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What's an Axiom?

So, just what is an axiom? Even if you know (or think that you know) it doesn't hurt to do an authoritative check. Let's start with a dictionary definition:

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 first of these two definitions is almost universally taught (generally in Euclidean Geometry, which is the only serious whole-brain math course that nearly all citizens in at least the United States are required to take to graduate from high school and which is therefore not infrequently the only math outside of a few courses in symbolic or predicate logic and maybe a course in algebra that a humanities-loving philosophy major is typically exposed to). 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)5.1.

This can best be illustrated by means of a simple example, well known to anyone who studies mathematics beyond the elementary level5.2. Everybody (as noted above) learns the geometry of Euclid, as the archetypical Axiomatic System. One begins with the Axioms of plane geometry and proceeds to derive Theorems (not Laws, which are something else entirely, if one actually bothers to call things by their correct names). Euclid for the most part (and his many overawed successors to a greater part) 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 only finally discovered in the mid to late 1800's (by Gauss, Riemmann, and a few others), geometry on (say) a curved surface such as that of a sphere is not the same as geometry on a plane. On a sphere, unique parallel lines always meet exactly twice. Triangles have more than 180${^\circ}$, with 180$^{\circ}$ 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, and that the difference is fundamentally connected to the differences in the axioms from which one reasons.

Different axioms, different theorems, different results, with all the axiomatic systems considered and their theorems equally empty in terms of ``meaning'', if by meaning you mean ``in some necessary relation to the real world''.

For a long time - that would be thousands of years - after the invention of axiomatic reasoning, this was the way the world worked. Philosophers (and a whole lot of mathematicians) continued to think of axioms as self-evident truths, laws of logic and mathematics, as it were, and a hundred-odd generations of students derived Euclid's theorems about triangle congruence without ever thinking too deeply about them. Even the belated discovery that there could be different axioms that led to different theorems left the sanctity of axiomatic and logical reasoning itself untouched, seducing many a philosopher to continue using the essentially classical reasoning processes that follow, in fact, from using a number of self-evident axioms that were rarely to never openly acknowledged and which were all unprovable assumptions, every one.

In the late 1800's and early 1900's, though, some fundamental cracks began appearing, this time in the theory of logic itself as increasingly brilliant mathematicians and physicists began examining it very critically indeed. This was motivated in part by the development of much that was startlingly new and different in mathematics. Suddenly it was not only not forbidden to challenge the masters such as Euclid, it became the very fashion!

This was almost entirely due to developments connected to the field of physics (one of Philosophy's great success stories and the father of quite a bit of mathematics). Iconoclasts showed that the Universe itself turns out, in plain fact, to be neither simple nor classical nor flat, and in fact to violate all sorts of ``self-evident'' principles to the point where human beings (with a few extremely well-educated and fairly brilliant exceptions, maybe) can no longer really understand it. Let's do a quickie review.

Einstein, Lorentz, and Minkowski discovered and wrapped up in a beautiful piece of new mathematics that space isn't flat after all, that time isn't a sacrosanct independent variable but is rather ``just another dimension'' not only on a par with spatial dimensions but one that mixes with them every time anything moves, and that Euclid's (and Galileo's) axioms where not, as it turned out, even the right axioms to describe the spatiotemporal structure of the Universe. I teach special relativity to both undergrads and graduate students, and it is quite literally a mind-expanding exercise to attempt to visualize and think in terms of four-dimensional, curved, space-time when your entire psychological perception of the Universe is very definitely of three apparently flat dimensions and an independent time5.3.

Consequently, every philosphical argument ever made that relies on an implicit temporal ordering of events or that is implicitly independent of the relative viewpoint of the observer (and there are arguments aplenty in this category, given the implicit ordering in modus ponens, if A then B) at least has to be reexamined and probably is just plain ``wrong'', if one has a criterion for correctness that includes using logic intended to apply to reality that is not egregiously inconsistent with the logic revealed in empirical observations of reality5.4. The broader lesson, though, is that such arguments, to have even provisional validity as the basis for some kind of rationalism, need to have a kind of ``invariance'' with respect to the space of possible fundamental axioms because tomorrow someone might well discover that four-dimensional spacetime is itself just a projective view of a structure that is much larger and more complex - or simpler - with different axioms and definitions that formulate the theory. If we aren't careful, we'll have to do the winnowing process all over again5.5.

Curved space is simple compared to quantum theory. By the end of the first or second year of physics grad school, most students5.6 have made a peace with special relativity theory as it is so mathematically elegant. Quantum theory takes years, decades even, to approximately understand. Feynman once said that ``Nobody understands quantum mechanics'' and Feynman was a card-carrying supergenius. Quantum theory is just a little bit too difficult for the human mind to fully comprehend, even when that mind can actually do computations with it and get correct answers.

Quantum mechanics can be developed axiomatically, and is usually taught at the introductory level by (at some stage) differentiating its axioms from and contrasting them with the axioms of classical mechanics. Perhaps the best example of a self-contained axiomatic development (one that avoids introducing the classical/quantum choice point until the geometry of the states of a generic ``system'' and the algebra of the measurement process are defined, making mathematically precise an issue that philosophers address in words) is Schwinger's Quantum Kinematics and Dynamics5.7.

As we'll discuss in future chapters, quantum theory pretty much destroys the implicitly classical conclusions of rationalist and idealist alike whereever those arguments implicitly rely on ``self-evident'' axioms that are classical in nature. It makes a hash of some of the supposedly inviolable fundamental premises upon which they argue, where a thing can either ``be or not be'' but not both. In quantum mechanics things are nearly always in a state that can only be called both, unless you look at them in which case they resolve into one or the other - it is impossible to speak in the abstract of the electron being in box A or box B, or of having passed through slit A or slit B unless you measure it and entangle its abstract state with your own unknown and unknowable state as an observer5.8. Even measurement doesn't get you out of the woods, as a measurement of property X often creates a state where property Y is no longer classically defined in accord with the naive ``Laws of logic''.

Note well that the point isn't that philosophical arguments should now all be consistent with quantum theory and we should all be logical positivists (more on that later). After all, quantum theory is likely enough not precisely correct and has yet to be properly unified so it can describe all the fields (especially gravity) within a relativistic framework where interactions are due to the curvature of spacetime and not the exchange of quanta of some underlying field. Even if physicists solve that problem (and they might, eventually) there is always, or so it seems, another box to be opened within the latest box we manage to find a key for. It is that philosophical arguments should begin by stating the axioms from which their conclusions are derived and should either be viewed as conditional truth that can be doubted and judged in accordance with those stated axioms or shown to be conclusions that are invariant with respect to classes of motion in ``axiom space''.

Whenever a physicist or mathematician starts talking like this5.9 you know you are in deep trouble. We actually were all in precisely this sort of trouble early in the last century, when a mathematician named Cantor was working out certain classes of infinity in set theory. Cantor was the guy who realized that while (for example) the count of the set of all rational numbers is a pretty big number - a countable infinity, in fact - the count of the set of all irrational numbers is a bigger number, an uncountable infinity. This little (very simple) observation had vast consequences in number theory and even in physics and calculus, where it is related to measure theory5.10.

It also had implications in the fields of computer science, where it could be related to the ``computability'' of various formal patterns and, as it turned out, to formal logic, the study of axiomatic systems! Our friend Bertrand Russell5.11 made an important contribution right about here involving just how a large set can be split up into smaller sets. This isn't a mathematics treatise, so we won't recapitulate these arguments in any detail but rather will get to the important point. The outcome of this line of reasoning is that by mapping ``axioms'' and ``propositions'' (things that can be considered true or false according to the axioms and logical deriviations therefrom) into a space of integers and applying the well-known logic of integer systems to them, the sanctity of axiomatic systems themselves was metaphorically whomped upside the head by Kurt Gödel5.12. What Gödel showed is important enough to warrant a chapter of its own (where we'll avoid the Evil of mathematical detail but demonstrate in fairly simple terms how verbalizable reasoning systems of nearly all sorts are either inconsistent (and mathematicians hate that) or incomplete (ooo, mathematicians hate that too).

Here is a summary of what you should take from this chapter and into the next. They are, I hope, a fair summary of the structure of modern mathematical logic as a system capable of examining itself and embracing modern physics and mathematics:

The last two elements - completeness and consistency - are fairly recent additions to logical and mathematical theory. In fact, there is a conflict of sorts between consistency and completeness, where a consistent system of more than a certain degree of complexity must be incomplete and contain statements that (for example) are true but cannot be proven, statements that are neither true nor false. Note that such a system can always be made to be complete by adding more axioms to specifically assign truth or falsity to these ``ambiguous'' or ``self-contradictory'' propositions but this, of course, generally can be done only at the expense of no longer being consistent.

This leads us in the most natural of ways to Gödel, who was the primary logician responsible for proving that logic is a tragically flawed tool even for the purpose of guiding abstract reasoning, let alone for fulfilling the rationalists' dream of deducing the True Nature of Being from Reason Alone.

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