I think the topic must be studied more carefully. In my opinion, structures such as numbers are fairly well studied (hopefully) as opposed to operations which remain the stumbling block of mathematics.
Their dynamics, interaction makes the structure of mathematics- invisible, so far, with few glimpses of a system existing behind the scenes.
In my opinion, hyperoperations does create new numbers as does any operation. Tetration is the original source of complex numbers, not sqrt(-1) and need to solve x^2+1.
So most likely, going up and in between in operations will reveal linkages to all already known numbers ( quaternions, octonions, sedenions, in all formats, other structures like groups etc.) . Some new hyperstructures may appear as well.
An illustration: if I and complex numbers are resulting from infinite tetration leading to infinite dimensional hypervolumes and hypersurfaces, then I+I will actually produce pentation, while I*I*I*I ..... will be hexation, I^I^I^I - heptation and I[4]n - octation of reals. And so on.
Most likely, infinite octation of reals leads to all kinds of quaternions, quaternion+quaternion = [9], quaternion*quaternion=[10], quaternion^quaternion=[11], quaternion[4]n= real[12]n leads to octonions.
And so on.
The trick would be to start from another end [infinity] and see what structure naturally would fit that purpose. In my opinion, it has to purely imaginary number, without any real element with infinite number of infinite orthogonal components.The speed of this operation will be the fastest possible which of course raises aquestion by itself- what does that mean- it can only mean that the speed of this operation is such that all infinite numbers of structures react to it simultaneously . It is omnipresent, that operation. That would be than totally undifferentiated number, which, when operation [infinity] or [infinity-1] is applied to it, will produce first real component.
In normal world, the higher opertations above tetration will then cycle with a cycle of 4, meaning in most cases 4 operations/interactions is good enough approximation,as all operations with increased speed work on structures with increaesed mathematical inertia.However, there is always pentation, [9], [13], [17],[21], etc. working behind summation, etc.
The existance of higher operations leads to fine structure of operations/interactions, which reveals itself only clearly if these opertations can be separated. The total value of fine structure constant is a result of some kind of summing of these smaller influences of higher operations/structures on the basic operations/interactions.
To close this, there is a need for organizational math, that regulates ( maps?) relations between structures ( numbers, number functions etc) and operations and operational mathematics. The prime candidates for this purpose are functions similar to Riemann Z.
One more thing is to accept that integers are not the simplest , but most complex number structures there are.
Ivars
Their dynamics, interaction makes the structure of mathematics- invisible, so far, with few glimpses of a system existing behind the scenes.
In my opinion, hyperoperations does create new numbers as does any operation. Tetration is the original source of complex numbers, not sqrt(-1) and need to solve x^2+1.
So most likely, going up and in between in operations will reveal linkages to all already known numbers ( quaternions, octonions, sedenions, in all formats, other structures like groups etc.) . Some new hyperstructures may appear as well.
An illustration: if I and complex numbers are resulting from infinite tetration leading to infinite dimensional hypervolumes and hypersurfaces, then I+I will actually produce pentation, while I*I*I*I ..... will be hexation, I^I^I^I - heptation and I[4]n - octation of reals. And so on.
Most likely, infinite octation of reals leads to all kinds of quaternions, quaternion+quaternion = [9], quaternion*quaternion=[10], quaternion^quaternion=[11], quaternion[4]n= real[12]n leads to octonions.
And so on.
The trick would be to start from another end [infinity] and see what structure naturally would fit that purpose. In my opinion, it has to purely imaginary number, without any real element with infinite number of infinite orthogonal components.The speed of this operation will be the fastest possible which of course raises aquestion by itself- what does that mean- it can only mean that the speed of this operation is such that all infinite numbers of structures react to it simultaneously . It is omnipresent, that operation. That would be than totally undifferentiated number, which, when operation [infinity] or [infinity-1] is applied to it, will produce first real component.
In normal world, the higher opertations above tetration will then cycle with a cycle of 4, meaning in most cases 4 operations/interactions is good enough approximation,as all operations with increased speed work on structures with increaesed mathematical inertia.However, there is always pentation, [9], [13], [17],[21], etc. working behind summation, etc.
The existance of higher operations leads to fine structure of operations/interactions, which reveals itself only clearly if these opertations can be separated. The total value of fine structure constant is a result of some kind of summing of these smaller influences of higher operations/structures on the basic operations/interactions.
To close this, there is a need for organizational math, that regulates ( maps?) relations between structures ( numbers, number functions etc) and operations and operational mathematics. The prime candidates for this purpose are functions similar to Riemann Z.
One more thing is to accept that integers are not the simplest , but most complex number structures there are.
Ivars
