On Monday, September 17, 2012 4:56:07 PM UTC-5, zuhair wrote: > This post concludes the subject I've raised here about reaching into a definition of mathematics. I'll only explain here what I wrote in the last post. "Mathematics is all of what can be faithfully formalized by a system extending logic whose axioms are justified a priori" logic here refers to technical logical language with proof systems like propositional logic and first order logic or second order logic with Henkin semantics,etc.. Now to explain this, I'll start by what do we mean by justification a posteriori: A statement is said to be justified a posteriori iff claim of its truth requires presentation of sufficient experiential evidence according to empirical quorum and if sufficient evidence was so presented. For example a statement like Most people have teeth, is a statement that demands experiential evidence and such evidence was supplied so it is justified a posteriori, the law of gravity is justified a posterior, etc.., Now a statement is said to be justified a priori iff claim of its truth do not require experiential evidence, and if sufficient non experiential evidence for its truth was presented. The a priori justified statements are of two types those that are Analytic and those that are Synthetic. Now the a priori analytic are statements that can be known to be true from mere analysis of their wording and its meaning, like for example the statement: All bald men are bald. Also the statement if a bald man is not bald then a woman is a man. Also the statement if x is taller than y and y taller than z then x is taller than z, etc.. Any statement or group of statements the truth of which can be decided upon from the mere analysis of their components are said to be Analytic and if they are so decidedly true then they are justified a priori. Now there is another kind of a priori statements that are not analytic, or at least we don't have any analytic proof of their truth, these are called "a priori synthetic" however they are thought to be true on intuitive basis or on some form of reasoning that shows informally the impossibility of their negation, what is called necessary basis, or at least impossibility to dispense with as far as understanding is demanded. I.Kant thought that such kinds of statements (i.e. a priori synthetic) are the ones that metaphysics and mathematics sprang from, and in a sense he is correct! On the other hand we can have a posterior synthetic statements. Now logic like in propositional logic and first order logic is analytic, so its validities are known a priori analytic. However the systems that are thought to extend it are either weak systems in which we have an analytic proof that they extend logic, or there is no analytic proof of that extension in which case we'll usually seek a synthetic argument to prove them extending logic, and most of the mathematical systems belong to the later kind, so they have axioms for which there is no analytic proof that they extend logic but yet the choice of those axioms is derived by "intuitive" reasoning that justifies their truth, so axioms are selected on a priori synthetic basis, and their selection is the arena of intuitive reasoning into extending logic. So mathematics is about extending logic in a priori manner. Now for the sake of discussion lets discuss what other fields of knowledge could be and their relation to this definition. What would be matters that are faithfully formalized by a theory extending logic whose axioms are justified a posteriori? Those will be empirical sciences. What would be matters that are faithfully formalized by a theory not extending logic whose axioms are justified a priori? This would be high reason theories like Transcendental logic and Aesthetic of I.Kant and the alike stuff, those lie prior to logic so they are not formalized by technical extensions of logic, extending logic means written in that logical language and being consistent. The high theories of reason are necessarily prior to the technical logical languages. Also mathematics that are not written in logical languages is also faithfully formalized by a theory not extending .... What would be matters that are not faithfully formalized by a theory extending logic whose axioms are justified a priori? This would be also empirical theories, and high reason. Now I said that mathematics is a priori extension of logic, a logic that is highly technical, but for what purpose? Mathematics is certainly an indispensable tool in understanding complexity in a rigorous manner. Logic alone cannot do that, it is only about securing non contradictory reasoning which can be very trivial, if not for mathematical machinery which enabled it to deal with complex situations and ramifications, so decoding complex situations is the arena of mathematics, structuralists may call it investigation of abstract "patters" or actually structures, which seems near to the truth of what mathematics is about, so accordingly mathematics succeeded in decoding complex patters and is indispensable for doing so. I personally see mathematics is about decoding complexity and it is necessary in doing so, and actually mathematics can be defined after this utility. However the definition above speaks about mathematics as how it has been successfully formalized in modern era, since this will enable it indeed to deal with complex structures. I think the definition presented here for mathematics is the correct one philosophically speaking. Zuhair
Mathematics is a little bird tweeting in a tree, which smells bad.