The sexagesimal foundation of mathematics

Why mathematics is easy to understand, easy to do, and easy to prove in base 60. The number theory behind the creation story and the beginning of time.

1. 1

The Sexagesimal Foundation of Mathematics:
The reason God does all Her mathematics in base 60.
M.W.R. Karskens


The last thing is simplicity. After having gone through all the difficulties, having played an endless
number of notes, it is simplicity that matters, with all its charm. It is the final seal on Art. Anyone who
strives for this to begin with will be disappointed. You cannot begin at the end.
Frederic Chopin

The reason is She based the Universe on a modulus six spiral in three dimensions. The continuum is
modelled by the natural numbers modulus 6. The ‘clock 6’ spiral is the foundation, the elementary
conduit connecting number theory and geometry. Spiral Mechanics from beginning to end, or “all the
way down”, dependent on how you choose to look at it.
Mathematics is easy to understand, easy to do, and easy to prove in base 60. The explanation is (in)
the structure underlying all the natural numbers: the prime numbers. Base 60 is built from the first
three prime numbers: 2 x 2 x 3 x 5 = 60.
In base 60 higher arithmetic equates geometry. The symmetry and continuity of the spiral is directly
coupled to all the prime numbers, the structure underlying the natural numbers. The elementary
symmetry, of the prime numbers in base 60, carries over to all the branches of mathematics. In
geometry through the coincidence of the factors of 60 with the smallest primitive Pythagorean triple
(abc). In arithmetic through computation with 4 factors as your base.
There is one single foundation that is proven true. Contrary to the Hellenic tradition, wherein the
axioms are derived from the mind of the philosopher, in the Sumerian and Babylonian tradition the
(single) axiom comes from the observation of certain facts. Thus the axiom is not established
deductively but inductively. The axiom is not taken to be true on the basis of faith, but proven to be
true. One starts from a proven premise or starting point for further reasoning and arguments.
The question puzzling the ‘old One’ is why?
Why do we tell time in base 60, in line with our linear experience of time. However, for all (other)
mathematics we go back to base 10.
Why do we consciously limit our ability in mathematics by doing computation with 2 factors less in
base 10? Or even with 4 factors less, when we restrict ourselves to binary? The Universe literally
hangs together from the multiplication of all the prime numbers. Everything at once all the time.
Empowered with 2 more factors in your base, mathematics is easy to understand, easy to do, and
easy to prove.
Base 60 is ‘complete’, it is a formal mathematical system based on one principle, with the foundation
build on reason. Therefor it is not prone to the inherent limitations of axiomatic formal systems
proven by Gödel. Base sixty is in essence ‘Mathematical Enlightenment’.

2. 2

Higher arithmetic/ number theory
The three-dimensional figure of the spiral is represented two dimensionally by a modulus 6 matrix.
The natural numbers in this matrix are ordered in rows of six.
As a result all prime numbers larger than 5 line up in column 1 and column 5; they are exclusively
congruent to 1 and 5 mod 6. See Table 1, Modulus 6 Matrix in base 10 and base 60, on page 3.

In base 60 the prime numbers (p ≥ 7) are divided in 16 archetypes.
 8 of the archetype prime numbers are congruent to 1 modulus 6:
o .07, .13, .19, .31, .37, .43, .01 and .49 .
 And 8 of the archetype prime numbers are congruent to 5 modulus 6:
o .11, .17, .23, .29, .41, .47, .53, and .59 .
These 16 archetype prime numbers compute into 136 different semiprime numbers (p x q = pq).
 72 semiprime numbers pq have a remainder of 1 modulus 6, and
 64 semiprime numbers pq have a remainder of 5 modulus 6.
In the sexagesimal number system the prime numbers do not seemingly sprout like weeds, instead
they flourish on the 16 perches of a Babylonian Garden.
In comparison, these same prime numbers are in the decimal number system divided in (only) 4 such
archetypes, i.e. all such primes’ last decimal digit is a 1, 3, 7, or 9.
To illustrate this point, all prime numbers ending on 1 in the decimal number system are subdivided
into .11, .31, .41, and .01 archetypes in the sexagesimal number system. The sexagesimal number
system is (literally) exponentially more refined than the decimal number system in terms of higher
arithmetic.

Table 2, Number base and number theoretical refinement.
Base Number theoretical
refinement
The place value of the last digit of
prime numbers in the number base
Prime number list
(factor(s) of the base)
2 1 1 (same as all odd numbers) 2, 3, 5, 7, 11, 13, 17, 19 , 23,
..
10 4 1, 3, 7, 9 2, 3, 5, 7, 11, 13, 17, 19 , 23,
..
30 8 1, 7, 11, 13, 17, 19, 23, 29 2, 3, 5, 7, 11, 13, 17, 19 , 23,
..
60 16 1, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43,49, 53, 59
2
2
, 3, 5, 7, 11, 13, 17, 19 , 23,
..
This exponential difference in refinement -of base 60 in comparison to base 10- leads to more
feasible results in all forms of computation: multiplication, division, (prime) factorization, matrices,
trigonometry, calculus, etcetera, when done in the sexagesimal number system as opposed to doing
them in the decimal number system or other less refined number bases.

3. 3

Table 1, Modulus 6 Matrix in base 60 and base 10.


1 2 3 4 5 6
7 8 9 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
25 26 27 28 29 30
31 32 33 34 35 36
37 38 39 40 41 42
43 44 45 46 47 48
49 50 51 52 53 54
55 56 57 58 59 1.0060
1.0161 1.0262 1.0363 1.0464 1.0565 1.0666
1.0767 1.0868 1.0969 1.1070 1.1171 1.1272
1.1373 1.1474 1.1575 1.1676 1.1777 1.1878
1.1979 1.2080 1.2181 1.2282 1.2383 1.2484
1.2585 1.2686 1.2787 1.2888 1.2989 1.3090
1.3191 1.3292 1.3393 1.3494 1.3595 1.3696
1.3797 1.3898 1.3999 1.40100 1.41101 1.42102
1.43103 1.44104 1.45105 1.46106 1.47107 1.48108
1.49109 1.50110 1.51111 1.52112 1.53113 1.54114
1.55115 1.56116 1.57117 1.58118 1.59119 2.00120
2.01121 2.02122 2.03123 2.04124 2.05125 2.06126
2.07127 2.08128 2.09129 2.10130 2.11131 2.12132
2.13133 2.14134 2.15135 2.16136 2.17137 2.18138
2.19139 2.20140 2.21141 2.22142 2.23143 2.24144
2.25145 2.26146 2.27147 2.28148 2.29149 2.30150
2.31151 2.32152 2.33153 2.34154 2.35155 2.36156
2.37157 2.38158 2.39159 2.40160 2.41161 2.42162
2.43163 2.44164 2.45165 2.46166 2.47167 2.48168
2.49169 2.50170 2.51171 2.52172 2.53173 2.54174
2.55175 2.56176 2.57177 2.58178 2.59179 3.00180
3.01181 3.02182 3.03183 3.04184 3.05185 3.06186
3.07187 3.08188 3.09189 3.10190 3.11191 3.12192
3.13193 3.14194 3.15195 3.16196 3.17197 3.18198
3.19199 3.20200 3.21201 3.22202 3.23203 3.24204
3.25205 3.26206 3.27207 3.28208 3.29209 3.30210
3.31211 3.32212 3.33213 3.34214 3.35215 3.36216
3.37217 3.38218 3.29219 3.40220 3.41221 3.42222
3.43223 3.44224 3.45225 3.46226 3.47227 3.48228
3.49229 3.50230 3.51231 3.52232 3.53233 3.54234
3.55235 3.56236 3.57237 3.58238 3.59239 4.00240
4.01241 4.02242 4.03243 4.04244 4.05245 4.06246
.... .... .... .... .... ....

4. 4

Arithmetic
Why would one consciously do mathematics with a self-imposed cognitive handicap? In sports we do
not have the convention that all players cover up one eye and play using only one leg, whenever they
are on the field. Nor would a general ever instruct his troops to tie an arm behind their back before
going into battle. Yet, in mathematics, that is primarily what we do when we discard two factors
whenever we start doing arithmetic in the decimal number system.
We limit ourselves for no rational reason whenever we revert to doing arithmetic in base 10 instead
of base 60. Base 10 (2 x 5) has 2 prime factors less than base 60 (2 x 2 x 3 x 5). For everything
mathematical this matters a great deal, because the Universe literally hangs together from the
multiplication of its factors. By doing math with two factors instead of with four, you lose out
exponentially every time.

Division and sexagesimal reciprocals
Regular sexagesimal numbers are natural numbers whose factors are only 2, 3, and/ or 5. In base 60
division by a regular number can be replaced by multiplication with the reciprocal of the number. In
the same way as we think of division by 2 as equal to multiplication by ½ or 0.5 . The sexagesimal
inverse of the number 2 is 30, because sixty (= one) divided by two equals thirty, as a half hour equals
30 minutes, and half a minute equals 30 seconds.

Hence, regular sexagesimal numbers are of the form:
n = 2
�
∗3
�
∗5
�


Examples
n = 2
3
∗3
0
∗5
0
= 8
n = 2
1
∗3
2
∗5
2
= 450

The reciprocal of n is given by:
ñ=
1
??????


Therefor in sexagesimal:
X
n
· Y
n
· Z
n
= XYZ
n
= 1
n

3
1
· (2
2
)
1
· 5
1
= 60
1
= 1
1


Sexagesimal reciprocal ñ =
60
�
2
�
∗3
�
∗5
� =
(2
2
∗3
1
∗5
1)
�
2
�
∗3
�
∗5
�

5. 5


Example
n = ñ

8 =
((2
2)
2
∗ 3
2
∗ 5
2
=) 60
2
2
3
∗ 3
0
∗ 5
0

= 2
1
∗3
2
∗5
2
= 450

Verification in base 10: 8 · 450 = 3600 = 60
2

Verification in base 60: 8 · 7.30 = 1.00.00

The reciprocal of 8 is 450 , because the reciprocal of 2
3
= 2
1
x 3
2
x 5
2
.
In sexagesimal, 8 times 7 minutes and 30 seconds is 1 hour , 1 hour = 2
4
x 3
2
x 5
2
.


Multiplication
Multiplication of two numbers is done with aid of the following algorithm.
ab = (
�+�
2
)
2
−(
�−�
2
)
2


Example,
1 x 7 = (
1+7
2
)
2
−(
7−1
2
)
2


= (
8
2
)
2
−(
6
2
)
2


= 4
2
– 3
2

= 16 – 9
A Babylonian scribe would instead of dividing 8 and 6 by 2,
multiply both with 30 (the reciprocal of 2).
1 x 7 = (8 x 30)
2
– (6 x 30)
2

= 240
2
– 180
2
(base10) , = 4.00
2
– 3.00
2
(base 60)
= 57.600 – 32.400 = 25.200 (base10) , = 16.00.00 – 9.00.00 = 7.00.00 (base 60)

6. 6

Note how 8 times 30 makes 4 minutes. The Babylonians used a floating point and therefor, the two
sexagesimal places of zero at the end would be left blanc. “This is the procedure.”
Before moving on to geometry, it is worth a minute of your time to consider the algorithm we are
taught for multiplication, and realize the procedure works for/ in all number bases.

100
11 x
100
1000 +
1100

In binary, it is the sum 4 x 3 = 12.
In decimal, it is the sum 11 x 100 = 1100.
In sexagesimal, it is the sum 11 x 3600 = 39600 (= 11 hours).

7. 7

Geometry

Geometry is easy to understand and easy to do in base 60. I will explain why this is the case and how
keeping to base 60 keeps your intuition and logic aligned. In base 60 geometry and arithmetic are
interchangeable. Fermat’s Last Theorem is proven directly by means of the Fundamental Theorem of
Arithmetic.

Theorem of Pythagoras �
2
+�
2
=�
2


For any Pythagorean triple (abc), the product of the two non-hypotenuse legs (ab), is always divisible
by 12, and the product of all three sides (abc) is divisible by 60. In addition, one side of every
Pythagorean triple is divisible by 3, another by 4, and another by 5. One side may have two of these
divisors, as in (8, 15, 17), (7, 24, 25), and (20, 21, 29), or even all three, as in (11, 60, 61).
For primitive solutions, one of ‘ a’ or ‘b’ must be even, and the other odd, with ‘c’ always odd, and
the Greatest Common Divisor (GCD) = 1. It is usual to consider only primitive Pythagorean triples -
also called "reduced" triples- in which ‘a’ and ‘b’ are relatively prime, since other solutions can be
generated trivially from the primitive ones.
The smallest primitive Pythagorean triple (a, b, c) is 3, 4, 5 . The legs of this triple are the factors of
the sexagesimal number system. Three times four times five equals sixty (3
1
· (2
2
)
1
· 5
1
= 60
1
).
Note that division by 3, 4, 5, 12, and 60 can be replaced by multiplication with the reciprocals, thus
with multiplication by 20, 15, 12, 5, and 1 respectively. Multiplication is significantly easier to do than
division. The result is however the same, as multiplication with the reciprocal will give you the exact
answer.
In base 60 geometry and arithmetic are interchangeable. You can switch between doing math with
numbers to doing math without numbers at will, because the smallest primitive Pythagorean triple
(abc) equals the factors of 60 (3, 4, 5). Thus, (higher) arithmetic = geometry, in base 60.
Spiral mechanics are easily split into dimensions:
2-dimensional
60 degrees �
2
+�
2
−��=�
2

90 degrees �
2
+�
2
=�
2

120 degrees �
2
+�
2
+��=�
2


3-dimensional
Diagonal (D) in a cuboid
1
�
2
+
1
�
2
+
1
�
2
=
1
�
2

8. 8

“One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are
absolutely certain and indisputable, while those of all other sciences are to some extent debatable
and in constant danger of being overthrown by newly discovered facts.
In spite of this, the investigator in another department of science would not need to envy the
mathematician if the laws of mathematics referred to objects of our mere imagination, and not to
objects of reality. For it cannot occasion surprise that different persons should arrive at the same
logical conclusion when they have already agreed upon the fundamental laws (axioms), as well as the
methods by which the other laws are to be deduced therefrom. But there is another reason for the
high repute of mathematics, in that it is mathematics which affords the exact science a certain
measure of security, to which without mathematics they could not attain.
At this point an enigma presents itself which in all ages has agitated inquiring minds. How can it be
that mathematics, being after all a product of human thought which is independent of experience, is
so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely
by taking thought, able to fathom the properties of real things.
In my opinion the answer to this question is, briefly, this: - As far as the laws of mathematics refer to
reality, they are not certain; and as far as they are certain, they do not refer to reality.”
- Albert Einstein, Lecture on Geometry and Experience, Berlin 1921.

9. 9

Sexagesimal Reciprocals and Fermat’s Last Theorem

Mathematics is easy to understand, easy to do, and also easy to prove in base 60.
In the sexagesimal number system there is a direct and intuitive proof of Fermat’s Last Theorem,
which is simple and short enough, it could have easily fitted the margin of Fermat’s book.
Fermat is proven with the help of the fundamental theorem of arithmetic by comparing Fermat with
the equation for sexagesimal reciprocal numbers.
The fundamental theorem of arithmetic (or unique factorization theorem) states that every natural
number greater than 1 can be written as a product of prime numbers, and that, moreover, this
representation is unique up to (except for) the order of the factors.
The two equations are nearly identical, the only difference is the addition sign ‘+’ in Fermat is a
product sign ‘x’ in the reciprocal formula. Because the Fundamental theorem of arithmetic is true,
Fermat’s Last Theorem is true.

Reciprocal equation - Fermat’s Last Theorem
�
??????
∗�
??????
=�
??????
�
??????
+�
??????
=�
??????


Fermat’s generalization of Pythagoras has integer solutions, only, for n= 1 and n= 2, and no integer
solutions for n≥ 3. The n= 1 solutions are all the additions of two integers, and the n= 2 solutions are
evidently all the Pythagorean triples. Integer solutions for n≥ 3 do not exist.
The reciprocal formula has integer solutions for all n (n≥ 1). The order of the factors X, Y and Z can be
rearranged. For every value of n the equation will give the correct pair of sexagesimal reciprocals.
You can change Z
n
to be the factor 3, 2
2
, or 5 since, as stated by the fundamental theorem of
arithmetic, the representation is unique except for the order of the factors.
By applying the theorem to both equations, we learn why Fermat can only have integer solutions for
n= 1 and n= 2, and the formula for reciprocals has integer solutions for all n. Every natural number
greater than 1 can be written uniquely as the product of prime numbers, and not as the sum of prime
numbers.

10. 10

n = 2
�
∗3
�
∗5
�


ñ=
1
??????


Reciprocal ñ =
60
�
2
�
∗3
�
∗5
�


n = ñ

2
�
∗3
�
∗5
�
=
(2
2
∗3
1
∗5
1
)
2
�
∗3
�
∗5
�

�


3
??????
∗(2
2
)
??????
∗5
??????
=(2
2
∗3∗5)
??????
=60
??????


�
??????
∗�
??????
∗�
??????
=���
??????
=1
??????


X
n
∗Y
n
=
(XYZ)
n
XY
n
=Z
n
=> X
n
∗ Y
n
=Z
n



X
n
∗Z
n
=
(XYZ)
n
XZ
n
=Y
n
=> X
n
∗Z
n
=Y
n



Y
n
∗Z
n
=
(XYZ)
n
YZ
n
=X
n
=> Y
n
∗Z
n
=X
n

11. 11

Sexagesimal Reciprocals - Fermat’s Last Theorem
�
??????
∗�
??????
=�
??????
�
??????
+�
??????
=�
??????


n = 0 n = 0
3
0
= (2
2
)
0
· 5
0
1 = 1 1
0
+ 1
0
≠ 1
0

(2
2
)
0
= 3
0
· 5
0
1 = 1 2
0
+ 3
0
≠ 5
0

5
0
= (2
2
)
0
· 3
0
1 = 1 X
0
+ Y
0
≠ Z
0

60
0
= (2
2
)
0
· 3
0
· 5
0
= n · reciprocal n

n = 1 n = 1
3
1
= (2
2
)
1
· 5
1
3 = 20 1
1
+ 2
1
= 3
1

(2
2
)
1
= 3
1
· 5
1
4 = 15 2
1
+ 3
1
= 5
1

5
1
= (2
2
)
1
· 3
1
5 = 12 3
1
+ 5
1
= 8
1

60
1
= (2
2
)
1
·3
1
·5
1
= n · reciprocal n

n = 2 n = 2
3
2
= (2
2
)
2
· 5
2
9 = 16 · 25 9 = 400 8
2
+ 15
2
= 17
2

(2
2
)
2
= 3
2
· 5
2
16 = 9 · 25 16 = 225 7
2
+ 24
2
= 25
2

5
2
= (2
2
)
2
· 3
2
25 = 16 · 9 25 = 144 11
2
+ 60
2
= 61
2

60
2
= 3600 = n · reciprocal n

n = 3 n ≥ 3
3
3
= (2
2
)
3
· 5
3
27 = 64 · 125 27 = 8.000 No solutions
(2
2
)
3
= 3
3
· 5
3
64 = 27 · 125 64 = 3.375 [ (2
2
)
3
+ 3
3
= 5
3
]
5
3
= (2
2
)
3
· 3
3
125 = 64 · 27 125 = 1.728 [ 64 + 27 = .. ]
60
3
= 216.000 = n · reciprocal n [ 91 = .. ]
[(7 · 13) = .. ]
Addition of 64 and 27 changes/ alters the
factors (instead of rearranging them)
from 2 and 3 to 7 and 13.

12. 12

n = 4
3
4
= (2
2
)
4
· 5
4
81 = 256 · 625 81 = 160.000
(2
2
)
4
= 3
4
· 5
4
256 = 81 · 625 256 = 50.625
5
4
= (2
2
)
4
· 3
4
625 = 256 · 81 625 = 20.736
60
4
= (2
2
)
4
·3
4
· 5
4
= 12.960.000 = n · reciprocal n
n = 5
3
5
= (2
2
)
5
· 5
5
243 = 1.024 · 3.125 243 = 3.200.000
(2
2
)
5
= 3
5
· 5
5
1.024 = 243 · 3.125 1.024 = 759.375
5
5
= (2
2
)
5
· 3
5
3.125 = 1.024 · 243 3.125 = 248.832
60
5
= (2
2
)
5
·3
5
· 5
5
= 777.600.000 = n · reciprocal n

n = 6
3
6
= (2
2
)
6
· 5
6
729 = 4.096 · 15.625 729 = 64.000.000
(2
2
)
6
= 3
6
· 5
6
4.096 = 729 · 15.625 4.096 = 11.390.625
5
6
= (2
2
)
6
· 3
6
15.625 = 4.096 · 729 15.625 = 2.985.984
60
6
= (2
2
)
6
· 3
6
· 5
6
= 46.656.000.000 = n · reciprocal n

n = ..

The Fundamental Theorem of arithmetic’ provides the explanation for solution for all n in the case of
the reciprocal formula, and only for n is 1 and 2 in Fermat:
Every integer greater than 1 either is a prime number itself or can be represented as the product of
prime numbers and this representation is unique up to the order of the factors.
Every integer is uniquely represented as the product of prime numbers only up to rearrangement,
and not the sum (addition) of prime numbers (x ≠ +).
�
??????
∗�
??????
=�
??????
, written unique as the product of primes for all n except for the order of the
factors, and never written uniquely as the sum of primes: X
n
+ Y
n
= Z
n
, for n larger than 2.

The factors of the sexagesimal number system 3, 2
2
, and 5 make it perfectly evident why Fermat’s
generalization of Pythagoras, X
n
+ Y
n
= Z
n
, only has integer solutions for n= 1 and n= 2. The n=1
solutions are all the additions of two integers, and the n= 2 solutions are the smallest primitive
Pythagorean triple. The same factors in the reciprocal equation give the integer solutions for all n.

Quod Erat Demonstrandum

13. 13

Science in antiquity
The Sumerians, from the observation of certain facts, had gotten the ‘(in) the beginning’ completely
right. Ever since, in the history of mathematics, everyone else has had the first part wrong. The
cradle of civilization was no matter of chance. The Sumerians invented the sexagesimal number
system, cuneiform script, literature, and irrigation.
The Austrian American mathematician and historian of science Otto Neugebauer in his seminal work
on astronomy and the exact sciences in antiquity (1952) wrote:
“ 21. Pythagorean numbers were certainly not the only case of problems concerning relations
between numbers. The tables for squares and cubes point clearly in the same direction. We also have
examples which deal with the sum of consecutive squares or with arithmetic progressions. It would be
rather surprising if the accidentally preserved texts should also show us the exact limits of knowledge
which were reached in Babylonian mathematics. There is no indication, however, that the important
concept of prime number was recognized.
All these problems were probably never far separated from methods we today would call ‘algebraic’.
In the center of this group lies the solution to quadratic equations for two unknowns. As a typical
example might be noted ..”
Indeed, there has seemingly been “.. no indication, however, that the important concept of prime
number was recognized.”, because everyone in mathematics since the ancient Greek has been
oblivious to the primary reason for the sexagesimal number system. An epic case of hiding in plain
sight, and mathematics’ very own historic version of the “invisible gorilla experiment”. The
experiment painstakingly reveals the hiatus in our knowledge of the fundamentals of mathematics in
comparison to that of the ancient Mesopotamians. Our minds don’t work the way we think they do.
We think we see ourselves and the world as they really are, but we’re actually missing a whole lot.
We are not aware we look at mathematics in a rectilinear fashion when we think of the standard
number line, nor are we aware of the impact on our view of the decimal prism. When we do
arithmetic with two less factors, we have no idea we’re missing a whole lot.

“Mathematics is, I believe, the chief source of the belief in eternal and exact truth, as well as in a
super-sensible intelligible world. Geometry deals with exact circles, but no sensible object is exactly
circulars, however carefully we may use our compasses, there will be some imperfections and
irregularities. This suggests the view that all exact reasoning applies to ideal as opposed to sensible
objects; it is naturals to go further, and to argue that thought is nobler than sense, and the objects of
thought more real than those of sense-perception. Mystical doctrines as to the relation of time to
eternity are also reinforced by pure mathematics, for mathematical objects, such as numbers, if real
at all, are eternal and not in time. Such eternal objects can be conceived as God’s thoughts. Hence
Plato’s doctrine that God is a geometer, and Sir James Jeans’ belief that He is addicted to arithmetic.
Rationalistic as opposed to apocalyptic religion has been, ever since Pythagoras, and notable ever
since Plato, very completely dominated by mathematics and mathematical method.
The combination of mathematics and theology, which began with Pythagoras, characterized religious
philosophy in Greece, in the Middle Ages, and in modern times down to Kant. Orphism before
Pythagoras was analogous to Asiatic mystery religions. But in Plato, Saint Augustine, Thomas
Aquinas, Descartes, Spinoza, and Kant there is an intimate blending of religion and reasoning, of
moral aspiration with logical admiration of what is timeless, which comes from Pythagoras, and

14. 14

distinguishes the intellectual theology of Europe from the more straightforward mysticism of Asia. It
is only in quite recent times that it has been possible to say clearly where Pythagoras was wrong. I do
not know of any other man who has been as influential as he was in the sphere of thought. I say this
because what appears as Platonism is, when analysed, found to be in essence Pythagoreanism.
The whole conception of an eternal world, revealed to the intellect but not to the senses, is derived
from him. But for him, Christians would not have thought of Christ as the Word; but for him,
theologians would not have sought logical proofs of God and immortality. But in him this is still
implicit. How it became explicit will appear.”
- Bertrand Russell The History of Western Philosophy (1945)

Thymaridas of Paros, the Greek Pythagorean number theorist, is said to have called prime numbers
rectilinear since they can only be represented on a one-dimensional line. Non-prime numbers, on the
other hand, can be represented on a two-dimensional plane as a rectangle with sides that, when
multiplied, produce the non-prime number in question.
Every Babylonian knew Thymarides’ statement, about prime numbers being ‘rectilinear’, is false.
Every Babylonian who knew how to multiply two numbers, could prove Thymarides’ wrong by
demonstration. The example of 1 times 7, given at the beginning, is a case in point. A rectangle with
sides of 1 and 7, is transformed to, the difference between two squares (the area between 4
2
and 3
2
).
This procedure had been applied routinely, as the standard method for multiplying two numbers, for
well over three millennia before Pythagoras or Thymarides were born.

The deductive method versus the scientific method
Thousands of years before Greek mathematics and deductive reasoning, inductive reasoning had led
to the creation of the sexagesimal number system. A coherent formal logical system based on one
single principle. A truly scientific theory of numbers at the beginning of history in the cradle of
civilization.

Bertrand Russell in The History of Western Philosophy (1945) wrote:
“The Greeks contributed, it is true, something else which proved of more permanent value to abstract
thought: they discovered mathematics and the art of deductive reasoning. Geometry, in particular, is
a Greek invention, without which modern science would have been impossible. But in connection
with mathematics the one-sidedness of the Greek genius appears: it reasoned deductively from
what appears self-evident, not inductively from what had been observed. Its amazing success in the
employment of this method misled not only the ancient world, but the greater part of the modern
world also. It has only been very slowly that scientific method, which seeks to reach principles
inductively from observation of particular facts, has replaced the Hellenic belief in deduction from
luminous axioms derived from the mind of the philosopher. For this reason, apart from others, it is a
mistake to treat the Greeks with superstitious reverence. Scientific method, though some few among
them were the first men who had an inkling of it, is, on the whole, alien to their temper of mind, and
the attempt to glorify them by belittling the intellectual progress of the last four centuries has a
cramping effect upon modern thought.

15. 15

There is, however, a more general argument against reverence, whether for the Greeks or for anyone
else. In studying a philosopher, the right attitude is neither reverence nor contempt, but first a kind of
hypothetical sympathy, until it is possible to know what it feels like to believe his theories, and only
then a revival of the critical attitude, which should resemble, as far as possible, the state of mind of a
person abandoning opinions which he has a hitherto held. Contempt interferes with the first part of
this process, and reverence with the second. Two things are to be remembered: that a man whose
opinions and theories are worth studying may be presumed to have had some intelligence, but that
no man is likely to have arrived at complete and final truth on any subject whatever. When an
intelligent man expresses a view which seems to us obviously absurd, we should not attempt to prove
that it is somehow true, but we should try to understand how it ever came to seem true. This exercise
of historical and psychological imagination at once enlarges the scope of our thinking, and helps us to
realize how foolish many of our own cherished prejudices will seem to an age which has a different
temper of mind.”

At the beginning of the 20th century mathematician had attempted to bring the whole of
mathematics under one single roof. They had believed they could give mathematics solid
foundations, but they had failed because of the inherent limitations of every formal axiomatic
system: inherently one starts from an unproven premise. In all types of formal axiomatic systems, the
axioms themselves are not proven, but taken to be true, and serve as a premise or starting point for
further reasoning and arguments. The word axiom comes from the Greek word axioma “that which is
thought worthy or fit” or “that which commends itself as evident.”
The axioms are derived from the mind of the philosopher. The axioms are not proven, they are
merely thought to be true and accepted as foundations. The validity of the deduction is not 100%
guaranteed. Because the validity of the deduction is completely dependent on the truth value of the
axiom from which it has been derived. You may believe the axiom to be true, but this believe
depends on faith and not on reason, as the axiom is not a proven truth. The intuition of the
philosopher is raised to the status of axiom.
In 1931 Gödel proved the ‘Greek type’ of formal axiomatic systems incomplete. In contrast, the
sexagesimal number system -spiral mechanics- is semantically and syntactically complete. The
Babylonian method is inductive as it follows from the observation of certain facts.

Eusebius of Caesarea in Praeparatio Evangelica ‘Preparation for the Gospel’ (313 AD) makes quite
clear ‘the one-sidedness of the Greek’ and their rectilinear perception of reality was well known
among the peoples of the fertile crescent. There was no discussion on their presupposed ‘genius’
and wisdom because they were not believed to have had much, all their knowledge was imported.
“In fact the said Pythagoras, while busily studying the wisdom of each nation, visited Babylon, and
Egypt, and all Persia, being instructed by the Magi and the priests: and in addition to these he is
related to have studied under the Brahmans (these are Indian philosophers); and from some he
gathered astrology, from others geometry, and arithmetic and music from others, and different
things from different nations, and only from the wise men of Greece did he get nothing, wedded as
they were to a poverty and dearth of wisdom: so on the contrary he himself became the author of
instruction to the Greeks in the learning which he had procured from abroad.”

16. 16

Alexanders’ solution to the intractable problem of the Gordian knot could mathematically be put in a
different perspective as more of the same ‘rectilinear thinking’, i.e. reasoning it doesn’t matter how
the knot is untied, and then ‘solving’ the problem with brute force.
Babylonian mathematics was only rediscovered in the late 19th century, when archeological
excavation revealed thousands of clay tablets in cuneiform script. Records of transactions, legal
contracts, and numerous (many thousands of) mathematical exercises. Only in the middle of the 20
th

century scholars started to become aware of (the level of) the mathematics on the clay tablets.
When the tablets were translated the mathematics on them puzzled both the archeologists and the
mathematicians. Indeed, it looked crazy to them. “I have added the circumference to the area of a
square. It is ‘45’.” To us this makes no sense, and for a long time scholars thought the Babylonians
only used numerical methods and had no geometry.

In one of his televised lectures Richard Feynman described the difference in mathematics between
the Greeks and the Babylonians as follows:
“ There are two kinds of ways of looking at mathematics which for the purpose of this lecture I will
call the Babylonian tradition and the Greek tradition.
In Babylonian schools in mathematics, the student would learn something by doing a large number of
examples until he caught on to the general rule. [..]
The Babylonian thing that I am talking about -which I don’t, really not Babylonian but- is to say I
happen to know this and happen to know that and I work out everything from there, and tomorrow I
forgot this was true but I remember that this was true, and then I reconstruct it again, and so on. I am
never quite sure where I am supposed to begin and supposed to end. I just remember enough all the
time so as the memory fades and the pieces fall out I re-put the thing back together again every day.
The method of starting from the axioms is not efficient in obtaining the theorems. In working
something out in geometry you are not very efficient if each time you have to start back at the
axioms. But if you have to remember a few things in the geometry you can always get somewhere
else. And what the best axioms are, are not exactly the same, in fact are not ever the same as the
most efficient way of getting around in the territory. In physics we need the Babylonian method and
not the Euclidean or Greek method. [..]”

Babylonians could effortlessly jump between arithmetic and geometry, because to them geometry
and arithmetic are not different fields of mathematics, they are not even different sides of the same
coin. They are both sides of the coin at the same time. Spiral mechanics combines duality and trinity,
the spiral embodies all prime numbers and products, everything at once. Geometry and arithmetic
are as inextricably entangled at the core as are space and time according to Albert Einstein.
“Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing.
The theory says a lot, but does not really bring us any closer to the secret of the ‘old one’. I, at any
rate, am convinced that He is not playing at dice. Waves in 3-dimensional space, whose velocity is
regulated by potential energy (for example, rubber bands)…. I am working very hard at deducing the
equations of motion of material points regarded as singularities, given the differential equation of
general relativity. - Albert Einstein (Letter to Max Born, December 4
th
1926)

17. 17

Einstein, Feynman, and Russell were all acutely aware of the inherent limitations of the Greek
axiomatic method. In 1901 Russell’s paradox had shown that some formalizations of Cantor’s set
theory led to contradiction. Gödel would in 1931 prove these type of axiomatic systems incomplete.
How well the Babylonians actually knew what they were doing and where they were going is shown
by their actions, there are examples abound of applying their knowledge directly to reality.
First, writing is applied spiral mechanics. We read this sentence left to right, and at the end of the
line, the script continues fluidly, with the first word one row below. Reading depends on our mental
construction of a continuous spiral in three dimensions. We can easily reconstruct an actual three
dimensional spiral from a page by rolling it up horizontally until the last word of the first sentence
lines up with the first word of the second sentence. The inventors of the sexagesimal place value
numbers system are also the inventors of the cuneiform script. Both were invented by the Sumerians
in what is today southern Iraq. The latter is the logical application of the former. A numerical place
value system in which the numbers have been replaced by words. Spiral mechanics converted to the
2-dimensional matrix.
Second, Archimedes’ spiral is as much Archimedes’ as Pythagorean triples are Pythagorean. The
Babylonians found an application for the spiral in irrigation. The knowledge bundled by Euclid in the
Elements was merely a derivative of the knowledge that had been discovered and developed in
Mesopotamia ‘In the beginning…”.
Last but not least I want to mention the ‘Bagdad battery’. The purposes the Sumerians and
Babylonians had for it remains a mystery, that it generates electricity (Volt), however, has been
repeatedly proven.

The advances in the sciences of the Sumerians and the Babylonians were induced by the sexagesimal
number system itself. The Babylonians worked from a number theory that was inductive, logical, and
complete. Everything from one single proven foundation. A scientific theory of numbers.
Bab-ilu, Babel, or Babylon, means ‘Gate to the Gods’, the Babylonians knew that was exactly the
mathematics the Sumerians had provided them with.

Conclusion - In Sum
The Babylonians adopted the sexagesimal number in full awareness of the marvelous discovery of
the Sumerians. The discovery that had stood at the heart of their invention of writing, the plough,
irrigation, and had provided guidance to their scientific advances. A scientifically based theory of
numbers. A formal logical system of mathematics based on one proven principle.
There is an old joke where someone is asked for directions. The person is answered: “Well, if you
want to go there, I wouldn’t start from here.” Indeed, mathematics outside base 60 is exactly like
this. You get off on the wrong foot. However, that is not a joke.
The mathematical Universe literally hangs together from multiplication: all natural numbers can be
represented uniquely as a product of primes. The question which has puzzled the ‘old One’, is why?
Why would one ever, rationally and in good conscious, do arithmetic with two less factors? While
one knows how to tell the time, knows how to count in base 60?

18. 18

For 2500 years people have been telling time in base 60, but for all things mathematical fall back on a
smaller number base, unaware of the inherent consequences of doing so, namely making
mathematics harder than it needs to be.
Stick to the sexagesimal number system and you will find, time is on your side.
The Babylonians always ended their mathematical work with a short phrase:
This is the procedure.

I would like to end with a last quote from chapter six of my book On the Theory of Numbers: Prime
Numbers and Enlightenment , the quote is from Saint Augustine of Hippo:
“Six is a number perfect in itself, and not because God created the world in six days; rather the
contrary is true. God created the world in six days because this number is perfect, and it would remain
perfect, even if the work of the six days did not exist.”