Faraday's Mathematics:
On Getting Along Without Euclid

Faraday Conference Lecture presented on 30 March 2001
St. John’s College
Annapolis, Maryland




1.  The Faraday Conference and the College

      Today marks, as we have just heard, the opening of a very special event at St. John’s, a conference devoted to one of our beloved authors, Michael Faraday.  It is not only unusual to have a conference devoted in this way to one author, but it is even more unusual to have that very author make a cameo appearance, as Michael Faraday has done this afternoon, to repeat a lecture out of the distant past — a children’s lecture, fresher with every successive generation.  This presents me with a special challenge, of course, as a lecture by Faraday himself is an impossible act to follow!  I can hope only to express my own appreciation of Michael Faraday, and to suggest he is a figure with whom St. John’s might well feel a special affinity.  He is, in a sense, “our kind of guy”.  Let me try to explain my understanding of that special bond. 

      Faraday, as we recognized this afternoon, is a very direct and acutely perceptive observer of nature: fact, rather than elaborate theory, is his domain — not however fact in the dry and objective style people call these days information, but fact in a very different sense, that of a warm and immediate appreciation of the creation.  He sees nature as an open book, a book to be read, as he himself read the scriptures, as simply, directly and fully as possible, with the least intervention of artful interpretation.  Public lectures were an integral component of his scientific life, because he very evidently felt that nature is a book meant to be read and enjoyed by everyone. Complicated theories and especially mathematical elaborations would only stand in the way; in their place, Faraday brings a new light to the sciences, illuminating their beauty, and encouraging a direct, intuitive sense of purpose and human meaning. 

      Now you might be surprised to hear that I would call this a dialectical reading of the Book of Nature, and that I sense a genuine relation to the direct and clearly dialectical way in which we at St. John’s approach the great books.  We embrace the conviction that these works were written, in whatever era, for us, and at the seminar table we engage in conversation with them as though their questions were directed to us.  In fact, we adopt their questions as our own, and pursue them, with the help of the authors, as if our lives depended on it.  We realize, of course, the respect in which this may involve an element of illusion — a thousand things and more than just two thousand years separate us from some of our most respected authors — but this is, like all good myths, more true than false.  The questions are still real ones, and our conversation with the authors about them is genuine and, in its best moments, still life-giving.  This seemingly naïve reading of the books, direct and unadorned with the apparatus of academia, turns us in the direction of a living light of genuine human loves, hopes, fears and earnest convictions.  Thus that simple and direct reading is, by the same token, a dialectical reading.  All of our hard work in the tutorials and laboratories is merely ancillary to it.  It is at the same time a genuinely free and democratic approach to education, since it is a reading appropriate for everyone, pursued because it is good in itself, not because it is useful for other goals.

      Faraday was a commoner, the son of a blacksmith, and thus had little formal education. He left school after the primary years to be apprenticed to a bookbinder, and he reputedly learned his first science by reading the article on electricity in the Encylopaedia Britannica as it went through the shop.  He was taught no mathematics beyond simple arithmetic, and he never felt that he had thereby missed anything that he needed, even as a physical scientist.  He thus turned directly to read the book of nature without adornment and went on to share it with all of us, as if confident it had been written for the very enjoyment of us all. It is in this sense that St. John’s shares with him a fundamental, life-giving dialectical conviction.  Faraday’s unmathematical science is fundamentally akin to our dialectical seminar.

      I want to carry this one stage further, to claim that we and Faraday are not merely on strikingly parallel tracks, but that the two concerns are finally one.  When what some of us still can think of as the “new program” in the liberal arts was introduced at St. John’s in 1937, the overall enterprise was characterized as the attempt “to recover the tradition of the liberal arts in the modern world,” and it was thought essential to this that the sciences be brought to the seminar table and fully incorporated in the body of liberal education.  That was the beginning of our long experience in reading mathematics together, and our never-ending experiments in constructing a liberal laboratory.  Thus the ongoing attempt at achieving a dialectical reading of the sciences is not incidental, but central to the purposes of the College.  It also happens to remain a central unsolved and increasingly critical problem in the world today.  That is why I have claimed that Faraday, with his direct and very human approach to the sciences, is “our kind of guy”.


2.  Dialectic in Euclid and the Meno

      Perhaps, since the term dialectic will be important tonight, it would be prudent to pause for a moment to reflect on this concept together.  Fortunately, we have our common experience of the seminar to refer to, and in turn, the seminar is modeled to a very good approximation by Plato’s dialogue, the Meno

      The Meno has three great movements, the tragic trilogy we recognize in many other forms as well — most literally, for example, in the three plays of Aeschylus’ Oresteia, but no less in Euclid’s Elements; or if Plato is right, in the very structure of LOGOS itself.  The first movement begins with the confidence and prospect of an opening question, which launches the conversation; though if it were not loaded as well with doom no story would ever emerge and nothing would ever be learned.  In Meno’s case, it is the prospect of gaining great human excellence or mastery — ARETE, translated “virtue” — with all the world’s rewards which he imagines will follow upon it.  Meno, who is full of words, rushes to the point of disaster, the APORIA, in which his emptiness and cowardice are exposed.  This is the second moment, the negation and the death of the argument.  In the tragedy it is the recognition of terrible error.  In Euclid it is the unveiling of the irrational, the recognition that the very triumph of Proposition I, 47 entails the stark consequence that if a number measures the side of the square, there can be no word to number the diagonal. Speech falls silent. The word for being silent in Greek is MUEIN, and this depth of negation is the still point of MYSTERY.  This is also the turning point, the birth of wonder, and the beginning of the third movement, the ultimate upward movement of the discourse. 

      Here, for Meno’s rescue, and hence of course for ours as well, Socrates performs a mystery play — a diagram, in a sense, within the dialogue.  So — appropriately for our Faraday lecture! — he brings forward one who has never studied geometry, one who is unscathed by education.  In Athens, this is a bright young slave; today, he would be Everyman.  Socrates guides this candidate down the same path Meno has just walked, to the same APORIA. With a drawing in the sand, the slave is shown a simple problem: to TELL the name of the side which will give the double square.  In the same way, Meno had demanded to be TOLD about virtue.  In both cases, the inherent limitation of LOGOS manifests itself, and a yearning springs up for some other source which will infuse truth and meaning which words and all their noble structures cannot alone contain.  This reversal, in the silence of the darkest moment, toward the fullness of light, is the crux of dialectic.  This is Eleusis.  Where Meno’s courage had collapsed in despair, his slave carries steadily on.  He sees and remembers, by the mystery called recollection, recognizing the truth as the new light dawns — as Socrates diagrams the answer which could not be told.  The slave sees that “yes,” the square on the diagonal must be the side of the double square.  It could not be said in LOGOS alone, but it could be revealed by the light of the intuitive intellect, NOUS. This is a diagram of the way virtue may be achieved.  All truth must be somehow like that.  Words spring to life, but only dialectically, to bring us to the point of recognition of truth of a sort which — if there is any truth at all — is never contained in words. 

      It is hard to escape the imagery which we meet in the Republic, in which, at the depth of the dialogue, that turn occurs from the firelights of the downward path in the cave (the downward path of deductive reason, of money and commerce of the Piraeus) to the upward path, back to Athens and the warm light of what is good and gives all things their purpose.  The downward path and the upward path are one and the same, only the light is of a different kind.  In the Oresteia this is the persuasion of the Furies and the foundation of a new concept of justice, in Athens.  In Euclid, we know, it is the mystic incorporation of the irrational within the rational, the nameless in the named, worked out by Theaetetus in the Tenth Book. 

      I trust that in this rather compressed account you can recognize something of the simple practice of our seminars here at St. John’s, and their essence as dialectical.  Questions become real and dialectical when the issue is truth.  That sounds a bit fancy, but it means only that a dialectical conversation is one which is serious, one about which we really care, because it really might affect our inner beliefs and actual lives. From LOGOS, then, to NOUS — the topmost of the four levels of the divided line, that thread of Ariadne which leads us out of the cave and into the light of the Republic.

      All this must seem a far cry from the quiet labors of Faraday’s workbench at the Royal Institution, but here the similarities may be deeper than the differences: Faraday, like Socrates and Meno, has an overriding concern for human virtue, and in a few minutes as well I think we will see Faraday asking himself an opening question which digs as deep and reaches as far as many we meet in the Dialogues themselves.  Faraday’s questions in the laboratory are truly dialectical: they go, albeit in his special way, to the sources of nature not in atoms and void but in beauty and truth, and hence its importance to us not as specialists, but as custodians and lovers of the natural order we have inherited.

      On the whole, as we know, the world does not support such a dialectical approach to the sciences.  The world has come to regard the modern sciences, which it thinks of as products of a scientific revolution, as autonomous sources of knowledge, managed by experts and successful precisely because they reject confusing questions of beauty, goodness or purpose and rely exclusively on hard tests of incontrovertible truth.  A certain conception of mathematical physics is often taken as the standard of such hard truth, and in general, the more nearly any science can imitate strict mathematical structure and quantitative measures, the more independent it becomes of any human critique.  Questions of beauty, meaning, and value are supposed valid for philosophy, but only cosmetic as far as science itself is concerned.  It is thought idle to question the meanings of the primary terms, or the overall criteria of truth itself.  In short, strict science is thought to be factored out of the dialectical project, and mathematics itself often appears to be the crucial issue.

      Now I claim that this is a fundamental mistake — and a dangerous situation. Science cannot be factored out of the dialectical search for truth and meaning, and indeed “science” is not science if it is not dialectical.  Science has no separate claim to truth.  Thus if Faraday can show us how to bring the study of nature back into the fold of human joy, understanding and purpose, how to reassert the human claim to know, manage and enjoy nature, we may want to watch closely how he goes about it.  Faraday begins by clearing the decks — necessarily ignoring the claims of the autocrats of an entrenched tradition he has never studied; though he has a set of careful and very different disciplines, perhaps it is a mathematics, of his own.  We here will continue to delight in the mathematical tradition we have had the rare privilege of learning, and learning to love, as he could not — but we may discover from Faraday how to see all this in a different, life-giving light — a light which is after all its birthright, and ours as well. 

      We have one interesting hint.  Although Faraday was untaught in mathematics, and proclaimed himself to be the unmathematical philosopher, there is an impressive witness on the other side.  This is James Clerk Maxwell, the one who put Faraday’s own thoughts on electricity and magnetism into mathematical symbols in the form the world knows and admires today as Maxwell’s equations.  Maxwell firmly and resolutely declared throughout his life that it had been Faraday all along who — despite appearances — was the real mathematician.  Perhaps we have been missing something about mathematics itself. 

      We are faced indeed with a very strange state of affairs: one of the most astute and productive scientists — indeed, physical scientists — of the first half of the nineteenth century not only had never studied algebra or trigonometry, but he had never studied Euclid.  (In that sense, the subtitle of our lecture tonight must be “On Getting Along Without Euclid!)  Not only had he never studied Euclid, but since Euclid is the master from whom our western tradition has learned the concept of a reasoned mathematical system, Faraday, although he was self-taught and widely read in many aspects of the sciences, never gained any clarity about the notion of a system which begins with axioms, definitions and postulates and then reasons logically downward from them to demonstrated conclusions.  He did not trust, make use of or perhaps even quite understand the idea of a scientific theory — the strategy of making hypotheses, reasoning from them to conclusions, carrying out experiments to test these conclusions, and thereby making systematic progress in the direction of establishing what passes for scientific truth — that is to say, saving the appearances and taking quantitative prediction as the test of truth.

      It appears to me therefore that Faraday is in effect challenging an accepted concept of science itself.  Admittedly, this will be a rather Quixotic reading of Faraday.  Faraday, I am claiming, is quietly but very deliberately entering the lists of dialectic, joining battle with the world of conventional wisdom, over the nature of scientific truth, and man’s role in relation to the natural world.  If all serious thinking is in some way dialectical, and the essence of the Socratic dialectic is the aporia or the negation which brings our minds up short, then we can expect serious works to set us thinking by incorporating such a note of polemic, even to the point of denying what might seem on the face of it to be their own major premise.  We seem to be learning, for example, that Newton’s Principia is itself dialectical, in truth a sharp polemic against mechanism — indeed, against what was confronting Faraday, and even us still today, as Newtonian mechanics.  I hope at the end of this lecture tonight to come back to the question of our reading of the mathematical sciences, when we ask what it is that Faraday in his complex simplicity may have to teach us about the world’s possible misunderstanding of mathematics and the sciences today

      Perhaps the best way to begin will be to observe Faraday at work at the Royal Institution, significantly perhaps at once both his place of work, and his home.  We can do this because he keeps an intense record of his closely interlinked thoughts and his labors, in the form of a detailed scientific Diary.  We pick this up at a point at which, late in life, he is undertaking a new series of researches. 





1.  Day 1: The goal for the day’s work

      The date is November 11, 1851, a Tuesday, and Faraday is beginning in his Diary an account of a fresh inquiry — “fresh” not in the sense that he has not been there before, but rather that he is making a new start on a long-familiar voyage.  This time he will make the journey even more thoughtfully and carefully, keenly aware now of the great prospects and the delightful nuances to look forward to, and skilled and wary in managing the shoals and intricate passages along the way.  Faraday numbers the entries in the diary in which he records the events of his life in the laboratory at the Royal Institution; each paragraph gets a number, and today especially most paragraphs will be accompanied by a marginal sketch or two.  Faraday is sixty years old, and today opens with the paragraph number 11666, and with three striking words: “Wanted to know”...

Wanted to know how the lines of force were disposed in and about magnets and Iron generally under certain circumstances of position…

      He is starting with a question which he is setting for himself: it is the topic sentence for the day’s work.  If this were a mathematical proposition, this would be the enunciation; if this were a dialogue, as perhaps it is, this would be the opening question.  As this is to be a day of hard work in the laboratory, we might have expected the paradigm to be that of making new discoveries or producing new effects, rather a construction or a search than a demonstration.  But no: the aim is not to make or do new things, but to know familiar ones in some new way.  He spells out exactly what he wants to learn, as if in some sense he must already have been there.  Here is the full enunciation of this day of learning:

Wanted to know how the lines of force were disposed in and about magnets and Iron generally under certain circumstances of position, and for this purpose used fine iron filings upon paper over the magnets, sprinkling them evenly and tapping the paper lightly. 

      Faraday chooses his words with care, though the resulting formulations are not always altogether transparent.  He will be studying the already well-known “lines of force” traced by iron filings in the vicinity of magnets, but he is setting a special standard for the day’s work by way of the terms generally and under certain circumstances of position.  These are, I think, mathematician’s standards; to know the pattern generally means to know the entire pattern, whole and for itself — as a mathematician might want to know not a segment of a circle, but exactly the whole circle for what it is, including of course its center.  Parallelograms, for example, can take many forms which look very different from one another; yet if we want to know the parallelogram generally, no particular instance will suffice.  So then magnets throw out many lines, but to know them generally perhaps means to know the pattern they make as a whole.  And when he adds under certain circumstances of position, the term certain means precisely defined, exactly reproducible configurations.  His methods will be those of the expert empiricist (as his underlining of fine filings already suggests), but his aims go beyond even a highly systematic collection of instances — which would be the way perhaps of natural history.  Faraday wants, not a merely descriptive account, even a careful and accurate one, but something more: just that true knowing which we may recognize, though he would not put it so, as the way of the mathematician.  This is to be the day, in Faraday’s laboratory, in which the familiar pattern of lines of force about a magnet will be rediscovered — reborn now as a mathematical object.  This is what I am proposing to call Faraday’s mathematics

      How indeed do we know when we are dealing with a mathematical object?  Let us give Euclid’s answer first, just for practice.  We recognize it by the fact that it is of an exact sort, timeless and in some sense fully knowable.  Euclid offers us the circle, for example, as an exactly known, precise and unchanging object.  We know that just as we cannot draw a mathematical point on a blackboard, we cannot construct a true circle on paper.  A true point has no parts, but every dot we can contrive is a horrendous blur of parts — thousands of different lines can join any pair of such dots, and no statement about the outcome could be precise.  Euclid’s mathematical objects are, therefore, creatures of the world’s mind: they are exact and timeless precisely because they are not contaminated with the obscurity which matter inherently brings with it.  We know the circle because it is a thing of the mind; it exists in the mind’s eye, and in the world’s mind, untouched by time or place, it simply is.  When we pretend to construct a Euclidean circle, nothing is happening, we are not making anything: Euclidean construction is simply a way of borrowing upon time to think something through — something which in itself has no taint of time or process.

      That was making and knowing in the world of Euclid.  But between the world of Euclid and the world of Faraday there has been a sea-change in the world of making and in the world of knowing.  The difference is, roughly speaking, God. 

      At St. John’s, as I have mentioned, we hold a certain secret of success, which is the myth — at once so true, and yet potentially so deceptive — that we can live in all the worlds: that all the books were written yesterday, and just for us.  Socrates says that he walked down “yesterday” to the Piraeus, and it is always true.  It is true as well that we could not be moderns if we did not know, thus intimately, how to be Greeks — and yet it is so easy, under the spell of this delusion, to forget how unlike the world of Euclid our world actually is.  We can almost forget that for the first year of the St. John’s program we live in a world without God.  Nothing could be further from the mind of Plato, Aristotle, Aeschylus or Euclid than the concept of a creator God — the opening question of the St. John’s seminar is likely to be about Fate; and in a world ruled by Fate there can be gods, many and wonderful, as Homer shows us, but it is a world of black death too, and there is no God in final command, no benign hand on the tiller. 

      Things are of course never, or hardly ever, as simple as they seem, and in order to make a point I’m vastly oversimplifying in this case.  But roughly speaking, it is true that if the God of Genesis does indeed create and rule the world, Necessity is vanquished and truth is no longer confined to a world apart, a world of mind alone.  Mathematical objects which dwelt in a separate and timeless world of Being now enter our immediate and material world, enter history and take on Existence in time, in place of mere Being.  Where the truth which made a circle timeless lay in the conception expressed in its definition, now the truth of an element of God’s world arises because God created it — because God said so

      Faraday’s mathematics is of this new kind, the mathematics of Existence rather than the mathematics of Being.  When he tells himself he wanted to know how the lines of force were disposed generally, I propose that he is demanding mathematical knowledge of this new and very different kind.  His day’s work may reveal to us what the mathematics of Existence will be like, and how it will be obtained.  We know that in place of such conceptual instruments as the ruler and compass, Faraday will perform his constructions in the laboratory, using his special skills with the iron filings as existential drawing instruments.


2.  The simple magnet

      Following Faraday’s enunciation of purpose, the work of this new day begins with paragraph 11667, with the words:

First a simple magnet, being a needle of about this size, well magnetized by a horseshoe magnet of power... [Slide 1]:

Slide 1

      It is as if Faraday were writing Chapter I of the book of the new mathematics, beginning most appropriately with the most elementary figure of this new learning.  We notice that it is defined ostensively, with the pronoun this, and a figure in the margin in which he depicts the needle and the pattern which emerges.  Here, then, is the way exact figures are to be introduced in Faraday’s mathematics — not by definition or conceptual grasp, as ideas, but by exact invocation, really here and now: not by Euclidean construction, which is an intellectual act, but by something closer to a conjuring trick — not by construction, then, but by invocation, or magic.  The little word this is so important!  This particular existent needle is the thing itself — this very size (quite tiny, apparently just a little under two inches) ennobled to become the foundation on which Faraday’s mathematical learning and teaching will be built.  Simple, humble, yet proud, because it is very real.  The thing itself.  It could be a sewing needle of Sarah’s, or a bookbinder’s needle, out of Faraday’s own trade.

      Of course, it is not as a needle that it is a mathematical object here, but rather as endowed with magnetism, conveyed by a horseshoe magnet of power.  A sprinkling of the iron filings quickly reveals the real mathematical object we will be dealing with — not matter, not even the steel of a good needle, but power [Slide 2].  Faraday alludes with special respect to power, and the great magnets which convey it.  Power is to be the subject matter of his mathematics; it is power that he wants to know about.  When he speaks of a horseshoe magnet of power, the phrase bestows an endowment or an accolade, as one might refer to a court of competent authority, a jewel of great price, of a knight of Arthur’s court.  The important point is that this figure, seated in this exact needle but made out of power, is not less mathematical than a figure of Euclid’s, only differently mathematical.

      Faraday delights in the result:  He crowds his description with praise:

It gave beautiful curves having perfect simplicity of form...

And it is clear that he is contemplating a truly mathematical object, for what else could be meant by the phrase perfect simplicity of form?  Perfection and simplicity exactly characterize a mathematical object, which he draws for his own contemplation.  We may be tempted to insist, “But what is this figure?”, wanting a definition in terms of ratios or intersections.  But that would be the wrong question.  Instead, Faraday is content to rest with the figure as he gets it, as God we are told rested with the completed creation.  Faraday’s mathematics includes a dimension which does not appear in Euclid’s text: Faraday says, It gave beautiful curves.  For conceptual definition, Faraday with humility substitutes delight in a form which he can know by seeing, but would not attempt to define or explain.  Knowing no longer means explaining.  In Faraday’s mathematics, delight perhaps outweighs definition.  It may be a good exchange.


      Faraday adds one little remark which is actually fraught with more significance than might appear:

...but it is to be remarked that N or S lines issued not from the ends of the needle but from down towards the middle.

      We have remarked that Faraday is not only in his own terms unmathematical, he is in a sense anti-mathematical.  He is not only engaged in his own researches, strictly disciplined but unmathematical, but is actively opposed to what he thinks of as the ways of the mathematicians.  Thus from the outset, Faraday’s inquiry in his new mode will be no mere contemplation of a perfect pattern, but will take quiet satisfaction in noticing that the pattern is of one sort, and not another. 

      Faraday has long had a problem with the mathematicians, who have been happy to dismiss his beloved lines of force by demonstrating that the same patterns can perfectly well be generated mathematically if one assumes that forces act at a distance from two points, the north and south poles, at or near the ends of the magnet.  The resulting curves then, though interesting, would have no special significance.  In particular, they would not indicate the existence of anything in the space.  The little filings are simply responding to forces which arise from the poles: take away the filings, and there is nothing present in the space.  But this, Faraday supposes, would require that the lines do emanate from such poles.  Therefore the fact that the lines come from other sources as well, down towards the middle, as he says, will be important information in his long struggle with the theoreticians — a struggle ultimately not against any particular theory, but against the very idea of theory itself.  It seems that when Faraday told himself he wanted to know, he meant he wanted to know fact.  He is embarking on a dedicated investigation of one important piece of the creation, of entities which are not theoretical constructs, however splendid, but instances, very simply, of fact.

      But Faraday’s little picture seems so innocent!  He perceives perfection of form, but he captures it quickly — as he must, for he has a long day with many more drawings ahead! — and without refinement.  Yet should we not say that in these few strokes are invested exactly that simplicity and perfection his words celebrate?  The Diary record is a visual trace of an act of perception, though as a rapid, selective sampling it cannot convey to our eyes what was actually before his own.  Would it get better by being more accurately or fully spelled out?  Faraday is in fact interested in capturing a fuller record of the patterns, and before the next day is out, he will have developed new techniques for recording the figures more fully and accurately.  As he does so, however, we might wonder whether in the process they become more mathematical.  He uses gummed paper and evidently draws on his old skills as a bookbinder to fix the actual iron filings in a permanent image.  Refinements of such techniques are important to him, and a detailed account of this new procedure in the Diary for Wednesday closes on this note of well-deserved satisfaction:

... pressure given by the hand or a 56 lb weight for half a minute or more.  On being taken up, all the filings in their proper places were attached to the gummed paper and when that was dried were fast attached to it.  No. 1 on the next page is the very specimen and the first made.

and here it is, just as Faraday preserved it in the Diary [Slide 3].

      We can perceive better now the “perfect beauty of form” to which Faraday himself is responding, though it must be a mistake to let this finer picture substitute for the true, underlying entity which Faraday is resolved to come, through these patterns, to know.  These are pictures of the figure, neither is the figure itself.

      Later that day, he says of yet another technique, verging on photography:

The results will do perfectly well as drawings to go with the paper and to the artist.

Slide 3

Slide 4

      The Diary may be a private place but Faraday is already intent as well on launching a public work.  Already, these modest seeds are destined to evolve into elegant engravings for the pages of the Philosophical Transactions and finally the Experimental Researches.  The innocent needle becomes a formal bar of steel [Slide 4]:

      Here, it might seem, is the graven image of a genuine, new mathematical form, represented in full glory.  Faraday defines it, however, only by telling us exactly how to conjure it.  Otherwise it comes without a pedigree — comes with the signature of no LOGOS — and if anyone were to contrive an equation for it, we can be confident that would not only be meaningless to Faraday, but irrelevant to the figure as Faraday knows and intends it.  To invoke once more that term always so important to Faraday, this new figure very simply represents a fact.  In place of an intellectual definition, we have a detailed recipe for its preparation.  Any scientist who follows the recipe with skill and care will meet that same fact.  In this sense, the Experimental Researches are, like Prospero’s Book, a great cookbook — an exacting manual for cooking up facts.  Facts are existential universals, exactly real existences.

      In the course of his scientific works, Faraday has had occasion in a similar way to certify the existence of many new and unique entities, and for each unique entity he has sought a correspondingly unique, made up name.  Over the years, he has long followed the practice of sending off for Greek names for them as they arose.  William Whewell, Master of Trinity College at Cambridge and expert in Greek, has obliged Faraday with Heracleitean terms for the upward road (anode), the downward road (cathode), and perhaps from the dialogue of the same name, Ion, the wandering one.  Now Faraday sends off his magnetic images to Whewell with the request for an appropriate new word.  Apparently — though we most unfortunately lack this one interesting letter — he has characterized them as figures of revolution about an axis in three-dimensional space.  And the answer has come back “SPHONDYLOID,” a word which he tells Faraday bears such suggestions in anatomy and botany (my dictionary reports a vertebra, as well as a certain beetle, he sphondule.  But Whewell always goes for the highest ground, and in a subsequent letter he has discovered a connection with the spindle of providence, a suggestion which seems in turn to appeal to Faraday. 

      Faraday adds the prized accolade of power, to yield the sphondyloid of power.  Though not destined to become one of Faraday’s best-selling scientific words, this is the name by which we shall refer to it for the rest of this evening.  Faraday himself introduces it into the literature only later, guarding it for a most carefully calculated rhetorical point, in the pages of the Philosophical Magazine, when the underlying figure has been fully certified as an existent entity.  But we need a term now, and may with apologies begin using it from this point on.  We should keep in mind that we have only a picture, with little clarity yet as to what it is that the term sphondyloid of power will ultimately prove to mean. 


3.  The needle broken

      We might, with the possession of this perfected image of the sphondyloid of power, imagine that our principal work is accomplished, and only interesting details remain.  The opposite is the case; these pictures are no better than the idyllic realm firmly rejected at the outset of Plato’s Republic as the naive City of Pigs.  Only a drastic, dialectical move will break the spell of rustic contemplation, and take us into the realm of serious knowledge to which Faraday, and we with him, are now committed.  Apart from this digression on the preparation of images, we are still only at the third paragraph of the First Day!  And Faraday’s immediate next move is to smash the beautiful sphondyloid: for he demands to know what is inside it.  

      Paragraph 11668 alludes, almost as in passing, to a decisive bisection of our once-perfect figure:

When the needle was broken into two parts, each part by itself acted well as above...

      The figure is not passive, and the result is the unleashing of a new action — the magic construction invoked now twice over [Slide 5]:

Slide 5

      The question immediately suggests itself to Faraday, can the broken needle be restored?  And indeed, we can be sure that Faraday already knew something of the irreversible path he is now very deliberately pursuing, a one-way bisection Euclid never knew — Euclid’s lines do not snap when you break them!  By breaking the magnet and pressing the parts together, Faraday is prompting the lines to spring to life — dare we say, almost literally?  He is making his way into the interior of the original needle, with the intention of demonstrating what he now describes:  that the lines run courses not only outside the needle, but within it as well.  He draws this picture, and concludes:

Now indeed it appears that certain of those curves which before were entirely within the body of the magnet are expelled into the air, because of the sudden diminution of conducting condition at that spot by rupture and want of continuity, and of those which thus come out through the sides of the magnet, part returns and is discharged at the nearest pole and part goes on and dips into the further half, the circuit being completed in space as in the unbroken magnet...

      This is high drama!  He captures a second picture.  In the first, the magnets had been “put together”, though they would no longer blend.  Now he draws them apart, stretching their inner lines over a longer path through the air [Slide 6]:

Slide 6

      He is exposing to the eye what he knew had been hidden inside.  What he is recognizing is that this system of lines, through this variety of forms, is revealing itself as a single entity, undergoing changes and transformations, but remaining through it all a single whole whose parts relate to that whole as organs might to a living body.  He increases the separation between the two halves, drawing conclusions as he goes:

Each having its own equator or maximum place of inner curves which would contain just as much force as the original equator...

      Is it, indeed, now simply one thing, or two, or even three as well?  Here one magnet has become two, but each he is convinced has just the power it had in the first place — the power is as before.  However we count, in Faraday’s arithmetic it is an entity which endures.  In his arithmetic modulo one, it is

One equals (One + One) equals (One + One + One) equals One.

      He now goes on to put this curious singleness through a sequence of transformations.  These are the “certain circumstances” he has had in mind, and through them he tracks the vicissitudes of the successive configurations carefully, at once by eye and in speech [Slide 7]:

Slide 7

He continues:

When placed at right angles... It is easy to see how this disposition arises from the former ... and how some of the curves of the third portion are now removed into the second, going across the air both at the poles in contact and at those which are now 90° instead of 180° apart.

Further [Slide 8],

Slide 8

The two halves were now brought together, as if one had made a further movement of 90°, so that they laid side by side with [un]like poles together ... the system is now weak and it must be remembered that by far the greater number of lines of force are passing directly across from magnet to magnet and are not sensible here without further inspection.

      We realize that Faraday is pursuing a continuously evolving motion picture bearing full intelligibility:  It is easy to see what is going on, he says.  And then...

To make these visible the little magnets were opened out slightly and then the lines were thus [Slide 9]:

Slide 9

... the equatorial ventrum of each small magnet begins to appear as the lines of force of each magnet tend to go back from one pole to the other, rather than across the intervening space...

     When the space was further increased [Slide 10]...  

Slide 10

      Can a mathematical figure be such a motion picture?  Faraday is connecting these stations in his mind in terms of the behavioral history of a single entity, a body of lively power reshaping itself through an entire ordered odyssey.  This is his vaunt:

All these cases are easily referable one to the other and flow as a very natural consequence from the nature and character of the lines of force.


5.  Insight

      All this while Faraday has been the observer, watching closely, and recording paragraph by paragraph the history the lines of force have been revealing in response to his astute promptings.  At this point, however, the flow suddenly reverses.  It’s remarkable: we are witnesses to Faraday’s moment of insight.  Whereas until now he has drawn, albeit with intuitive understanding, what his eyes have shown him, he now grasps what is happening with the lines of force, and can confidently dictate to his hand, to draw a figure his eyes cannot show him.  One previous figure had been hesitant [Slide 11]...

Slide 11

      But the next is altogether confident and clear.  This is a picture not of what he sees, but what now he knows that he knows:  He has endowed the needles with thickness enough to show forth what he now knows is going on inside [Slide 12]: 

Slide 12

      His eyes never saw this, but his mind did.  Every line is continuous, and every line has had its own history through each of the stations of these motions.

In the last case for instance, the lines through the magnet and outside generally as represented.  Some pass through both magnets, while some turn short round and do not; and the maximum internal effect is resident within the equator of either magnet.

      But this was only a “for instance” — he could as well draw, with the same x-ray of his mind’s eye, the full story of the lines of any of the patterns he has drawn.


      Faraday has found the mathematical entity of which he has been in pursuit.  He has seen the system of the lines of force as a single body, in its completeness.  Every line within it is continuous and has a continuous history, and so it is with the whole body of lines as well.  I think he already had it in view when he asked the opening question of this Day; but he has had to see it: if this is a ritual, it is all the fresher if he has seen it all, clearly, in every paragraph of the unfolding.  Yet not one of its positions or arrangements is the entity, nor even all of them together in the aggregation of their transformations through time. 

      Rather, it is what he has called, remarkably, the nature and character of the lines of force.  Nature —  PHYSIS — Aristotle tells us is a principle of motion within a self-moving beingIn the unadorned candor of his perceptive vision, this seems to be exactly what Faraday has seen.  He does not emphasize this, but this is what he says.  He is perceiving the sphondyloid of power as if it were a living thing, and finds its intelligibility in the consistency of its underlying nature.  We are calling his sphondyloid a mathematical object, yet he is treating it as a living thing.  Is this a contradiction?  Perhaps we should not be so surprised — after all, Ptolemy sees the planets as mathematical objects precisely because they are intelligences, while on the other hand the term PHYSICS has always referred especially to living material!  I would suggest, incidentally, that the term has that meaning for Newton as well, and that his famous Principia — the Mathematical Principles of Natural Philosophy — is fully intended not merely as a text in what we now reductively call physics, but as indeed, the Book of Life. 



The scientist as gardener

      Bacon, in his essay on Gardens, reminds us that “In the beginning, God Almighty first planted a garden,” adding that, “Indeed, it is the rarest of pleasures”; and we have seen that Faraday, too, delights in the works of the Creation.  It is a nice suggestion to think of the natural order as a garden, and as Faraday has such a sense of the life of his materials, to imagine Faraday as God’s Gardener, and the Diary as capturing the special wisdom of a gardener’s notebook.  Bacon’s voice is surely the opposite of Faraday’s — Bacon, the Lord Chancellor, certainly thinks of God as giving directions and having His staff do the digging; Faraday by contrast is the hard worker, loving, knowing and managing the earth and all its creatures.  The magnetic pattern is not only perfect, but primarily and fundamentally beautiful: to know it is to know it as beautiful, while to make it is rather to cultivate lovingly, than to construct.  The scientist’s reward is not as creator, but as cultivator or gardener — one who opens to us in full clarity the truth and beauty of a work which belongs first to another.  It is a role of beautiful humility.  Faraday’s mind is, as we shall see, constantly inventive and busy — but it is always the wise and understanding mind of the master-gardener.  It is never the role of the gardener to explain or create life itself.  Euclid knows his figures thoroughly because they are the works of mind not unlike his own.  Faraday knows the wondrous magnetic patterns as works of a Creator far greater than himself.  Yet this garden is our inheritance, and Faraday is taking full charge on behalf of us all!




A.  The Moving Wire Writes: Faraday on Quantity


1.  A Second Approach to the Magnetic Lines of Force

      Faraday might seem at this point to have completed his study of the magnetic lines of force.  We believe we have seen, beneath their patterns’ apparent variations, an underlying wholeness.  But can we be sure?  Faraday recognizes that a further step is essential.  He has from the outset had in mind a second, independent method of examining these same lines of force — one which will move from a qualitative study of their spatial forms, to a new explicitly quantitative point of view.  They have not yet been revealed as strictly knowable — we have not yet taken the full measure of Faraday’s mathematics, or his new physics. 

      What quantity can Faraday measure, to determine whether the lines of force constitute, though all their evolutions, a single unvarying system?  Just as we know the parallelogram fully only insofar as we can measure its magnitude, and thereby gain the concept of area, so perhaps we can truly know the sphondyloid only insofar as we can find a corresponding measure of its magnitude.  What, then, will be the megethos of a sphondyloid?  We have seen how they pass through many apparent forms, but are these not varying manifestations of a single measurable entity?  Faraday’s word for this underlying, Protean magnitude is power — and only some mode of measuring it will give his mind rest.  He often speaks of the number of lines of force, but they aren’t quite like pennies in a jar or sheep in a pen.  Yet it may help that the only relation he seeks is that of equality.  The only number he can really use is one

      Not surprisingly, perhaps, he has already been contriving the very instrument he will need to make this measurement, in the form of a simple loop of wire, connected to a meter — a special form of galvanometer — one which is either very sensitive, or not sensitive at all, depending on how you look at it — to detect a flow of electricity through the wire.  Twenty years before, early in his career as discoverer, he introduced to the world the principle on which this new test of the lines of force will rest.  We shall see in a minute just how this method of the moving wire works, and how it will constitute the second, quantitative approach to the lines of force which he requires.  


2.  The Moving Wire as search engine

      Faraday’s probe for his quantitative investigation of the magnetic lines of force will be the moving wire.  It can take any form that a wire can take, but for present purposes Faraday tends to use a loop, one form of which he draws in his Diary in this way [Slide 13]: 

Slide 13

      You see that it is a simple loop with a long tail which will go off to connect to the galvanometer; the loop is of a size to fit neatly over the end of one or two of Faraday’s magnets, in this way [Slide 14]:

Slide 14

      The loop is only really closed when the far end of the tail is attached to a meter which will detect a flow of current through it.  I hesitate to show you what Faraday’s meter actually looks like — for the moment we can suppose that it indicates when a current flows in the loop: the more the current, the more the deflection.  We will refine that statement in a moment. 

      I will not try to explain the principle on which this works, which was perhaps Faraday’s most famous discovery, which he had made some twenty years earlier — if I tried to explain it in Faraday’s way, we would be here a long while and I for one would soon be lost.  People are still trying to explain it.  But it can be summarized in this little drawing of Faraday’s — a beautiful example of a simple topological figure, showing the relationship of two shapes [Slide 15]:

Slide 15

      Here we have two interlocking loops. The one on the left is our electrical circuit consisting of our loop capable of carrying an electrical current, with the meter for measuring it — one circuit.  The other is a closed magnetic line of force: we see how important Faraday’s earlier insight is, that each line of force makes a closed loop of this sort.  There are two ways of looking at this.  If the wire, the loop on the left, is carrying a current, it gives rise to a line of magnetic force which runs around it in the direction shown.  On the other hand — and this had been Faraday’s astonishing discovery twenty years ago — if we start with a magnet but no current, and then move the wire in such a way that it cuts through the magnetic line of force, a current will suddenly be induced in the electrical loop and it will show up on our meter.  So we can use the loop in this second way to detect the magnetic lines — and our meter will tell us when we are cutting through them and, hopefully, in some sense “how many” we have cut.  We are in contact with the power!

      In this one simple interlocking figure, then, we can see the beautiful symmetry and deep correlation Faraday is increasingly perceiving between electricity and magnetism — an electrical current gives rise to magnetic lines of force entwined with the circuit; while the same loop cutting through a line of force will have a current induced in it.  Faraday is now realizing that when the loop is used in this second way, it becomes the very instrument he needs to measure the magnetic lines of force quantitatively.  The sketches in his Diary soon show him making all sorts of investigations with his moving wire, and the tables accompanying them show him equally carefully recording sets of quantitative results.  Here we see how he has stood up the little magnetic needle in a block of wood, and is bringing the loop down over the top, while reading his meter to see what how many lines he has cut [Slide 16]:

Slide 16

      He tries it with north up, and then with south up; and he tries it with the loop carried down to various levels.  He finds that north and south readings tend to be the same, and that the best reading comes when you go all the way to the equator and stop there.  Lifting the loop off gives the opposite reading to carrying it down.  He tries it with two equal magnets, side by side, and finds he indeed gets twice the reading.  All is well, he keeps saying [Slide 17]:

Slide 17

      He is relating the concept of cutting the lines of force, and thereby measuring them, to the image he has had of the underlying form of the sphondyloid of power.  Each line of the sphondyloid makes one closed path.  If the exploring loop comes down upon the sphondyloid from above and makes its way down to the equator, it will have cut each and every line exactly once, and will thus have measured exactly the total power of the entire sphondyloid.  The problem of measuring sphondyloids is solved, in principle.


3.  The Escher meter

      To avoid details which in fact might not have interested Faraday much, I have given a merely suggestive account of his current meter, or galvanometer.  It is often instructive, however, to consider what Faraday is not interested in.  An entire era of electrical technology was springing from Faraday’s discoveries, and he has had a role in pursuing many of these complexities; but as a philosopher he tends to avoid the very things which most captivated the technologists.  For instance, a sensitive meter will show that the current in the loop depends at any moment on the instantaneous direction and velocity of motion of the wire, as well as on its resistance to the flow of current.  There are laws for these sorts of things, and a whole era of elementary physics textbooks — and even it may be a St. John’s lab manual or two — have filled our heads with them.  A meter needle which is sensitive and responsive to what is going on would be jumping all over the place as the loop made its way through the sphondyloid of power — a situation too complicated to be of any importance.  If on the other hand the meter, though sensitive, were very sluggish it wouldn’t notice all that detail, but sum up the whole experience in a single overall reading.  This is the sort of meter after Faraday’s own heart.  He has little interest in these details at this point — he is on the track of something far more global and more important.  So he has devised an instrument which, though sensitive to small currents, is utterly insensitive to the detail of events.  It will report one single number measuring no more or less than the underlying power itself.

Slide 18

      To contrive a meter of such simplicity requires something like demonic intricacy, and as this reveals in a sense the other side of Faraday’s mathematical mind I can’t resist showing you what he has done.  We have to start with a standard galvanometer of the sort that was being used by proper German scientists of a completely different frame of mind to make accurate measurements.  This one was devised by Wilhelm Weber, who with Gauss had been measuring the earth’s magnetic field [Slide 18].

      When current flows in the vertical loop, it creates a magnetic force at the center, and there a compass needle is mounted to detect that field.  When there is no current, the compass needle is held firmly in place by the magnetic field of the earth.  But the loop is oriented so that when a current does flow, it tries to point the needle at right angles to the earth’s field.  There is a battle between these two orthogonal fields, and the needle adjudicates between them — the result being decided by trigonometry.  I’m sure Faraday doesn’t know anything about trigonometry, but he sees what’s wrong with this kind of meter, and how he can make one much better for his purposes.  Weber’s meter is quick to take a reading, so it notices details you’d rather not see.  It has a lot of wire in it, which has resistance, so you have know algebra and take into account something called Ohm’s Law, and since it also has to work against the strong field of the earth, it is not very sensitive.  Faraday wants to make a meter which is exactly the opposite on all counts: very simple, very sluggish, and at the same time very sensitive.  He has a special galvanometer carved out of a solid block of copper, of which this is his delighted drawing [Slide 19]:

Slide 19  Slide 20


      He has the mathematical mind of an Escher!  You will see why it’s shaped like this in a moment.  Using the solid block of copper, he effectively creates a circuit of virtually no resistance, thereby swallowing up all possible current and eliminating any need for bothering with pesky things he doesn’t much care for, like Ohm’s Law.  To increase sensitivity immensely, he adopts a trick called the astatic principle for his compass needle.  Everybody knows if you have a compass needle it will point north.  But by stacking a double-decker pair of needles, pointing in opposite directions, and joining them so that can only move in tandem, you make a wonderful compass needle which will not point at all — what a triumph! — thereby canceling any constraint by the earth’s field, so that takes care of that limitation [Slide 20].

      On the other hand, you can run your current in a twisted-back double-reversed pattern so that whereas the earth’s field cancels, the currents combine to turn each needle in the same direction.  Here is Faraday’s sketch of that concept [Slide 21]:

      The only restraint on the needles will come from an extremely delicate suspension thread of cocoon silk.  And by making the current climb an Escher stairway of interrelated loops with, one suspects, some connection through the fourth dimension he makes one current work twice on each needle, and keeps everything in symmetric balance.  When he got it built he found it was so delightfully delicate it took 26 beats of his watch to swing from one extreme to the other.  You could practically go out to lunch — or combine a whole series of successive experiments, as Faraday in fact does — while it was taking a single reading. 

Slide 21

      Indeed, this wondrous meter is a very picture of Faraday’s mathematics — so delightfully mathematical that practically all mathematics, and physics as well, is eliminated, and so distant from the flow of time and the detail of the daily world that it could perceive only the most inclusive forms, the most global, and hence only the most intuitively interesting, measures.  It’s like NOUS shaking off the fetters of LOGOS.  Might it not be a truly Socratic instrument: attending to the one, and turning away from the many?  It’s Alcibiades and the Flute Girls!  It is in any event the perfect instrument for measuring sphondyloids!  Faraday knew virtually nothing of so-called Newtonian mechanics, but he realized very well why he did not care for, or have any need for, that sort of thing.


B.  Trial:  The Search for the Unchangeable Sphondyloid


1.  Getting started


      Now we have the instrument, what shall we do with it?  In one way or another, the underlying sphondyloid must be pursued through all the strange variations and distortions we have seen it undergo, and must be shown to be in truth constant in magnitude under every vicissitude to which it can be subjected.  Faraday must show that the sphondyloid is a real and inviolable entity, no matter how you challenge it.  The first step is easy; we have already seen it.  We will measure the undisturbed sphondyloid by carrying the loop over the magnet and down to the equator.  That gives us the benchmark, underlying measure to which all the others must conform: whatever the outward changes of appearance, the measured power must the be same.  All we want to get from now on, is equality, to keep getting the same number again.  So step one is easy. 

      The next thing Faraday did with his original magnetic needle was, as we saw, to break it.  The lines, as we saw, were vented into a variety of different systems.  We got a pattern like this [Slide 22]:

Slide 22

      Trying to account for all these, to prove the total was constant, would be quite a mess!  This looks like a hopeless case.  Faraday actually toyed with the situation, bringing in the loop sideways in the equatorial plane, to measure parts of this array bit by bit [Slide 23]:

Slide 23

      But this is definitely not Faraday’s style: Again, it is a question of taking a complexity and turning it into a simplicity

      Suppose instead, we re-construe our figures in an entirely different way?  Instead of looking at the broken magnet as one hopelessly distorted sphondyloid, let’s think of it as two separate sphondyloids, the patterns revealing a dynamic action going on between them.  This same picture, of two magnets with “unlike poles” together, becomes a case of strong attraction [Slide 24]:

Slide 24

      The lines between them now are seen as lines of attraction, as if they were stretched elastic, tending to contract.  If we reverse the right-hand magnet, bringing like poles into proximity, we get a corresponding image which in turn looks like repulsion [Slide 25]:

Slide 25

      With this clue, Faraday is ready to set up a simple system for the testing of sphondyloids [Slide 26].  We can regain simplicity if we focus on measuring just one of the two sphondyloids.  The loop need only be used to measure the strength of DE in the usual way.  A second, stronger magnet A is brought up solely in order to put DE to the test.  In this attracting mode, A will tend to strengthen DE, and the question becomes, whether DE and its sphondyloid will remain inflexible, or yield to the persuasions of the stronger magnet. 

Slide 26                                                                          Slide 27

      When the dominant magnet is reversed, it will be repelling, and tend to decrease the strength of DE; which in turn looks like repulsion [Slide 27].

      This is the format in which the very existence of the sphondyloid will be put to a series of crucial, quantitative tests — a formal ordeal, the trial by torture of the true sphondyloid.  This formal structure of the test, with the formal stations at which judgment is to be made, is in a sense the negative image of a fact.  We have a fact only as consequence of a judgement.  Here, we can point to a true sphondyloid as a fact, rather than a mere concept, only insofar as we can be sure that at least some of our magnets are truly permanent, so that we can truly say in provable instances, one magnet, one sphondyloid.  So now the question of the existence of the sphondyloid has turned into the quest for a truly permanent magnet — for if no actual magnet can be rigidly permanent, the sphondyloid must remain no more than a fascinating image — a tantalizing picture, more or less akin to table-lifting, a concept with no existential home — thus not an existence, not a fact on which to build.


2.  Defining the canonical trials

      The investigation, a true Inquisition, becomes a highly formal structure — and we see how much Faraday already intended from the outset when he spoke of certain circumstances of position.  This formality makes the test a matter of record, strictly reproducible anywhere, by any experimenter, anytime.  There are four strengthening positions [Slide 28]:

Slide 28

and four diminishing positions [Slide 29]:

Slide 29

making a complete ordeal of eight positions [Slide 30]:

Slide 30

      The true sphondyloid must remain constant under all temptation, standing fast to yield neither an increase when under seduction, nor a decrease when attacked.  Euclid’s mathematical figures were never called upon to endure such cruel treatment!

      In each series it is the third position which exerts the most severe test.  From the original power of 16.3°, DE is tempted up to 18.73°, while under extreme repression it is forced down to 15.37°.  Witches have burned at stakes for failures less than these!  Faraday may seem more forgiving, yet he reveals some evidence of discomfort.  He says of the second set:

All these are a little below the original force of DE as they ought to be, and show how little this hard bar-magnet is affected. 

      Faraday is actually not at all forgiving of weak and yielding magnets, so perhaps at this point he is rather testing his faith in his own insight, than in the magnetic rigidity of available steel.  He tells himself firmly:  Unchangeable magnets are therefore required; and we can understand his frustration when we read the trouble he has gone to in preparing his still imperfect magnets D and E for their test.  In Faraday’s world, which is quite possibly ours, it is altogether appropriate for a good mathematician to be, in the line of his regular duty, a good chemist and metallurgist as well — Athena, Circe, but above all, Hephaestus.  Listen to this:

I therefore procured some plates of thin steel twelve inches long and one inch broad, and making them as hard as I could — [we know he is working here with severities of fire and water!] — afterwards magnetized them very carefully and regularly, by two powerful steel bar-magnets, shook them together in different and adverse positions for a little while, and then examined the direction of the forces by iron filings.  Small cracks and irregularities were in this way detected in several of them; but two which were very regular ... were selected for further experiment and may be distinguished as the subjected magnets D and E.

So this was the origin of our valiant pair, DE ! 

      Faraday continues his relentless search for a truly hard steel, but in his own mind is now ready to extrapolate to conclusions which would only be valid for perfect, unchangeable magnetsFact, insight, and sheer faith in some way coalesce at this point. For Faraday, the evidence has become decisive that beyond the fallibility of even the best of achievable steel, the underlying sphondyloid is present, constant and enduring.  Such a fact, to which all else has been ordained, is the perfect diamond of truth on which Faraday’s science builds.  It may be a diamond still in the rough, but he has seen through to a diamond nonetheless and that is what makes all the difference.  Faraday now knows that magnetic power consists of interactions of these real entities — they fill space, and it is they which constitute the tremendous powers of magnetism.  What had been empty space, even the most total of vacuums, is suddenly revealed as charged — not with matter, but with something far more real and interesting, with vitally active electro-magnetic power. 





      One of the more interesting books I’ve seen on Hegel is titled “Why Hegel? Why Now?” — and it has struck me that a book about Hegel almost by definition has to be about Hegel now — it would be contradictory to write seriously about Hegel otherwise.  I realize that I have the same feeling about Faraday and world history — his message is inherently for us, especially here at St. John’s, today.  So in closing I ask, “Why Faraday? Why Now?”  That sounds like a problem for the Question Period, but let me suggest a line of approach which I myself would take. 

      I want to go back to our unbroken needle — miraculously restored [Slide 31]:

Slide 31

      This time when we snap it open, it will prove to be full of light!  Let us break it again, placing the two halves beside one another, this time thinking in terms of the interaction of two sphondyloids, and the transfer of power as one acts upon the other [Slide 32]:

Slide 32

      The lines of force are taut!  When sphondyloids encounter, each reveals the life in the other, and something like a process is coming about.  If we reverse one of the magnets, they now repel and we have to persuade them to stay together [Slide 33].

      This looks like opposing powers.  You say these are only tiny needles, but remember — here the only magnitude is one — there is no scale.  They may be as big as the cosmos. 

Slide 33

Slide 34                                                                        Slide 35

      Here they are as graven images, patterns in full splendor, of powers coming together — Faraday calls this coalescing [Slide 34]... and diverging [Slide 35].


      Here are all the dynamos and motors of the world to come, all the cell phones and all the signals and, as it turns out, all the light, the heat, all the life.  And yet what most interests Faraday may be something deeper, which lies behind all this.

      Faraday had won through to the fact of the sphondyloid’s real existence only by virtue of this one figure, probably the fullest image of Faraday’s mathematics — at once the least, and yet the largest of his images (as I once heard said of a Rembrandt etching) [Slide 36]:

Slide 36

One circle of magnetism, dynamically linked to one circle of electricity, constituting together — should we say? — one sphondyloid of electro-magnetic power.  In Faraday’s mathematics,

(One plus One) equals One

What is this figure, composed as if of insight, telling us?  Faraday is sure he sees the implication of transfer in time, and invests some of his last energies as a discoverer in an essentially hopeless attempt to measure the time of magnetism — which might be the velocity of light — across the backyard of the Royal Institution.  It was an experiment doomed to fail in the first instance, but doomed as well to succeed in the long run. 


      The cascade of successive publications which sprang from this first day’s work of Faraday’s, really of course one lifetime of work, were gathered in turn and entered into Volume Three, the last, of Faraday’s Experimental Researches in Electricity.  They were no sooner published than they were bought by a young Scotsman, curiously out of place at Cambridge: James Clerk Maxwell, who immediately read them and fell in love with them — at once with Faraday and with Faraday’s insights.  From then on, it was a labor of love, frequently attested to by Maxwell over the years, to translate Faraday’s enticing diagrams into the symbolic structures of modern field physics. 

      Why such love?  I am saying that it was by no means merely the fascinations of objective science — but primarily the very spirit of Faraday’s dialectical search for the true unity beyond all the apparent diversity which was leading this way into modern physics.  A sense of the beauty, vitality and wholeness of nature, and an open delight in all of these, gave birth to Faraday’s figures, and thence to Maxwell’s devotion to them as well.  So if NOUS or its modern descendants characterizes this spirit of Faraday’s and Maxwell’s, then it was NOUS and not just LOGOS which was leading the way on a dialectical highway to field theory, relativity theory and thence at least through this one portal into our modern world.

      Maxwell’s interpretation confirmed what Faraday had already sensed, that the linkage in his figure harbored a prophecy: the power must not merely exist, but propagate in space.  A new ordeal of measurements, not unlike Faraday’s lonely quest for the true sphondyloid, confirmed for Maxwell that indeed its velocity would be that of light.  If it is light, it is all light.  If we go out to that point half-way to our Sun, source of all our light, heat, life and hope, we find this figure is there our lifeline, in what would else be cold space.  Faraday’s figure bears it all.  Thus Maxwell came to maintain not only that the mathematics of electromagnetism had all along been Faraday’s, but the electromagnetic theory of light, as well. 

      The bond exhibited in this figure might be interpreted in many ways.  As I suggested earlier, I think it confronts us with a serious issue today, namely the claim of the sciences to hold in autonomy the keys to objective reality.  This might almost be symbolized by yet one more little example, the discussion in the Meno of the definition of color, a model for Meno of the problem of defining virtue.  Socrates asserts that color is that which always follows figure.  “How silly!” is Meno’s response, while he is delighted with an alternative Socrates then offers — a sophistic account in terms of, essentially, certain splendid intricacies of photons and rhodopsin molecules.  Meno does not notice that these molecules have no color.  In the same way Newton insists that though he may sometimes speak of rays as red, he must be understood to mean ribrifick, or red-making, as neither those rays, nor any other body in Newton’s cosmos has any color.

      The Socratic science, which is that of Faraday and so attractive to Maxwell, begins in effect with color and with life, for it is founded in beauty and, above all, springs from wholeness as primary.  The current sophistic of reductive science, which presumes to tell us what is really real, treats qualities and values, indeed love and life itself, as epiphenomena.  The issue is drawn for us by that great myth, more modern than all the rest, the Timaeus — with its own version of what may be essentially the same diagram: this time, of the cosmos.  The cosmos, Timaeus tells us, is a ZOON, a living creature, incorporating soul and intelligence.  Timaeus illustrates the world soul with a figure strangely similar to Faraday’s.  Many of you know it: two circles are drawn on the sphere of the cosmos, one the equator of the daily motion, the other, inclined to it, the ecliptic on which the sun travels: the motion of the same, and the motion of the other, from which together spring life and mind.  Is Faraday’s less an image of soul, than Timaeus’? 

      Plato always manages to turn the opening question back on us.  The Meno has left us with an unresolved dilemma.  When the slave boy says “yes” to the steps in Socrates’ argument, is he simply following Socrates’ leading, or does he really mean “yes”?  There is always a skeptic in every seminar — maybe they’re in the majority now, in this age of disillusion — to insist that he’s just saying that, and it doesn’t prove anything.  Maybe; mysteries are notably hard to track down.  But we see that the fate of the cosmos hangs forever in the balance as we wait for an answer!  For Faraday, however, it is never a question.  “It gave beautiful curves having perfect simplicity of form,” he says; and that is no epiphenomenon or mere talk, but a primary insight on what has turned out to be a high road to our modern physics.  He shows us what is beautiful and interesting, and how by the very same token it becomes a seat and cause of life and power as well.  We borrow more metaphors from our physicists than we often realize, and we will do well to consider the implications of the better images Faraday is still offering us.  He takes always the largest view.  Not aggregates and resultants of tiny competing atomic forces, but larger, whole patterns founded in unity and beauty.  His images read in many ways, and in all, they bespeak the realm of a good gardener, in easy command of fields of beauty and life — not strife and confusion.  Real physics could perfectly well go either way, it could today be more vital and teleological than Aristotle’s Physics ever was — our reductive readings have simply left us the victims of a tyranny of bad scientific habits.  If the time has indeed come to return the sciences to the seminar table in greater earnest, then this is a most appropriate time indeed to be holding a conference with Michael Faraday on the prospects of bringing that renewed conversation about, and we might hope as well that St. John’s may prove an auspicious place to be doing it.



      And here is the seat of power itself, the Great Royal Institution Electromagnet.  This is the one, we are told, Faraday loved at the Christmas lecture to throw a loaded coal scuttle at, and the fire tongs as well.  I’m confident it never failed to catch them both [Slide 37]:

Slide 37