User:Thepigdog/Origins of logic

The deterministic model

The deterministic model is that the world is a state machine. At every instance in time, the current state of the universe is used to look up a huge table of states. The look up returns a single state which is the next instance in time.

The deterministic model is no longer accepted as correct according to the laws of physics. It is presented here as a starting point for logic.

Reasoning forward

From facts and theories, reasoning may be applied forward to give predictions and answers. Mathematics starts with the deterministic model. A current state determines a single future. This is analogous to an expression, with no free variables, always representing a single value. A calculation always represents a single value.

Reasoning Backward

From a present state, or result, reasoning backwards finds states that lead to this future. Multiple pasts, may lead to the same future. This is analogous to equation solving in mathematics, where there may be multiple solutions to an equation.

Basis of Mathematics

The starting point for mathematics is reasoning about objects. The axioms of ZFC are the axioms of the object model, abstracted into the language of mathematics.

Cardinality

Because of identity objects may be counted.

Classification

Because of properties, and identity, objects may be classified into sets.

Logic as decomposition

Instead of looking at the whole universes state, mathematics looks at isolated aspects of the universe state and makes implications. Implications are made based on the properties of objects. So instead of one giant state table the behavior of the universe is described by many rules, each describing some isolated aspect of the behavior of the universe.

An obvious means of data compression is the representation of common piece of data as a symbol. These symbols are the inspiration for variables in mathematics. Variables are used to construct general implication rules. General rules describe many individual rules in a single general form, so allow the representation of the rule to be compressed.

Boolean logic reduces implication to a simple calculation involving the Boolean values true and false. This insight removes the need to consider the world as a state machine. Instead mathematics regards the whole universe as a single state, which is gradually revealed through the passing of time.

Equation Solving

If the universe was regarded as a deterministic state machine, then equation solving is analogous to determining what previous state resulted in this current state. More generally, equation solving is determining what value, given as a parameter to a function, gives a particular value.

The solution to this problem may have zero, one, or many values. A variable may implicitly represent these multiple values. However in mathematics, an expression, without free variables, represents a single value.

Infinity

In mathematics we use the object model to describe a continuum as an infinity of points. Uncountable infinity is the consequence of forcing the object model to model a continuum.

Infinity can never be represented as an instruction in the thought process. No program may repeat a particular calculation an infinite number of times, if it wishes to finish. Infinity is then a symbol.

In practical terms, delta means as small as it needs to be, and infinity means as large as it needs to be. The whole of mathematics may be reduced to a single limit. A single delta. In this model a real number has a fixed number of digits. Its size is bounded by a single maximum value.

This is not the way mathematics has proceeded. But, for implementing intelligence, there can never be any infinities of calculation. Only symbols that represent infinity.

Functions may have inverses. An inverse function is only possible if the mapping is one to one (by analogy, only one past leads to this future state). Inverse functions may be created by adding conditions to constrain the inverse to a single vale.

Using inverse functions all equations, with a single instance of the variable being solved for, may be solved.