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 Goddoes all Her mathematics in base 60.
M.W.R.Karskens
The last thing is simplicity.After having gonethrough allthedifficulties, having played an endless
numberof notes,it is simplicity that matters,with all its charm.It is thefinal seal on Art. Anyonewho
strives forthis to begin with will be disappointed.You cannotbegin attheend.
Frederic Chopin
The reasonis She basedthe Universe ona modulus six spiral inthree dimensions. The continuumis
modelledbythe natural numbers modulus6. The ‘clock6’ spiral is the foundation, the elementary
conduitconnectingnumbertheoryandgeometry.SpiralMechanics frombeginningtoend,or“all the
waydown”,dependentonhowyouchoose tolook at it.
Mathematicsiseasyto understand,easytodo,and easyto prove inbase 60. The explanation is(in)
the structure underlyingall the natural numbers:the prime numbers. Base 60is builtfromthe first
three prime numbers:2x 2 x 3 x 5 = 60.
In base 60 higherarithmeticequatesgeometry. The symmetry andcontinuityof the spiral isdirectly
coupledtoall the prime numbers,the structure underlyingthe natural numbers.The elementary
symmetry,of the prime numbersinbase 60, carriesoverto all the branchesof mathematics.In
geometry throughthe coincidenceof the factorsof 60 withthe smallestprimitive Pythagoreantriple
(abc). In arithmeticthroughcomputationwith4factorsas yourbase.
There isone single foundationthatis proven true.Contraryto the Hellenictradition,whereinthe
axiomsare derivedfromthe mindof the philosopher, inthe SumerianandBabyloniantraditionthe
(single) axiomcomesfromthe observationof certainfacts. Thusthe axiom isnot established
deductivelybutinductively.The axiomisnottakentobe true on the basisof faith,butprovento be
true.One starts froma provenpremise orstartingpointforfurtherreasoningandarguments.
The questionpuzzlingthe ‘oldOne’iswhy?
Why dowe tell time inbase 60, inline withourlinearexperience of time.However,forall (other)
mathematics we goback to base 10.
Why dowe consciouslylimitourability inmathematics bydoingcomputation with2factors less in
base 10? Or evenwith4 factors less,whenwe restrictourselvesto binary? The Universe literally
hangstogetherfromthe multiplicationof all the prime numbers.Everythingatonce all the time.
Empoweredwith2more factors inyour base, mathematicsiseasytounderstand,easytodo,and
easy to prove.
Base 60 is‘complete’,itisaformal mathematical system basedonone principle, withthe foundation
buildonreason.Thereforitisnot prone to the inherentlimitations of axiomaticformal systems
provenbyGödel. Base sixtyisinessence ‘Mathematical Enlightenment’.

2. 2
Higher arithmetic/number theory
The three-dimensionalfigure of the spiral isrepresented twodimensionally byamodulus6matrix.
The natural numbers inthismatrix are orderedinrowsof six.
As a resultall prime numberslargerthan5 line up incolumn1 and column5; theyare exclusively
congruentto 1 and 5 mod6. See Table 1, Modulus6 Matrix in base 10 and base 60, onpage 3.
In base 60 the prime numbers (p≥ 7) are divided in16 archetypes.
 8 of the archetype prime numbers are congruentto1 modulus6:
o .07, .13, .19, .31, .37, .43, .01 and .49 .
 And8 of the archetype prime numbers are congruentto5 modulus6:
o .11, .17, .23, .29, .41, .47, .53, and .59 .
These 16 archetype prime numberscomputeinto136 differentsemiprimenumbers (px q= pq).
 72 semiprime numbers pqhave aremainderof 1 modulus6, and
 64 semiprime numbers pqhave a remainderof 5 modulus6.
In the sexagesimal numbersystemthe prime numbersdonotseemingly sproutlikeweeds,instead
theyflourish onthe 16 perchesof a Babylonian Garden.
In comparison,these same prime numbers are inthe decimal numbersystem dividedin(only) 4such
archetypes,i.e.all suchprimes’lastdecimal digitisa1, 3, 7, or 9.
To illustrate thispoint,all primenumbersendingon1in the decimal numbersystemare subdivided
into.11, .31, .41, and.01 archetypesinthe sexagesimal numbersystem.The sexagesimalnumber
systemis(literally)exponentiallymore refinedthanthe decimal numbersystemintermsof higher
arithmetic.
Table 2, Numberbase andnumbertheoretical refinement.
Base Numbertheoretical
refinement
The place value of the lastdigitof
prime numbersinthe numberbase
Prime numberlist
(factor(s) of the base)
2 1 1 (same asall 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
22
, 3, 5, 7, 11, 13, 17, 19 , 23,
..
Thisexponential differenceinrefinement -of base 60in comparisontobase 10- leadstomore
feasible resultsinall formsof computation:multiplication,division,(prime)factorization,matrices,
trigonometry,calculus, etcetera, whendone inthe sexagesimal numbersystemasopposedtodoing
theminthe decimal numbersystemorotherlessrefinednumberbases.

3. 3
Table 1, Modulus6 Matrix inbase 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 1.00 60
1.01 61 1.02 62 1.03 63 1.04 64 1.05 65 1.06 66
1.07 67 1.08 68 1.09 69 1.10 70 1.11 71 1.12 72
1.13 73 1.14 74 1.15 75 1.16 76 1.17 77 1.18 78
1.19 79 1.20 80 1.21 81 1.22 82 1.23 83 1.24 84
1.25 85 1.26 86 1.27 87 1.28 88 1.29 89 1.30 90
1.31 91 1.32 92 1.33 93 1.34 94 1.35 95 1.36 96
1.37 97 1.38 98 1.39 99 1.40 100 1.41 101 1.42 102
1.43 103 1.44 104 1.45 105 1.46 106 1.47 107 1.48 108
1.49 109 1.50 110 1.51 111 1.52 112 1.53 113 1.54 114
1.55 115 1.56 116 1.57 117 1.58 118 1.59 119 2.00 120
2.01 121 2.02 122 2.03 123 2.04 124 2.05 125 2.06 126
2.07 127 2.08 128 2.09 129 2.10 130 2.11 131 2.12 132
2.13 133 2.14 134 2.15 135 2.16 136 2.17 137 2.18 138
2.19 139 2.20 140 2.21 141 2.22 142 2.23 143 2.24 144
2.25 145 2.26 146 2.27 147 2.28 148 2.29 149 2.30 150
2.31 151 2.32 152 2.33 153 2.34 154 2.35 155 2.36 156
2.37 157 2.38 158 2.39 159 2.40 160 2.41 161 2.42 162
2.43 163 2.44 164 2.45 165 2.46 166 2.47 167 2.48 168
2.49 169 2.50 170 2.51 171 2.52 172 2.53 173 2.54 174
2.55 175 2.56 176 2.57 177 2.58 178 2.59 179 3.00 180
3.01 181 3.02 182 3.03 183 3.04 184 3.05 185 3.06 186
3.07 187 3.08 188 3.09 189 3.10 190 3.11 191 3.12 192
3.13 193 3.14 194 3.15 195 3.16 196 3.17 197 3.18 198
3.19 199 3.20 200 3.21 201 3.22 202 3.23 203 3.24 204
3.25 205 3.26 206 3.27 207 3.28 208 3.29 209 3.30 210
3.31 211 3.32 212 3.33 213 3.34 214 3.35 215 3.36 216
3.37 217 3.38 218 3.29 219 3.40 220 3.41 221 3.42 222
3.43 223 3.44 224 3.45 225 3.46 226 3.47 227 3.48 228
3.49 229 3.50 230 3.51 231 3.52 232 3.53 233 3.54 234
3.55 235 3.56 236 3.57 237 3.58 238 3.59 239 4.00 240
4.01 241 4.02 242 4.03 243 4.04 244 4.05 245 4.06 246
.. .. .. .. .. .. .. .. .. .. .. ..

4. 4
Arithmetic
Why would one consciously domathematics withaself-imposedcognitive handicap?Insports we do
not have the conventionthatall players coverupone eye andplayusingonlyone leg,wheneverthey
are on the field.Norwouldageneral everinstructhistroopstotie anarm behindtheirbackbefore
goingintobattle.Yet, inmathematics,thatis primarily whatwe dowhenwe discardtwofactors
wheneverwe startdoingarithmeticinthe decimal numbersystem.
We limitourselvesfornorational reasonwheneverwe reverttodoingarithmeticinbase 10 instead
of base 60. Base 10 (2 x 5) has 2 prime factorslessthanbase 60 (2 x 2 x 3 x 5). For everything
mathematical thismattersagreat deal, because the Universeliterallyhangstogetherfrom the
multiplicationof itsfactors.Bydoingmathwithtwo factorsinsteadof withfour,youlose out
exponentially everytime.
Division and sexagesimal reciprocals
Regularsexagesimal numbersare natural numberswhose factors are only2, 3, and/or 5. In base 60
divisionbyaregularnumbercan be replacedbymultiplicationwiththe reciprocal of the number.In
the same way as we thinkof divisionby2 as equal tomultiplicationby½or 0.5 . The sexagesimal
inverse of the number2 is30, because sixty(=one) dividedbytwoequalsthirty,asa half hourequals
30 minutes,andhalf a minute equals30seconds.
Hence,regularsexagesimal numbersare of the form:
n = 2 𝑎 ∗ 3 𝑏 ∗ 5 𝑐
Examples
n = 23 ∗ 30 ∗ 50 = 8
n = 21 ∗ 32 ∗ 52 = 450
The reciprocal of n isgivenby:
ñ=
1
𝑛
Thereforinsexagesimal:
Xn
· Yn
· Zn
= XYZ n
= 1n
31
· (22
)1
· 51
= 601
= 11
Sexagesimal reciprocal ñ=
60 𝑑
2 𝑎∗3 𝑏∗5 𝑐 =
(22∗31∗51) 𝑑
2 𝑎∗3 𝑏∗5 𝑐

5. 5
Example
n = ñ
8 =
((22)2∗ 32 ∗ 52=) 602
23 ∗ 30 ∗ 50 = 21 ∗ 32 ∗ 52 = 450
Verificationinbase 10: 8 · 450 = 3600 = 602
Verificationinbase 60: 8 · 7.30 = 1.00.00
The reciprocal of 8 is 450 , because the reciprocal of 23
= 21
x 32
x 52
.
In sexagesimal, 8times7minutesand30 secondsis1 hour, 1 hour = 24
x 32
x 52
.
Multiplication
Multiplicationof twonumbersisdone with aidof the followingalgorithm.
ab = (
𝐴+𝐵
2
)
2
− (
𝐵−𝐴
2
)
2
Example,
1 x 7 = (
1+7
2
)
2
− (
7−1
2
)
2
= (
8
2
)
2
− (
6
2
)
2
= 42
– 32
= 16 – 9
A Babylonianscribe wouldinsteadof dividing8and 6 by 2,
multiplybothwith30(the reciprocal of 2).
1 x 7 = (8 x 30)2
– (6 x 30)2
= 2402
– 1802
(base10) , = 4.002
– 3.002
(base 60)
= 57.600 – 32.400 = 25.200 (base10) , = 16.00.00 – 9.00.00 = 7.00.00 (base 60)

6. 6
Note how8 times30 makes4 minutes. The Babyloniansusedafloatingpointandtherefor,the two
sexagesimal placesof zeroatthe endwouldbe leftblanc. “Thisisthe procedure.”
Before movingontogeometry, itiswortha minute of yourtime toconsiderthe algorithmwe are
taught formultiplication, andrealize the procedureworksfor/inall numberbases.
100
11 x
100
1000 +
1100
In binary,itisthe sum4 x 3 = 12.
In decimal,itisthe sum11 x 100 = 1100.
In sexagesimal,itisthe sum11 x 3600 = 39600 (= 11 hours).

7. 7
Geometry
Geometry iseasyto understandandeasytodo in base 60. I will explainwhythisisthe case andhow
keepingtobase 60 keepsyourintuition andlogicaligned.Inbase 60 geometryandarithmeticare
interchangeable.Fermat’sLastTheoremisprovendirectlybymeansof the Fundamental Theoremof
Arithmetic.
Theoremof Pythagoras 𝑎2 + 𝑏2 = 𝑐2
For any Pythagoreantriple (abc),the productof the twonon-hypotenuse legs(ab),isalways divisible
by 12, andthe productof all three sides(abc) is divisible by60. In addition,one side of every
Pythagoreantriple isdivisibleby3,anotherby 4, and anotherby 5. One side mayhave two of these
divisors,asin(8, 15, 17), (7, 24, 25), and (20, 21, 29), or evenall three,asin(11, 60, 61).
For primitive solutions,one of ‘ a’or ‘b’ mustbe even,andthe otherodd,with‘c’always odd,and
the GreatestCommonDivisor(GCD) = 1. It isusual to consideronly primitivePythagoreantriples -
alsocalled"reduced"triples- inwhich‘a’and‘b’are relativelyprime,since othersolutionscanbe
generatedtriviallyfromthe primitiveones.
The smallestprimitive Pythagoreantriple (a,b,c) is3, 4, 5 . The legsof thistriple arethe factorsof
the sexagesimalnumbersystem.Three timesfourtimesfiveequalssixty (31
·(22
)1
·51
= 601
).
Note that divisionby3,4, 5, 12, and 60 can be replaced bymultiplication withthe reciprocals,thus
withmultiplicationby 20,15, 12, 5, and 1 respectively.Multiplicationis significantly easiertodothan
division. The resultishoweverthe same,asmultiplicationwiththe reciprocal will giveyouthe exact
answer.
In base 60 geometryandarithmeticare interchangeable.Youcanswitchbetween doingmathwith
numberstodoingmath withoutnumbersatwill, becausethe smallestprimitivePythagoreantriple
(abc) equalsthe factorsof 60 (3, 4, 5). Thus, (higher) arithmetic=geometry,inbase 60.
Spiral mechanicsare easilysplitintodimensions:
2-dimensional
60 degrees 𝐴2 + 𝐵2 − 𝐴𝐵 = 𝐶2
90 degrees 𝐴2 + 𝐵2 = 𝐶2
120 degrees 𝐴2 + 𝐵2 + 𝐴𝐵 = 𝐶2
3-dimensional
Diagonal (D) ina cuboid
1
𝐴2 +
1
𝐵2 +
1
𝐶2 =
1
𝐷2

8. 8
“One reason why mathematicsenjoysspecialesteem,aboveallothersciences, is thatits lawsare
absolutely certain and indisputable,whilethoseof all other sciencesare to someextentdebatable
and in constantdangerof being overthrown by newly discovered facts.
In spiteof this,the investigatorin anotherdepartmentof sciencewould notneed to envy the
mathematician if the lawsof mathematicsreferred to objectsof ourmere imagination,and notto
objectsof reality. Forit cannotoccasion surprisethatdifferentpersonsshould arriveatthe same
logical conclusion when they havealready agreed upon thefundamentallaws(axioms),aswellasthe
methodsby which the otherlaws areto be deduced therefrom.Butthereis anotherreason forthe
high repute of mathematics,in thatit is mathematicswhich affordstheexactscience a certain
measureof security,to which withoutmathematicsthey could notattain.
At this pointan enigma presentsitself which in all ageshas agitated inquiring minds.How can it be
thatmathematics,being afterall a productof human thoughtwhich isindependentof experience,is
so admirably appropriateto theobjectsof reality?Is human reason,then,withoutexperience,merely
by taking thought,ableto fathomthepropertiesof real things.
In my opinion the answerto this question is,briefly, this: - As faras the lawsof mathematicsreferto
reality, they are notcertain; and as faras they are certain,they do notrefer to reality.”
- AlbertEinstein, Lecture on Geometry and Experience,Berlin 1921.

9. 9
Sexagesimal Reciprocals and Fermat’s Last Theorem
Mathematicsiseasyto understand,easytodo,and alsoeasy to proveinbase 60.
In the sexagesimal numbersystemthere isadirect andintuitive proof of Fermat’sLastTheorem,
whichis simple andshortenough,itcouldhave easilyfitted the marginof Fermat’s book.
Fermatis proven withthe helpof the fundamentaltheoremof arithmeticbycomparingFermatwith
the equationforsexagesimal reciprocal numbers.
The fundamental theorem ofarithmetic (or unique factorizationtheorem) statesthateverynatural
numbergreater than 1 can be writtenas a product of prime numbers,and that, moreover,this
representationisunique up to (exceptfor) the order of the factors.
The two equations are nearly identical,the onlydifference isthe additionsign ‘+’ inFermatisa
productsign‘x’ in the reciprocal formula.Because the Fundamental theoremof arithmeticistrue,
Fermat’sLast Theoremistrue.
Reciprocal equation - Fermat’sLast Theorem
𝑋 𝑛 ∗ 𝑌 𝑛 = 𝑍 𝑛 𝑋 𝑛 + 𝑌 𝑛 = 𝑍 𝑛
Fermat’sgeneralization of Pythagoras hasintegersolutions,only, forn=1 andn= 2, and no integer
solutions forn≥3. The n= 1 solutionsare all the additionsof twointegers,andthe n=2 solutionsare
evidently all the Pythagoreantriples. Integersolutionsforn≥3 do not exist.
The reciprocal formulahasintegersolutionsforall n(n≥ 1). The order of the factors X, Y and Z can be
rearranged.Forevery value of nthe equation willgive the correctpairof sexagesimalreciprocals.
You can change Zn
to be the factor 3, 22
, or 5 since,as statedbythe fundamental theoremof
arithmetic, the representationisuniqueexceptforthe orderof the factors.
By applyingthe theoremtobothequations,we learnwhy Fermatcanonlyhave integersolutionsfor
n= 1 and n= 2, andthe formulaforreciprocalshasintegersolutionsforall n.Everynatural number
greaterthan 1 can be written uniquely asthe productof prime numbers,and notasthe sumof prime
numbers.

10. 10
n = 2 𝑎 ∗ 3 𝑏 ∗ 5 𝑐
ñ =
1
𝑛
Reciprocal ñ =
60 𝑑
2 𝑎∗3 𝑏∗5 𝑐
n = ñ
2 𝑎 ∗ 3 𝑏 ∗ 5 𝑐 =
(22 ∗ 31 ∗ 51)
2 𝑎 ∗ 3 𝑏 ∗ 5 𝑐
𝑑
3 𝑛 ∗ (22) 𝑛 ∗ 5 𝑛 = (22 ∗ 3 ∗ 5) 𝑛 = 60 𝑛
𝑋 𝑛 ∗ 𝑌 𝑛 ∗ 𝑍 𝑛 = 𝑋𝑌𝑍 𝑛 = 1 𝑛
Xn ∗ Yn =
(XYZ)n
XYn
= Zn => Xn ∗ Yn = Zn
Xn ∗ Zn =
(XYZ)n
XZn
= Yn => Xn ∗ Zn = Yn
Yn ∗ Zn =
(XYZ)n
YZn
= Xn => Yn ∗ Zn = Xn

11. 11
Sexagesimal Reciprocals - Fermat’s Last Theorem
𝑋 𝑛 ∗ 𝑌 𝑛 = 𝑍 𝑛 𝑋 𝑛 + 𝑌 𝑛 = 𝑍 𝑛
n = 0 n = 0
30
= (22
)0
· 50
1 = 1 10
+ 10
≠ 10
(22
)0
= 30
· 50
1 = 1 20
+ 30
≠ 50
50
= (22
)0
· 30
1 = 1 X0
+ Y0
≠ Z0
600
= (22
)0
· 30
· 50
= n · reciprocal n
n = 1 n = 1
31
= (22
)1
· 51
3 = 20 11
+ 21
= 31
(22
)1
= 31
· 51
4 = 15 21
+ 31
= 51
51
= (22
)1
· 31
5 = 12 31
+ 51
= 81
601
= (22
)1
·31
·51
= n · reciprocal n
n = 2 n = 2
32
= (22
)2
· 52
9 = 16 · 25 9 = 400 82
+ 152
= 172
(22
)2
= 32
· 52
16 = 9 · 25 16 = 225 72
+ 242
= 252
52
= (22
)2
· 32
25 = 16 · 9 25 = 144 112
+ 602
= 612
602
= 3600 = n · reciprocal n
n = 3 n ≥ 3
33
= (22
)3
· 53
27 = 64 · 125 27 = 8.000 No solutions
(22
)3
= 33
· 53
64 = 27 · 125 64 = 3.375 [ (22
)3
+ 33
= 53
]
53
= (22
)3
· 33
125 = 64 · 27 125 = 1.728 [ 64 + 27 = .. ]
603
= 216.000 = n · reciprocal n [ 91 = .. ]
[(7 · 13) = .. ]
Addition of 64 and 27 changes/alters the
factors (instead of rearranging them)
from2 and 3 to 7 and 13.

12. 12
n = 4
34
= (22
)4
· 54
81 = 256 · 625 81 = 160.000
(22
)4
= 34
· 54
256 = 81 · 625 256 = 50.625
54
= (22
)4
· 34
625 = 256 · 81 625 = 20.736
604
= (22
)4
·34
· 54
= 12.960.000 = n · reciprocal n
n = 5
35
= (22
)5
· 55
243 = 1.024 · 3.125 243 = 3.200.000
(22
)5
= 35
· 55
1.024 = 243 · 3.125 1.024 = 759.375
55
= (22
)5
· 35
3.125 = 1.024 · 243 3.125 = 248.832
605
= (22
)5
·35
· 55
= 777.600.000 = n · reciprocal n
n = 6
36
= (22
)6
· 56
729 = 4.096 · 15.625 729 = 64.000.000
(22
)6
= 36
· 56
4.096 = 729 · 15.625 4.096 = 11.390.625
56
= (22
)6
· 36
15.625 = 4.096 · 729 15.625 = 2.985.984
606
= (22
)6
· 36
· 56
= 46.656.000.000 = n · reciprocal n
n = ..
The Fundamental Theoremof arithmetic’providesthe explanationforsolutionforall ninthe case of
the reciprocal formula,andonlyforn is1 and 2 inFermat:
Everyintegergreaterthan1 eitherisaprime numberitself orcanbe represented asthe productof
prime numbers andthisrepresentationis uniqueup to the orderof the factors.
Everyintegerisuniquelyrepresented asthe productof prime numbersonly upto rearrangement,
and not the sum (addition) of prime numbers (x ≠+).
𝑋 𝑛 ∗ 𝑌 𝑛 = 𝑍 𝑛 , writtenunique asthe productof primes forall n exceptforthe orderof the
factors, andneverwrittenuniquelyasthe sum of primes: Xn
+ Yn
= Zn
, for n largerthan 2.
The factors of the sexagesimal numbersystem 3,22
,and 5 make it perfectly evidentwhyFermat’s
generalizationof Pythagoras,Xn
+Yn
= Zn
, onlyhas integersolutionsforn=1 and n= 2. The n=1
solutionsare all the additionsof twointegers,andthe n= 2 solutions are the smallestprimitive
Pythagoreantriple. The same factorsinthe reciprocal equation givethe integersolutionsforall n.
QuodErat Demonstrandum

13. 13
Science in antiquity
The Sumerians,fromthe observation of certainfacts,had gottenthe ‘(in) the beginning’completely
right. Ever since,inthe historyof mathematics,everyoneelsehashadthe firstpart wrong. The
cradle of civilizationwasnomatterof chance.The Sumeriansinventedthe sexagesimal number
system, cuneiformscript, literature,andirrigation.
The AustrianAmericanmathematicianandhistorianof science OttoNeugebauerinhisseminalwork
on astronomyandthe exact sciencesinantiquity (1952) wrote:
“ 21. Pythagorean numberswere certainly notthe only case of problemsconcerning relations
between numbers.Thetablesfor squaresand cubespointclearly in the samedirection.We also have
exampleswhich deal withthe sumof consecutivesquaresorwith arithmetic progressions.Itwould be
rathersurprising if the accidentally preserved textsshould also show usthe exact limits of knowledge
which were reached in Babylonian mathematics.Thereis no indication,however,thattheimportant
conceptof prime numberwasrecognized.
All theseproblems were probably neverfarseparated frommethodswetoday would call‘algebraic’.
In thecenter of thisgroup lies the solution to quadraticequationsfortwo unknowns.Asa typical
examplemightbe noted ..”
Indeed,there hasseemingly been“..no indication,however,thattheimportantconceptof prime
numberwasrecognized.”, because everyoneinmathematicssince the ancientGreek hasbeen
oblivioustothe primaryreasonforthe sexagesimal numbersystem.Anepiccase of hidinginplain
sight,andmathematics’ veryownhistoricversionof the “invisiblegorillaexperiment”.The
experimentpainstakinglyrevealsthe hiatusinourknowledgeof the fundamentalsof mathematics in
comparisontothat of the ancientMesopotamians.Ourmindsdon’twork the waywe thinktheydo.
We thinkwe see ourselvesandthe worldastheyreallyare,butwe’re actuallymissingawhole lot.
We are not aware we lookat mathematicsinarectilinearfashion whenwe thinkof the standard
numberline,norare we aware of the impacton our view of the decimal prism.Whenwe do
arithmeticwithtwolessfactors,we have noideawe’re missingawhole lot.
“Mathematicsis, I believe,the chief sourceof thebelief in eternaland exact truth,aswell asin a
super-sensibleintelligible world.Geometry deals with exact circles, but no sensible objectis exactly
circulars, howevercarefully wemay use ourcompasses,therewill besome imperfectionsand
irregularities. This suggeststheviewthat all exact reasoning appliesto ideal as opposed to sensible
objects;it is naturalsto go further,and to arguethat thoughtisnoblerthan sense,and the objectsof
thoughtmorereal than thoseof sense-perception.Mysticaldoctrinesasto the relation of time to
eternity are also reinforced by pure mathematics,formathematicalobjects,such asnumbers,if real
at all, are eternal and notin time. Such eternal objectscan be conceived as God’sthoughts.Hence
Plato’sdoctrinethatGod is a geometer,and Sir JamesJeans’belief that He is addicted to arithmetic.
Rationalistic asopposed to apocalypticreligion hasbeen,ever since Pythagoras,and notableever
since Plato,very completely dominated by mathematicsand mathematicalmethod.
The combination of mathematicsand theology,which began with Pythagoras,characterized religious
philosophy in Greece, in the Middle Ages,and in modern timesdown to Kant.Orphismbefore
Pythagoraswasanalogousto Asiaticmystery religions.But in Plato,SaintAugustine,Thomas
Aquinas,Descartes,Spinoza,and Kantthereis an intimate blending of religion and reasoning,of
moralaspiration withlogical admiration of whatistimeless, which comes fromPythagoras,and

14. 14
distinguishestheintellectual theology of Europefromthe morestraightforward mysticismof Asia.It
is only in quite recent timesthat it hasbeen possibleto say clearly where Pythagoraswaswrong.Ido
notknowof any other man who hasbeen as influentialashe wasin thesphere of thought.Isay this
becausewhatappearsasPlatonismis,when analysed,found to bein essence Pythagoreanism.
The wholeconception of an eternalworld,revealed to the intellect butnot to the senses,is derived
fromhim. Butfor him,Christianswould nothavethoughtof Christasthe Word;but forhim,
theologians would nothavesoughtlogicalproofs of God and immortality.But in him this is still
implicit. Howit becameexplicit will appear.”
- Bertrand Russell The History of Western Philosophy (1945)
Thymaridas of Paros,the GreekPythagoreannumbertheorist,issaidtohave calledprime numbers
rectilinear since theycanonlybe representedonaone-dimensionalline.Non-primenumbers,onthe
otherhand,can be representedonatwo-dimensional plane asarectangle withsidesthat,when
multiplied,produce the non-prime numberinquestion.
EveryBabylonian knew Thymarides’statement, aboutprime numbers being‘rectilinear’,isfalse.
EveryBabylonian whoknewhowtomultiplytwonumbers, couldprove Thymarides’ wrongby
demonstration.The exampleof 1 times7,givenat the beginning,isacase in point.A rectangle with
sidesof 1 and 7, istransformedto, the difference betweentwosquares(the areabetween42
and32
).
Thisprocedure hadbeenapplied routinely, asthe standardmethodformultiplyingtwonumbers,for
well overthree millenniabefore PythagorasorThymarideswere born.
The deductivemethod versus the scientific method
Thousandsof yearsbefore Greekmathematicsanddeductive reasoning, inductivereasoninghadled
to the creationof the sexagesimal numbersystem.A coherentformal logical systembasedonone
single principle.A trulyscientifictheoryof numbersatthe beginningof historyinthe cradle of
civilization.
BertrandRussell in The Historyof WesternPhilosophy (1945) wrote:
“The Greeks contributed,it is true,something elsewhich proved of morepermanentvalueto abstract
thought:they discovered mathematicsand theartof deductivereasoning.Geometry,in particular,is
a Greek invention,withoutwhich modern sciencewould havebeen impossible. But inconnection
with mathematicsthe one-sidednessofthe Greek genius appears:it reasoned deductivelyfrom
what appearsself-evident,notinductivelyfrom whathad been observed. Its amazingsuccessin the
employmentof thismethod mislednot onlythe ancientworld,but the greater part of the modern
world also.It has onlybeen very slowlythatscientificmethod,which seeks to reach principles
inductivelyfromobservationof particularfacts,has replacedthe Hellenicbelief indeductionfrom
luminousaxiomsderivedfromthe mindof the philosopher. Forthisreason,apartfromothers,it is a
mistaketo treat theGreeks with superstitiousreverence.Scientific method,though somefew among
themwere the first men who had an inkling of it, is, on the whole,alien to their temper of mind,and
the attemptto glorify themby belittling theintellectual progressof thelast fourcenturies hasa
cramping effectupon modern thought.

15. 15
There is, however,a more generalargumentagainstreverence,whetherfortheGreeks or foranyone
else. In studying a philosopher,therightattitudeis neither reverence norcontempt,butfirsta kind of
hypotheticalsympathy,untilit is possibleto know whatit feels like to believe his theories,and only
then a revival of the critical attitude,which should resemble,as faras possible,thestate of mind of a
person abandoning opinionswhich hehasa hitherto held. Contemptinterfereswith thefirst part of
this process,and reverencewith thesecond.Two thingsare to be remembered:thata man whose
opinionsand theoriesare worthstudying may bepresumed to havehad someintelligence, butthat
no man is likely to havearrived atcomplete and finaltruth on any subjectwhatever.When an
intelligent man expressesa viewwhich seemsto usobviously absurd,weshould notattemptto prove
thatit is somehowtrue,butwe should try to understand how itevercame to seemtrue. This exercise
of historical and psychologicalimagination atonceenlargesthe scopeof our thinking,and helpsusto
realize howfoolish many of ourown cherished prejudiceswill seem to an agewhich hasa different
temperof mind.”
At the beginningof the 20th century mathematicianhadattemptedtobringthe whole of
mathematicsunderone single roof.Theyhadbelievedtheycouldgive mathematicssolid
foundations, buttheyhadfailed because of the inherentlimitationsof everyformalaxiomatic
system:inherently onestartsfroman unproven premise. Inall typesof formal axiomaticsystems,the
axiomsthemselvesare notproven,buttakentobe true,and serve asa premise orstartingpointfor
furtherreasoningandarguments.The wordaxiomcomesfromthe Greek wordaxioma“thatwhichis
thoughtworthyor fit”or “that whichcommendsitself asevident.”
The axiomsare derivedfromthe mindof the philosopher.The axiomsare notproven,theyare
merelythoughttobe true and acceptedas foundations.The validityof the deductionisnot100%
guaranteed. Because the validityof the deduction iscompletely dependentonthe truthvalue of the
axiomfromwhichithas beenderived.Youmaybelieve the axiomtobe true,butthisbelieve
dependsonfaithandnot onreason,as the axiomisnot a proventruth. The intuitionof the
philosopherisraisedtothe statusof axiom.
In 1931 Gödel provedthe ‘Greek type’of formal axiomaticsystemsincomplete. Incontrast,the
sexagesimal numbersystem -spiral mechanics- issemanticallyand syntactically complete.The
Babylonianmethodisinductive asitfollowsfromthe observationof certainfacts.
Eusebiusof CaesareainPraeparatioEvangelica‘Preparationforthe Gospel’ (313AD) makesquite
clear‘the one-sidednessof the Greek’andtheirrectilinearperceptionof reality waswell known
amongthe peoples of the fertile crescent. There wasnodiscussionontheirpresupposed ‘genius’
and wisdombecause theywere notbelievedtohave had much,all theirknowledge was imported.
“In factthe said Pythagoras,whilebusily studying thewisdomof each nation,visited Babylon,and
Egypt,and all Persia,being instructed by the Magiand the priests: and in addition to these he is
related to havestudied underthe Brahmans(theseareIndian philosophers);and fromsomehe
gathered astrology,fromothersgeometry,and arithmeticand musicfromothers,and different
thingsfromdifferentnations,and only fromthewise men of Greece did he get nothing,wedded as
they were to a povertyand dearthof wisdom:so on the contrary hehimself becametheauthorof
instruction to theGreeks in thelearning which he had procured fromabroad.”

16. 16
Alexanders’solutiontothe intractable problemof the Gordianknot couldmathematicallybe putina
differentperspectiveasmore of the same ‘rectilinearthinking’,i.e.reasoningitdoesn’tmatterhow
the knot isuntied,andthen‘solving’the problemwithbrute force.
Babylonianmathematicswasonlyrediscoveredinthe late 19th century,whenarcheological
excavationrevealedthousandsof claytabletsincuneiformscript. Recordsof transactions,legal
contracts, andnumerous (manythousandsof) mathematical exercises. Onlyinthe middle of the 20th
centuryscholars startedto become aware of (the level of) the mathematicsonthe claytablets.
Whenthe tabletswere translatedthe mathematicsonthempuzzledboththe archeologistsandthe
mathematicians.Indeed,itlookedcrazytothem.“I have addedthe circumference tothe areaof a
square. It is‘45’.” To usthismakesno sense,andfora longtime scholarsthoughtthe Babylonians
onlyusednumerical methodsandhadno geometry.
In one of histelevisedlecturesRichardFeynmandescribedthe difference inmathematicsbetween
the Greeksand the Babyloniansasfollows:
“ There aretwo kindsof waysof looking at mathematicswhich forthepurposeof this lecture I will
call theBabylonian tradition and theGreek tradition.
In Babylonian schoolsin mathematics,the studentwould learn something by doing a largenumberof
examplesuntil hecaughton to the generalrule. [..]
The Babylonian thing thatIam talking about -which Idon’t,really notBabylonian but- isto say I
happen to knowthisand happen to know thatand I workouteverything fromthere,and tomorrowI
forgotthiswastrue butI remember thatthis wastrue,and then I reconstructit again,and so on.I am
neverquite sure whereI am supposed to begin and supposed to end.Ijustrememberenough allthe
time so asthe memory fadesand thepieces fall out I re-putthething back togetheragain every day.
The method of starting fromtheaxiomsis not efficientin obtaining thetheorems.In working
something outin geometry you arenotvery efficientif each time you haveto startback atthe
axioms.Butif you haveto remembera few thingsin the geometry you can alwaysgetsomewhere
else. And whatthe bestaxiomsare,are notexactly the same,in fact are notever the sameasthe
mostefficient way of getting around in the territory. In physicswe need the Babylonian method and
notthe Euclidean orGreek method. [..]”
Babylonians couldeffortlesslyjump betweenarithmeticandgeometry, because tothemgeometry
and arithmeticare notdifferentfieldsof mathematics,theyare notevendifferentsidesof the same
coin.Theyare bothsidesof the coin at the same time. Spiral mechanicscombinesdualityandtrinity,
the spiral embodiesall primenumbersandproducts,everythingatonce.Geometryandarithmetic
are as inextricablyentangledatthe core as are space and time accordingto AlbertEinstein.
“Quantummechanicsiscertainly imposing.Butan inner voice tells me thatit is notyet the real thing.
The theory saysa lot,but doesnotreally bring usany closer to the secret of the ‘old one’.I, at any
rate,am convinced thatHe is notplaying at dice. Wavesin 3-dimensionalspace,whosevelocity is
regulated by potentialenergy (forexample,rubberbands)….Iamworking very hard atdeducing the
equationsof motion of materialpointsregarded assingularities,given thedifferentialequation of
generalrelativity. - Albert Einstein (Letter to Max Born,December 4th
1926)

17. 17
Einstein,Feynman,andRussellwere all acutelyaware of the inherentlimitationsof the Greek
axiomaticmethod. In1901 Russell’sparadox hadshownthatsome formalizationsof Cantor’sset
theoryledtocontradiction.Gödel wouldin1931 prove these type of axiomaticsystemsincomplete.
How well the Babylonians actually knewwhattheywere doing andwhere theywere going isshown
by theiractions,there are examples abound of applyingtheirknowledge directlytoreality.
First,writingisappliedspiral mechanics.We readthissentencelefttoright,andat the endof the
line,the scriptcontinuesfluidly,withthe firstwordone row below.Readingdependsonourmental
constructionof a continuousspiral inthree dimensions.We caneasilyreconstructanactual three
dimensional spiral fromapage byrollingitup horizontallyuntil the lastwordof the firstsentence
linesupwiththe firstwordof the secondsentence.The inventorsof the sexagesimal placevalue
numberssystemare alsothe inventorsof the cuneiformscript.Bothwere inventedbythe Sumerians
inwhat istoday southernIraq.The latteristhe logical applicationof the former.A numerical place
value systeminwhichthe numbershave beenreplacedbywords. Spiral mechanicsconvertedto the
2-dimensional matrix.
Second, Archimedes’spiral isasmuch Archimedes’asPythagorean triplesare Pythagorean. The
Babyloniansfoundanapplicationforthe spiral inirrigation. The knowledge bundledbyEuclidinthe
Elementswasmerelyaderivative of the knowledgethathadbeendiscoveredanddevelopedin
Mesopotamia‘Inthe beginning…”.
Last but not leastIwant to mention the ‘Bagdadbattery’. The purposes the Sumeriansand
Babylonians hadforitremainsa mystery,that it generateselectricity(Volt),however, hasbeen
repeatedlyproven.
The advances in the sciences of the Sumeriansand the Babylonianswere inducedbythe sexagesimal
numbersystemitself.The Babylonians workedfromanumbertheorythatwas inductive,logical,and
complete.Everythingfromone singleprovenfoundation. A scientifictheoryof numbers.
Bab-ilu,Babel,orBabylon, means‘Gate tothe Gods’,the Babylonians knew thatwasexactly the
mathematics the Sumerianshad providedthemwith.
Conclusion - In Sum
The Babyloniansadoptedthe sexagesimal number infull awarenessof the marvelousdiscoveryof
the Sumerians.The discoverythat hadstoodat the heart of theirinventionof writing,the plough,
irrigation, andhadprovided guidance totheirscientificadvances.A scientificallybasedtheoryof
numbers.A formal logical system of mathematicsbasedonone proven principle.
There isan old joke where someone isaskedfordirections.The personisanswered:“Well,if you
wantto go there,I wouldn’tstartfromhere.” Indeed,mathematicsoutside base 60isexactly like
this. You getoff on the wrongfoot. However, thatisnot a joke.
The mathematical Universe literallyhangstogether frommultiplication:all natural numberscanbe
representeduniquelyasa productof primes.The question whichhaspuzzledthe ‘old One’, iswhy?
Why wouldone ever,rationallyandingoodconscious,doarithmeticwithtwolessfactors?While
one knowshowto tell the time,knowshow tocountin base 60?

18. 18
For 2500 yearspeople have beentellingtime inbase 60,but forall thingsmathematical fall backona
smallernumberbase,unaware of the inherentconsequencesof doing so,namelymaking
mathematicsharderthanit needstobe.
Stickto the sexagesimal numbersystemandyouwill find,time isonyourside.
The Babyloniansalwaysendedtheirmathematical workwithashortphrase:
This is the procedure.
I wouldlike to endwitha lastquote fromchaptersix of mybook On the Theory of Numbers:Prime
Numbersand Enlightenment ,the quote isfromSaintAugustine of Hippo:
“Six is a numberperfectin itself, and notbecauseGod created the world in six days;ratherthe
contrary is true. God created theworld in six daysbecausethisnumberis perfect,and it would remain
perfect,even if theworkof the six daysdid notexist.”