What is an axiom?
From Webster's Revised Unabridged Dictionary (1913) [web1913]: Axiom, n.-- L. axioma, Gr.; that which is thought worthy, that which is assumed, a basis of demonstration, a principle, fr.; to think worthy, fr.; worthy, weighing as much as; cf.; to lead, drive, also to weigh so much: cf F. axiome. See Agent. 1. (Logic and Math.) A self-evident and necessary truth, or a proposition whose truth is so evident as first sight that no reasoning or demonstration can make it plainer; a proposition which it is necessary to take for granted; as, ``The whole is greater than a part;'' ``A thing can not, at the same time, be and not be.'' 2. An established principle in some art or science, which, though not a necessary truth, is universally received; as, the axioms of political economy.
These definitions are the root of much Evil in the worlds of philosophy, religion, and political discourse. These two definitions are both universally taught (generally in the first form, generally in Euclidean Geometry that is the only serious math course that nearly all citizens in at least the United States are required to take). A relatively few students may move on and hear the term used in the second, ``wishful'' sense (wishful in that by calling an established principle an ``axiom'' one is generally trying to convince the listener that it is indeed a ``self-evident and necessary truth'').
Alas, they are both fundamentally incorrect (although the second is closer than the first). When I say incorrect, I mean that they are completely, formally, and technically incorrect, not just a little bit wrong in detail. Neither of these is what an axiom is, in mathematics (from which technical usage the term's definition is derived).
This can best be illustrated by means of a simple example, well known to anyone who studies mathematics beyond the elementary level. Everybody (as noted above) learns the geometry of Euclid, as the archetypical Axiomatic System. One begins with the Axioms of plane geometry and proceed to derive Theorems. Euclid (and his many overawed successors) did indeed hold the axioms to be self-evident truths, although one should carefully note that the Latin root means ``that which is assumed'' and not ``that which is self-evidently known''.
Well then, what about non-Euclidean geometry?
As was long ago discovered, geometry on (say) the surface of a sphere is not the same as geometry on a plane. Unique parallel lines always meet exactly twice. Triangles have more than 180, with 180 being a strict lower bound for ``small'' triangles that lie approximately in a plane. That isn't to say that there is no geometry on the two-dimensional surfaces of spheres, or hyperboloids, or ellipsiods, or arbitrary amoeba-like-bloboids, only that it is different from geometry on the plane.
Different axioms, different theorems, different results, with all the axiomatic systems equally empty in terms of ``meaning''.
This leads one to dangerous, convoluted reasoning. The axioms of mathematics themselves can become variables in a higher order application of mathematics. In some cases those axioms can be quantified (parametrically mapped into numbers) so that one can ``dial a theory'' by selecting a set of numerical parameters. In others the axioms cannot be sensibly denumerated (numerated in an ordinal sense with any sort of metric of ``closeness'') and are unique, disjoint, random.
Worse, once mapped into numbers, the axioms themselves can become self-referential. One can write axioms and reason axiomatically to derive theorems about the axioms. Alas, as Gödel observed many years ago while working all of this out, the resulting mathematical systems can easily become fundamentally conflicted, with true but unprovable propositions and propositions that ``sound'' like meaningful hypotheses which in fact cannot be proven true or false and somehow appear to be neither.
Why should questions (including this one) have answers? For any of a wide class of questions, especially including questions that might in any way direct or indirect refer to themselves (like this one) they don't. That is, it is perfectly possible to formulate expressions that look like questions, sound like questions, fool the mind into thinking that they are questions to the extent that all sorts of time and energy are expended attempting to answer them, but that are not questions (or more generally, hypotheses, propositions, other entities whose truth or falseness or relationships we might wish to explore).
Everybody is probably familiar with the old chestnut:
(reformulated as a ``question''). If you answer the question yes, it should have been no, which means that it should have been yes (ad infinitum) if one orders the answer sequence in a temporal order. Of course there is no reason to consider an answer and follow a sequential chain, especially in mathematics where logic should transcend sequence. This isn't a question, it is a pseudoquestion. It has no answer because it isn't a question. The answer isn't ``yes'', or ``no'', or ``because'' - it is a great, rushing silence in response to a set of disjoint meaning fragments that, when integrated, have no meaning at all...
Unfortunately, all propositions concerning the state of existence can be formulated as pseudoquestions.
Not necessarily self-referential ones - psuedoquestions can easily appear to reference external ideas like ``God'' or ``reality''. They are pseudoquestions in the sense that they have no deriveable answer. The only answers possible are then none at all or an axiomatic answer, and an axiomatic answer may or may not lead to an non-conflicted, consistent, complete deductive system when combined with other axioms.
It is amusing before moving on to recall a couple of the many times pseudoquestions like this have been used to destroy Evil Computers in books and movies. The Prisoner, for example, asking ``The General'' the one word question ``Why?'' The very question above causing an Evil Robot to melt down in the sequential cycle in the old television version of Lost in Space. Harlie (in Gerrold's When Harlie was One) concluding that all one needs to answer this sort of question is an infinite amount of time and awareness, as it sets out to perpetuate its own, greatly augmented, existence for that purpose.
Hah. Good Luck Harlie.
All of this digression is really only intended show that axioms, far from being ``self-evident truths'' or even the gentler ``established principles'' are, in both mathematics and derived usage in physics, science, philosophy and other disciplines nothing greater than assumptions. There is nothing more dangerous or powerful in the philosophical process than selecting one's axioms. There is nothing more useless than engaging in philsophical, religious, or social debate with another person whose axioms differ from one's own.
An axiom is at heart something that cannot be proven. It is something whose truth or falsehood cannot even be addressed (except, of course, in any of a variety pseudoquestions and additional axiomatically derived answers that will soon have all the participants melting down in a puff of smoke or writing grants for the purpose of perpetuating existence while working out the ``answers''). An axiom is a free choice, a selection out of an infinite space of possibilities, upon the back of which which we will choose to derive our system of so-called reasoning, dealing with contradictions and inconsistencies as best we can - or just ignoring them.
How to convince you of the importance of coming to a full, conscious realization of the truth of this observation in real human affairs? We have to take a journey of two parts. The first is through a historical exploration of fundamental axioms of The Cosmic All, with David Hume as our tour guide (accompanied by his two clowns, Decartes and Berkeley). The second is through a very much current exploration of genetic optimization, self-organized structure, and social geneto-memetics.
Yes, sorry, this may or may not be easy for you (depends, for example, on whether or not you are fully sentient or just pretending) but there is little alternative. If you've been paying attention and haven't already figured out where I'm going with everything to the point where you are horribly bored, you should now be intellectually poised above a pit of existential despair.
I hate to leave you poised there (so I'll write on and possibly give you something a bit more comforting than existentialism at the end of it all). However, I equally well hate for you to be going ``huh'' and scratching your head, when you are supposed to be poised and screaming at glimpse of philosophical Nothingness beneath, so permit me to get out the block and tackle and tie this rope around your feet - there, comfy now? Now we'll just swing you out over this pit, hold on to your loose change. There. Now look up - errr - down.
Note that all the questions that you hold most dear (no matter what they are) have just had their legs kicked out from under them - if all the fundamental questions are really pseudoquestions, is it not the case that we can build little chains of sensible-sounding questions leading from any tiny question to one of the biggies? It is.
Ah, you begin to see the pit? Don't whimper too much, now. Consider:
How can we ask whether we should have a reuben or a grilled cheese sandwich for lunch when we can't answer the pseudoquestion at the base of the whole question chain of why eat lunch in the first place (to stay alive), why stay alive (because god wants me to, because I'm evolved want to), why did I evolve, why are there laws of physics, why is there a God? Psuedoquestions. There is basically no reason to choose between reuben, grilled cheese, or a hot shit on marble sandwich for lunch until we find a pragmatic way of dealing with this issue.
Note that a variety of smart-ass solipsictic answers work, but only if you are already living on the bottom of the pit of existential despair. After all, the solipsists get to choose whether or not to continue imagining that they're reading all of these words that they were clever enough to think up in third person. Surely they can imagine some way to make hot shit tasty.
The rest of us have no Good Reason for Doing Anything without first having a good reason for being, and right about there we run into trouble as reason and being in a single proposition is totally pseudo. Yet we generally do choose a sandwich, and that choice sometimes even works out well for us. We only get in trouble if we think too deeply about it and the waiter starts to fidget and look around longingly at his other tables. And of course there are days that you choose grilled cheese but the cook prepares grilled shit...
This is more than a bit of a shame; so much so that sensitive souls literally go mad over it. What is the suicide's standard apology? I have no reason for living. Surrounded by choice, a mentally ill person often has little to no free will. Confronted by an endless parade of choices big and small we make them, badly and well, for better or worse, yet most humans never work out why they do what they do from the beginning (the most fundamental of questions that underlie those decisions) to the end (the choice of lunch today). At best their choices work if not examined too closely or deeply, lest they fall into the Pit.
We should be able to do better.
This is why it is important for you to clearly recognize the True Nature of the most fundamental factors that you do use as the deepest basis of all of the casual decisions you make throughout the course of the day. Ultimately, they are founded on your axioms, both philosophical and memetic. If your philosophical axioms include a belief in God, and your memetic axioms include the particular interpretation of Leviticus that prohibits pastrami and provolone or bread made with milk in the same bite, well, the reuben is out. If your personal axioms also include the laws of temporal continuity and causality (and hence, physics, biology, and all the rest), you might well conclude that hot shit on marble isn't likely to be either tasty or nutritious, leaving you with grilled cheese. This decision would be even more soundly based (given these same axioms) if you both have memories of enjoying toasted cheese sandwiches past and are silly enough to believe that something as ephemeral as a memory has any bearing whatsoever on the Now.
We leave it as an exercise for the reader to work out how to resolve a set of axioms that includes God, Leviticus, temporal continuity and causality into a system of reasoning and decision making that doesn't have too many internal contradictions.
You may not agree, of course, but according to my axioms, the Truth Shall Make You Free. In particular, simply having the epiphinaic insight that axioms are ultimately both a matter of totally free choice (as in they are neither true nor self-evident in any sense of the word, and cannot even be judged for consistency or esthetics without more axioms to tell you how to do so) and that most of the axioms you do have are very probably geneto-memetic social imprinting and not something you ``chose'' at all (providing that we agree on enough axioms for us to be able to continue a discussion at that point) might just empower you to, perhaps for the first time in your axiomatically surpressed and conditioned existence, to choose your axioms as a matter of absolute, conscious, meta-reasoned choice.
I will, of course, offer up a set that I find particularly lovely and useful, and even moderately consistent (although Gödel teaches us not to take that horribly seriously). They come with a free shave and haircut, metaphorically speaking, as we sloppily adopt as an esthetic standard (not really an axiom) a wee bit of the William of Ockham's single contribution to Western Thought. Once accepted these axioms can form the irrational basis for a reasonably rational view of the Universe, and can even provide at least some poetic meta-answers to some of the unanswerable pseudoquestions, which is the best one can ever hope for.
In the meantime, I'll have the reuben on rye, chips on the side, with a frosty cold beer. Mmmmm. Don't worry, my axioms permit it. As long as I get enough exercise, so does my wife...