By Bruce Director

Most people, because of an apparent childhood recollection, associate whole numbers with the counting of objects, and, as such, the numbers 1, 2, 3, ... to whatever large number imaginable seems to be the natural order of things, hence the name "natural numbers." But as St. Paul says in the oft-cited 1 Corinthians 13, "When I was a child, I spake as a child, I understood as a child, I thought as a child; but when I became a man, I put away childish things." And Plato in the Republic teaches that for warriors to become mature leaders, they must learn to "see the nature of numbers with the mind only". In the following weeks' pedagogical discussions, we will begin to learn to think of the "nature of numbers," instead of "natural numbers".

The association of whole numbers with the counting of objects seems plausible in the small, but, as we begin to think of larger numbers, the association with the counting of objects seems less and less valid. It is clear we think not in terms of single numbers, but in terms of groups of numbers, say groups of 10. When we reflect back, we realize that even with small numbers this is the case, and what had seemed plausible, was wrong.

In "On Conjectures" Cusa discusses the "Natural Progression" of numbers this way:

"To contemplate the nature of number is more acutely useful to you, the more deeply you attempt to investigate the rest in its similitude.At first, however, concern yourself with its progression, and you shall confirm that it is accomplished by the quaternary. Indeed, one, two, three and four added together produce ten, which unfolds the natural power of simple unity.mNow from this same ten, which is the second unity, the quadratic unfolding of the root is achieved through a similar quaternary progression: 10, 20, 30 and 40 added together are one hundred, which is the square of the denary root. Likewise, the hundred exerts the thousand as unity through the same movement: 100, 200, 300, and 400 added together are one thousand. Yet do not proceed further on this path, as if something still remained.

"However, not only after the ten--as with eleven, where after the ten a regression to unity occurs--but also in a similar manner after the thousand, the repetition is not denied; in the natural influx, there are therefore no more than ten numbers, which are contained in a quaternary progression. And beyond the one thousand, the sum of the cube of the denary root, there is no variation in the repetition, since this arises through the triply repeated quaternary progression in the denary order. Consider also, that the quaternary, the unfolding of unity, contains the power of the total number...."

We move closer to knowing the "nature" of numbers, when we think of larger numbers, such as the number of years in the procession of the equinox (The change of the constellation of the zodiac which rises with the sun at the equinox.) This is an approximately 26,000-year cycle, reflecting the juxtaposition of the cycle of one rotation of the earth around the sun, and one rotation of the direction of the earth's axis with respect to the stars. Both cycles are seen in the mind only, for we only see the changes in position of the sun in the sky, and the changes in position of the stars. We never see the actual motion of the earth around the sun, or the rotation of the earth's axis itself. When we juxtapose in our mind, these two mental concepts, the number 26,000 arises. No one has ever, nor could ever, arrive at this number by counting. The number arises solely from the construction of a metaphor, composed by juxtaposing two distinct ideas, associated with physical processes.

All numbers we can think of, are associated with metaphors in a similar way, as the metaphor of Gauss' Easter algorithm beautifully illustrates. No number is ever thought of by itself, but only in relation to some metaphor, even though the underlying metaphor is not always apparent. To discover the deeper implications of this fact, we should try some experiments, using Gauss' "Disquisitiones Arithmeticae," as our guide.

First, we must abandon all together, any mental dependence on the "natural" order of whole numbers. (This is easier said than done, as this childish idea--which doesn't come from children--is a very deeply embedded axiomatic assumption.) Instead, think of numbers with respect to Gauss' concept of congruence.

If the difference between two numbers is divisible by a third number, they are said to be congruent with respect to the third number. The two numbers are called residues of each other, the third number is called the modulus.

Sound simple? Try some examples. I'll give you a couple to get you started. 13 and 138 are congruent relative to modulus 5, but non-congruent relative to modulus 11. -9 and +19 are congruent relative to modulus 7.

Under this concept of congruence, the relationship of three numbers, defines the relationship of all numbers to each other. No number is self-evident, but are related to one another by the relationship of the {interval} between them to the modulus. Each interval, defines under what moduli, these numbers are congruent or non-congruent. This concept of number, as the juxtaposition of two ideas, an interval and a modulus, more truthfully reflects the way numbers actually arise in the mind.

By a simple experiment, one can see how deeply embedded the so-called natural order of numbers is. Think of any number as a modulus, called m. Now take a series of numbers, begin with any arbitrary number, a, and write down a sequence of numbers a, a+1, a+2,...a+m-1. The number of numbers in this sequence, will equal the number of 1's in the modulus. Now take another number not in the sequence, called A. This number will be congruent to only one number in the sequence, relative to the modulus m. Now take A+1. What number is that congruent to? Then A+2, until you get to A+m-1. See what happens when you continue to add 1 to A after this point.

Try this with different numbers for a modulus. Does your naive imagination insist on thinking of the natural sequence of numbers a, a+1, a+2, ... as primary, and the concept of congruence as intrusive? Do you find yourself becoming angry at the way the concept of congruence disrupts the simple "natural" ordering of whole numbers? If, as we illustrated above, that in the real world, all numbers arise from metaphor, why is your mind so insistent on protecting, the seemingly childish idea of the natural ordering of numbers.

Now have some more fun. Instead of adding 1 to a, add 2, or 3, or 4. See what happens.

Recently a good friend, who resides temporarily in Baskerville, Va. suggested the following real life example of counting by congruence with respect to a modulus, known as the "Baskerville Modulus Problem". In a prison with three buildings, each with two dorms A and B, each dorm rotates going to lunch first. If dorm A in Building 1 goes to lunch first on Monday, how long will it be before that same dorm eats lunch first on a Monday? Once you figure that out, chicken is served every third Thursday. If the dorm doesn't get called first, the chicken gets cold and rubbery. How often does dorm A of Building 1 get hot chicken?

We left you last week in the mind of a prisoner contemplating the timing and quality, (or lack thereof) of his future meals, so as to confront the contradiction between the naive imagination's belief in the "natural progression" of numbers, and, the "nature" of numbers, as seen with the mind only. If you took into your mind the prisoner's plight, the real life implications of this contradiction should have become clear to you.

In the first part of the "Baskerville Modulus Problem", our prisoner is caught in a six-day cycle, in which he goes to lunch first once every six days. Count this rotation according to the "natural" order of numbers, 0, 1, 2, 3, 4, 5, with 0 counted for eating lunch first, 1 counted for eating lunch second, etc. But, counting according to the "natural" order of numbers, and, the reality of the prisoner's lunch schedule begins to diverge on day 6, for on the sixth day, the prisoner eats lunch first again, counted as 0 in the order of eating lunches. On the seventh day, the prisoner eats lunch second, or 1 in the order of eating lunches, and so forth.

Now think of this in terms of Gauss' conception of congruence. On day 1, the prisoner eats lunch first--0 in the order of lunches. On day 6, the prisoner eats lunch first again--0 in the order of lunches. So, there is something the same between 1 and 6. Obviously, they are not equal, but they are {congruent} relative to modulus 6. Because the interval between 0 and 6 is divisible by 6. (Gauss introduced a new symbol to distinguish congruence from equality. That symbol is an equal sign with an extra line. Due to technical considerations we can not reproduce that symbol here.)

Continuing on to the 12th day, the prisoner eats lunch first again. So, 0 is congruent to 6 and 12 relative to modulus 6. Again, the intervals between 0 and 6, 6 and 12, and 0 and 12 are all divisible by the modulus, 6. Similarly on day 7, the prisoner eats lunch second (counted 1). 7 and 1 are not equal, but are congruent relative to modulus 6. This relationship, of congruence with respect to modulus 6, reflects the specific ordering principle, of the domain of the prisoner's lunch schedule, which diverges from the simple linear counting of the days according to the "natural progression" of numbers.

This concept of congruence, is analogous to Kepler's concept of congruence in the first book of the Harmonies of the World. There, Kepler notes that the word congruence means to Latin speakers what the word harmony means to Greek speakers. The Greek word for harmony and arithmetic, come from the same Greek word-stem which means, in English, "to fit together." Kepler denotes as congruent, polygons which can be fitted together. For example, in a plane, triangles, squares, and hexagons, can be fitted together perfectly, while pentagons cannot. On the other hand, in a solid, triangles, squares, and pentagons, can be fitted together, while hexagons cannot. What polygons are congruent and what are not, is dependent on the domain in which the congruence is formed.

This relationship of congruence with respect to domain, can be seen in our prisoner's situation. There are 7 days of the week, which can be counted according to the "natural progression" by Sunday=0, Monday=1, etc. Here again, the "natural progression" and reality diverge when we get to the second Sunday, which is the 7th day (since we started counting with 0 not 1). Again, this second Sunday, is similar to the first, but it is obviously not equal, it is {congruent}--0 is congruent to 7 modulus 7.

The days of the week are ordered according to congruences relative to modulus 7 and the prisoner's lunch schedule is ordered according to congruences relative to modulus 6. The numbers 0-6, relative to modulus 7 and 0-5, relative to modulus 6 were called by Gauss the "least positive residues" of the modulus, because they are the smallest positive numbers, with which every other number, no matter how large, will be congruent.

Now combine these two ordering principles. The prisoner's eating schedule (modulus 6) with the days of the week (modulus 7). If, the prisoner eats first on Monday (counted as day 1) how long will it be before he eats first again on a Monday? If you worked through this problem after last week's pedagogical discussion, you would have found, that our prisoner ate lunch first on Monday after 42 days (6x7) or every 6 weeks, and, in the intervening period, the prisoner ate lunch first once on every day of the week.

But now look at the second part of the problem: Chicken is served every third Thursday, and if the prisoner doesn't get there first, the chicken is cold. How often does he get hot chicken? Count the weeks, 0, 1, 2, 3 etc. On week 0, chicken is served on Thursday, on week 1 and 2 no chicken is served, but on week 3 chicken is served again. 3 and 0 are congruent relative to modulus 3. So, chicken is served on those Thursdays which fall on weeks that are congruent to 0 relative to modulus 3.

What does this mean for our prisoner? Well, since he eats lunch first on the same day of the week, once every 6 weeks, he'll eat lunch first on every sixth Thursday. Or, on those Thursdays which occur on weeks whose numbers are congruent relative to modulus 6. So he'll get hot chicken every 6 weeks. Right? Not necessarily. If he happens to be lucky enough, that he eats lunch first on a Thursday that serves chicken, he'll get hot chicken every 6 weeks, but, if on the Thursday he eats lunch first, no chicken is served, he'll never get hot chicken. In fact, only two groups of prisoners will ever get to eat hot chicken.

What has happened to our numbers? First, the natural progression diverged from reality, but we discovered a more real ordering principle. But now it seems to diverge again. There must be another ordering principle involved, of which we were unaware.

Look back at the difference between the two parts of the problem. In the first part, we combined two cycles, one relative to modulus 6 and one relative to modulus 7. In the second part, the two cycles were relative to the moduli 6 and 3. What's the difference?

One way to discover from whence this paradox arises, is to shift gears, in a direction discussed by Larry Hecht in a pervious pedagogical discussion concerning the discovery of Gauss' contemporary Louis Poinsot. (See Pedagogical Discussion 97136lmh01). Think of 7 points (or vertices of a polygon) placed roughly on a circle. Number those points, 0-6. Now connect the points in order 0, 1, 2, 3, 4, 5, 6, 0. Now connect the points, skipping every other one: 0, 2, 4, 6, 1, 3, 5, 0; then skipping every third one; 0, 3, 6, 2, 5, 1, 4, 0; and so forth, for all the numbers 0-6. (This is best done using a different 7 points for each drawing, rather than super-imposing each one on top of each other.) What are the differences and similarities between each figure created by this method? Now do the same with 6 points. Compare these orderings, with the sequences of least positive residues of the series: a, a+1, a+2...; a, a+2, a+4, a+6...; a, a+3, a+6, a+9,...; a, a+4, a+8, a+12...; etc. with respect to modulus 6 and 7.

After performing the above experiment, we can now see more clearly the paradox of the "Baskerville Modulus Problem." The ordering principle, of which we were previously unaware, is the concept of prime numbers. (See Pedagogical Discussions 97056bmd02 & 97066bmd001.) The number of vertices, is one modulus, and the number skipped, is another modulus. When the two moduli are combined, a polygon is created. The shape of the polygon, and the order in which the points are connected, is dependent on which two moduli are combined. In the case where the two moduli are prime relative to one another, (i.e., they have no common factors), all the points are connected, and a complete polygon is formed. In the cases where the moduli are not prime relative to one another, such as 3 and 6, the polygon will not be complete, leaving some vertices unconnected.

So unless, the modulus of the lunch schedules, and the modulus of the schedule when chicken is served, are relatively prime, there is no guarantee that our prisoner will get hot chicken.

What the above demonstration proves, without a doubt, is, that, only by freeing our minds from the prison of linear thinking, and effectively organizing so as to bring about the complete exoneration of LaRouche, can we insure that our prisoner will get hot chicken, whenever he wants it!

by Bruce Director

We last left our prisoner confronting the divergence of, on the one hand, the endless succession of days, one after the other, and, on the other hand, the actual ordering of those days according to the physical events that occur in them. This clash, between two concepts of number, sparks our prisoner to embark on a journey to discover the nature of number, beyond the realm of sense-certainty. Reflecting back on his childhood education, he realizes that his thinking about number is confined to a rigid set of rules and operations, mere manipulations of numbers as external objects, memorized, not discovered, to be re-called on command.

Suddenly the liberating words of Nicholas of Cusa from {On Conjectures} come to his mind: "The essence of number is therefore the prime exemplar of the mind.... In that we conjecture symbolically from the rational numbers of our mind in respect to the real ineffable numbers of the Divine Mind, we indeed say that number is the prime exemplar of things in the mind of the Composer, just as the number arising from our rationality is the exemplar of the imaginal world."

He begins to recall some happier memories of his childhood quest for knowledge, reminiscing how he once playfully discovered hidden relationships among numbers, while secretly exploring their nature with his mind only. Little things, oddities he kept to himself. Once, he had ventured to tell his teacher about one such discovery, only to be discouraged by the response, "Don't be an oddball. That has no practical application. You won't need that in later life."

Now such canons and dogmas memorized in youth are of no use, if they ever were. He finds himself free to inquire anew, beginning first with those elementary principles, which, never simple (except to the simple-minded), unfold a rich bounty of profound ideas, if the underlying, seemingly subtle, paradoxes are sought out.

He takes out a paper and pencil, and unfolds a series a numbers with the following construction:

Begin with a unit *

and add a unit **

and add another unit ***

and another unit ****

and so forth.... *****

******

*******

********

*********

**********

***********

************

It seems apparent enough, from the method of construction, that each number is unique, differing from all others by its relationship to the process of adding one, just as each day follows another. But, seeking to shed the shackles from his mind, our prisoner tries to discover what is behind the numbers, by looking into the numbers on a different level, besides the succession of adding one. He tries the following experiment:

With each number, he alternately marks one unit from each end, beginning with the first and last unit, then proceeding to the second and second to the last unit, until he can go no further.

(The reader is required to make his or her own drawings by hand, rather than rely on computer generated images. Hand drawings, even crude ones, contain within them the cognitive process, whereas the computer images suppresses same.)

What emerges, from this process, is that numbers are distinguished from one another by more than just adding one. Some numbers, (every other one) has a unit left unmarked in the middle, while in the others, no unit remains in the middle.

Again the words of Nicholas of Cusa come to mind: "It is established that every number is constituted out of unity and otherness, the unity advancing to otherness and otherness regressing to unity, so that it is limited in this reciprocal progression and subsists in actuality as it is. It can also not be that the unity of one number is completely equal to the unity of another, since a precise equality is impossible in everything finite. Unity and otherness are therefore varied in every number. The odd number appears to have more of unity than the even number, because the former cannot be divided into equal parts and the latter can be. Therefore, since every number is one out of unity and otherness, so there will be numbers in which the unity prevails over the otherness, and others in which the otherness appears to absorb the unity."

A smile comes across the prisoner's face as he now sees the once familiar concept of even and odd numbers (thrust at him as an almost trivial distinction in his youth), in a new light. His joy is mixed with a tinge of anger, as he realizes this new light is not new at all, but, in fact, an ancient discovery, he should have relived as a youth. Unlike what he was taught in school, the concept of even and odd, is not a mere description about a particular number, but a concept associated with the {nature} of number itself. The doctrine he was taught in school seemed to work, but because of it, his mind didn't.

His anger abates as he turns back to his inquiry. He leaves it to others to uncover how these ancient discoveries were written out of the curriculum.

The infinity of all numbers, has now been divided by two, according to the nature of the individual numbers, when each of them is divided by two. The discovered principle of even and odd, divides the infinity of numbers into two {types} -- those numbers in which "otherness prevails over unity," and those numbers in which, "unity prevails over otherness."

From the original construction of all numbers by adding one, no number is equal to any other number. But now, he discovers some numbers are alike but not equal to others. To bring this discovery into a One, the prisoner is taken to Gauss' concept of congruence. All numbers of the same {type} are congruent to each other, and those of a different {type} are non-congruent. There are two {types}. So under Gauss' concept, all even numbers are congruent to each other relative to modulus two. Likewise, all odd numbers are congruent to each other relative to modulus two. And, all even numbers are non-congruent to all odd numbers relative to modulus 2.

Seeing this, the prisoner desires to continue the exploration, dividing the numbers again. This time, he starts with the even numbers only, taking the parts created from the first division, and marking off the units from each end until he can go no further. (The reader is required to complete this step for himself.)

Now the even numbers have been divided into two {types}; those whose parts when divided in this way, leave no unity -- called even-even, and those whose parts, when divided this way, still have a unity in the middle of the part -- called even-odd. The odd numbers, in turn, are divided into two {types}. Those which have even numbers on each side of the unity left in the middle -- called odd-even and those which have odd numbers on each side of unity that was left in the middle -- called odd-odd. The infinite has now been divided four times!

Again, numbers of the same type are not equal, so we go to Gauss' concept of congruence, to bring this new discovery into a One. Each number of any of these four types -- even, odd, even-odd, even-even, odd-even, odd-odd, is congruent to all other numbers of that type, relative to modulus 4.

Yet there is nothing self-evident, from the construction of numbers by the addition of one, from which the now-discovered distinction between even, odd, even-even, even-odd, odd-even, and odd-odd, logically follows. To be able to envisage, from this small distinction among numbers, a different domain, other than the linear domain of adding one, the prisoner must free himself from the constraints of his formal thinking. That domain is characterized, not by linearity, but by curvature, of which the principles of even and odd are but a reflection. The nature of that curvature will be further discovered, by new investigations to which the prisoner looks forward.

As the prisoner contemplates his next experiments, he's interrupted. Oddly enough, it's time for lunch.

Back from lunch, our prisoner eagerly digs deeper into his investigations of the nature of number, fueled by enthusiasm from his recently demonstrated capacity to discover truth by his own powers of reason. He's determined to avoid the various textbooks lying around (not wanting to fraternize with the enemy), relying instead on a well-worn copy of Euclid's Elements, whose text contains the footprints of some classical Greek discoveries. The more profound nature of these discoveries are not explicitly stated in Euclid's Elements, but the profound nature of these ancient thoughts are, neverthelss, reconstructible in the mind. Combining centuries of discoveries in his mind simultaneously, he turns to Book 9, Propositions 21-34 to reconstruct for himself, the indicated discoveries concerning even and odd numbers, pondering these Propositions, in dialogue with the more advanced standpoint of Nicholas of Cusa's "On Conjectures." (As noted last week, "The odd number appears to have more of unity than the even number, because the former cannot be divided into equal parts and the latter can be. Therefore, since every number is one out of unity and otherness, so there will be numbers in which the unity prevails over the otherness, and others in which the otherness appears to absorb the unity.")

From Cusa's standpoint, the indicated principles of Euclid can be stated as follows: When two even numbers are added, the otherness still prevails over unity, producing an even number. When two odd numbers are added, the unities from each one are combined, making otherness prevail over the unity, and producing an even number. When an even and an odd number are added, the unity of the odd number remains, producing an odd number. In sum when like numbers are added, the otherness prevails over unity, and an even number is produced. When unlike numbers are added, unity prevails over otherness, and an odd number is produced.

When an even number is added an even number of times (i.e., multiplied by an even number), otherness continues to prevail, resulting in an even number. When an odd number is added an even number of times (i.e., multiplied by an even number) otherness prevails and an even number results. When an odd number is added an odd number of times (i.e., multiplied by an odd number), unity still prevails over otherness, and an odd number results.

Different from addition, unlike numbers, when mulitplied, produce even numbers, and like numbers preserve their type. (The reader will find it liberating to demonstrate for yourself, that this is true in all cases; also the reader should discover the similar principle for subtraction and division, and for the second order types of even-even, even-odd, odd-even, and odd-odd, with respect to addition, multiplication, subtraction and division.)

Having discovered so much from the construction of linear numbers, the prisoner extends his experiments into a new domain and now investigates the construction of polygonal numbers. He begins with the polygon with the smallest number of sides, the triangle.

*

* * *

* * * * * *

* * * * * * * * * *

* * * * * * * * * * * * * * *

* @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @

and so on.

Unlike linear numbers, triangular numbers are constructed not by adding one, but by adding the linear numbers themselves. Each successive triangle, contains within it, all previous triangles, plus the next linear number. The added part is called a Gnomon, (denoted in the above figures by the symbol @) which in Greek geometry, means a shape, which when added to a figure, yields a figure similar to the original one. The word Gnomon is derived from the Greek word to know. (The triangular pillar on a sun-dial, which casts the shadow that marks the time, is also called a Gnomon.) In the above representation, each Gnomon is represented by a different symbol.

(The reader is again urged to make your own hand drawings of the construction of triangular numbers, instead of relying on these computer generated representations. Hand drawings are an efficient means of unfolding the cognitive process. When you make these drawings, locate for yourself, the preceeding triangles, in the successive one.)

Triangular numbers are constructed by adding all previous linear numbers together. 1; 1+2; 1+2+3; 1+2+3+4; ...; resulting in the series of triangular numbers, 1, 3, 6, 10, .... The differences (intervals) between each triangular number forms the series, 2, 3, 4, 5, .... The difference between the differences is always 1. Here, unity is found, not in the construction of the numbers, but in the differences of the differences.

Intrigued by this discovery, he extends the experiment to the next polygon, the square. Square numbers are constructed thusly.

* @ & $ # %

* * * * @ @ @ & $ # %

* * * @ * * * * @ & & & $ # %

* * @ * * * @ * * * * @ $ $ $ $ # %

* @ * * @ * * * @ * * * * @ # # # # # %

* @ @ @ @ @ @ @ @ @ @ @ @ @ @ % % % % % %

and so on.

Again, each square contains within it all previous squares, plus the addition of a Gnomon. (The Gnomon with respect to each square is denoted by the symbol @. The last figure represents each Gnomon with a different symbol.) The square numbers increase by adding every second linear number, to the previous square number, 1+3; 1+3+5; 1+3+5+7; resulting in the series of square numbers, 1, 4, 9, 16, 25, 36,... The differences (intervals) between each square number, forms the series of odd numbers, 1, 3, 5, 7, 9, ..., and the difference between any two odd numbers is always 2, or is always divisible by 2.

The prisoner can now prove, why these difference are always odd, by looking at the nature of each Gnomon, from the standpoint of his previous discoveries about the nature of even and odd numbers. (When making your hand drawings, distinguish each successive Gnomon and see how each square contains, nested within it, all previous squares. Then look at each Gnomon from the standpoint of the nature of adding even and odd numbers.)

The prisoner now thinks, "Under what conception can I bring the generating principle of the square numbers into a One." The square numbers are obviously not equal to one another, so equality is not the right conception. But, congruence is not self-evident, as no modulus can be found, relative to which all square numbers are congruent. But the differences (intervals) between the square numbers, (i.e., the odd numbers) while not equal, are all congruent to unity relative to modulus 2. Here the unity is found, not in the formation of the square numbers, nor in the differences between the square numbers, or even in the difference between the differences. Unity is found, as that to which all the differences between the square numbers, are congruent, relative to the modulus of the difference of the differences. (In this case, modulus 2.)

Excited by the ability of his mind to increase its cognitive power, by discovering a congruence, not on the surface, but in the underlying generating principle, he drives the process further. By extending his experiments to polygons of increasing number of sides, the prisoner seeks to force new anomalies to emerge, so he can find what new ordering principles he can discover.

So on to pentagonal numbers. Which he constructs thusly:

*

* *

* * * *

* * * * * *

* * * * * * * * *

* * * * * * * * * * *

* * * * * * * * * * * * *

* * * * * * * * * * * * * *

* * * * * * * * * * * * * * *

and so on.

Here again, each pentagonal number contains nested within it, all previous pentagonal numbers. (Here the reader must make his own hand drawings, as this computer is utterly incapable of doing the work for you, let alone the thinking.) Each pentagonal number increases over the previous pentagonal number by the addition of every third linear number. 1+4; 1+4+7; 1+4+7+10; 1+4+7+10+13; resulting in the series of pentagonal numbers, 1, 5, 12, 22, 35, .... The differences (intervals) between the pentagonal numbers forms the series, 4, 7, 10, 13.... The difference between any two differences is always 3, or is divisible by 3.

Like the square numbers, and the triangular numbers, the pentagonal numbers are not equal, and no modulus can be found, relative to which all pentagonal numbers are congruent. But, when the prisoner looks to the generating principle of pentagonal numbers, a modulus can be found under which the ordering principle can be thought of as a One. The differences between the pentagonal numbers are all congruent to unity relative to modulus 3. Again, unity is found, as that to which all the differences between the pentagonal numbers are congruent, relative to the modulus of the difference of the differences. (In this case, modulus 3.)

This process can be extended to polygons of ever-increasing numbers of sides, forming hexagonal numbers, heptagonal numbers, octagonal numbers and so on. The prisoner spends some time, carefully drawing each series of polygons, so as to bring the generating principle of each polygonal series into a One in his mind. (The reader is well advised to do the same.)

Having done this, a new, more profound question comes before the prisoner's mind. "What is the generating principle, under which the generating principle of all polygonal numbers can be brought into a One?"

With each new polygon, a new series of numbers is constructed. Unlike linear numbers, which increase by adding one, the polygonal numbers, increase by an increasing amount each time. Each polygonal series, is unified, not with respect to each number of the series, but by the differences between those numbers, which are all congruent to unity relative to a modulus formed by the differences of the differences. (The reader will see that the modulus is always two less than the number of sides of the polygon.)

The prisoner has discovered a generating principle, of a generating principle.

(These discoveries, some of which were embodied in classical Greek knowldedge, were subsequently investigated by Pascal and Fermat, formed a basis for Leibniz' discovery of the differential calculus, and were reworked by Gauss' in the development of the complex domain.)

The prisoner steps back and looks at his work, taking a deep breath of fresh air. He feels as though he's climbed a high peak, on a path whose direction and steepness has changed along the way. The path began with the simple step of adding one, to construct the linear numbers. The path became more curved and the angle of ascent changed, as the concept of numbers was extended into the domain of polygons. Now, at the summit, the change in curvature, and changing angle of ascent, are thought of as a One, under a principle that is congruent with the principle which he started, thought of in an entirely new way. Now the addition of unity, is found, not in the generation of the numbers themselves, but in the generation of the moduli, under which the differences between each polygonal number series are themselves made congruent to unity.

In seeing, with his mind, this whole process from the summit, he asks himself, "What curvature is all this a reflection of?"

His free-thinking is suddenly interrupted by the sound of footsteps. He looks up to see a well-dressed, slightly paunchy baby boomer, with an air of self-importance about him. The man is clutching a very large heavy textbook. As he comes close, the prisoner looks quizzically at the stranger, who sticks out his hand, saying, "Dr. Crumbbucket here. Glad to meet you. I'm a visiting professor, of applied and theoretical bullshit. I understand you're in need of instruction."

The prisoner stares for a moment, as the fresh air seems to rush out of his head. His lunch gurgles in his stomach. He prepares to defend his mind.

by Bruce Director

"I have information that you've been playing around with numbers," Dr. Crumbbucket inquired of our prisoner." Perhaps I could help you to learn the ropes."

"Well," our prisoner says slowly, trying to buy some time to collect his thoughts, "I was just sort of making some experiments."

"Experiments!" Crumbbucket shrieks. "With numbers? No one experiments with numbers. There are well-established rules for the proper manipulations of the figures. Rules which have been handed down from professor to professor, generation to generation. Complicated rules, intricate rules. These take years to learn. Either you can learn these rules, or we give you an electronic calculator with pictures on it. No one can learn by experiments with numbers. There's nothing to experiment with. Besides, you can't do experiments in the mind."

"Not {in} the mind," the prisoner corrects, "{About} the mind. These experiments are to discover how my own mind thinks."

"Whatever," the professor mumbles, after a short pause.

"Do you know all the rules?" The prisoner is still trying to collect his thoughts.

"Virtually all of them. And as soon as a new one is invented, I learn that one, too."

"Is this what you had to do to get your PhD.?" the prisoner asks.

"Yes. I had to memorize, aggrandize, temporize, fantasize, eulogize, surmise, bastardize, etymologize, generalize, syllogize, tautologize, ventriloquize, analyze, brutalize, formalize, legalize, socialize, symbolize, agonize, fraternize, tyrannize, plagiarize, Anglicize, summarize, and vulgarize, but, I haven't, at least not yet, had to hypothesize. If you want to learn, we can begin the lessons immediately."

Crumbbucket's face is getting redder as he speaks, and small beads of sweat are forming on his forehead and on his chin. The prisoner has a sinking feeling that his whole day is about to be wasted. With no place to go, he has to think fast. Suddenly, a discovery, that, until now was only half-formed in his mind, comes into view. He decides to put the Professor to a test.

"Let me first show you what I've discovered by experiment," the prisoner says.

"Okay, but don't take long. We have a lot of work to do, if you want to learn what I have learned."

The prisoner quickly reviews his experiments and discoveries with even, odd and polygonal numbers, to set the professor up for the test.

"That's no big deal. We have rules for all those things. If you knew the rules, you wouldn't have had to go through all those manipulations with lines, and dots, and all those drawings. Let's get on with it."

"Before we go on, dear Professor, let me put to you a series of questions, so you can better understand the results of my experiments. Are you agreeable to this?"

"If it doesn't take too long," the professor answers, shifting his weight from side to side, while one of his knees vibrates quickly back and forth.

"Okay," the prisoner begins, "Since I've already discovered some things about linear and polygonal numbers, I now ask what happens when I add areas?"

"Areas?"

"Yes. Areas. If I have an area whose magnitude is one, and I add another area whose magnitude is one, what is created?"

"Well, that's obvious. 1 + 1 = 2."

"And, if I add an area whose magnitude is two to an area whose magnitude is two, what is created?"

"The same: 2 + 2 = 4."

"And, if I add an area whose magnitude is four and I double it, what happens?"

"The same thing. 4 + 4 = 8. Of course, 2 x 4 = 8 is the same thing. As with 2 + 2 + 2 + 2 = 8. Likewise the same with 2 x 2 x 2 = 8. Or 2^3=8."

"Okay. Well, let's try drawing these areas and see what happens?"

"Why do you waste time with drawings?" growled the Professor. "I just showed you how you can add, multiply, or take the powers to get the answer. Why in the devil's name do you want to waste time with drawings?"

"Just try it. It won't take long. Here," the prisoner gently hands the professer his pencil and paper.

"Me? Draw?"

"Yeah, please. Just try it."

"Whatever," grumbles the professor, as he reluctantly takes the pencil and paper.

"Now, draw a square whose area is one," instructs the prisoner. The professor complies, drawing a small square in the middle of the paper. (As usual, the reader is urged to make your own drawings.)

"Now draw another square of the same size, attached to the previous square," comes the next instruction.

"What has been created?" the prisoner asks.

"A one by two rectangle," replies the professor.

"And what is the area of the rectangle?"

"Two."

"See, we've added two squares, and we've gotten a rectangle," the prisoner says proudly.

"What's the difference?" says Crumbbucket dismissively, "I got the same answer following the rules: 1 + 1 = 2. And that was much quicker."

"Keep going," says the prisoner, ignoring the professor's insolence.

"Please, draw another one by two rectangle attached to the one you've already drawn. Now what have you created?"

"A two by two square."

"And what is the area of that square?"

"Four," the professer responds. "But so what, I already figured the answer. 2 + 2 = 4. Also 2 x 2 = 4."

"Please. Can we continue?" The prisoner coaxes the professor to continue the drawing. Dr. Crumbbucket draws another four by four square attached to the previous one, making a two by four rectangle whose area is eight. And continuing, drawing another two by four rectangle attached to the previous one, making a four by four square whose area is sixteen.

"See," the prisoner says excitedly, "First you had a square, and you added a like square, making a rectangle whose area was double the square. Then you added a like rectangle, and you got a square whose area was double the rectangle. Then you added a like rectangle, and you got a square whose area was double the area of that rectangle. As you proceeded, you got another square, then a rectangle. The first addition made a rectangle, the second addition made a square, the third addition made a rectangle, the fourth addition made a square, and so on."

"But I got the same answer this way, 1x2=2x2=4x2=8x2=16..., or alternatively, 2^0=1, 2^1=2, 2^2=4, 2^3=8, 2^4=16..." answers the professor.

Grinning from ear to ear, the prisoner rejoins, "But from your way, you didn't discover the series of alternating squares and rectangles. Now you've discovered that the odd-numbered additions of areas make rectangles and the even-numbered additions make squares."

The professor snorts, shrugs his shoulders and says, "Are you ready to learn the rules?"

"Can we try one more series of questions?" asks the prisoner.

"Just one more," agrees the professor hesitantly, his curiosity getting the better of him.

"Try this," instructs the prisoner. "Draw a square whose area is the same as the first square, one. Next to that, draw a square whose area is the same as the one by two rectangle, two. And next to that, draw the two by two square, and next to that a square whose area is the same as the two by four rectangle. Do this for all the areas you created by the first series of drawings."

The professor makes a neat drawing of squares, one next to the other with areas one, two, four, eight, sixteen, and so forth.

"Now, Dr. Professor. What is the length of the side of the first square whose area is one?"

"One, of course," the professor answers.

"And what is the length of the side of the second square whose area is two?"

"The square root of two," the professor states matter of factly.

"And what is the square root of two?"

"It's the length of the side of the square whose area is two, and is denoted with a symbol thusly," the professor responds without blinking, tracing a radical sign in the air with his finger.

"But," replies the prisoner, "I already know the area of the square is two. You are simply repeating yourself, to tell me that the length of the side, is `The length of the side of the square whose area is two.'"

"The square root of two," the professor repeats, more emphatically than before.

"But that doesn't say anything. What's the square root of two?" the prisoner asks again. "Can we continue? What is the length of the side of the next square, the one whose area is four?"

"Two," answers the professor.

"Very fine. And what is the length of the side of the next square whose area is eight?" asks the prisoner.

"The square root of eight." This time the professor's pride in his ability to answer is tinged with trepidation, anticipating the prisoner's response.

"There you go again. You have only repeated the question as the answer. I ask, `What is the length of the side of a square whose area is eight?' and you answer, `The length of the side whose area is eight.' That is not an answer. From that, we have discovered nothing."

Perceiving the professor's obvious distress, the prisoner tries to be gentle, hoping that his prodding will liberate the professor's mind.

The professor stares for a moment in disbelief at the resistance of the prisoner to accept his answer.

The prisoner asks again, "What is the length of the side of the square whose area is two or eight? Or in your words, what is the square root of two, or eight?"

"Here, hold this," the professor hands back the pencil and paper after the briefest moment's pause, and picks up his heavy textbook, wildly flipping the pages. "I know it's in here somewhere," he says as he balances the book in one hand, turning the pages with the other. The prisoner stands mute with a wry smile on his face.

"Just a minute. I'll find it," begs the professor. "Damn it! Wrong book. Hang on a minute. Don't go away, I'll be right back. I have to get my other book."

"I shall return," the professor calls, his voice trailing off. The prisoner watches the professor scurry down the hall, half of his shirt-tail hanging out of his pants, the sound of clanging chains diminishing as he gets further away.

Free from the immediate encounter with the professor, the prisoner turns his thoughts back to the drawings just created. He spends some time making similar drawings, in which he increases the area by three each time, then by four, then by five. Each time he creates an alternating series of squares and rectangles, with the first addition being a rectangle, the second a square, the third a rectangle, and so forth. The rate at which the areas grow, changes, but the type of change in each case is the same; the odd-numbered additions (powers) make rectangles, the even-numbered additions (powers) make squares. He has discovered even and odd, in a new domain.

When he added linear numbers, thus forming polygonal numbers, the rate of growth changed for each type of number, but remained the same within each series. Among linear numbers, he discovered congruences, between even and odd, between even/even, even/odd, odd/even, and odd/odd. Among the polygonal numbers, he discovered congruences with respect to the change between each number. Areas (geometric numbers), reflect an entirely different type of change, as the numbers are increased.

These two-dimensional geometric numbers, reflect a new domain. Congruence with respect to even and odd remains, but in an entirely different way. Here, evenness reflects squares and oddness reflects rectangles. When these magnitudes are transformed into only squares, the sides of the even ones are commensurable with the area, while the sides of the odd ones are incommensurable with the area. The concept of number cannot be seperated from the content of number, which is a reflection of the domain in which that number is situated. Even something as seemingly simple as even and odd, is different in different domains. The poor professor didn't even suspect, that from the method he used, he really didn't know the area of half the squares he drew, even though he seemed to be able to draw them. "There's probably some hope for him," the prisoner thinks to himself, "if he'll only try to discover, rather than just learn."

The prisoner asks himself again, "What curvature do these processes reflect?" In his mind's eye, he sees, with respect to each type, different principles of growth, which are reflected as a series of nested curves; a circle, an Archimedean spiral, and an equiangular spiral. Each type of curvature, is reflected simultaneously, yet distinctly, in each process.

Now he thinks of a new, most important project: "What is the nature of the curvature, which bounds these curves?"