by Bruce Director
The following is provided to provoke some thinking, with respect to matters raised in previous pedagogical discussions, and to lay a conceptual basis for subjects to be taken up in the near future.
Plato’s dialogue of the Laws, continues in the short appendix known as the Epinomis:
“Let us then first consider what single science there is, of all those we have, such that were it removed from mankind, or had it never made its appearance, man would become the most thoughtless and foolish of creatures. Now the answer to this question, at least, is not overhard to find. For, if we, so to say, take one science with another, ’tis that which has given our kind the knowledge of number, that would affect us thus, and I believe, I may say that ’tis not so much our luck as a god who preserves us by his gift of it….
“But we must still go forth a little on our argument, and recall our very just observation, that if number were banished from mankind, we could never become wise at all. For a creature’s soul could surely never attain full virtue, if the creature were without rational discourse, and a creature that could not recognize two and three, odd and even, but was utterly unacquainted with number, could give no rational account of things, whereof, it had sensations and memories only, though there is nothing to keep it out of the rest of virtue, valor, and sobriety. But without true discourse, a man will never become wise, and if he has not wisdom, the chiefest constituent of full virtue, he can never become perfectly good, and, therefore, not happy. Thus there is every necessity for number as a foundation, though to explain why this is necessary, would demand a discourse still longer than what has gone before. But, we shall be right, if we say of the work of all the other arts which we recently enumerated, when we permitted their existence, that nothing of it all is left, all is utterly evacuated, if the art of number is destroyed.
“Perhaps when a man considers the arts, he may fancy that mankind need number only for minor purposes — though the part it plays even in them is considerable. But could he see the divine and the mortal in the world process — a vision from which he will learn both the fear of God and the true nature of number — ….
“Well then,… How do we learn to count?… There are many creatures whose native equipment does not so much as extend to the capacity to learn from our Father above how to count. But in our own case, God, in the first place, constructed us with this faculty of understanding what is shown us, and then showed us the scene he still continues to show. And in all this scene, if we take one thing with another, what fairer spectacle is there for a man, than the face of day from which he can then pass, still retaining his power of vision, to the view of night, where all will appear so different? Now as Uranus never ceases rolling all these objects round, day after day, and night after night, neither does he ever cease teaching men the lore of one and two, until even the dullest scholar has sufficiently learned the lesson of counting. For any of us who sees this show will form the notion of three, four and many. And among these bodies of God’s fashioning, there is one, the moon, which goes its way, now waxing, now waning, as it lights up one day after another, until it has fulfilled fifteen days and nights, and they, if one will treat its whole orbit as a unity, constitute a period, such that the very slowest creature, if I may say so, on which God has bestowed the capacity to learn, may learn it…. [W]hen God made the moon in the sky, waxing and waning, as we have said, he combined the months into a year and so all the creatures, by a happy providence, began to have a general insight into the relations of number with number. `Tis thus that earth conceives and yields her harvest so that food is provided for all creatures, if winds and rains are neither unseasonable nor excessive; but if anything goes amiss in the matter, ’tis not deity we should charge with the fault, but humanity, who have not ordered their life aright….
“… So we must do what we can to enumerate the subjects to be studied, and explain their nature and the methods to be employed, to the best of the abilities of myself who am to speak and you who are to listen — to say, in fact, how a man should learn piety, and in what it consists. It may seem odd to the ear, but the name we give to the study is one which will surprise a person unfamiliar with the subject — astronomy. Are you unaware that the true astronomer, must be a man of great wisdom? I don’t mean an astronomer of the type of Hesiod and his like, a man who has just observed settings and rising, but one who has studied seven out of eight orbits, as each of them completes its circuit in a fashion not easy of comprehension by an capacity not endowed with admirable abilities. I have already touched on this and shall now proceed, as I say, to explain how and on what lines the study is to be pursued. And I may begin the statement thus.
“The moon gets round her circuit most rapidly, bringing with her the month, and the full moon as the first period. Next we must observe the sun, his constant turnings throughout his circuit, and his companions. Not to be perpetually repeating ourselves about the same subjects, the rest of the orbits which awe enumerated above are difficult to comprehend, and to train capacities which can deal with them we shall have to spend a great deal of labor on providing preliminary teaching and training in boyhood and youth. Hence there will be need for several sciences. The first and most important of them is likewise that which treats of pure numbers — not concreted in bodies, but the whole generation of the series of odd and even, and the effects which it contributes to the nature of things. When all this has been mastered, next in order comes what is called by the very ludicrous name mensuration, but is really a manifest assimilation to one another of numbers which are naturally dissimilar, effected by reference to areas….”
Plato presents the irony, of a connection between the study of “pure numbers not concreted in bodies,” and the mastery, in the mind, of the motion of the heavenly bodies — astronomy. As we discovered by our previous investigations into linear, polygonal, and geometric numbers, and Gauss work on the calendar, this connection is in the realm of Higher Arithmetic — Gauss’ re-working of classical science.
In our previous studies, we quickly learned the foolishness of thinking of numbers in connection with objects or bodies. Instead, we began to discover, that knowledge lies in investigating the relations between numbers, not the numbers themselves. We discovered how to begin to distinguish these relations as different {types} of differences (change) among numbers. Numbers, related to one another by the same {type} of difference, are congruent relative to that {type} (modulus). These {types} of differences, can be distinguished from one another, either by magnitudes, as in the case of linear and polygonal numbers, or by incommensurability, as in the case of geometric numbers. As we discovered with the application of Higher Arithmetic to the determination of the Easter Date, when the mind abandons all foolish fixation on objects, and focuses instead on the relations between them, an extremely complex many, can be brought into our conceptual ken.
A similar approach can be taken with respect to the issues Jonathan raised in last week’s pedagogical discussion. Nothing can be discovered about the astrophysical, by, as Plato indicates, simple observations, like the methods of Hesiod. Instead, one must look to the {type} of change, (relations), of which those observations are only a reflection.
Think of two objects, representing two observations of a planet in the sky. What is the relationship between these two objects? What one must be investigate, is the type of difference (change) between those objects. Or, under what curvature (modulus) are the relations between these objects congruent.
For example, if those two objects are related to each other by a straight line, then the type of difference is measured by rectilinear action, no matter how small the interval between them. If, however, they are related to one another by a circular arc, the type of difference will be characterized by constant curvature, not rectilinear action, no matter how small the interval between them. Or, if they are related by an elliptical arc, the type of difference is characterized by changing curvature, no matter how small the interval between them. The mind must distinguish, the type of change, rectilinear, constant curvature, changing curvature, or types of changing curvature. The determination of which type of change, is related to these specific observations, is not a formal question, but a matter of discovery.
By the time he was 16 or 17, Gauss had already discovered a new type of difference, congruence in the complex domain, which he applied to his work throughout his life. Not until 37 years later, in his second treatise on biquadratic residues, did Gauss begin to elaborate the metaphysical principles behind this discovery.
We can gain some insight, into Gauss’ thinking, from the following fragment, taken from one of Gauss’ 1809 notebooks:
Questions to the Metaphysics of Mathematics
1. What is the essential condition, that a can be thought of, to combine concepts with respect to a magnitude?
2. Everything becomes much simpler, if at first we abstract from infinite-divisibility and consider merely discrete magnitudes. For example, as in the biquadratic residues, points as objects, intersections, therefore relations as magnitudes, where the meaning of a + bi – c – di is immediately clear. (This is accompanied by a grid in the complex domain. See “The Metaphysics of Complex Numbers” Spring 1990 21st. Century Magazine)
3. Mathematics is in the most general sense the science of relationships, in which one abstracts from all content the relationships.
Assume a relationship between two things, and call that the simplest relationship, etc.
4. The general idea of things, where each has a two-fold relationship of inequality, are points in a line.
5. If a point can have more than a two-fold relationship, the image of it, is the position of points that are connected by lines, in a surface,. But, if one should investigate here all possibilities, it can only concern the points, which are in a three-fold reciprocal-relationship, and giving a relationship between relationships.
6. It were extremely important, to bring the theory of differences to clarity without magnitudes. As occurs, for example, in the series differences in a plane leveller. The position of the bubble in the glass pipe is determined to be at rest by the geometrical axis of the pipe, and a line through the plane of the feet.
* * *
In this brief fragment, we can see the complete unity in Gauss’ mind, between mathematics, metaphysics, and physics. To help grasp this, the reader should perform the following demonstration with a carpenter’s level, while thinking of the above discussion:
Hold the level on a surface so that the bubble is a rest in the middle. Now rotate the level around a line perpendicular to the surface. The bubble will not move. Now rotate the level along an axis, in the direction of the glass tube. The bubble will still not move. Now rotate one end of the level up and the other end down, on an axis parallel the surface, but perpendicular to the level. The bubble moves. Movement of the bubble back and forth, is inseparably connected with movement of the level in a second direction. These two actions back-forth and up down, are not the same thing in two directions, but One two-fold action.
(If you are self-conscious, while thinking about this demonstration, you should be able to discover where the gremlins of Newtonian mysticism might be lurking in your mind.)
Acutely aware that only metaphor can adequately convey an idea, Gauss wrote to his friend Hansen on December 11, 1825:
“These investigations lead deeply into many others, I would even say, into the Metaphysics of the theory of space, and it is only with great difficulty can I tear myself away from the results that spring from it, as, for example, the true metaphysics of negative and complex numbers. The true sense of the square root of -1 stands before my mind (Seele) fully alive, but it becomes very difficult to put it in words; I am always only able to give a vague image that floats in the air.”
In upcoming weeks, we will re-construct some of Gauss’ metaphors. We leave you today, with the following from the Epinomis:
“Now the proper way is this — so much explanation is unavoidable. To the man who pursues his studies in the proper way, all geometric constructions, all systems of numbers, all duly constituted melodic progressions, the single ordered scheme of all celestial revolutions, should disclose themselves, and disclose themselves, they will, if, as I say, a man pursues his studies aright with his mind’s eye fixed on their single end. As such a man reflects, he will receive the revelation of a single bond of natural interconnection between all these problems. If such matters are handled in any other spirit, a ma, as I am saying, will need to invoke his luck.”