The Formal Problem with the Laws of Thought

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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
level^{5.2}. 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, 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, 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
time^{5.3}.

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*
reality^{5.4}. 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
again^{5.5}.

Curved space is *simple* compared to quantum theory. By the end of
the first or second year of physics grad school, most
students^{5.6} 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 Dynamics*^{5.7}.

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
observer^{5.8}. 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
this^{5.9} 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 *un*countable infinity. This little (very simple) observation had vast
consequences in number theory and even in physics and calculus, where it
is related to *measure* theory^{5.10}.

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 Russell^{5.11} 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
Gödel^{5.12}. 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:

*Propositions*are objects that we wish to rationally analyze and assign a value of true or false to. Note that these are*algebraic*or*symbolic*objects. A ``penny'' is not the right kind of object as it cannot be true or false or ``future cloudy, try again later^{5.13}; a statement such as ``All men are mortal'' is a proposition.*Axioms*are*not*self-evident truths in any sort of rational system, they are*unprovable assumptions*whose truth or falsehood should always be mentally prefaced with an implicit ``If we assume that...''. Remembering that ultimately ``assume'' can make an*ass*out of*u*and*me*, as my wife (a physician, which is a very empirical and untrusting profession) is wont to say. They are really just assertions or propositions to which we give a special primal status and exempt from the necessity of independent proof.*Definitions*basically specify the objects upon which the axioms act or the nature of that action. They are purely descriptive and hence also unprovable, but they are also not assumptions. You cannot*prove*that ``penny'' stands for slivers that might be copper, zinc, or whatever, produced by an authorized governmental institution, with one of several possible classes of history and morphology, you can only assign the word to refer to that class of*actual*objects each of which is a unique individual with its own specific*differences*) by means of a sufficiently precise definition. This definition itself is expressed in words that require definition. Ultimately any given dictionary is*circular*- it defines words in terms of other words in the dictionary and cannot be understood unless you*already understand*those words.How then can we group objects into a class and name the class ``penny''? It is one of the miracles of human consciousness, this ability to generalize and construct symbolic algebras and languages, and is clearly built in human functionality as most other animals lack it altogether and even in humans it is remarkably fragile and dependent on developmental stimulation at just the right time.

*Rules of Logic*that we've already discussed above. For thousands of years it was thought that the rules of logic were universal and beyond question - axioms in the sense of being*manifest truth*. It was discovered less than a hundred years old, however, that the Law of the Excluded Middle is not, in fact, a universal ``law'' but rather an assumption. It can be*left out*of certain classes of logical systems and the resulting system still works to support ``reason''. Certain interpretations of quantum theory similarly suggest that the Law of Contradiction is essentially classical in nature and cannot be naively applied to classical statements in a quantum theory.A particle cannot be ``be at position '' and ``not be at position in classical theory - to assert this would be a contradiction. However, in quantum theory there is a third alternative - that its wavefunction has nonzero support at and the particle can

*neither*be said to be*or*not to be ``at position ''. The English words make perfect classical sense but are not valid forms for quantum reasoning, and making naive classically formulated statements about the particle and its position will lead one to all sorts of classical paradoxes.Even the law of identity (which is by far the strongest of the three) gets a bit shaky in a world where a positron/electron pair can be anihillated to produce photons, or created from photons in the inverse process, especially when the electrons themselves are

*always*being described by relativistic wave functions that are microreversible and the electron, the positron, and the photons are quantum mechanically entangled and smeared out over space and time.The moral of the story isn't that logic is somehow invalid, it is that we need to be very cautious about our belief in absolute truth, especially when those beliefs concern the system by which we decide on truths. History is full of cases where the human mind was trapped by its own preconceptions. In this case we are linquistically trapped by the

*classical language*learned at a young age by our*classically evolved brains*where things can be ``seen'' only in three or fewer dimensions and it gives one a headache to try to draw or imagine objects in four or more, where propositions cannot be true and false and must be one or the other. It is interesting to note that even a child's toy like the Eight Ball is smart enough to answer ``maybe'' or ``try again later'' but logicians for*thousands of years*insisted on ``yes'' or ``no'' with no middle ground!- Axiomatic systems can be
*consistent*(where none of the axioms directly or indirectly contradict themselves). They can also be*inconsistent*. Easily, as it turns out. Almost inevitably, really, especially if you are careless and start throwing in too many propositions as axioms. There may be only one way to solve any given mathematics problem correctly but there*always*an infinity of ways to get it wrong, and getting it wrong usually arises from a student using some axiom or theorem incorrectly, de facto introducing a*new and inconsistent*axiom into the problem. - Axiomatic systems can be
*complete*(where all propositions that can be sensibly framed can be determined to be either true or false by developing the axioms with logic) or they can be*incomplete*. There can actually be propositions that are sensibly framed and whose semantic content is understandable in human language whose dualistic truth or falsehood*cannot be determined*within an otherwise sensible and well defined set of axiomatic reasoning.

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.

Gödel

The Formal Problem with the Laws of Thought

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