Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


fom
Posts:
1,968
Registered:
12/4/12


Re: Infinity: The Story So Far
Posted:
Feb 28, 2014 10:15 AM


On 2/28/2014 2:07 AM, mueckenh@rz.fhaugsburg.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 DedekindPeano 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 16element affine plane representing 2 8element blocks and the remaining 8element block partitioned into the 5element line at infinity for the affine plane and a 3element complement.
It is a theorem that every sufficiently large odd integer is the sum of 21 5thpowers of primes,
http://en.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem#Relevant_results
So, reconciling incommensurables associated with the representation of 4fold multiples of primes by odd numbers leads to a representation odd numbers by 21fold 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.



