Friday, December 23, 2011

The Logic of Mathematics

Mathematics, has never seemed too logical to me and this is due to (1) pedagogy, and (2) the use of the "equals" sign. 
   Now for pedagogy, most math teachers or at least their students, seem to conceive that math is about real objects -marbles, pyramids, train tracks, etc. and that math is about countable things. However most mathematicians and aristotle would state that math is just symbols made up by the mind and arranged logically with only an incidental application to reality. This response obviates some of the obstacles to learning math that the "realist" view builds up; for instance, that negative or imaginary numbers don't exist and yet math deals with them. Additionally it also makes the theory of equations much more sensible since equations are not like see-saws where, by the operation of weight, each side of the saw tends to balance w/the other side but are rather logical relations of equality such that if one side changes, then the other side must, logically, change. A non-realist view also has the advantage of circumventing the biggest problem in realist math: simultaneous determination; for in mathematics the parts of an equation do not cause the other parts of the equation or expression. 1+1 does not cause 2 in the sense that a man throwing a ball causes it to fly, for math is not physics. So mathematical objects have at best, only a mental causal relationship, and the inputs and outputs of expressions are always stationary since every way one manipulates an expression makes it either equal to itself or it completely changes the definition of the expression so that no matter the expression or the changes done to it, that expression is always itself. This law of identity is always and everywhere the same as the law of simultaneous determination.
   The second reason is that when one solves an equation one uses the fact that it is equal to something in order to prove that it is equal to some other thing. However, because of the use of the equal sign, everything is explicitly stated to be equal prior to the proof that everything is. And so, to solve an equation is basically the same thing as to reason circularly. But circular reasoning is illogical so that raises a question, "is it possible to base mathematical proof (equation solving) on a logical foundation?" The answer to this is yes for several reasons but let me just sketch out my preliminary conclusions first.

1,2,3, etc. are defined like this: 1=--1, 2=--2,3=--3 or each number is the complement of its complement.
3-1 is just the difference b/t the complement of 3’s complement and the complement of 1’s complement. And this is just the complement of the set “3&1” in the set of 3 things. And this is 2.

Clearly I intend to use set algebra to deduce normal algebra. But set algebra is basically logic, so normal algebra will be established logically. 

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