Nonsensical philosophical musings on indirect proof and paradoxes

Kittens & Gorillas 153 Notice I not only prove I am not guilty I also prove the opposite: I am innocent. When a mathematician uses this trick, it is called an indirect proof and works the same way as the alibi. Assume the opposite is true of some theory you want to prove (I am guilty). If it generates a contradiction or paradox (can't be in two places at once) you can deduce the opposite must be true (innocence). Mathematicians use this all the time. It assumes, of course, mathematics is consistent and that true and false are opposites. Some mathematicians argue this is too strong an assumption. Why should we assume consistency and recognize only two logical states, true and false? These mathematicians believe the only way to prove a theorem is with positive argument rather than using the opposite of a negative argument. They don't allow indirect proofs in their mathematical models. This type of mathematics is unsurprisingly called positivism. It’s a pure theory but, unfortunately, if you try to follow it you lose much of our current mathematical knowledge and understanding. Most modern mathematicians think it a historical curiosity, but it does pop up from time to time. Modern mathematics is founded on the axioms that true and false are the opposite of each other and that inconsistency is forbidden within the system. Mathematical proofs submitted to journals are not permitted to contain inconsistencies or result in paradoxes. Paradoxes - When Logic Fails “T would not be a member of any club that would admit me. Groucho Marx Paradoxes occur when a state- ment makes no sense, or results in an internal contradiction as with Groucho Marx’s famous quote. They are widely used in mathematics to implement indi- rect proofs. To do this, we sup- pose something is true, and if it results in a paradox then the Groucho Marx HOUSE_OVERSIGHT_015843