable when unnaturally reduced to their elements.
We return, however briefly, to the examination of the Principia in our classes. It can be useful for today’s student to see how the material and the spiritual (as ordinarily understood) can be somehow combined, even if only temporarily. That is, this is not a combination that tends to recommend itself to Newton’s successors, who are inclined to be more “mechanical” than Newton ever thought it sensible to appear to be.
It can also be useful for our students to see how intuition has to be relied upon by the most serious scientists. This may be easier to observe in Galileo than in Newton (although a kind of intuition can be said to be drawn upon in the recourse by Newton to the Divine in the concluding Scholium of the Principia). I myself find particularly instructive that working of intuition in Galileo which kept him from accepting what his experiment seemed to have “demonstrated,” that the movement of light from one place to another is instantaneous. (See my Introduction to Douglas A. Ollivant’s Jacques Maritain and the Many Ways of Knowing collection [Catholic University of America Press, 2002], which should be available on the anastaplo.wordpress collection.)
Newton’s use of the Divine may even be understood as his way of “humanizing” his scientific speculations. Perhaps his considerable reliance upon geometry may also have that effect. But it is an effect lost upon contemporary physicists, who routinely confess themselves intimidated by the geometry that Newton employs. (Mr. Chandrasekhar’s last book was, in effect, a translation of Newton’s geometry into a mathematical mode that contemporary physicists could more readily read. See my review of that book in Volume 1997 of The Great Ideas Today.)
It can be instructive to consider, however superficially, what the shift from geometry to algebra has meant in recent centuries. A greater facility, especially in adapting “theory” to “practice,” does seem to come with algebra, something evident perhaps in our remarkable technology, a technology which can assure us that we are getting quite a bit right in physics and chemistry. But are we misled into believing as well that we, in our reliance upon the algebraic mode, understand more than we do?
Here our familiarity, in the Basic Program, with Plato’s Meno can be helpful. The Boy is asked to provide the line necessary for the production of an eight-foot square. Today, we can speak of the square root of eight. Then, the Boy could be helped to recognize that the line sought is the diagonal of a four-foot square.
Differences between the geometric approach and the algebraic approach can be instructive. That is, one offers us something that may be seen, the diagonal of a four-foot square; the other offers us something that may be heard, the square root of eight–but such hearing is not as reliably graspable as it may seem, inasmuch as the number produced is an irrational number–in this case it is 2.828+, which is, when squared, 7.997584+. That is, even though this is a number provided for one’s hearing, it is ultimately unsayable.
It is perhaps not accidental that we do tend to say, when we understand something that is explained to us, “I see what you mean.” (We can also speak here of being enlightened.) In so doing we can be understood to be tacitly ratifying Newton’s reliance upon geometry as the primary means for recording and transmitting his greatest discoveries, even though more “advanced” mathematics had been used in making those discoveries. (See the Isaac Newton entry in the Dictionary of National Biography, p. 386.)
We can, like Newton in his Principia, begin to close on a pious note by recalling an apt endorsement of geometry, even though by someone who may not have known either it or divine things as well as he believed that he did. I refer here to Thomas Hobbes’s characterization of geometry as “the only science that it has pleased God heretofore to bestow on mankind.” (Leviathan, chap. 4) That is, is geometry, more than its elaborate successors (such as algebra), somehow natural to the way human beings see and think? Indeed, astrophysicists today can even be heard to speak of the geometric properties of the universe, thereby returning (at least in speech) to the Newtonia view of things.
*George Anastaplo is Lecturer in the Liberal Arts, The University of Chicago; Professor of Law, Loyola University of Chicago; and Professor of Political Science and of Philosophy, Dominican University. Among his publications particularly relevant here are 1 The Bible: Respectful Readings (Lexington Books, 2008), p. 327; 2 an edition (with Laurence Berns) of Plato’s Meno (Focus Publishing Company, 2004) (this includes a detailed reconstruction of the geometrical exercise displayed in the dialogue); 3 “Some Perhaps Presumptuous Questions About Sir Isaac Newton” (Hyde Park Women’s Society, Chicago, Illinois, May 5, 2011), to be posted on the anastaplo.wordpress Internet website). See, also, John A. Murley, ed., Leo Strauss and His Legacy: A Bibliography (Lexington Books, 2005), pp. 733f.