## Tuesday, August 25, 2015

### Unconventional Arithmetic

THE COLOPHON VARIABLE - [
￼]. Meaning ‘finishing touch’. Named after a Greek territory known for making decisive differences in battle.  A more typical usage of this name is as the section of a book used for displaying technical data about publishing. I have created one specific symbol, which plays a similar role for mathematics. However, in mathematics its role is mathematical. It is used for showing missing data or extrapolations, particularly in the expansion of math to logical systems. For example,

[Mod 4] 4 colophon colophon colophon￼￼￼ = {4,4,4,4} or {4,8,12,16} or just [Mod 4 : 16]

However, in different contexts it has a different relationship.

| colophon ￼: +4| will multiply everything else by 4.

|￼colophon : +1/5| will divide everything else by 5.

| +10 : ￼colophon | will apply an operation such that the result is always 10.

Hence, we can say | {x OR y OR z OR ...} : colophon ￼| = God variable ￼ when the variables are free.

| ￼colophon : (1/ sigma) = God ￼| = x : colophon ￼ could mean that x is a trans-finite number > infinity + 1.

On the other hand, if | colophon ￼: (1/S) = God ￼| = x : colophon ￼= akhenaton, then we might propose  ‘God ￼i’ as a solution. This is a case where numbers greater than infinity are still incomplete, in other words, where a standard has not been adopted.

The new symbols are the Akhenaton mentioned earlier, and the God variable, as I define it later in a not-inherently-religious sense.

THE DEMI-GOD VARIABLE:

[ ‘
￼’ ]. Also called the Hell variable. This is what God equals when the conditions of God cannot be met. For example,

| {x OR y OR z  OR …} /: colophon ￼| = Demi-god variable￼. Note the slash before the colon, meaning in the case of the colophon, that x, y, and z cannot be produced at all. This a way of creating true irrational numbers (numbers that cannot be produced). This is also potentially a coherent view of ‘i ’, although such a coherent i might require infinite variables, and is therefore incomplete as a way of guaranteeing its completeness----it is ‘completely incomplete’.

Similarly, in the opposite case:

Colophon ￼: {f(x) OR f(y) OR f(z) OR …} = 1 / Demi-god ￼, because  ￼Demi-god = 1/ ￼God i . and Colophon ￼: {f(x) OR f(y) OR f(z) OR …} =  ￼God i.   Therefore,  Hell God ￼￼i =  1, because God i =  1 / Demi-god ￼.

Another way to interpret the  ￼Demi-god (Hell) variable, is that it is  hard worker. It tends to emerge as an isolation from conglomerates of other variables. For example, ￼Hell =  1 /God￼ i, where God￼, i, and 1/x are all isolations themselves. In this way, it is the ultimate concept of an exceptional variable that lies before absolute irrationality. It is, so to speak, the gateway to irrationality.

GOD VARIABLE - [
￼]. My theories on trans-finites predict that infinity is the opposite of the digit one. One is located in the middle of the new Cartesian Coordinate System, thus permitting all symmetric structures to be uneven in value, and thus, permitting these structures (in my system) to exist as entities on the trans-finite level. Even numbers, on the other hand, are compared with reversing the counting motion, returning the value to zero. But, one may ask, if ‘infinity’ is the opposite of ‘one’ --- then what is the opposite of zero? The answer, it appears, is that there is some entity --- real or imagined --- that might be called God (or, in mathematics, the God variable). This entity is the one entity which is beyond infinity, and the one entity that genuinely responds to zero. Discovering a significance for the God variable might be difficult, but tentatively it could involve impossible solutions or unexpected results. The God variable can thus be used to refer to mathematical exceptions. It’s something like if i were a trans-finite number. If there were multiple values of 1i beyond 1 and -1, then we get a concept of the ‘divine zero’. Similarly, if infinite i had values other than -infinity and +infinity, we would get a sense of the divine infinity. The God variable in this sense is about perfect definitions regardless of all conventions whatsoever. The God variable can be understood in its most extensive basic sense as the intensive value-laden product of the most rigorous proof theory. For example, instead of Godel’s Incompleteness, we begin with a free variable, a variable-only calculus. The variable is then defined as a strong variable (whatever that means), and the strong variable is allowed to construct further variables according to the rule that the constructed system is stronger than all available proofs. Although this may seem impossible in the sense that the construction defines its provability, if the system is understood to be arbitrary, exclusive, and stringent, then the structure is stronger as a system than it is as a calculation. By adopting a stronger system than calculation, the God variable appears to hold. Thus, the first property of the God variable is that it is arbitrary or perfect. The second property of the God variable is that it is stronger as a system than it is as a calculation. Beyond that point, it is possible to conceive of systemic variables which meet the criteria. For example, equations, conjunctions, and exclusions fit into the God-variable system. Also, a large class of trivial or qualified variables meet the criteria. These can then be used as a platform for a divine theory of probabilistic occurrence. A third criteria appears to be that the system is not restricted to mathematics. For example, a God variable might project that if there were a God variable, then other variables could be strong. Or, if variables are absolute, then the God variable is rational, thus leading to the conclusion that irrationals are rational. These kinds of deductions serve a proof theory which is broader than mathematics, but which, through the idea that God is a variable, continually refer to mathematics for their idea. Summation in this theory could be seen as related to the theory of being unable to sum, which then relates to the God variable, which then refers to the full definition of the logic. In this way, mathematics becomes purely logical, and logic appears to reduce to the God variable. The God variable can be seen as the principle of rationality, but also the principle of creativity. Working with the God variable involves much more general, and initially much less restrictive considerations. The difficulty is in making the God variable matter for proof theory. For most purposes, it is an Akhenaton, a third wheel. But when mathematical assumptions are put into question, the point is, there is something beyond infinity. Something very small, but very significant. Something involving rationality and creativity. Even if it doesn’t exist, it can be used as a hypothesis to get something from nothing. If it doesn’t exist, then the somethings from nothing are pure formality. But if it does exist, then its legitimate to expand beyond irrationals.

All of the above from The Dimensional Mathematics Toolkit (not yet published as of August 2015, scheduled to be published in the next four years or so).