Mind Over Mathematics:
Can You Solve This Paradox?
Elementary Arithmetic

Printed in The New Federalist, May 24, 1997


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Series Introduction:
Mind Over Mathematics.

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With this column, the New Federalist is inaugurating a regular series of pedagogical demonstrations, designed to show that, within today's academic professions, most of what is taught as scientific fact, is an out-and-out fraud. As the method of the 18th-century Enlightenment has increasingly won out against the scientific method of the Italian Renaissance, true scientific method has disappeared.

As Lyndon LaRouche has often put it, if you have an academic degree, you should sue your university for consumer fraud.

But we don't want you just to take our word for it. Beginning with this column, we will be presenting a series of pedagogical paradoxes, designed to show you that formalism in any field leads to the denial of fundamental truths--starting with the nature of man as being a creative individual made in the image of God. In one issue, we'll present the paradox; in the next, the method by which a solution can be obtained. The point is not to get the right answer, per se, but to uproot habits of linear thinking, to actually provide access to the kind of creative thinking upon which all scientific breakthroughs depend.

We begin with what amounts to a prologue to the prime number paradox. It should get the juices flowing.
--Nancy Spannaus


Mind Over Mathematics
Gauss' Method for Adding 1-100

by Bruce Director

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Most people today, be they scientists, economists, corporate planners,
TV  weathermen, or ordinary citizens, make  life-and-death decisions based on
a  fraudulent conception of mathematics,  even though, for the most part, they 
are totally unaware of it. In modern times, this state of affairs can be  traced
to the fraud perpetrated during the mid-1700s by the Swiss mathematician
Leonhard Euler, who at  the time was in the service of the pro-British oligarch
Frederick the Great of Prussia. By sheer political  thuggery, Euler attacked the
work of  Gottfried Leibniz, whose discoveries in  science, philosophy, law, and 
economics, were the inspiration of the Founding Fathers of the United States, 
and the basis for all progress in  science and technology to this day. 
      Against Leibniz, Euler insisted on reducing all human knowledge to 
mathematical formalisms, a set of rules  to be obeyed, no matter what. In 
Euler's world, if the mathematical formalism said the world should work a 
certain way, the mathematics was right,  even if the evidence contradicted it. 
With the popularization of the modern  digital computer, this slavish 
adherence to mathematical formalism,  has taken on the proportions of a 
dangerous mass psychosis. How many  times have you heard someone say,
``According to the computer...''? 
      All creative thinkers, including  the greatest mathematical scientists, 
were not as stupid as Euler and his  followers. A famous anecdote from the 
childhood of the greatest mathematical  scientist of the 19th century, Carl 
Friedrich Gauss, illustrates the point. 
      One day, when Gauss was about 10, his teacher asked the students to
find  the sum of all the integers from 1 to  100. (In those days, before the 
Internet, teachers had to find other  ways to dull the minds of students, and 
keep them busy at the same time.) 
      After being given the assignment, the boys immediately set to work,
doing  their calculations on little slates.  The usual procedure was for the first 
student who finished, to place his  slate on a table in the middle of the  room.
The next to finish placed his  slate on top of the first, and so on,  until all were
done. 
      Scarcely had the teacher stated  the assignment, than Gauss proudly 
walked to the front of the room, placed  his slate on the table, and sat back 
down at his desk.  The other students  labored for more than an hour, while 
the teacher looked skeptically at Gauss  the whole time. When the time came
to  check the answers, Gauss's slate had  only one number on it, which was the 
correct answer. 
      Can you figure out what Gauss did?  Next week, we'll show you Gauss's 
method, and introduce you further into  the world of Mind over Mathematics. 
  
 

Mind Over Mathematics Part II
Gauss' Method for Adding 1-100

by Bruce Director

CAN YOU SOLVE THIS PARADOX?

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Last week in this column, we recounted a problem encountered by the
great scientist Carl Friedrich Gauss when he was 10 years old. His arithmetic
teacher had assigned the students, the mind-busying task of adding up the
whole numbers, from 1 to 100. The students were to do their calculations at
their desks, on small slates, and, when finished, place their slates on a table in
the middle of the room, so the slate of the first one finished, would be at the
bottom of the pile. 
      The teacher had barely given out the problem, when the young Gauss,
wrote one number on his slate, walked to the front of the room, placed his
slate on the table, and returned to his desk. For more than an hour, the other
students labored over the calculations, while the young Gauss sat quietly,
enduring the skeptical stares of the teacher and his fellow students. When it
came time to check the answers, Gauss's slate had written on it the number
5,050, which was correct. How did he do it? 
      Gauss was able to arrive at the solution so quickly, because he didn't
allow his mind to be bound by the formal rules of mathematics, implicit in the
instructions of the teacher. When the teacher asked the students to add up the
numbers from 1 to 100, all the students, and most of you, ``instinctively''
thought: Begin with 1, then add 2, then add 3, and so on and on. This type of
thinking is not really an ``instinct,'' but merely an example of how the mind is
governed by underlying assumptions, of which most people are unaware. In this
case, the underlying assumption is that the so-called ``natural'' order of
numbers, 1|+|1|=|2, +1|=|3, + 1|=|4, ... is somehow fixed by
nature--when in fact, this ordering of numbers, and the numbers themselves,
are creations of the mind. If your thinking is governed by this underlying
assumption, you will believe, falsely, that the only way to solve the problem,
is to add the numbers one by one. 
      But human beings need not be slaves to their underlying assumptions.
Instead, human beings, unlike other creatures, are capable of making
themselves conscious of the underlying assumptions which govern their
thinking, and changing them. This is what the young Gauss did. 
      Instead of being chained to the ``natural'' order of numbers, Gauss
thought of all the numbers between 1 and 100 all at once, as a whole. Then he
reordered the numbers, in his mind, into pairs. The first number, 1, and the
last number, 100, when added together, total 101; the same for the second
number, 2, and second to the last number, 99; It is immediately clear that there
are 50 pairs of numbers which equal 101, making a total of 50|@ms|101,
which equals 5,050. 
      Once Gauss ``saw'' this ordering in his mind, the problem was solved.
He didn't have to add up each number, or even each pair. It was the ordering
of the numbers on which Gauss focussed his mind, not the numbers
themselves. This is real mathematics. Here the numbers work for the mind, not
the mind for the numbers. 
      A valuable lesson can be learned from this simple example. When
encountering any problem, the first question to ask is: ``How am I thinking
about this problem? What assumptions are implicit in the way I'm thinking
about the problem? How can I reorder the problem so that I am no longer
constrained by those underlying assumptions?'' 
      This, of course, is not the way we're accustomed to thinking, but it is
the way all creative minds think. Most people assume that creativity is not
something they can master. Not true. If we make the effort to replicate, in our
own minds, those creative discoveries made by others, we'll soon learn to be
creative ourselves. 
      Next week, we'll investigate another underlying assumption about
numbers, in examining the question, ``From what are whole numbers made?'' 

PEDAGOGICAL DISCUSSION

Odd and Even Numbers
-- Thank God for the Odd One

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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. [bmd] 

Pedagogical Discussion

An Exporation of the Relationship -
Among Number, Space, and Mind -

by Laurence Hecht

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        I can conceive in the mind of six objects, whose
relationship to one another I wish to investigate.  Their
character as real objects does not interest me, but only that
quality which makes them distinct, thinkable.  They are, thus,
objects in thought.  I will label them with the number
designations 1 to 6, though I might equally denote them by
letters, or any other symbols which allowed me to keep them
distinct in my mind.  I am interested in discovering the number
of different ways these six distinct objects can be formed into
pairs.  Their representation by numbers, allows a convenient
means of investigating this.  I first list all the pairs of 1
with the other 5, then all the pairs of 2, and so forth.  The
result is summarized in the table:

        12
        13   23
        14   24   34
        15   25   35   45
        16   26   36   46   56
        ==   ==   ==   ==   ==
         5    4    3    2    1

Counting the number of pairs in each column and summing them,
produces 5 + 4 + 3 + 2 + 1 = 15 pairs.
        In another form of representation, I can imagine the six
objects as points on a circle, and portray their pairing as the
straight lines connecting any two.  Drawing them produces a
hexagon, and all the straight lines that may be drawn between its
points.  Counting all the connecting lines, we find 15, the same
as the number of pairs above!  The mind rejoices in the discovery
of the equivalence of the two representations.
        Closer examination of the second form of representation, now
reveals also a difference with the first.  In the first, nothing
distinguished one pair from the next, except the symbols used to
designate them.  In the second, we discover three distinct
species of relationships among pairs, each characterized by a
different length of connecting line.  We have (i) the six lines
forming the sides of the hexagon; (ii) the six somewhat longer
lines connecting every other vertex (i.e., 13, 24, etc.); (iii)
the three longest lines connecting diametrically opposite
vertices (14, 25, 36).
        Where, before, the mind celebrated the sameness, it now
rejoices at the difference of the two forms of representation,
and is impelled to look for its cause.  We hypothesize that the
difference must reside in a property of the spatial mode of
representation.  We may reflect that, from the manifold ways we
might have chosen to arrange our six points in space, we chose to
place them on the circumference of a circle, equally spaced.  An
arbitrary arrangement of six points in a plane would have
produced another, less-ordered relationship among the pairs.
Another arangement, a spiral perhaps, would have produced a
richer ordering.
        Thus, from the positing of relationship among things in the
mind, we moved to two modes of representation of that
relationship, then to their sameness and difference, then to the
causes of that difference.  Having hypothesized that the latter
is the result of the spatial form of representation, we are next
led to explore the variety of such representations.
        Of the great variety of possibilities, we choose now to rise
above the plane, in order to examine the relationship among six
points in three-dimensional space, the familiar backdrop for our
visual imagination.  Just as the circle aided us in ordering the
points in the plane, here its counterpart, the sphere, comes to
our aid.  Six points, spaced evenly around the surface of a
sphere, form the vertices of the Platonic solid known as the
octahedron.  We can picture two of its six points at the north
and south poles of a globe, and four more forming a square
inscribed in the circle of the equator.  Connecting each point to
its nearest neighbor, we find the 12 lines which form the 8
equilateral triangles, which are the octahedron's faces.  But we
have not yet connected the six points in all the ways which space
allows.  Each point can yet be connected to its opposite, forming
3 more lines, which are diameters of the circumscribing sphere.
Behold, again, the 15 paired relationships of six objects, now
clothed in a new ordering, this time of two species!
        We may now compare the three modes of representation our
mind has invented to investigate these pairings:
        1) By number, which produced the series 1 + 2 + 3 + 4 + 5 =
15.
        2) In planar space, using the circle, which produced the
three species of lines connecting the points of the hexagon.
        3) In space, using the sphere, which produced the two
species of lines connecting the vertices of the octahedron.
        In turn, each of these modes of representation suggests new
investigations.  For example, with respect to the first (i.e.,
number), we may inquire into the pairwise combinations of other
numbers of things, from which we soon discover that, in general,
for "n" things, the number of pairs that can be formed is equal
to n(n-1)/2, and we may next inquire, what is the expression for
combinations three-wise, four-wise, or n-wise?
        With respect to the second (the distribution of points on a
circle and their combinations), we discover that there exist
species beyond the regular polygons, which are known as the star
(or Poinsot) polygons.  These cannot be generated out of any
arbitrary number of points, but only when the number of points,
and the order in which we take them, are relatively prime to each
other (that is, have no common divisor).  The first of the star,
or Poinsot, polygons, appears when we take five points on a
circle, and connect every second one until the figure closes
(that is, 1 to 3, 3 to 5, 5 to 2, 2 to 4, and 4 to 1).  The
result is the star pentagon, or pentagram, which is conveniently
described as 5/2.  We can then discover the 7/2 and 7/3, the 8/3,
the 9/2 and 9/4, and so forth.
        With respect to our third mode of representation of the
pairwise combination of things (the distribution of points on a
sphere), a new ordering principle arises: that a perfectly even
distribution is only possible in the cases of 4, 6, 8, 12, and 20
points.  When we investigate these, we find species of pairwise
combinations called edges, diagonals, diameters, and some others,
the greatest variet of species occurring in the 20-point figure.
        Now, let us reflect on the higher ordering principle: All of
the representations we have given, even the spatial, are
creations of mind, products of the arithmetic or visual
imagination.  Yet, so real do these creations of the mind seem to
us, we may be tempted to marvel at them as if they had some
existence outside of the mind.  ("But Platonic solids are {real}.
I can build them!" you say.  Perhaps you never have.  Anyone who
has tried, soon discovers a, sometimes gooey, massiness where
massless points are supposed to be, a very finite thickness to
the infinitely thin lines of the edges, and a, sometimes wrinkly,
bulk to the massless surfaces.  Even three-dimensional space, the
forgiving medium of all our constructions, which seems so
certain, so real, is only the ingenious work of the mind, the
visual imagination.  All are products of the mind.)
        But when, in nature, the mind discovers forms just like
these we have just created (thought), put there not by us, but by
something like to us in mind, yet much vaster, then may we truly
marvel, and reflect: What makes nature makes us.  What we make in
mind, think, is then nature -- and may be so in a higher form
than what we perceive outside us.  (The proof of this truth,
well-known to readers of this publication, need not be repeated
here.)  So in the ordering, number, space, and mind, the mind
stands at both ends of the series, as both creator of its own
images, and perceiver of others; the one is called imagination,
the other, reality.  Yet they are both real, as we just showed,
and even both imagined, in so far as the perceived external is
{known} only through the images of mind.
        With such considerations, true science begins.  [lmh]


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The preceding article is a rough version of the article that appeared in The American Almanac. It is made available here with the permission of The New Federalist Newspaper. Any use of, or quotations from, this article must attribute them to The New Federalist, and The American Almanac.


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