What's an Axiom?

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Pursuing the mathematical study of axiomatic system themselves leads one
to some dangerous, convoluted conclusions, conclusions that would have
more than sufficed to get you burned at a metaphorical (or quite
possibly a real) stake if they'd even been proposed during Euclid's
time, or during the 1300 or so years in which the Church dominated
philosophical discourse with its iron hand, its inquisition, and its
very ``special'' axiomatic system described in later
chapters^{6.1}.

To make a long story short (and relatively simple!), propositions
(mathematical statements) can be mapped into *numbers*, usually
called *Gödel numbers*. For example, they can be encoded by the
ASCII^{6.2} string that
represents the statement. Some propositions are used to determine
certain arithmetical mappings, for example the truth value of other
propositions; these presumed true propositions are then the axioms of
the ``theory'' consisting of the set of all enumerated propositions.

We can then write down propositions about themselves - propositions
that refer to their own Gödel number, and a strange thing happens.
*Either* the axiomatic system is *inconsistent* (not all the
axioms can be true, although Gödel's theorem of course cannot tell you
which ones are true and which ones are false) *or* there are
propositions the truth or falsehood of which cannot be determined by
applying the consistent set of axioms - the axiomatic system is thus
*incomplete*. Axiomatic systems with an enumerable set of
propositions that can be made self-referential are thus either
incomplete or inconsistent.

It is tempting indeed on the philosophical side to make too much of
this, just as it is *equally* tempting to the mathematicians and
computer scientists (for whom the theorem makes some very practical
statements about computability) to make too *little* of it. We'll
try to come in ``just right''.

First of all, it *does* impose some fairly stringent limitations on
what we can know from science, but most of those limitations are *irrelevant* to the use of science as a tool for human understanding.
The primary lesson I will emphasize below is that it should force us to
think carefully about the axioms underlying Natural Philosophy whether
or not those axioms are openly or even covertly acknowledged in most
philosophical or scientific discourse. This is a primary failure of
Russell, who *did* know some mathematics and should have known
better in his philosophical discussions of e.g. inductive reasoning than
to do anything but identify the validity of inductive reasoning as an
axiom and hence beyond analysis for anything but consistency or
completeness. Second, it *is* an immensely important theorem in
reference to languages of all sorts and hence to the most common form of
expressive human thought.

In fact, all sentences can be framed as propositions. They can all be
mapped into unique *Gödel* numbers by means of humble ASCII. All
human written or spoken language can be encoded/transcribed into
sentences, and sentences (like this one) can easily refer to themselves.
It therefore seems perfectly reasonable that one can easily get into
Gödelian knots when analyzing the truth or falsehood of any statement
that can be written or spoken, including all philosophical reasoning,
all computer programming, and all statements of physical or natural law,
although in the latter case the language of mathematics is perhaps not
so simply trapped in the ASCII web.

To put it another way, mathematical and logical and semantic systems
that can be written in such a way that they can refer to themselves 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 all questions (including this one) have answers?

Here is where we *can* draw some very useful conclusions from
Gödel. 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 statements in English (or any other language) 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).

Here's fun mental game that has been around for a rather long while one way or another:

- The Law of Contradiction tells us that any given statement cannot
be true
*and*false at the same time. - The Law of Excluded Middle tells us that any given statement must
be true
*or*false. - The following question is false.
- The preceding statement is true.

All attempts to parse out a consistent answer instantly kick one into a
loop. Somehow the two statements contradict each other, yet the
sentences clearly exist and are *independently* sensible. In fact,
the sensation that parsing the language leaves us with is that neither
of these statements is true *or* false and somehow they are both
true *and* false at the same time. Both the Law of Contradiction
and the Law of Excluded Middle crash to the ground, and with it nearly
everything we thought we ``knew'' on the basis of logic applied to
statements that can be written in English.

So much the worse for reason. In addition to things that can be True or
False, there can be things that are Neither. Or Both. Or ``Answer
Cloudy, Try Again Later''. This is in the context of *trivial
syllogisms*: simple sets of two or three statements that are supposed to
be *easy* to take to an unambiguous conclusion. If reason fails us
here, what can we expect when we ask a question like ``Should abortion
be legal'' or ``does God send humans who commit murder-suicide to
nominally defend their faith to heaven or hell''?

Fundamentally, asking if these statements are true or false isn't a
question, it is a ``pseudoquestion''^{6.3}. It looks like a question (or proposition) semantically and
grammatically, but because it has no answer it isn't, really, a
question.

Note that it doesn't have an answer in the sense that we don't *know* the answer or that we might hypothesize an answer and have it turn
out to be correct or incorrect. The answer isn't ``yes/true'', or
``no/false'', or ``maybe'' or ``I don't know'' or an oscillating
sequence of true/false values or even the much beloved ``because'' - it
is the great, rushing *silence* that results in response to a set of
mutually self-referential sentence fragments that *seem* to mean
something individually but that, when logically integrated, have no
meaning at all. They are the ``undefined'' operations of the algebra,
so to speak, the one divided by zero of common discourse.

There are lots of self-referential pseudoquestions or pseudostatements
that have logical values that are ``odd'' and lead one to conclude that
*even in systems of mathematics and logic* our ability to create
*complete* axiomatic systems is very much limited. For example,
meditate on the statement:

This sentence is unprovable.

Suppose it is false. Then it is, in fact, provable. However, provable
things are necessarily true, which is a contradiction. By the good old
law of contradiction, it exists and is not false so it must be true.
From this Gödel was able to conclude that in (sufficiently complex)
axiomatic systems, there exist statements that are *true but
unprovable*, which means that the axiomatic systems cannot be *complete* (if we accept the Laws of Contradiction and the
Excluded Middle, at any rate).

Of course as we just demonstrated with a fundamental anti-syllogism, and
have suggegsted in several contexts before, neither of these are
unquestionable truth in any logical system that *also* contains the
statement:

which exists, seems to make ``sense'' semantically, but is neither true nor false. It is (to borrow an idea from a previous chapter) - no-thing. Not-trueThis sentence is false.

All axioms (with a narrow definition of axiom that precludes them in
fact being provable from other axioms as theorems), as it turns out, are
pseudostatements. This is a tough statement, and those of you who are
alertly following the discourse should be saying to yourselves ``Wait a
minute, pseudostatements aren't things which *could* be true or
false and we just don't know it, they are things for which it are not in
some fundamental sense either true or false. Surely there are
statements that we could make as axioms (and hence are assumed to be
true) that in fact *could* be correct, aren't there?''

Well, let's think about that. Let's leave out the Laws of Thought for the moment - we've already seen that two of them appear to be pseudostatements and the Law of Identity I'm perfectly willing to accept as intrinsic truth since it damn near defines the notions of truth and being themselves - a thing is what it is, whatever that might be, if it isn't no-thing.

How about the axioms of mathematics? Clearly these are all
pseudostatements. It is neither true nor false that parallel lines
never intersect. Rather, it is a statement that we all agree upon as a
prior basis for further reasoning, and if we assume that they *always* intersect all we get is *different* conclusions, not
``true'' conclusions or ``false'' ones. Looked at this way, it seems
closer to being a definition in a mutually (we hope) consistent language
than an actual assumption.

How about the axioms of science? As we will see in great detail below,
the axioms of science are (among other things) the axioms of mathematics
plus such window-dressing as an Axiom of Causality through Natural Law,
an Axiom of Spatiotemporal Conservation of Natural Laws, and more and
even less rarely stated axioms. Couldn't the Universe in fact *be*
causal? Might it not be the case that Natural Laws exist as a truth
independent of whether or not we can ``prove'' it?

Here our knowledge of set theory and number theory comes in extremely
handy. Suppose that the Universe is ``like'' one of three kinds of real
number. It could be like a rational number, all perfectly ordered
internally to the extent that the algorthm that generates its digit
string eventually repeats. It could be like an irrational number such
as or - a digit string that never repeats but that is
nevertheless just as tightly bound to an algorithm that generates its
digits as any rational number. It could be like an irrational number
with *truly random* digits - a digit sequence that *cannot* be
generated by *any* algorithm or iterated map.

The latter class *includes* the two former classes. Any given
rational digit string, however long, has exactly the same chance of
being randomly generated as any particular irrational one. How can one
resolve the difference between these latter two? It is literally
impossible to distinguish a Universe that is completely causal and
ordered in its internal structure (like a rational number is causal and
ordered) from a universe that is intrinsically completely random but
that happens to be completely ordered^{6.4}. In a very real sense, the question of which sort of Universe it
is makes no sense, because no matter what the pattern of organization
``within'' the Universe itself, we cannot extrapolate the pattern into
the mechanism that produced the Universe. If we try, we merely extend
the boundaries of what constitutes ``in the Universe'' according to
whatever answer we decide upon or observe (or don't). An utterly causal
Universe can itself be the result of causality in a larger meta-Universe
or can have *no* cause (whatever that means) in the larger
meta-Universe, and so on ad infinitum.

Ultimately, we are left pretty much with the Law of Identity. The
Universe is what it is, at any or all meta-levels. Beyond that, we can
assume that it is causal. We can assume that it is acausal. (Or better
yet, we can define the patterns that we observe with our senses to be
causal or acausal). Either answer can only be ``proven'' with
additional axioms or possesses *even in existential reality* the
same general arbitrariness as mathematics, where the notion of ``truth
value'' of an axiom does not hold.

How about the axioms of sensate being, of psychology, of perception?
Again, identity is fine. Each instant of individual perception (both
self-perception and the input from the sensory stream we identify as
connecting to the ``real world'') is what it is. That unnamed thing
that perceives exists every instant that perception exists. This is not
an *axiom*, it is the essence of empirical observation, it is *identity*. We *are* our instantaneous perceptions of the sensate
and self.

From this it begins to be clear that *all propositions concerning
the state of existence except this one (which is really the essential
statement of identity, the identity of your being and your perception as
an empirical truth- whatever they ``are'' beyond that according
to your beliefs) can be formulated as pseudostatements*.

Not necessarily self-referential ones - psuedostatements can also
easily appear to reference external ideas like ``God'' or ``reality'',
or can throw into conflict inconsistent ideas such as ``omnipotence''
and ``omnibenevolence''. As we will show below (recapitulating the work
of the masters, but with a bit more attention paid to
mathematical/logical rigor): with *one* exception questions
concerning reality are pseudoquestions in the sense that they have *no self-evident, rigorously provable answer*. They fall into an even
*more general* class of pseudostatements - those that have truth
values that *vary* as one varies the fundamental axioms that
underlie the logical system one uses to assign truth values. They are
questions like ``How many times do parallel straight lines meet'' - the
answers depend on your *assumptions*, on your *axioms*. The
answers to such a question range from none to an infinity of possible
answers or to an *axiomatic* answer, and an axiomatic answer may or
may not (according to Gödel) lead to an non-conflicted, consistent,
complete deductive system when combined with other axioms!

One is tempted to meditate upon an axiomatic system containing the
axiomatic proposition "Statements that refer to themselves directly or
indirectly *except this one* are not a part of this axiomatic
system". ``This statement is false'' is therefore not true or false
because it is not a statement, it is a pseudostatement and explicitly
excluded from the class of permissible propositions. The laws of
Contradiction and the Excluded Middle are then recovered, but only for a
relatively tame subset of the set of all propositions.

It is amusing before moving on to recall a couple of the many times
Gödelian 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 example question
above causing an Evil Robot to melt down trying to resolve 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. One can only imagine Harlie in the
infinitely distant future being asked if there is a God and replying
``There is now...''

Hah. Good Luck Harlie.

All of this digression is really only intended to 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 neither more nor less than
*unprovable assumptions* - indeed in many cases pseudostatements
where we don't really know what it means to assert that the statement is
*true*. Furthermore, *even* with one's axioms in hand,
axiomatic reasoning is far less powerful than was imagined until the
1900's, and there is a very serious risk, nay, a near certainty, that
any nontrivial axiomatic system proposed as a basis for understanding
``everything'' will be neither complete *nor* consistent and that in
any event it cannot be both.

There is nothing more dangerous or powerful in the philosophical process than selecting one's axioms, especially given that they are nearly invariably expressed in sloppy old human language. There is nothing more useless than engaging in philosophical, religious, or social debate with another person whose axioms differ significantly from one's own.

To reiterate, an axiom is at heart something that *cannot be
proven*. It is itself a *pseudostatement* whose truth or falsehood
*cannot even be addressed* except, of course, with any of a variety
of *other* pseudostatement axioms and their associated axiomatically
proven or disproven propositions that will soon have all the
participants in any debate melting down in a puff of smoke if the
resulting system of ``reason'' is inconsistent or, like Harlie, writing
grants for the purpose of perpetuating existence for the rest of
eternity while working out the complete ``answer'' if it is incomplete.
An axiom is a *free choice*, a denumerable selection out of a *non*denumerably infinite space of possibilities, upon the back of which
we *choose* to derive our system of so-called reasoning, dealing
with contradictions and inconsistencies as best we can - or just
ignoring them.

How can I convince you of the *importance* of coming to a full,
conscious realization of the truth of this observation in real human
affairs? For philosophies, expressed as social and religious memes,
have an enormous impact on our daily lives and indeed are the very
forces of history that have led the world to have the shape it has.
Problems within our world very often have their origin in the
fundamental problems - inconsistencies and incompletenesses - in our
underlying axioms, along with our *memetic* tendency to treat our
social and religious axioms as being ``true'' beyond all examination or
consideration.

To understand this, we have to take a journey of two parts. The first
is through a (partly historical) exploration of the development of the
fundamental axioms of The Cosmic All, with David Hume^{6.5} as
our highly skeptical bus driver, Bertrand Russell^{6.6} as our
tour guide, and accompanied by the Three Stooges:
Descartes^{6.7}, Berkeley^{6.8}, and Kant^{6.9}. We might talk a wee bit about Plato
(respectfully) and a few others (not so respectfully at all) but I
wouldn't inflict a reading of even the Republic on anyone else who
wasn't totally into it, in which case you've probably already read it
five times and annotated all the margins and highlighted the significant
passages.

The second is through a very much *current* exploration of genetic
optimization^{6.10}, complex systems^{6.11},
and social geneto-memetics^{6.12}.
This is some more of the new math stuff that hasn't really been
understood by modern philosophers and metaphysicians. Naturally, I
won't omit using a wee bit of modern *physics* as well, being as how
I *am* a physicist, and physics in the twentieth century did such a
*lovely* job of utterly destroying so *much* supposedly rational
nineteenth century philosophical reasoning about our Cosmic All.

Yes, sorry, pretty technical, but I *will* do everything I can to
make it all very simple, just as Russell did in his little book.

Paradoxes

What's an Axiom?

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