Philosophy - Mathematics

Computation is the processing of mathematical-objects. And before, one might handle such things well. He must have understanding of them: the fuller, the better. Take time and read Philosophy of Mathematics. For those with a solid-background and an basic understanding of set theory. It should all be self-explanatory.

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This field of mathematical philosophy begun when early man began counting, aggregating, and arranging objects. And, it is from those underpinnings that the foundation of such a philosophy should be organized and carefully fitted in place. So, it might support the development of the "floors above it" within the field of mathematics: arithmetic (number theory), algebra, topology, calculus, and ecetera. Hence, mathematics true primitive-fundamental is the singular-object, also known as the implement.

Interestingly enough, within the field of engineering computational-devices; the process of declaring, defining, and organizing the components and their interactions is called "implementation". And, it is that implementation that is the solution for a "problem". Likewise, the resolution of a problem in mathematics is ultimately the completion of a number of processing steps. Which organize an initial-set of implements described by the problem specification. This design of this solution-process is an "implementation".

And a well-crafted set of axioms providing the foundations of this philosophy most probably would be best-fashioned with the enumeration of a primitive-set of compositional-rules such as:

• The (null, non-existent, non-present, empty, zero)-implement is denoted by symbol "0".
• The (singular, unitary, individual, atomic, primary, basic)-implement is denoted by the symbol "1".
• The composition of a "0" and "0" by aggregation is still "0". Nothing combined with nothing is simply that.
• The composition of a "0" and "1" by aggregation results in a group of "1". Nothing combined with something is simply that something.
• The composition of a "1" and "1" by aggregation results in a group of {"1","1"}. Something combined with something is simply that, a grouping of a something and another something. This quantity might be denoted with another symbol distinguishing it from the first singular implement.
• The pattern of compositions might be extended indefinitely.

And, such a list of definitions might be used for the enunication of the four fundamental operations: {+,-,*,/}. It is from these that all of the other mathematical-operations on implements might be defined. Also, the operations of mathematical-logic might be derived from the above, based upon the rules of {+,*} for {or,and}. Plus, one might develop a systems of aggregations or collections of implements defining rules and delineating various categories of groupings and object containers.

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Finally, much like any mathematics-proof with a set of given premises, one might add any other premise which he chooses that augments the "original-set" in the resolution of the proof. As long as, those new statements do not violate any of the previously established ones. As such, mathematics is very much an art. Wherein, one might produce new, never before seen "hues" (defintions and operations) by blending the existing colors on the palette of mathematics. And from such, he might paint "novel" mathematical-masterpieces on his canvas.

When all is said and done, it might be seen that more than one system of axioms might be sufficient for providing a sound foundation for mathematics. And, each of these might be "equivalent".