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I caution you in advance, some concentration will be required over the next few minutes, in order to capture the germ of Gauss's creative genius. So stick with me, and you will be greatly rewarded.
I have chosen to look into a subject, which most of you, and even your children, think you know something about: arithmetic. Plato, in the Seventh Book of the Republic, says that all political leaders must study this science, because arithmetic is the science whose true use is ``simply to draw the soul towards being.'' Arithmetic, Plato says, is never rightly used, and mostly studied by amateurs, like merchants or retail traders for the purpose of buying and selling. Instead, political leaders, must study arithmetic, ``until they see the nature of numbers with the mind only, for their military use, and the use of the soul itself; and because this will be the easiest way for the soul to pass from becoming to truth and being.''
By now, you may already get the hint, that what Plato and Gauss, meant by arithmetic, is not, the buying and selling arithmetic, you and your children were taught in school. Gauss called his, the ``higher arithmetic,'' and in 1801, he published the definitive study into higher arithmetic, called by its Latin name, Disquisitiones Arithmeticae. As you will see, if your teachers had been interested in training true citizens of a republic, higher arithmetic, is what you would have learned; not the amateurish calculations, needed by fast-food cashiers, stock-brokers, derivatives-traders, or calculating methods of the statisticians and actuaries who determine what lives are cost-effective, for HMO's and insurance companies.
Unfortunately, the Enlightenment dominates the thinking of most people today, comprising an Empire of the Mind, where people instinctively stick to simple addition, even though a higher ordering principle is discoverable. They labor under the illusion, that adding the numbers one-by-one, is their only choice, simply because the creative powers of their minds are unknown to them. It is the underlying assumptions, of which people are not even aware, which determine their view of the world. Only by becoming conscious of these underlying assumptions, and then changing them, can any scientific discovery be made.
Now let's catch a further glimpse of Gauss's genius, (and a little of our own) by turning to another, more profound application of higher arithmetic.
Being barely literate, Gauss's mother could not remember the date of her first son's birth. All that she could remember, was that it was a Wednesday, eight days before Ascension Day, which occurs 40 days after Easter Sunday. This was not necessarily an unusual circumstance in those days, as most parents were preoccupied with keeping their infant children alive. Once the struggle for life was secured, the actual date of birth might have gone unrecorded.
Twenty-two years later, the mother's lapse of memory, provoked the son to employ the principles of higher arithmetic, to measure astronomical phenomena, with a discovery grounded in the principle that the cognitive powers of the human mind are congruent with the ordering principles of the physical universe.
In 1799, Gauss determined the exact date of his birth to be April 30, 1777, by developing a method for calculating the date of Easter Sunday, for any year, past, present or future.
A lesser man, wanting to know such a bit of personal information, would have relied on an established authority, by looking it up in an old calendar, or some other table of astronomical events.
Not Gauss! He saw, in the riddle of his own birth-date, an opportunity, to bring into his mind, as a unified idea, the relationship of his own life, to the universe as a whole.
The date that Easter is celebrated, which changes from year to year, is related to three distinct astronomical events. Easter Sunday, falls on the first Sunday, following the first full moon (the Paschal Moon), following the first day of Spring, (the vernal equinox). Because of Easter's spiritual significance, and its relationship to these astronomical phenomena, finding a general method for the precise calculation of the date of Easter, had long been a matter of scientific inquiry.
Let's look at what's involved in the problem of determining this date. You have the three astronomical events to account for. First, is the year, the interval from one vernal equinox to the next, which reflects the rotation of the earth around the sun. Second, the phases of the moon, from new moon to full moon to new again, which reflects the rotation of the moon around the earth; and third, is the calendar day, which reflects the rotation of the earth on its axis.
Of course, you can only ``see'' these astronomical events in your mind. No one has ever seen, with their eyes, the orbit of the earth around the sun, or the orbit of the moon around the earth. Not until modern space travel had anyone ever seen the rotation of the earth on its axis. We see with our eyes, the changes, in the phases of the moon, the changes in the position of the sun in the sky, and the change from day to night and back to day. We see ``with our minds only'', the cause of that change. This type of knowledge is outside the Enlightenment's straight jacket. Each of these astronomical phenomena is an independent cycle of one rotation. The problem for calculation, is that, when compared with each other, these rotations do not form a perfect congruence (Fig. 1).
There are 365.2422 days in one year; 29.530 days in one rotation of the moon around the earth, called a synodic month; and 12.369 synodic months in one year. These figures are also only approximate, as the actual relations change from year to year, depending on other astronomical phenomena. Here we confront something, which was known to Plato, and specifically identified by Nicholas of Cusa, in his On Leonard Ignorance. There exists no perfect equality in the created world. Perfect equality, exists only in God. But since, man is created in the image of God, through his creative reason, man can rise above this limitation, and see the world with his mind, ever less imperfectly, as God sees it.
As Leibniz says in the Monadology, man can reflect on God with his reason only; and ``we recognize, what is limited in us, is limitless in Him.''
So if there is no equality in the created world, we need a different concept. Our mathematics must be concerned with some other relationship than equality, if we are to successfully measure the created world.
This is real mathematics, not the Enlightenment's dead mathematics of Leonhard Euler and today's illiterate computer nerds, like Bill Gates, who think a computer is the same as the human mind. Their mathematics is no more than a system of rules to be obeyed. The Enlightenment imposes a false separation between the spiritual and physical realms. If the physical world doesn't conform to the mathematics, the Enlightenment decrees, there is something wrong with the physical world, not the mathematics! And, if the creative mind rebels against the dead mathematics of Euler and Gates? The Enlightenment demands that the mind submit to the tyranny of mathematics.
Gauss's higher arithmetic begins with a concept different from simple equality. The concept of ``congruence.'' Here again, you see how you and your children have been lied to by your teachers. Most of you have been taught, that congruence is the same as equality, when applied to geometrical figures, such as equal triangles. Not true.
Gauss's concept of congruence, follows the concept of congruence developed by Johannes Kepler, in the second book of the Harmonies of the World. The word congruence, Kepler says, means to Latin speakers, what harmonia, means to Greek speakers. In fact the word harmonia, and arithmetic, both come from the same Greek root. Instead of equality, congruence means harmonic relations.
Here are some examples of what Kepler means by congruence (Fig. 2). As you imagine, in the plane, I can increase the size and number of sides, of each polygon, without bound. But, when I try to fit polygons together, with one another, I bump into a boundary. Triangles, squares, and hexagons are perfectly congruent. Pentagons, for example, are not (Fig. 3). In some cases, when I mix polygons together, such as octagons and squares, I can make a mixed congruence.
However, when I go from two to three dimensions, and try to form a solid angle, the boundary conditions for congruence change. For example, pentagons, which aren't congruent in a plane, are congruent in a solid angle (Fig. 4). Hexagons, which are congruent in a plane, are not congruent in a solid angle.
So you see, the type of congruences which can be formed from polygons, is dependent on the domain, in which the action is taking place.
Gauss carried this concept of congruence over into arithmetic, using whole numbers alone. Two whole numbers are said to be congruent, relative to a third whole number, if the difference between them is divisible by that third number. The third number is called the modulus (Fig. 5). Gauss designated the symbol @id to distinguish congruence from equality (=).
You may recognize a similarity between the concept of congruence with the idea of musical intervals. In higher arithmetic, it is the interval between two numbers, and relationship between those intervals, which concern us. Just as in music, it is the intervals, and the relationship between the intervals, which communicates the musical ideas, not the notes themselves.
Another property of congruent numbers, is that they leave the same remainder when divided by the modulus (Fig. 6). These remainders are called least positive residues. For example, 16 and 11 are both congruent to 1 modulo 5. In higher arithmetic, numbers are related, not by their equality, but by their similarity of difference, with respect to a given modulus.
There are other important relationships among congruent numbers. For example, if two numbers are congruent relative to a given modulus, they will be congruent to a modulus which divides that modulus. For example, if 1,997 is congruent to 1,941 modulo 28, they will also be congruent, relative to modulus 4 and modulus 7, as 4 X 7 = 28 (Fig. 7).
Here we are ordering the numbers, not according to their ``natural'' given order, but according to a mental concept of congruence. In this way, we make the numbers work for our mind, not enslave our minds, to the order of the numbers.
By 1582, the Julian calendar was off by ten days. The vernal equinox, the first day of Spring, was occurring on March 10th or 11th instead of March 21. Easter, therefore was also occurring earlier in the year. Both the material and spiritual world, had gotten out of whack.
So, in 1582, Pope Gregory XIII, put a new calendar into effect; ten days were dropped out of that year. In addition, the leap year skipped three out of four century years, and every fourth century year, would be a leap year; for example, the year 2000 will be a leap year, but 1900, 1800, and 1700, were not.
Thus, in order to calculate Easter Sunday, and thus determine his own birthday, Gauss had to make congruent, three astronomical phenomena, and two imperfect states of human knowledge!
He accomplished this by reference to two other cycles, or moduli. Because the synodic month and the calendar year, are unequal, the phases of the moon occur on different calendar days, from year to year. But every 19 years, the cycle repeats. So, for example, if the Paschal Moon occurs on say, March 23, in one year, it will occur on March 23, 19 years later. If the Paschal Moon occurs on April 11, the next year, it will occur on April 11, again in 19 years.
If we call the first year in this cycle ``year 0,'' the next year, ``year 1,'' the last year will be ``year 18.'' In this way, the calendar years in which the phases of the moon coincide, will be congruent to each other relative to modulus 19. So, if you divide the year by 19, those years with the same remainder, will have the same dates for the phases of the moon.
The calendar days on which the days of the week occur, also change from year to year. Today is Sunday, February 16. Next year February 16, will be on a Monday. Since there are seven days of the week, this cycle would repeat every seven years, but because every four years is a leap year, this cycle repeats itself, only every four x seven, or 28 years.
However, in the Gregorian calendar, this cycle is thrown off, by the century years. This cycle is called the solar cycle.
Take the number of the year, divide by 19, call the remainder a. For 1997, a=2. In the language of higher arithmetic, 1997 is congruent to two, modulo 19. This tells you where, in the 19-year cycle of the phases of the moon, and the calendar day, the year 1997 falls.
Divide the year by four. Call the remainder b. For 1997, b=1. 1997 is congruent to one, modulo four. This tells you the relationship with the leap year cycle.
Divide the year by seven. Call the remainder c. For 1997, c=2. 1997 is congruent to two, modulo seven. This tells you the relationship between the calendar day, and the day of the week.
The next step is a little more complicated (Fig. 10): Divide (19a + M=24) by 30; call the remainder d. For 1997, d=2. This gives you the number of days, after the vernal equinox, that the Paschal Moon will appear. M changes from century to century, and is calculated from the cycle of dates on which the Paschal Moon occurs, in that century. For the 18th and 19th century, M=23. For the 20th century M=24.
Finally, divide (2b + 4c + 6d + N=5) by seven and call the remainder e. For 1997, e=6. This gives you the number of days from the Paschal Moon, to the next Sunday. This formula takes into account the relationship of the year to the solar cycle. N also changes from century to century and is based on the cycle of the days of the week on which the Paschal Moon occurs in that century. Sunday being 0, Monday being 1, Saturday being 6. For the twentieth century, N=5.
Gauss calculated the values of M and N into the 25th century, and derived a general method for calculating these values for any century. Unlike some people today, Gauss, was not planning on the ``end times.''
Therefore, Easter Sunday is March 22 + d + e. For 1997 that is March 22 + 2 + 6 or March 30, 1997 (Fig. 11).
Gauss's method, obviously has applications, far beyond the determination of his birthday, or the date of Easter Sunday, for any year. In his later work, Gauss brought even more complex astronomical observations into congruence, by use of these same powers of the mind. But, this little example gives you a sense of how a universal creative mind can take any problem, and see in it an opportunity to extend human knowledge beyond all previous bounds.
Of course, we too can learn a lesson from this. The next time a child asks you a question about how the world works, something like, ``why does the moon change from day to day?'' or, ``why does the sun change its place during the day and over the course of the year?,'' don't tell that child to look up the answer in a book, or log onto the Internet. Help that child to discover how, as Plato says, to see the nature of numbers with the mind only.
Then, take that child, with this newly acquired discovery, outside and show him the night sky. Then, that child will be able to see, in that night sky, the image of the workings of his or own mind, and to see also, reflected back, in that image, an imperfect, yet faithful, image of the Creator, Himself.
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