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Gödel

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 chapters6.1.

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 ASCII6.2 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:

So, if we assume as premises the first two statements (which, we must carefully note, are two of the Laws of Thought and it would be very bad indeed for ``reason'' should they prove incorrect), are either of the last two statements true or false?

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:

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

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 $\sqrt{2}$ or $\pi$ - 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 ordered6.4. 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 nondenumerably 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 Hume6.5 as our highly skeptical bus driver, Bertrand Russell6.6 as our tour guide, and accompanied by the Three Stooges: Descartes6.7, Berkeley6.8, and Kant6.9. 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 optimization6.10, complex systems6.11, and social geneto-memetics6.12. 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.


next up previous contents
Next: Paradoxes Up: Reason and its Limitations Previous: What's an Axiom   Contents
Robert G. Brown 2007-12-17