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Topic: Infinity: The Story So Far
Replies: 7   Last Post: Feb 28, 2014 10:15 AM

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fom

Posts: 1,968
Registered: 12/4/12
Re: Infinity: The Story So Far
Posted: Feb 28, 2014 10:15 AM
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On 2/28/2014 2:07 AM, mueckenh@rz.fh-augsburg.de wrote:
> On Thursday, 27 February 2014 23:15:28 UTC+1, fom wrote:
>
>

>>
>> What exactly do you mean when you distinguish "+1" in
>>
>> the manner you have given above?

>
> Either you know it or you don't. It is the first step of mathematical understanding. It cannot be formalized without assuming other primitives which however are not as primitive, in the development of the human race as well as in the development of the single individual. ( Ontogenesis repeats die phylogenesis, haeckel)


Well, I disagree with that.

But, it is ridiculously difficult.

Euclid uses the part relation in three
senses in his axioms. He uses it in
the discussion of multitudes, magnitudes,
and points.

It is the concept of "equals" in relation
to "even numbers" where addition is to
be found. So, the primitives are

1 + 1

2 + 2

3 + 3

and so on.

In criticizing the mathematics of his day,
Brouwer argued that there was a fundamental
"twoness" in mathematics.

My arithmetical axioms assume both '1' and
'1 + 1' in the same axiom. And, this is done
because the general identity criterion is
given by

AxAy( x + 1 = y + 1 -> x = y )

Euclid says that odd numbers do not measure
equals, but differ by 1 from even numbers.
So, the identity for odd numbers is given
by

1 + 1 -> 1 = 1

2 + 2 -> 3 = 3

3 + 3 -> 5 = 5

etc.

while the identity for even numbers is
given by

3 = 3 -> 2 = 2

5 = 5 -> 4 = 4

7 = 7 -> 6 = 6

etc.

Because odd numbers differ from even
numbers by a unit.

Then this gets tied together when you add
the axioms asserting that multitudes are
composed of units. That is when everything
is given a successor and everything different
from a unit is a successor of something.

If you consider the theorem that 4 times
every odd prime is representable as the
sum of 4 odd integral squares, then the
measure of even numbers by equals,

1 + 1 -> 1 = 1

2 + 2 -> 3 = 3

3 + 3 -> 5 = 5

etc.

gives you all of your odd prime numbers.
And, you already have '1 + 1' since that
had been in the original existence assumption
to support

1 + 1 -> 1 = 1

The well ordering given by Dedekind-Peano
succession then yields the fundamental
theorem of arithmetic.

Where the real complexity arises is in
figuring out how to put the statements

2 + 2 -> 3 = 3

5 = 5 -> 4 = 4

together. This seems to involve quadratic
forms.

Hurwitz proved that

1 * 2^k and 5 * 2^k for k = 0, 1, 2, ...

cannot be the lengths of an interior
diagonal of a cuboid with natural edges.

Now, if you compare with Pythagorean triples
written as

a, b, c^2

Then, you see that

1, 2, 5

is of that form. I am not calling this
a Pythagorean triple. I am just making
the observation that these numbers are
in that relation. If you write it as

a, b, c

instead, you obtain

1, 2, surd(5)

These are the components for the golden
ratio. Moreover, the golden ratio has
a relation to a successor function when
its denominator 'b' is taken as a unit,

(a + b)/a = a/b

(a + 1)/a = a/1

a + 1 = a^2

The other relation to cuboids is that if
all of the edges and face diagonals of a
cuboid are natural, then one of the edges
is divisible by 11. Notice that this is
a slight strengthening of the statement
above.

Now,

4 * 3 = 12 = 1^2 + 1^2 + 1^2 + 3^2

and

12 = 6 + 6 -> 11 = 11

These numbers do relate to surd(5) through
a great deal of finite geometry. The
golden field is a quaternionic number system
constructed from the rationals extended by
surd(5),

http://en.wikipedia.org/wiki/Icosian

This number system can be used to form the
Leech lattice,

http://en.wikipedia.org/wiki/Leech_lattice#Using_the_icosian_ring

which has a covering radius of surd(2).

This in turn is related to the sporadic permutation
groups, M_11, M_12, M_22, M_23, M_24

http://en.wikipedia.org/wiki/Mathieu_groups

of which M_24 is the automorphism group for the
Steiner system (5,8,24)

http://en.wikipedia.org/wiki/Mathieu_groups#Automorphism_groups_of_Steiner_systems

Investigations of the S(5,8,24) had been facilitated
by the use of the Miracle Octad Generator devised by
Curtis. The underlying structure had been that of
a 16-element affine plane representing 2 8-element blocks
and the remaining 8-element block partitioned into
the 5-element line at infinity for the affine plane
and a 3-element complement.

It is a theorem that every sufficiently large odd
integer is the sum of 21 5th-powers of primes,

http://en.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem#Relevant_results

So, reconciling incommensurables associated with
the representation of 4-fold multiples of primes
by odd numbers leads to a representation odd
numbers by 21-fold sums of prime powers.

This is all finite geometry.

But everyone thinks that mathematics is about
counting on their fingers and looking cool
at cocktail parties talking about Russell's
voodoo.









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