Appendix 3
Reductio ad absurdum: an unacceptable method of negatively-proving a proposition in math
In this traditionally-accepted method of negatively proving a proposition (for example, that the square-root of 2 is an irrational number) one begins with a proposition which one seeks to prove is false via a path of accepted mathematical reasoning or 'steps', eventually reaching an 'absurd conclusion' thereby disproving the original proposition.
In a proof that relies on the reductio ad absurdum methodology, the 'internal steps' or statements (within the proof itself) are accepted as logically 'following on from' the one before (i.e. assuming the veracity of each successive step) in a kind of 'if...then' manner. Proceeding thus, if a contradiction or 'absurd conclusion' is reached, then this is considered sufficient grounds to disprove the original proposition. Hence the name for this method, reductio ad absurdum.
As a contemporary 'method of proof', this reductionist technique proves nothing except its own inadequacy and absurdity because it rests upon reasoning that is flawed in a number of ways.
The first problem is perhaps subtle and not a problem at all to some, but it too often reductio ad absurdum 'sets sail' in the hope of not finding any 'new ground' at all. It is thus aptly named, in my view, reductio...
Secondly, it is 'reductionist' in the sense that it largely (or entirely) rests upon a world of black-and-white, of binary systemic thinking: in terms of 1's or zeroes', yes-or-no, 'either/or—and-nothing-more'. There does not seem to be any room for 'what-ifs...' or 'in-betweens', or polycorrect and/or seemingly-contradictory solutions in reductio ad absurdum so-called proofs. For example, what if we put to one side our previously prejudicial 'rules' concerning the square roots of negative numbers, and accept for a moment that you can in fact take the square root of a negative number? (Let i = √-1. This was certainly a conclusion which would have ranked in the 'absurd category' not so many years ago, but now we have a huge branch of mathematics based on such 'complex numbers', and it turns out they have practical applications, and are not merely the whimsical or absurd invention of certain number theorists.)
The third problem is that the steps in such proofs now require too many 'qualifications and footnotes'. These in turn require further 'qualifications and footnotes'. This turns the reductio.. into reductio ad absurdum ad infinitum. If you are at odds with this 'proposition' and have not come across Kurt Gödel's famous Incompleteness Theorem, look up Gödel's Incompleteness Theorem, then you may understand the problems surrounding the reductio ad absurdum orthodoxy.
'Curtainly', there exist gaps (paraphrasing Gödel)
[Paraphrasing the spirit of his theorem is very hard since it is very complex and difficult to put into non-mathematical terms. But with the help of a little whimsy i shall try..]
There exist in the universe numerous asides, sidetracks, detours, backtracks, wormholes, and squirm-holes in logic. In short, all these arise from the profound interconnectedness of the universe. So if you come across a conclusion that appears in some sense 'closed', the curtain is nonetheless probably snagged on some plete, inadequate, or 'incomplete'. Furthermore, logic is full of wholes, and these wholes are also full of holes.
The Universe reveals itself as 'wholy whole', and yet aloof;
A human (being averse to 'holes') attempts to get a proof,
or feels for One wholly by degrees...
But an elf conceals 'the holes' for fun (and, by 'Universal Decrees'.)