This book is dedicated to the giants of mathematical and scientific philosophy upon whose backs it stands: Plato, Hume, Descartes, Gödel, Bayes, Shannon, Cantor, Cox, Jaynes, and many more, too many to count, actually. I do wish to explicitly acknowledge Cox's The Algebra of Probable Inference, Jaynes' Probability Theory: The Logic of Science, and MacKay's Information Theory, Inference, and Learning Algorithms, which collectively establish what is very likely ``the'' rigorous basis for knowledge expressed as a contingent degree of belief and many of its connections to worlds both concrete and abstract.
It is also dedicated to my philosophy professor and guru at Duke, George Roberts (a disciple of Bertrand Russell), who had an enormous impact on me as I pursued an ``invisible'' philosophy major at Duke to accompany my physics major (invisible because at the time Duke had no way of acknowledging the completion of a Bachelor of Science in one discipline and a Bachelor of Arts in another).
Finally, it is dedicated to my good friends and colleagues in the Duke Physics Department, especially Richard Palmer (for teaching me about Jaynes, Bayes, maximum entropy, and complex systems in general way back in Statistical Mechanics in grad school) and Mikael Ciftan, who has been as a second father to me for nearly thirty years now.
No book is written in a vacuum. I have been extraordinarily fortunate to have had the support and encouragement and love of many, many people over a lifetime. My family, my friends, my colleagues (who are also my friends) on the beowulf list, and my many, many students: This book is for you all.
Although this book inevitably contains a certain amount of mathematics and science - often expressed as ``natural philosophy'' or ``mathematical philosophy'' - it is not intended to be a mathematical or scientific treatise. Indeed, its basic subject is not physics but metaphysics, our basis for knowledge itself rather than any particular thing that we ``know'' (or rather, that we believe very strongly to be true) about the world. It is written to be as accessible as possible to as general an audience as possible. So don't be intimidated - you can read this, and understand it, even if you aren't terribly good at ``math''.
This is a work on meta-axiomatic metaphysics. By this, I mean specifically that it is a work on the axioms one might use to choose axioms, specifically the personal axioms required to make sense of the marvelous Universe we find ourselves living in.
It might surprise you to know that you can choose your personal axioms for any of several reasons. For example, you could be asking yourself ``What are my personal axioms and why should I care about them?''
An axiom is a belief. In more precise terms, it is an assumption, usually an assumption made as part of the foundation of a set of conclusions arrived at by deductive logic, or as one of the premises of an argument leading to a state of conditional knowledge. So this is a work on how to choose what you believe, and hence what you ``know''.
Of course, to make a choice one has to already have some basis for that choice, and that basis is itself a set of one or more axioms! That leads one to a bit of a bootstrapping problem - how can one sensibly decide what to believe without already having some beliefs? That's the ``meta'' bit - this is a work on the axioms required to, among other things, choose axioms. It presents a way of consistently bootstrapping a set of beliefs about the Universe.
As you will see, it will also show that this bootstrapping, while arguably the best possible self-consistent solution to the fundamental problem of metaphysical philosophy in a mathematically precise sense, is not unique nor can it (or any other candidate set) be proven to be true in any way that should be taken seriously. Indeed, one fundamental conclusion of this work is that metaphysical philosophy is (at its heart) bullshit in the precise sense that it cannot ever achieve one of its design goals - to allow you as a real human to acheive a state of absolute certainty about anything at all except the undeniable reality of your own instantaneous perceptions (whatever they might be).
The wise man built his house upon the rock,
The wise man built his house upon the rock,
The wise man built his house upon the rock,
And the rains came tumbling down!
The foolish man built his house upon the sand,
The foolish man built his house upon the sand,
The foolish man built his house upon the sand,
And the rains came tumbling down!
This is obviously a book about axioms. If you're a mathematician or logician, you probably have a very good idea what an axiom really is. Nearly everybody else (including many scientists or engineers, alas) has an idea, but it probably isn't precisely correct.
This is doubtless because the first and only time many people encounter the term in anything like its correct form is in high school geometry, and even there many a high school geometry teacher fails to make the true definition of the term clear. Afterwards if anybody uses the term at all (outside of logic, math, computation, or science), they are probably trying to sell you something in a pseudo-erudite way1.1.
This simply won't do. Many a ``deep'' philosophical disagreement arises just because the two sides are using the same term in different ways and don't realize it. In that case the real disagreement or point of conflict is the meaning of the term and not ``reason'' (in the sense of a logically analyzable argument) at all.
As it happens, the term ``axiom'' has at least three completely distinct meanings. One is its (correct, literal, historical, Euclidean) meaning in logic and mathematics, with a few minor variances in meaning depending on strict context or adjectival modifiers, and the other two are colloquial meanings used in common discourse and sometimes (incorrectly) used in mathematics and logic as well.
Unfortunately, these latter two are nearly opposites of the first in a critical way. So even if you know (or think that you know) what an axiom is, let's review some common dictionary definitions and precisely indicate which sense of the term we are going to use throughout this work. From Webster:
These two commonly accepted definitions for the term axiom are the root of much evil in philosophical and mathematical discourse because they are both basically incorrect. Wikipedia has the correct definition1.3 :
...an axiom is any starting assumption from which other statements are logically derived. It can be a sentence, a proposition, a statement or a rule that enables the construction of a formal system. Unlike theorems, axioms cannot be derived by principles of deduction, nor are they demonstrable by formal proofs - simply because they are starting assumptions - there is nothing else they logically follow from (otherwise they would be called theorems). In many contexts, ``axiom,'' ``postulate,'' and ``assumption'' are used interchangeably.
As seen from definition, an axiom is not necessarily a self-evident truth, but rather a formal logical expression used in a deduction to yield further results....
To be absolutely clear, it is only this latter mathematical or logical meaning of the word ``axiom'' that is used throughout this work. To us an axiom will always be neither more nor less than an unprovable assumption upon which a process of reasoning is based. We will indeed spend a fair bit of time hammering home the point that there are very few ``self-evident truths'' at our disposal as human souls experiencing a complex sensory stream1.4, and that therefore nearly all so-called knowledge is based on axioms in the sense of assumptions that permit us to transform our instantaneous sensory perceptions into a conditional knowledge of a presumed objective world.
This observation leads, of course, to an acute rational-existential crisis. If almost all of our knowledge is based on unprovable assumptions, if rationality itself is fundamentally irrational in the sense that it cannot be proven, only guessed or assumed, then are we not all in the position of the ``wise'' man who built his house upon the solid rock of the floodplain of absolute truth only to learn that solid rock doesn't percolate, while the ``fool'' on the uncertain, sandy hill stays nicely dry when the inevitable floods come1.5?
Wisdom, foolishness, and knowledge itself all depend on many assumptions, and reality is far too complex and interesting for us to mindlessly rely on the foundation for knowledge we have been indifferently taught by a largely ignorant and uninformed society. If history teaches us anything, it is that ``truths'' taught with the force of law (often backed by deadly threat) by our elders, by religious and political leaders, by our peers, or even inferred from our own necessarily limited experience have often turned out to be diametrically incorrect, usually because that knowledge was based upon the wrong assumptions.
What then, are the right assumptions? What are the axioms that we should use as the foundation of our edifice of knowledge? In fact, how can we even choose one set of axioms as being ``better'' than another without axioms to enable us to ordinally rank axiom sets in terms of ``goodness'', and what should they be? The rest of this book is devoted not precisely to answering these questions (as there is obviously no unique and correct answer) but rather to providing you with all that you need to choose a set of personal axioms that works for you as a deliberate act of free will.
In order for you to become free to choose, you will have to do a wee mental exercise. If you refuse for any reason to do it, reading the book is largely going to be a waste of time1.6. You don't have to do it all at once, but as you read you should be working on it.
What you have to is to perform an autobeliefectomy - to psychologically ``operate'' on yourself to stop believing pretty much everything you have been taught or think that you know. The point of this exercise is that you have (up to now) been programmed with many of your most fundamental beliefs. A great deal of this programming occurred when you were a small child and had no ability to choose to believe or disbelieve what you were being taught. Some of it is even biological and built right into you by genetics and evolution. You have to become self-aware of this programming and come to doubt it and all knowledge derived from it.
Don't worry, the knowledge itself won't go away, and later you can come back to it and deliberately decide what to keep and what to reject.
So, if you believe in God, stop and permit yourself to frankly doubt God's existence. If you believe that there is no God, stop and permit yourself to openly acknowledge that God might exist. If you are a True Believer in the theory of gravitation, evolution, creationism, or some particular political platform permit yourself the sheer luxury of imagining that your beliefs might be incorrect.
Don't stop there - work on disbelief until you entertain at least a tiny bit of doubt about the perfect truth of what you see with your own eyes, what you hear, what you taste, what you touch. Are your senses perfect? Is the world precisely as it seems to be? Are your memories of the past always correct representations of what you actually saw, heard, said1.7? Or could you be, have you ever been, mistaken?
One point of this exercise is that it will gradually help you to understand that most of your beliefs fall into one of the following categories:
Logicians tend to be concerned with the first of these as it embraces the framework of reason itself, for example the so-called Laws of Thought. Mathematicians tend to be more interested in the second pair - fundamental axioms and theorems derived therefrom. Normal human beings, however, live almost all of their lives basing most of their decisions from moment to moment on reasoning driven by assumptions of the latter sort - things we believe to be true that are not ``first principles'' themselves nor provable from first principles (fundamental axioms).
These axioms are in many cases the things we call our opinions, as in it is my opinion that chocolate ice cream is better than vanilla or it is my opinion that it is morally acceptable to eat meat. Can I prove either of these assertions from a commonly held set of fundamental axioms? Of course not. Oh, I can ``justify'' them - pick out an few assertions that do lead to these statements as conclusions, but you might well not agree that those assertions are ``true''.
If I were to be totally honest, I might not really think they were either. In fact, I could almost certainly well pick out different (but still reasonable-sounding) assertions and argue that the opposite conclusions are true, that vanilla is better than chocolate when it comes to flavors of ice cream. Socrates (apparently) used to have great fun doing just that - first proving something to be true based on accepted principles and then turning around and proving it to be false on equally accepted principles (proving, if anything, the probable but highly occult inconsistency in those principles).
This would be fine if those normal human beings reasoned ``gently'' with the mish-mosh of beliefs, instincts, biological imperatives, superstitions, folklore, and conditioning that make up this overcomplete, self-contradictory, self-referential axiomatic ocean of opinions and ad hoc presumed truths. Unfortunately, that is not the case. It is always: Abortion is wrong. Sunni Islam is right. Capitalism is evil. The sun and stars revolve around the earth, which is manifestly flat. We have the right to invade your land, kill you, and take its wealth for ourselves. We have the right to life, liberty and the pursuit of happiness. The flag is a sacred object. Animals have no souls. The word ``perhaps'', the phrases ``I think that'' or ``it might be the case that'' never, ever appear.
All too often in human affairs, the more uncertain our collective and individual axioms are, the more they seem to lead to contradictory conclusions, the more passionately one or the other of those conclusions is embraced as absolute truth or the work of the devil. All too often it becomes the devil's work indeed as the conflict born of a difference of opinion, unresolvable even in principle by means of reason, leads to theft, to rule by brute force, to murder, to war. A difference of opinion concerning a piece of ancient history1.9is currently responsible for countless deaths in the middle east with more coming every day, with an absolute intolerance of those whose unprovable opinion differs from one's own.
This is why it has never been more essential for human beings who are not logicians or mathematicians to understand the arbitrariness of the axiomatic basis of reason itself and to fully grasp the uncertainties in all derived knowledge. Reason is a powerful tool, but the answers it gives in human affairs are rarely drawn in black and white but rather in shades of grey. Given this uncertainty, the social or personal egotism of righteousness seems to be out of order - we learn to respect the opinions of others when they differ from our own and demand the same respect from them for our own opinions, even while trying to find a common axiomatic ground we can agree on and base a society on.
Ultimately, as we will examine in detail below, reason is based on faith - faith that some fundamental core of axioms concerning our perceptions are true and that the laws of reason correctly applied to those axioms will lead us to a system of knowledge of ourselves and an inferred Universe in which we seem to live. The scientist with a belief in the law of causality and the validity of the experimental inductive process and the priest with a belief in God as the self-consistent cause of both causality and induction themselves are in one very important sense equally irrational in their beliefs. In both cases their axioms cannot be proven, only accepted on faith.
Even to judge between these two ``generic'' sets of fundamental axioms on grounds less strong than boolean certainty, or to conclude that one is ``better'' than the other on some scale of ``goodness'' or ``desirability'' requires meta-axioms to guide the judgement, and they in turn are equally arbitrary and unprovable. The rock of human reason, examined closely at its very foundation, turns out to be a fog of quicksand so tenuous that it hardly can be expected to support our own weight, let alone that of the Universe, come the rains of every human's personal experience of life: pain, suffering, and inevitable death with nothing but an imperfect knowledge of what it's all about.
Is it any wonder that, seeing that pain, suffering and death, intuitively understanding the apparent arbitrariness of human beliefs and having no good way of choosing amongst the vast, confusing, and contradictory tangle of possibilities (all touted as self-evident truth by their proponents) many people turn away from reason altogether and choose to embrace either a form of intuitive mysticism that rejects science and established religion alike or slavishly accept the literal truth of some piece of scripture regardless of how much it contradicts common sense and everyday experience? In both cases it is vastly easier not to have to figure things out, much easier not to live in a state of perpetual doubt.
To make sense of that fog and transform it once again into something capable of supporting knowledge and understanding, let us begin by formulating one of the fundmental ``theorems'' of meta-axiomatic reasoning. This theorem will look odd, as one might expect an object associated with meta-reasoning to look. After all, before we can reason we have to reason about reasoning, which involves a bit of a bootstrapping problem because without reason how can we reason about reasoning, and yet without doing so how can be be sure that any conclusions drawn from reason are reasonable?
Ahem. Don't worry. Reasonable or not, it all ends up making sense.
Aristotle maintained that women have fewer teeth than men; although he was twice married, it never occurred to him to verify this statement by examining his wives' mouths. -- Bertrand Russell
This is a book about knowledge. We will look very carefully at what it means to know something, what can be known and what must at best be assumed or guessed or hoped for. ``Knowledge'' all by itself seems a bit lonely and pointless, or perhaps too ``zen'' to be of any use, sitting around just knowing, knowing, knowing. We'd like to put that knowledge to use so (among other things) we don't die of starvation or walk off of a cliff while we're too busy just knowing to think about feeding ourselves or reasoning out that if we walk off of cliffs we'll go splat at the bottom.
For better or worse, we lack what might be called ``direct knowledge'' of the world. As we'll see, pretty much everything we think we know is either a sensory impression or is the result of a process of reasoning, one so automatic that we may not even be aware that it is occuring but that is critical to the process of developing knowledge and using it for whatever purpose seems good to us, such as staying fed and un-splatted by the many hazards of thoughtless existence.
This is therefore also a book about reason. To come to understand the interplay between knowledge and reason and see how they make us what we are, we have to bootstrap the process - begin with one or the other and then iterate reason and knowledge to a full comprehension of both. It seems wise to start with reason, since reason is active and interpretive where knowledge alone is somehow ``passive'' and unlikely to take us anywhere. If you like2.1, knowledge is a map of sorts - possible a map with errors, maybe even a map of an imaginary country, but a map nonetheless. Reason is our navigator, our fearless explorer, that uses the map with all its imperfections to discover the world, perhaps filling in and correcting the map as it does so.
Let us begin, then, by taking a look at symbolic deductive reason, as it is in some very deep sense the foundation of ``rationality'' itself. If humans are thinking beings, the rules that govern reason and thought seem like they must be the tools that will lead us from a state of ignorance to one of knowledge of our selves and the world we appear to be living in. They are the means by which we move, in small steps from here to there as we fill in our personal maps, correcting errors, coping with inconsistencies, exploring terra incognita, imagining what might lie beyond the horizon of our immediate perceptions.
This book is not going to be anything like a textbook in logic or a rigorously developed mathematical treatise. I'm assuming that you, dear reader, are intellectually curious but that you may have had little exposure to formal logic or math beyond algebra and some geometry. It will suffice for our mutual purpose for me to present a few key ideas that illustrate in the most general way the essence of the process of reason (and some problems associated with that process) without getting bogged down in its endless algebraic machinations. In a few places I'll put a bit more detail than in others, places you can use to initiate a ``wiki-romp'' through wikipedia2.2 to learn more.
A few of my readers may, of course, be mathematicians or logicians, far more expert than I in those algebraic machinations. They will note that I've omitted this, glossed over that. I beg their indulgence as I make sweeping statements such as the following.
All symbolic logical arguments that are not various trivial
manipulations of tautology ultimately boil down to something like the
following, a rule called modus ponens2.3 .
Suppose 
and 
are assertions. It doesn't really matter what they
are -
might stand for ``Pigs have wings.''
might stand for
``Pigs can fly.'' We can then formulate a new assertion such as ``If
pigs have wings, then pigs can fly.'' or, symbolically,
.
We can then formulate a logical argument according to rules that are literally thousands of years old at this point:
(If pigs have wings, then they can
fly.)
(Pigs have wings.)
(Pigs, therefore, can fly.)
Note that logic has nothing whatsoever to say about whether or not pigs do have wings, or whether or not a winged pig can in fact fly. As far as this argument goes, they are axioms - things we are assuming to be true to develop the argument. For the moment do not worry about how we might decide if they are really true or not, because that seems like it is knowledge and we're still working on the processes of reasoning that seem crucial to the development of logic.
The way that demonstrative logic works is that if
is true and if
is true then one cannot logically
doubt that the conclusion
is true. It is a theorem, or
contingent truth, a conclusion derived from the axioms.
I am not suggesting, of course, that logic doesn't have more symbolic
operations, other ways of deriving theorems, only that non-trivial
logical arguments always have two components2.4, one that
we can think of as data, and the other that we can think of as
rules for transformation, and that both are axioms of any
given logical demonstration that leads to a conclusion not already
included in the data.
and
don't have to be single asserted
truths, either one could be a whole set of asserted truths (including
definitions).
doesn't have to be a single rule for
taking one asserted truth to another, it can be any of a whole set of
such rules. The point is that modus ponens captures the essential process of deduction as a way of beginning with a set of
things we ``know'' (or rather, assume to be true) to a larger set. We
begin with
, end with the union of
and
where in the
``interesting'' cases
is not a subset of
.
Even the addition of classes of things and more complex predicate assertions, for example the famous Aristotelian syllogism:
We can show that modus ponens is always the general form of a deductive argument (one that does not merely restate the data) in the following cute self-referential formulation2.5:
where we don't even
care what
is or if it is ``true'' or ``false'' or can or cannot be
doubted. This is precisely what I meant above when I asserted that
``knowledge'' is empty, a list of presumed tautologies, data that just
``is'' - things get interesting only when one adds reason.
For a long time - indeed, thousands of years - the premises for most philosophical arguments weren't really presumed to be true - they were thought to be obviously true, so true that not further argument was necessary to demonstrate their truth, and many, many conclusions were ever so rigorously derived from them. Only in the last four hundred years has attention been properly paid to the uncertainty and indeed potential variability of these premises, and only in the last century or so have cracks appeared in the self-consistent logical foundation of pure reason itself.
Here is my own little ``logical argument'' as it will be developed througout a goodly part of this book. You will note that it has a striking similarity to the argument structures illustrated above (especially the last one) - it makes certain observations or assumptions and uses them to draw a conclusion. In particular, it is like the last one - it is a logical argument about logical arguments, a syllogism about syllogisms, so it implicitly refers to itself.
This argument is not really unique, although this particular amusing, abstract, and self-referential formulation may be. It simply highlights a serious problem with logic. How can we ever tell if the premises of a logical argument are ``true''? There seem to be two general ways, one of which leads to mathematics, systems of consistently manipulating symbols, the other of which leads to knowledge, the establishment of a semantic relationship between symbols and - ``something else''.
Mathematics is developed by never knowing, or caring, if the premises,
the axioms of any particular mathematical theory, are really ``true''.
They are simply defined to be true, end of story. Afterwards, the
laws of thought and process of deductive reason are all about
consistency and inconsistency, completeness or incompleteness, the mechanical analysis of a system of relationships that permit the
symbolic formulation of assertions according to rules and the consistent
association of contingent true, false, or undecideable values to each
assertion in the entire Universe of assertions including the original
axioms themselves (which might be self-contradictory, as would be the
case if we start with both of the axioms:
is true; and,
is false
(for any
), or self-referential, as in ``This statement is false.'').
Pure mathematics is beautiful and devoid of meaning because there is no logically necessary connection between the systems of symbols being consistently manipulated and any thing at all. It is a peculiar form of knowledge, because it isn't knowledge of anything at all, it is a knowledge of contingent relationships between symbols, the essence of ``abstract'' knowledge. This intangible quality of mathematics is so elusive that it can easily trick us into granting it ``mystical'' properties, and indeed at various times in history mathematics has been something of a religion, one founded on the religious belief that one or another set of axioms, say those concerning numbers2.6 or geometry2.7 , were fundamental truths and related in some way to the metaphysical basis for all things, beyond all doubt or variation.
They have further seduced philosophers into writing immense bodies of learned discourse (the bane of all students) attempting to ``prove'' that reality is nothing but the symbol, or the symbol the real, that matter is really mind or the other way around. We'll have none of that here, don't worry. This work is not about the arcane.
Well then. What about that something else? How do we reason about our experience and memory, how can we use reason to develop a knowledge of our selves and something that appears to be an objective external reality?
Some three hundred and fifty years ago Hume2.8 observed that:
That conclusion should have spelled the end of an era of philosophy - Hume had showed that some of the fundamental goals of reason were in fact unreasonable because logic and rational processes could only be extended to be about anything at all on the back of fundamentally irrational2.10 assumptions that have to be made outside the system of logic and reason used to arrive at the conclusions. In other words, to put it bluntly, no matter how pretty and logically rigorous an argument applying to the real world may appear, its conclusions can always be doubted. Indeed, if one changes one of the arbitrary assumptions upon which it is founded, one will equally logically arrive at different conclusions.
Hume thus teaches us that all human knowledge about anything but logical systems themselves (for example, set theory or mathematics) is thus revealed to be conditional on certain assumptions that cannot themselves be logically derived or proven. Unfortunately we already have showed that logical systems themselves are no better off. They, too, are conditional on data, and rules that cannot be logically derived as truth, but instead are defined to be true. We can extend Hume's argument by adding the observation that even if we could know truths about the real world, there is an immense ``space'' of possible axiom sets that might define the relationships and rules for deduction that would permit us to extend those truths and we cannot know that any given set of those axioms is ``true'' or ``false'' save by assumption or definition.
Goodness! Looks like we have some work to do before we can end up with anything like ``knowledge''. For one, if you go over the entire argument as presented thus far, you will see that we have one possible ``out''. In one crucial part of our definition of the logic of thought we used the word ``doubt''. Doubt infers a state that is neither true nor false - it is in between. We will find marvelous uses for doubt. For another, we already can see that Hume's argument can be reduced to the provable conclusion:
(Almost) anything that we think we know can be doubted. (Almost) everything we think that we know is contingent. (Almost) nothing we think that we know can be proven beyond all doubt. Therefore Philosophy itself (as a means of arriving at certain knowledge) is bullshit!In other words, philosophy has been able to prove that as long as philosophy's goal is to provide certain knowledge it, and we, are just plain shit-outta-luck (SOL). Ain't happenin'. Forget it.
Did Hume's startling observation (as originally presented or as extended here) stop philosophers from philosophizing? Of course not. Philosophers have to eat, and if they don't get paid for philosophy they might have to work for a living2.11.
Did it even cause them to stop philosophizing badly, and at the very least state their basic assumptions along with their arguments. Hardly. You have to understand that if the premises of an argument are correctly stated, reason can be reduced to algebra (as was originally shown in a few interesting analyses by George Boole). The only way two algebricians can arrive at different conclusions by the mechanical process of generating a consistent chain of logic leading from premises to conclusions is if their premises differ. Or, well, I suppose, if they make a mistake, but we're talking about two good logicians here.
Ultimately, then, disagreement about the conclusions of some argument in a forum of consistent reasoning is absolutely equivalent to disagreement about the premises. Semantically, all logically valid arguments that arrive at distinct conclusions are precisely equivalent to the following argument:
Almost without exception, philosophers subsequent to Hume have (perhaps) paid homage to Hume's demonstration that one cannot use deductive and/or induction to obtain certain knowledge about anything at all and then proceeded to use deduction (often in the company of all sorts of implicit induction) to arrive at ``certain'' conclusion after ``certain'' conclusion, never openly admitting that these conclusions are devoid of any sort of logically necessary relation to the real world.
The one exception to this rule (with which I am familiar - although exceptions may well be as common as dirt these days and I might well not know it) appears to be Bertrand Russell, who unsurprisingly was as much a mathematician as a philosopher, and whose lovely book Problems in Philosophy is still today one of the most perfect deconstructions of the philosophical process ever written. Even Russell, however, fails to openly acknowledge the importance, and arbitrariness, of axioms in this work, and while he actually writes down in his chapter on induction statements that are very nearly the axioms used by Cox to derive a formal system of plausible reasoning he does not pursue them.
This book will spend considerable energy exploring one of Russell's most important contributions to mathematical philosophy because it is entirely relevant to the process of arriving at an axiomatized theory of knowledge. This is his work on the paradoxes of set theory and self-referential statements, which culminated in the formal derivation of Gödel's theorem. Logical systems of sufficient complexity have certain - problems - that are extremely relevant to both my self-referential arguments above and the following question:
How can we choose our axioms?
The short answer is that in order to choose axioms out of an infinity of possible axiom sets, we, uh, need axioms to help us ordinally rank other axioms so some are ``better'' than others. Which axioms should we use to ordinally rank axioms? Well, we need axioms to help us with that. We in fact need an infinite chain of meta-axioms, meta-meta-axioms, and so on to tell us how to choose the axioms to choose the axioms to choose our axioms that we're going to use as the basis of a system of knowledge that might, just might, be relevant to the real world. Alas, neither you nor I would have the patience to follow all the meta's, and after the first or second one I tried to write out (and get you to pay for, by the page) we'd all get bored. Besides I couldn't afford the infinite amount of paper required to print out an actual draft of the book.
For better or worse, this book will therefore break tradition with earlier works on philosophy in two ways. One, it will do its very best to actually write down a set of meta-axioms and a set (maybe even more than one) of fundamental axioms one might use to reason about the real world. In a nutshell, it will introduce a sort of a ``measure'' on the space of axioms, define a kind-of-center to that space, and judge axioms on the basis of the distance of the resulting deductions from it. It will all be very sloppy because there isn't any point in not being sloppy - we're working to build a foundation that will hold up on the sand, and it has to be able to shift and change with the tides and winds. However it may prove to be enough to help mankind keep its balance and reason well about everything that matters, even when we have just shown, fairly convincingly I hope, that reason itself (especially reasoning ``about'' the real world) is fundamentally unreasonable!
Socrates - And surely this instinct of the dog is very charming;-your dog is a true philosopher.
Glaucon - Why?
Socrates - Why, because he distinguishes the face of a friend and of an enemy only by the criterion of knowing and not knowing. And must not an animal be a lover of learning who determines what he likes and dislikes by the test of knowledge and ignorance?
Now, if you've been paying attention you should now be intellectually poised above a Pit of Existential Despair (PED). This is deliberate.
However, there is a distinct possibility that you are instead going ``huh'' and scratching your head, when you are supposed to be dangling out there screaming at the glimpse of Philosophical Nothingness that underlies All Things. This won't do. So permit me to get out the block and tackle and tie this rope around your feet - there, comfy now? Now - mmmph - we'll just crank you up and swing you out over the PED, hold on to your loose change and try not to lose your eyeglasses, if any. There. Now look up - errr - down.
We've just learned that hundreds of years ago David Hume made the observation that one cannot deduce anything about the real world without making assumptions that cannot themselves be deduced; they must be inferred from making observations about the real world. Unfortunately, the process of inference cannot be deduced either, nor can induction be induced. To this I've added the observation that even ``math-y'' things that one usually thinks of as being ``deductively pure'' and hence knowable as absolute truth are in fact deduced from a set of assertions, called the axioms of a theory, that cannot themselves be proven.
So what's left? We cannot know anything certain about the external
world. We cannot even know anything ``certain'' about math - if we
change the axioms of, say, plane geometry we might end up with curved
space geometry! In the first one can prove that the sum of the interior
angles of a triangle is always 
radians. In the second one
can prove that this is generally not the case, that one recovers
the plane rule in the limit that the space is flat.
Which one is ``true''? Either? Neither? Both?
For a brief, dizzying moment, it seems like we know nothing! Forget difficult questions, like whether or not God exists. It seems that even things we have always taken for granted, such as the objective existence of the book in which you're reading these words, are not certain to be true or unconditionally true. Suddenly all the questions that you hold most dear (no matter what they are) have just had the legs kicked out from under any possibility of finding true answers. If Life is going to have any reason, you're going to have to find a way to put that reason there yourself, because nobody else knows anything more about how everything works than you do - and you don't know much!
Ah, you catch a glimpse of the bottom of the pit, with its waiting demons of despair and confusion? Stop that whimpering! I'm not going to drop you in. In fact, the purpose of this whole work is to fill in this pit so that it is no longer lurking as a trap beneath your every step in life. But first we have to face the pit and even embrace the pit. So life is uncertain, what else is new? Truth be told, you already knew that and have always known it.
To banish the PED forever from your life, you will need at least one sure thing, one thing that cannot be doubted, one thing that is clearly, without question true. Once that one true thing is found, you can choose axioms that will permit you to conditionally extend your knowledge and fill in the pit.
So where can we find that one true thing? To answer that, we think back to our discussion of deduction, we concluded that if the premises of a well-formed deductive argument are true, the conclusion cannot cannot be doubted. Doubt describes a state of uncertainty. It describes our degree of belief, with certainty of truth or falsehood being the opposite poles where doubt vanishes or becomes complete.
Let us, then, follow in the footsteps of Reneé Descartes3.1 and use doubt as a tool in our quest for truth.
Descartes, as you probably know, was in some sense the ``father of modern rationalism''; indeed, we are all de facto cartesian rationalists even though, as we've seen above, rationalism itself is fundamentally irrational. Descartes made many important contributions to mathematics, to the birth of ``natural philosophy'' (science), and to philosophy proper. He (like Hume) was a giant of the European Enlightenment and we are all immensely in his debt, for all that most of his philosophical conclusions were wrong.
One, however, was very right indeed. Descartes, like all humans that are smarter than a piece of lawn furniture, had moments of youthful rebellion and existential crisis that fuelled a desire to discover the truth of all things. He ran away and joined a mercenary army (Europe at the time was an eternity of warfare) to see the world and look for his own personal way out of the PED. Early on, he met one of the early natural philosophers (Isaac Beekman).
This meeting gave him a purpose he was to pursue for the rest of his life. In a dream he invented coordinate systems and analytic geometry, then worked on the application of coordinates to the development of mathematics and physics and a natural philosophy where all things, including God, where known with the certainty of mathematics.
To achieve this goal, he began his most famous philosophical exploration, based on methodological skepticism, to arrive at a wonderful conclusion. Let us apply this method to ourselves.
Just for the moment, pretend that you don't know anything at all. This should be fairly easy, given our arguments thus far. Doubt everything!
Did man really land on the moon? Maybe, maybe not. Maybe it was all done in Hollywood. You remember the sun rising yesterday - does that mean it really happened? Not necessarily. Anybody who's ever been married is well aware of how even your most convincing memories of something only 24 hours ago do not necessarily coincide with those of someone else's equally convincing memories. Do you know the sun will rise tomorrow? Well, it hasn't happened yet. The sun might well explode before tomorrow.
We actually find it easier to doubt all this than Descartes did, because we've had the advantage of reading books, seeing movies that vividly portray the doubtability of the objective reality of that which we perceive with our senses, or remember, or imagine. Our senses can be fooled or mistaken. Our memories are even more fallible. Our imagination, even of things such as mathematical truths, has a kind of an ephemeral quality and besides are always contingent truths at best.
For example, in James Gunn's The Joy Makers3.2 we are shown a world where humans are cocooned by a vast computer charged with making humanity ``happy'', which it manages by completely controlling their sensory input. This kind of theme was reprised in the Matrix3.3 movie trilogy, where Neo is awakened from a ``reality'' that turns out to be a computer simulation. Three movies later, it isn't clear that the reality he's been awakened to is all that real either.
Working a bit harder, Descartes, too, eventually decided that he could doubt just about everything. He could doubt that which he saw, smelled, tasted, heard, felt, because sometimes a dream has the force of reality but turns out to be a dream. He could easily doubt his memory, as it is fallible and deceiving. As he considered each thing that he thought that he knew, he discovered that he could doubt it - everything but one thing.
When Descartes attempted to doubt his own existence, he ran into a bit of a problem. It was impossible to divorce the act of doubting from something that was doing it. Try as he might, Descartes couldn't doubt the existence of the doubter unless he existed to do the doubting. He had found one true thing: He existed!
You too, have this same true thing. Perhaps you are ``a unit in the Matrix''. Perhaps the Universe is all an illusion. Perhaps you were created only yesterday by a powerful and malicious mad-scientist complete with a full set of memories of an apparent past, and that same scientist plans to terminate the experiment in the next five minutes. However, as you muse about this, as you consider alternative hypotheses that might explain whatever it is that you are feeling and remembering and seeing, it is impossible to deny that you are there doing the feeling, remembering and seeing.
Descartes summarized this with his famous ``I think, therefore I am''. Although it should be carefully noted that this isn't quite the same thing as ``I am doubting, and therefore existing'' it is a catchy little sound-bite3.4.
Thus far, Descartes has, through his doubt, achieved agreement with one of the oldest of philosophies, the one underlying Hinduism, and so have we, riding along. He has discovered the Atman, or Self, as an undeniable ``truth''. In Hinduism the Atman is that thing which sits ``within'' each of us3.5 and has the essential existential property. It Is, with a capital I. For convenience, in the rest of this work we will refer to the self-that-cannot-be-doubted, the thing-that-is-existing at the heart of all of our sensory experience, memory, and mentation, as the Atman in deference to the truly ancient and nameless philosophers who first conceived it and wrote of it.
Let us spend precisely one paragraph on existence, because philosophers great and small have wasted much breath on the idea, deciding whether existence is or isn't a predicate and just how existence factors into logic and ontological arguments. All of it bullshit, mind you. We will not worry about existence as a property of things because so far there aren't any things, at least things that can be doubted. There is only one Thing, the Atman (of each of us - so far I can doubt that you, dear reader, exist just as you can doubt that I do). That the Atman is existing cannot be doubted by the only thing that gives a rat's ass about ``existence'' in the first place - the Atman itself, and e.g. Russell's objections to the ``I'' in the statement, or objections to its logical or set theoretic validity are simply irrelevant.
This is the one point where Descartes and Hume agree. There is no possibility of doubting that you (whatever ``you'' may turn out to be) are experiencing the sensory flow of your instantaneous awareness. This is an empirical truth, not a logical one, and indeed it is the only empirical truth that is beyond all doubt. We can even leave the role of ``doubt'' behind. At any instant that we are Self-aware, as long as our Atman is perceiving, we exist and any attempt to assert the contrary may be logically permissible but is just plain stupid and inconceivable and obviously false to the conceiving mind making the assertion. We may not know what we are, but we know that we are not nothing
We can now use pure reason to deduce a tiny handful of related truths that (as is always the case with logical deductions) are little better than restatements of the obvious. We've already just made one of them; here is a short list of it and other correlary truths associated with the empirical truth of the Atman, stated as you should read them (in first person, as they are true only to the perceiving mind that is reading them:
``I exist'' should already be clear. If you are reading these words, musing on what they mean, thinking, imagining, dreaming - if you are doing anything at all and are aware of the doing then you exist as an awareness even if you are mistaken in all other respects about the true nature of that which you are thinking about. The Atman is, even if all else is illusion, because the Self cannot experience an illusion without being.
``Nothing does not exist'' is less clear. Epistemologists and logicians are doubtless already sharpening their knives merely to read the words. What I mean to convey by this is the following. Let us make the following two mutually exclusive, and universally sweeping propositions. These are really serious metaphysical statements, mind you - I'm not talking about any finite or embedded versions of them.
So when I say that nothing does not exist, I'm asserting that it turns that that the absolute Void of utter nonexistence is not, in fact, the case. Something exists. Given the dichotomy between absolute, ``universal'' non-existence (a state in which yes, the proposition itself could not be formulated) and one where ``something exists'', well, we've formulated the proposition and are thinking about it because we exist and we are something and so nothingness, the ultimate Void, does not.
Here, now3.6. The Atman serves to perfectly fill the Pit of Existential Despair in the here-now of each being's self-aware existence. No matter what your doubts about the world, about its or your own reason for being, you have a place to metaphorically stand, a fundamental empirical truth. You Are.
Take comfort in that for a moment. Relax, go get yourself a beer or glass of water, maybe have a snack. Take a walk. Enjoy a few moments of quiet pleasure - even if we eventually conclude that the world is all an illusion, a dream, the Universe isn't nothingness - at the very least the dreamer exists.
All set? Then let's continue. I exist without doubt or question (you should be thinking as you read this). What about everything else? How do I get from knowledge of ``just me'' to where I can have some degree of confidence that the beer I'm drinking is real and not just a dream, an illusion, a simulation being generated by an Evil Supercomputer and sent to a Unit in the Matrix?
The answer is simple. We have to start to make assumptions, assumptions that build a bridge between our Self and the outside world. We must always keep in mind that our assumptions might be incorrect - they are doubtable. We will not know that they are true, so we will have to believe in them. We will have to have faith.
Even faith, however, should have some foundation in reason. As we prepare to pick the axioms of our personal existence, it will really help us a great deal if we start with a set of meta-axioms - axioms to help us pick axioms.
You exist. Your Atman is the precise center of your being. It may not be terribly clear at first what ``you'' are, but don't worry too much about it - no answer that can be framed in symbols such as words can be correct. It seems silly to define the one certain thing we know in terms of things we are less sure of.
As you sit there, reading, still in a state of doubt-shock from the process of denying the possible objective reality of pretty much everything outside of yourself, we can at least hope that certain questions float to the top of your awareness. In case they don't, permit me to help them along.
If all I'm certain of is that I am, what is all of this?
The correct answer, if you think about it, is that ``all of this'' is pretty much entirely a sensory stream of information. Let's break down some named components of your awareness. You are aware of some mix of:
Some part of those sensations have a certain ``intensity'' that you have associated up to now with an outside world. As you read this book, you experience a very powerful and immediate sensation that you interpret as sight, an experience that is highly structured and that visually carries symbolic information that you may well be ``hearing'' inside your mind in a way that is less intense than real sounds you are hearing at the same time.
Some of those auditory echoes loop back still further. If I suggest that you imagine a ``red, red rose'' you see the words on the page, hear them in your mind, and visualize, if only for a fleeting moment, an appropriately red rose. Perhaps the words trigger a memory of a real red rose you've seen, evoking a variety of sensory impressions that again are less intense than those you associate with what you are experiencing now but may be more intense, or more specific, than a ``generic'' imagining of a red rose.
There are some mysteries associated with this process. Your Atman seems to have volitional control over your immediate sensory stream. By willing it in certain ways, you can create impressions of external things you identify as parts of ``your body'' moving around as you direct them to, your sensation of touch responds accordingly, your visual field and auditory sensations vary as you reach for your beverage, and your taste and smell kick into play as you take a sip and flavors explode into your mouth and nose.
Yet throughout ``you'' in some sense remain unmoved, watching, listening and experiencing the sensation of motion and action and touch and taste as a sensory parade unfolds before your eyes.
This sensory stream is ``true'' in that it is a part of your own immediate truth. You can doubt that the objects of your sense are really there, but it is difficult to doubt that the sensations themselves are not. It is, in fact, data. It is now the job of your Atman to make sense of that data. Unfortunately, it is not as easy as it looks, especially since you've agreed to doubt things until you make a personal decision as to what can sanely be believed. Let's see why.
It is easiest to see why the Atman might well be mistaken about almost anything associated with the senses if we go over the Allegory of the Cave4.1 as written by Plato. The following is a modest adaptation of the cave in the half-remembered words of my teacher, George Roberts, who was my primary guru in undergraduate philosophy back in the 70's at Duke and a disciple of Bertrand Russell. Note well that my version is a bit embellished (as was George's) to better illustrate the essential point I wish to make - I'm a storyteller, not a historian.
Imagine a cave inhabited by prisoners who have been chained there since birth, fastened into infernal devices that only permit them to see the blank cave wall in front of them. As sensory input is a major concern here, we can imagine further that their sense of touch is for all practical purposes deadened by immersion in an immobilizing gel bath held at a uniform temperature. They wear filters so that they cannot smell. They are fed intravenously. They experience through their senses what we permit them to experience, no more, no less.
Behind them there is a raised platform and a light. As objects are carried back and forth on the platform, the shadows of these objects are cast onto the wall before them and are all that the prisoners see. Voices from those that carry the objects and sounds that the objects might make while being carried are reflected off the wall so that they too appear to come from the shadows thus cast. The prisoners chatter back and forth (where we needn't examine too closely how it is that they end up with a language for chattering in) trying to make sense of that which they see.
Some objects cast very similar shadows, so they are given a common name by the prisoners. Some objects are always carried by in the company of other objects or always are associated with certain sounds (the sound of being dragged down the platform, for example), and so relationships between the objects themselves, or between the objects and certain sounds, are inferred. The prisoners concoct elaborate theories to describe their little ``world view'' and get into violent disagreements when their conclusions differ, even though they are oriented in such a way that none of them sees exactly what the other prisoners see. Fortunately, no prisoners are injured in these ``violent'' disagreements as they are, after all, confined - the disagreements are strictly verbal4.2.
Leaving aside in its entirety from this point on Plato's purpose in introducing the allegory or the rest of the story (which has little to do with this book) let us examine just one question - what do these prisoners ``know'' and how do they know it? There is such a wealth of things to learn about this question from even this simple example that it is difficult to know where to begin.
First of all, it is clear from the way the allegory is set up that nearly everything that they think they know is mistaken. We are gifted with a bird's eye view of the whole scenario as created by these words, so that you are able to note that there are actually three different objects that cast circular shadows being carried back and forth. One is (for example) a large black wheel being rolled along that is used on tractors, one is a giant spherical inflated weather balloon being floated along on a string, and a third isn't round at all - it is the housing of a complicated generator and only accidentally has a round projective cross section if it is carried across in one orientation.
Yet the prisoners name all three things with the same word. They see the same generator housing being carried by in a different orientation and name it something altogether different. They are effectively blind to the color, the true shape, the density, the function, the texture, and the smell of the objects - most of the relevant properties of those objects have been ``erased'' by the process of casting only a projective view of them on the wall: their shadows. Even the beliefs of the prisoners concerning the sources of the noises are incorrect, as the track is oriented in such a way that the real sources of the noises (the people carrying the objects, the carts and so on that they drag them through with) are invisible even as a shadow. The squeaky round becomes a female round seeking a mate - a silent round is hunting, the round that makes grating noises is a male, clearly.
The shadows cast on the wall are not real objects, they are only projective images. Yet they are the only experience of those objects, or of objects of any sort, that the prisoners have ever known! Our prisoners are imprisoned by their bonds to be sure, but they are even more tightly bound by the restrictions we have placed on their sensory input. Their visual field is restricted to where binocular vision is of no use to them and it is very doubtful that they have any visual concept of depth. Their visual universe is two dimensional, and there is little reason for them to believe that a third even exists.
This allegory can be carried to even greater extremes, and has been in a number of fictional works, notably Flatland, a Romance of Many Dimensions4.3 by Edwin Abbott, the aforementioned The Joy Makers by James Gunn4.4 and the also previously mentioned Matrix film trilogy4.5 . Let's think about these three examples (all highly recommended reading, or watching).
In Flatland the consequences of living in a two dimensional manifold (plus time) that is, in fact, embedded in a three dimensional manifold (plus time) are explored. The theme of the Cave is clearly echoed as a two dimensional being struggles to make sense of the projection of a three dimensional being that can apparently travel through walls and appear and disappear at will, who can be heard even when it is literally out of the (two dimensional) world.
Naturally, at first this is extremely disturbing to the inhabitants of flatland - black magic, violations of causality (as they see it), miracles - it simply appears senseless to them at first because it is so outrageous and beyond their experience. However, as time passes a few flatlanders come to mathematically grasp the truth, and one of them is even forceably ``uplifted'' to live henceforth as a three dimensional entity.
This is a version of Plato's cave where the prisoners live on the two dimensional wall of their mental cave, where the bemused guard in three dimensions comes to try interact with them and ends by freeing one. Indeed, it works both ways - lineland and pointland are successively more constrained, but all unite in doubting the existence of the higher dimensions where (we see from outside the story) reality actually dwells. Even the three-dimensional being who can clearly see the succession of embedded realities of lower dimension cannot fathom how he could live in a reality embedded in a higher one.
The Joy Makers, in contrast, isn't a mathematical moral tale but is rather a classic of speculative science fiction with human characters and a complex plot. In it, hedonism becomes the ruling ethos of society, to live for pleasure the greatest good. Since work is unpleasant, society automates more and more, delegating the work that still needs to be done to machines operated by intelligent computers. Since it is unpleasant to be frustrated in one's desires, neuromechanical interfaces that can simulate reality to any precision desired are developed and operated by a massive controlling computer to deliver the precise realization of every individual's immediate desires instantly (while caring for their otherwise immobile and irrelevant bodies). They in fact live in Plato's Cave with computers controlling what they perceive in a sort of direct-connection ``massively multiplayer role playing game''4.6 virtual reality in which no truly bad thing is ever permitted to happen to them and which in no way reflects the supposed real ``reality'' in which their bodies and the controlling computers reside.
Readers should note well that this anticipates the main theme of The Matrix trilogy by some thirty years, just as Plato anticipated it (in a low-tech realization) by several thousand. However, The Joy Makers proceeds to a more macabre conclusion, as the computer that ends up controlling the dreams of the entire remaining population of the world is forced by the protagonists (who seek to escape the bondage of perfect hedonism and live a ``real life'') to realize that it too is a ``living being'' and subject to the law requiring it to be happy. Ultimately the last humans escape the bondage of illusory pleasure as the computer realizes the satori that only way to avoid the pain of its own life is to seek out the cessation of that pain in its own prolonged amusement leading to death.
However, the book's protagonists can never be sure that they have ``won'' and actually escaped in the end. Naturally they wished strongly to win, and the computer is programmed to provide them with the perfect illusion of having fulfilled their strongest desires. Did they really win, or has the computer ensnared them after all and placed them back in a simulated reality where they appear to be living free? It is, of course, impossible to tell.
The Matrix series explores this theme still further and extends it across multiple layers of supposed reality. Neo (the main character) is living a perfectly normal life in a perfectly normal city in a perfectly normal society - on the surface - but he has odd dreams. One day he is awakened and discovers that he has really been a biological power unit in a vast machine (the weakest part of the story, actually, as it illustrates a profound ignorance of the second law of thermodynamics) and that the ``reality'' he has known his whole life is just a collaborative simulation not unlike World of Warcraft is today but wired directly into his brain. However, as the story unfolds (over three movies) it gradually becomes equally clear that the new reality he has been ``freed'' into is no more real than the one he left. It too is some sort of projective simulation and exists, ultimately, only in the mind as a sensory stream in interaction with whatever constitutes Neo's ``self''. If we need any further irony, we can always take note that the whole story is a movie, creating yet another level of reality in our minds.
The common theme to these diverse examples is that even though we can never doubt our own existence, we are all, always, ``prisoners in the cave''. What we ``see'' might or might not correspond to an actual external reality. Perhaps what we perceive is is only a projective simulation, shadows cast on the wall of the cave of our senses, generated by an external intelligence (godly or diabolical or merely natural as you prefer). Perhaps it is only a projective view of an objective reality with no guiding hand, but important things happen in the dimensions we cannot ``see'', things that make some occurrences in our restricted manifold seem like ``magic'' or appear to have one set of correlations that we interpret as relationships of ``cause'' and underlying order where in fact they have a completely different order in the higher order reality.
The well-read reader - one who has for example read Michio Kaku's Hyperspace4.7 or is otherwise passingly familiar with e.g. quantum string theory or the question of hidden variables in physical theories in general - will recognize that these are not idle speculations, they are perfectly permissible conjectures in real science. Indeed the entire history of Natural Philosophy (science) since the Enlightenment has largely consisted of figuring out how to look beyond the narrow limits of our biological senses with e.g. telescopes and microscopes and controlled experiments to untangle some of the projective magic of our mundane experience.
At this point, it should be clear to you that constructing a set of axioms to apply to your sensory stream so that it can ``make sense'' to you is obviously a process that has no unique solution. There are infinitely many ways to explain it all, and so far we have no way of choosing one over the others. You can freely choose to be a paranoid and believe in an evil genius joymaker or to be religious and believe in a benevolent omnipotent diety and no one can prove you wrong. You can choose to believe that you are the evil genius, and that you are actually dreaming up everything you perceive, and no one can prove you wrong. You can choose to believe that everything is just as it seems, and that the book you appear to be holding is just that - a real book, located in a real space-time manifold - and no one can prove you wrong.
But which of these possibilities (or the infinitely many more one can imagine that differ in details great and small) is right? We do not know for certain. We cannot know for certain, ever. If someone tells you that they are certain that Jesus is Lord, they are mistaken. They may believe it; they cannot be certain. If someone else asserts that physical science proves that there is no God, they are incorrect. They, too, may believe it; they, too, cannot be certain.
Thus we arrive at the first, extremely important conclusion of this book. Since we - none of us - can be absolutely certain if our beliefs about the shadows cast on the walls of our personal caves are correct, it would be a really good idea to develop a certain amount of tolerance for the beliefs of others4.8. Allow me to apologize beforehand for hammering this point home repeatedly throughout this book, but I fully plan to hammer this point home repeatedly throughout this book. Self-righteousness isn't just obnoxious and unpleasant - it is actively stupid.
Still, we feel intuitively that even though we cannot prove that any given interpretation of our sensory experience is correct by means of pure deductive reason from the one certain truth of our own existence - Descartes' original attempt along these lines having been tragically flawed in an uncorrectable way - some explanations seem somehow better than others. In fact, some seem to much better that it is literally difficult to imagine that they could be completely false!
We should recognize this as both a trap and an opportunity. The trap is fairly straightforward. When you say ``better'' you mean, literally, ``more believable'', since certainty left town for good and won't be back. Unfortunately, we have no way to rank-order propositions in order of believability; it seems as though we'd need propositions to tell us how to rank-order propositions, just as a considerable number of axioms are required to define the concept and action of ``greater than'' and ``less than'' in number theory or geometry. And what tells us that these propositions (that rank-order propositions) are better than other propositions about propositions? Ooo, looks like a deadly logic loop with no possible resolution.
The opportunity is to recognize that you are, in fact, standing on the one piece of absolutely solid ground in the entire Universe, the one that can hold you up even when you make mistakes - your Self. You are free to choose pretty much any set of beliefs, including a set that rank-orders beliefs themselves, without any need for further justification. If we can somehow define what is meant by ``more believable'' with mathematical rigor, then perhaps believability, plausibility, can take the place of ``truth'' as we seek to untangle the evidence of our senses. What we end up may not be ``knowledge'' in the same way that you interpreted the word a day or two ago, but perhaps it will do. At the very least it might give you a reason to believe your memory that a day or two ago actually happened!
Let us, then, endeavor to come up with a set of meta-axioms that will help us to rank-order axioms in such a way that these axioms are no longer held to be certain truths - beyond all doubt - but instead are merely plausible, with some more believable than others. We seek to frame a metaphysical theory of knowledge that can cope with the certainty of uncertainty and still function in a way that ``works well enough'' to deal with our current sensory input whether or not it ultimately reflects an objective reality or is the illusory product of a diabolical genius or even is our own demented and somewhat schizophrenic imagination.
The flickers on the walls of our personal cave may only be shadows, but inside of the empirical truth of the Atman that exists, in a way, as their perception, they're all we've got! We must choose a way to make the best of them.
Logic has long been associated with knowledge. Our sense that we ought to be able to arrive at a completely rational description of all things derives in no small part from the immense success of logic as a methodology for determining consistent sets of statements, usually with clearly stated order of contingency.
I'm being quite deliberate when I avoid using the word true sets of statements in that last sentence. Logic does not generate truth, it generates consistency in assertions of truth. Truth per se, as we have just learned, is quite elusive. Without it logic is essentially an algebraic game with no logically necessary connection to the dizzying stream of sensory data that we are experiencing.
Nevertheless it is an important game. This work should not in any way be interpreted as a blanket condemnation of logic or rational thought, but rather as a way of addressing its admitted weakness by providing some sort of criterion for evaluating truth where logic alone cannot. In this chapter we are going to do much of the preliminary work on meta-axiomatic rules required to help us ``boot'' into a system of axioms that will, eventually, restore you to a state of sane knowledge in which knowledge of the limitations of your knowledge accompanies the knowledge itself.
This is an extraordinarily difficult chapter to write, in part because it needs to be extraordinarily clear and articulate in some mix of human language and the language of logic, which doesn't look or work quite the same way that English does. To achieve the requisite clarity I'm going to begin with a simple list of meta-axioms that we will use as constraints on the axioms that we will use to build a worldview5.1 . Some are (I think) almost self-explanatory - the meta-axioms that define the Universe to which all candidate worldviews must apply, for example, need little amplification or justification. Others I spend whole chapters on later, as there is much to say about why they are selected.
As we begin, remember to always keep in mind that these axioms are ``meta'' because each of them applies to the propositions themselves that we might tentatively advance to establish and develop a worldview. We will show quite clearly in later chapters how there exist whole sets of axioms, some that differ radically from one another, that can completely and consistently explain everything in our experience. There are, if you like, an infinity of ways of building a worldview!
In other words, there is no uniqueness property even in principle for theories of everything. This is easy to prove mathematically. One can always form an outer product5.2 of everything we know (on an individual, one Atman at a time basis) and a literal infinity of consistent logical systems that are disjoint from our experience to arrive at a distinct logical system that explains our actual experiences as well the one we started with did. For example, I could assert that there exists a completely disjoint space-time continuum where the world described in Tolkein's Lord of the Rings series of books exists in objective reality, and by definition the assertion will not affect anything that happens in this space-time continuum. The two space-time continua need not even be completely disjoint - interactions that are rare (for whatever reason) might leave little trace in either one, and that trace might have many possible causes.
How can we ``reject'' such fanciful assertions from the realm of acceptable discourse we seek to establish as we collectively try to build a common worldview, so that we don't waste valuable mental energy treating them as if they really matter? We need meta-axioms that establish a ``bullshit scale'' that can rank axiom sets that contain excessive amounts of bullshit ``lower'' than ones that do not. Yet (as we will see, and indeed establish on a quantitative basis) we cannot be draconian in this - there are plenty of examples in real science where fanciful hypotheses that started out almost as ``intuitively unbelievable'' as this one turned out to be very believable indeed when one day, a researcher did the moral equivalent of trap an elf in their back yard in the company of a bunch of orcs (effectively demonstrating that a hypothesized Tolkein middle-earth that is not completely disjoint from our own is ``suddenly'' quite plausible where before it seemed as if it was complete fantasy).
Our meta-axioms will need to be logically flexible, in other words, to accomodate changes in our state of knowledge, not ``rigid'' the way boolean logic is now. We (fortunately) can learn and figure things out and arrive at new contingent truths through a creative process that requires a bit of freedom to work. Although we will need to begin with Aristotelian or Boolean logic, an algebra of truth and falsehood, we should end up with something that describes the actual dynamic state of human knowledge, including a variety of speculative extentions, which almost never achieves anything like the certainty represented by the words ``true'' or ``false''. We will start with the two integers 0 and 1 (to represent false and true) and add axioms to uncover the infinity of real numbers in between them, and then use the entire range to describe our state of knowledge.
Also keep in mind that the set of meta-axioms itself is no more demonstrable as logical truth than anything else. We have no good reason to consider it unique, and indeed I will point out at least a couple of de facto common extensions (and the difficulties associated with its extension or alteration). Its ultimate justification will be that it ``works'' to do what we want it to do - build a path from our One True Thing to contingent knowledge of as much of ``everything'' as we can manage.
Ultimately, the decision by human society to adopt the set of meta-axioms given below is de facto establishing a social contract concerning the kind of discourse that will be called ``reason'', and introducing a measure of quantitative separation from the kind of discourse we will consider ``fiction'', ``fantasy'', ``unsupported belief'', ``mythology'', and ``religion''. We desperately need such a social contract, because as things stand reason literally means different equally arbitrary things to different humans and people get killed by other people in acts of war and violence every day as a direct consequence. Different axioms often lead to different conclusions, and people don't understand the fact that the axioms themselves are never self-evident truth!
With that clearly understood, here is a list of meta-axioms, loosely grouped into categories by purpose. Most of these categories are not themselves ordered in any logical way - one should not think of existential meta-axioms as ``prior'' to rational meta-axioms, for example just because I present the one before the other. The exceptions are the meta-axioms of certainty and doubt. We will start with them because they are prior and will apply to all meta-axioms and axioms in our list.
As we have seen, we are fresh out of self-evident truth. Our knowledge is provisional, built out of assumptions and the shadowy impressions of our senses, our memory, and some sort of reasoning process.
Yet without some foundational ``truths'', we cannot proceed towards knowledge of any sort at all. We clearly need meta-axioms to formally establish these principles in a way that permits some sort of rational process to proceed, and yet simultaneously explicitly acknowledge the limitations of that process that are inherited from our original uncertainty. I offer several.
This set of meta-axioms is to be assigned the provisional value of true or certain in all logical analysis.
I can almost hear you objecting: That's not a meta-axiom, that's a religious statement! ``Thou shalt not have any sets of meta-axioms before me'', so to speak. ``Why does he get to specify truth instead of me.'' ``What's so special about his meta-axioms.'' And finally, ``Isn't that a Gödellian loop?''
Let me answer the last one first, as it's the easiest. Yes, this meta-axiom refers (among other things) to itself; it is self-referential and hence all sorts of fun. As for the rest, I'm not trying to be arrogant or suggest that this list is, in fact, perfect. If you think about it, however, you'll realize that of course it is a necessary meta-axiom for any axiomatic system and (if you accept it) terminates all sorts of requirements for meta-meta-axioms and so on. Even if we didn't explictly state it, it would be implicit so let's write it down at the very beginning.
By the way, you're quite right, it is just like one of the axioms of organized scriptural religions, that use it for exactly the same reason we do - to terminate the critical thinking loop somewhere to we have a place to start. I personally think that doing it inside a religious scripture is a bad thing, but we'll get to that later.
This meta-axiom is not intended to suggest that this list of meta-axioms is unique, complete, or really correct. In fact, we'll add a few more meta-axioms that explicitly give us permission to change the list carefully and thoughtfully to make it work better. All that it states is that (whenever we are ready to proceed to build a worldview on the basis of the meta-axioms and axioms we select) we must accept the complete correctness of our set of meta-axioms as our ultimate working hypothesis and use its rules to assess the axioms in the various proposed systems or we'll be left with axiomatic chaos5.3, where quite literally one proposed axiomatic description, be it incomplete, inconsistent, or ``apparently'' incorrect will be literally as good as any other. There are an infinity of Fairy Theories5.4 lurking to sweep us away into unreason, and only this meta-axiom allowing us to assert sets of meta-axioms stands against them.
The axioms of a worldview can always be doubted - and should be!By now you should have a pretty solid understanding of the fact that axioms are not self-evident truth, they are working hypotheses. The simple middle-earth example above demonstrates that they can never be unique - there is a practical infinity of worldviews that will lead to precise consistency with our experience.
We have, in fact, proven by an explicit example that our knowledge of the Universe is necessarily incomplete, or at least cannot ever be known to be complete on the basis of any set of observations, any more than the inhabitants of Plato's cave could ever be certain of the true nature of the objects casting the shadows they perceived on its walls, let alone obtain a knowledge of the entire world that lay outside of the cave altogether.
There are additional sources of uncertainty that should lead us to doubt. Very shortly we will put down meta-axioms that act on whole systems of axioms in ``fuzzy'' ways - axioms that encode the importance of flexibility in the process of reasoning in the absence of certain truth or falsehood. These axioms can be mistaken in the way they rank one system of reason up or down relative to another, and yet still be valuable as we often have to proceed through intermediate states of incomplete knowledge where we are drawn to what we later feel to have been incorrect worldviews. Indeed, the history of philosophy consists of little else.
As we proceed, it is absolutely essential to remember the parable of the cave. Even though the projection of the shadows of the Universe on our personal experience might be extraordinarily consistent with some given set of ``simple'' rules that allows us to predict their behavior, the true reality of the objects that cast them can be vastly more complex and interesting, but forever hidden from our view5.5.
This meta-axiom has a fairly obvious corrolary. Since all axioms can be doubted and are at best provisional truth, we would all do well to:
Keep an open mind about absolutely everything.This is (if you like) the meta-axiom of critical thinking. It is the lack of critical thinking that leads to pretty much every single one of this sorry planet's self-inflicted human ills: war (especially religious war), political war, sustained poverty, global plagues and disease. There are solutions to every single one of these problems that can easily be found by a systematic process of critical thinking and experimentation, but this process only works for people who can give up certainly and abandon self-righteousness.
The meta-axiom of the open mind (and doubt, and critical thinking) tells us that we should be very, very wary of ever claiming certain knowledge of pretty much anything. We should constantly reconsider our most cherished beliefs - religious, political, ethical, scientific - and see if experience and reason still support them. There is a need for personal intellectual humility implicit in this. We should never, ever, tell another human that their beliefs are necessarily wrong, only that they seem implausible and here is why we don't share them.
Both sides in any serious debate need to fully understand that they don't have to share their axiomatic beliefs. On the other hand, both side in a debate need to understand that a disagreement over axioms will lead to disagreement about conclusions, so that agreement on a common set of axioms is a necessary prior to engaging in reason together instead of mere pointless argument.
Doubt is, as we are rapidly discovering, a powerful meta-axiom indeed. Make it your own. If someone comes up to you and tells you that they have the magic key to perfect knowledge, something beyond the One True Thing that we learned from doubt that they assert that they know beyond all doubt: No matter if that knowledge is based on science, religion, or anything in between - put your hand on your wallet and back away slowly. They want to sell you something.
With the exception of the contents of this book, of course, which are meta-axiomatically true, after all. Axioms contains the magic key to imperfect (but very, very good, high quality, USDA Prime Choice) knowledge, and hopefully I've already sold it to you5.6.
These are meta-axioms that define the Universe of discourse for our worldview. Our one undoubtable truth from a couple of chapters ago will of course be one of them. We will add two more primarily to exclude certain classes of bullshit theories, for example solipsism5.7 and non-Universal definitions of the Universe.
This is the only thing in this book that is not, really, an axiom. It is the ultimate empirical truth, the One True Thing we learned from doubt, the thing that plugs the existential void, the place we must stand to reach out and touch the world. We're going to create a set
from this one thing, and place ``I'', your Atman, in it.
It is, recall, beyond doubt. If (and only if) you are self-aware, experiencing ``something'' (such as the reading of these words), capable of doubting your own existence and thereby demonstrating it, then your Self is in a state of certain being. As previously noted, I will call the essential self-aware Self the Atman in homage to its non-Western discoverers, although I will frequently refer to it as Self (capitalized) as well, to help emphasize their approximate equivalence.
As something that is literally a Self-evident truth we can certainly include ``I exist'' as a true statement in all of our reasoning. By making this a meta-axiom and requiring logical consistency (see below), we will save ourselves all sorts of trouble with nonsense at the hands of idiots that want to assert or try to prove using ``reason'' that we don't exist. Maybe they don't exist, but while I'm writing these words I damn well do, and if these same words are happenin' in your head as you read them, so do you.
As I'm sure you can see (literally, as you read these words in a book containing information that is not yet a part of your Self, but becoming a part of your Self as you read), your sensory impressions seem to correspond to an actual objective external reality. As we have already learned from our use of doubt, we cannot be certain that this reality exists. Nevertheless, if we try to assess our level of doubt (using a scale of uncertainty introduced meta-axiomatically below) in units that for the moment we will consider to range from ``complete, utter, bullshit'' to ``cosmic satori'', we're going to reject from consideration all axiomatic theories as being various forms of bullshit if they allege that the external reality we seem to perceive isn't objectively real.: Something Other than just my Atman exists.
We will admit axioms into our worldview only if they do not contradict the possibility of knowledge of an actual ``something'' that exists outside of our Self. Sayonara, Solipsism! If you wish to do solipsistic reasoning, make up your own damn set of meta-axioms (or read what I have to say about it in the chapter devoted to solipsism).
Note well that I've created and labelled a set
containing Other.
Note that I do not insist on including the axiom that
is not the empty set. I only insist on not including axioms that
state or imply as a deductive conclusion that
is the empty
set. There is actually a point to this - it permits the difference
between
and
(defined next) to be as small as you like as long as
it isn't empty.
Thus the real set that contains everything is the union of the set containing only my Self and the set containing everything that is not my Self. I'm calling this union Brahman, again in homage to philosophers who worked out this partitioning of knowledge in considerable detail long before Descartes or anyone else..
The meta-axiomatic insistence on this partitioning is not religious, and the use of these names is not intended to endorse Hinduism or Buddhism or Jainism. If anything, the partitioning is mathematical or set-theoretic. If the ``religious'' terminology bothers you, think Self, Other, and Universe.
Set theorists will doubtless be bothered by our creation of a possibly infinite set Universe. This is a step fraught with peril in mathematics and logic and set theory. However, it is utterly harmless when considering real Universes for reasons that are left for the doubter to work out as an utterly pointless exercise in considering the difference between semantic sets (manipulations of sets of symbols) and sets of things that actually exist. Note well the necessary lack of paradox in the latter, however easy it is to generate paradoxes in ill-formed or self-referential predicates in the former.
The reason that I proceed in this way is to positively exclude anything that exists outside of the Universe. Two contexts where this occurs are ``Multiple Universe'' theories of quantum mechanics and "God, creator of the Universe'' theories of religion. In the former, what is meant is that the Universe is really a lot bigger than ``just'' what we see. Infinitely bigger, as the real total Universe, Brahman, contains all of those multiple universes (small `u') that make it up. The whole point of the ``Uni'' in Universe is that there is only Just One.
In the latter, the idea is that God exists, um, well, ``someplace''. It's a place that isn't a place, where in time that wasn't time, God created the Universe of Space and Time and Matter out of something that wasn't anywhere at all before because there was nowhere but where God was which wasn't a place at the time...
Ptui! Let's just put a stake right through the heart of bullshit such as this from the very beginning. Brahman is all things, all spaces, all times, period! It is the Universe! There is no place, no time, no being, but it. Even if reality consists of an infinity of utterly disjoint space-time-continua, each of them made up of an infinity of ``multiple universes'', Brahman is the whole thing. Even if the Universe is a veritable Hilbert Hotel5.8 there is still only one of them.
This eliminates so much truly pointless argument over infinities; please adopt this meta-axiom and make it your own. You'll be glad you did!
Note well that very shortly we will assert that Brahman is all information, as the most compact way of dodging still more nasty dogfights over the number of mental angels that can dance on the heads of material pins. It also has the convenient effect of transforming ``space'' and ``time'' into information. Information is sort of like coordinates, only better because it need not be semantic.
Now that we have an assertion of truth/existence and some sets to reason
about that we can imagine being (empty) and not being (empty), we need
to establish that these are meta-axioms of reason. We truly do
want to use reason to establish consistency across our system of
axiomatic beliefs and maximal correspondance between those beliefs and
the sensory impressions and thoughts and memories that we will
eventually axiomatically assume to be correlated with
, with Other,
as the experiencings of
, my Atman.
We are going to begin with Boolean/Aristotelian logic to bootstrap our development of reason. This will give us, for example, pretty much all of mathematics to use as the basis for further reasoning (and a whole slew of sets of axioms that are the basis of the different kinds of mathematics that we can use as ``plug-ins'' for the worldview we are trying to build where it suits us to do so). With mathematics in hand, we will meta-axiomatically quantify knowledge on a better scale than True and False.
Using these meta-axioms we will be able to consistently choose a set of axioms that will allow us to create an algebra of probable reason that will turn out to be a superset of Boolean algebra. We thus could have consistently begun with this algebra from the beginning as our algebra of reason and demonstrated its self-consistency as the basis for a theory of partial reason and knowledge, which is - surprise - what we really have!
Philosophers could have made so much progress so much earlier, if they'd taken nearly every one of Aristotle's conclusions and said, ``I doubt that'' and inverted it and tried that instead. Not too surprising for a philosopher who asserted that women had a different number of teeth than men - and was married several times!
Axiom sets leading to complete and consistent theories of symbolic reason are acceptable for inclusion in our theory of knowledge. Note that this meta-axiom does not allege that a mathematical theory itself is knowledge of anything at all, but it is entirely permissible to postulate an association between some set of symbols and patterns of experience and thereby ``inherit'' a system of consistent contingent truths from the mathematics that one can compare to those patterns.
Our first use of the meta-axiom of mathematics is to replace the 0 and 1, false and true, non-being and being of ordinary logic with the ordered set of real numbers in the range [0,1], with the ordered set of ``degree of belief'' ranging from definitely false, through all shades of maybe, to definitely true, from absolute knowledge of non-being through all shades of certainty to absolute knowledge of being. Note that this mapping is not unique and that we don't care. All we need is an inexhaustible ordinal set of values, but this particular range is sufficient and convenient.
We have to ``bootstrap'' our theory of knowledge, which will eventually come to include Aristotelian reason as an embedded algebra, with Aristotelian reason, so we will simply acknowledge that the so-called ``laws of thought'' are meta-axioms that define the permissible development of certain classes of axiomatic theories. Yes, this is arguably already encompassed by the meta-axiom of mathematics above, but we are shooting for clarity here, not compactness, and mathematics itself is derived using the algebra of logic. We will (for excellent reasons that will be revealed as we go along) express them in their Boolean algebraic form (and in the process neatly sidestep all the issues associated with e.g. intuitionism and so on as being essentially irrelevant to our meta-axiomatic purposes).
We know we can write down sets of axioms with internal
contradictions; it is as simple as asserting
and 
as both
being true for any given axiomatic assertion
. Often, of course, it
is much more difficult to detect contradictions. Theories with internal
contradictions are inconsistent and it is easy to show that any assertion can be proven in such a theory using the laws of formal
logic, which one should correctly interpret as the statement that no theorem can be proven in such a theory.
It is equally simple to write down self-referential assertions such as ``This assertion is false'' whose truth value cannot cannot be decided, or to create ``paradoxes'' of various sorts (especially when using human language). Gödel basically has shown us that certain axiomatic systems are not both complete - established so that all possible theorems can be proven true or false - and consistent (free from internal contradiction).
We will rank axiom sets that are free from contradiction above (better than) axiom sets that are either inconsistent or consistent but incomplete because they lead to undecidable propositions, at least as far as reason-based knowledge is concerned. We already know that our personal knowledge of Brahman, the Universe, is incomplete, and not just in the sense of logic. However, we'd very much like whatever we ``know'' to be consistent - to make sense.
Whenever we encounter two sets of axioms that lead to the same state of objective knowledge of the real world (through our sensory stream and axiomatically driven rational thought process) we will consider the smaller set (in an information-theoretic sense to be later defined) to be the better one. This allows us to eliminate non-essential axioms, and to put strict limits on any suggestion that middle-earth exists as a disjoint reality. This meta-axiom is historically known as Ockham's Razor5.9 the unprovable but entirely reasonable principle that a simpler theory that leads to the same conclusions is better, all other things being equal.
Given two otherwise equally effective theories, one of which possesses certain symmetries or correspondances that make it more beautiful, we should accord the axioms that lead to the more beautiful theory as being somewhat better than the axioms leading to the ``ugly'' one. Often, but not always, a beautiful theory is also optimally parsimonious, but in the event that it is not, we should once again be able to use our judgement or intuition to adjust our ranking of theories, even if that judgement is inevitably ``human'' and ``subjective''. This meta-axiom simply recognizes that esthetics in fact does play a signficant role in the process of scientific discovery. The prediction of the existence of the positron in physics (for example) followed by its experimental verification shows its entirely practical importance; similar considerations drive the search for the Higgs particle even today.
Every occurence in our human experience is unique. A necessary step to the construction of any sort of parsimonious theory concerning the Universe requires the projective reduction of this experiential uniqueness, the construction of classes of experiences, and the ability to assign symbols and hypothesize rules that ``compress'' those classes. In order for logic, reason, mathematics, or language to be able to serve as a basis for knowledge, we must first establish a correspondance between each symbol in our theory and objects or operations over these projective classes.
That we can do so and that it works is a true miracle! In one sense, we never step into the same river twice, but if we wish to be able to reason about rivers in general or the Mississippi in particular in order to successfully avoid drowning we need to be able to do so anyway. This core meta-axiom is needed in order for us to be able to assert the most fundamental axioms of the reasoning process that leads us to a non-immediate knowledge of the real world, so do not undervalue its importance.
This will be a dangerous, error-fraught process. We will do well to remember that the map is not the territory5.10 , and keep in mind all the lessons learned from Plato's Cave where our projection process erased important details and led us to what amounted to a functional mythology, a false description of the Other that lay outside the prisoners' Selves. Nevertheless, without admitting this meta-axiom, it is quite easy to see that our system of knowledge will devolve to one kind of philosophical disorder - an instantaneous awareness of experiencing, with no ability to differentiate our sensory stream into past or present, left or right, back or forward, up or down, or identify spatially located, temporally persistent ``things'' at all!
Perhaps it is true that the semantic map we build in our minds is not the territory we expect it to represent. On the other hand, it is a territory all by itself! This ``map'' is, in fact, all that is real to us. It is us.
This observation represents (as always) opportunity and trap. We can
exploit it to gain tremendous insight, or misuse it and find ourselves
trapped in a religious war over an arbitrary assertion. The particular
assertion at issue here is: Is the Universe ``mind'' or is it
``matter''? This is one reason that the meta-axiom of semantics is so
important. It asserts that there can, at the very least, exist maps
between a set of symbols and something else (which may well be another
set of symbols, another map, or may be instead an objective reality, a
territory). Afficionado's of map more or less state that all
territories are maps and exist as a state of knowledge in the mind, and
then not infrequently take leaps off into solipsism or deism to provide
the ``necessary'' mind. Fans of territory argue that nature provides us
with a a physical reality that supports all of our so-called ``maps'' My
imagining of any symbolic proposition
is simply a peculiar and
transient set of electrical impulses, metastructures that have
arisen as a self-organized state out of the potential complexity
inherent in the physics of the ``necessary'' matter of the Universe.
At this point you should be wise enough to realize that neither viewpoint can exist without unprovable axioms that support both. It is undeniably true that your worldview will exist in your mind, but the worldview that makes the most sense is one where that mind is in turn consistent with an external world. There is no possibility of disentangling the two from the point of view of our Selves.
We will therefore make a meta-assertion that eliminates the differentiation and (we can hope) renders the issue moot. We will consider the fundamental reality to be information. This is not inconsistent with physics and materialism - physics assigns to each ``thing'' in the material Universe coordinates, and indeed makes the coordinate system itself an integral part of that which is. Physicists understand that when they solve a problem using symbols, that the map is not the territory, but at the same time to them the reality of the territory is indistinguishable from a perfect symbolic representation of it. That's the whole point of the game.
Idealists should also be able to live with it (or rather build an axiomatic worldview within it) because information is (if they think about it) what they've been asserting is ``mind'' all along. The trick, then, is to come up with a consistent set of axioms that allow us to understand information, in particular systems of information that self-consistently contain themselves. We've taken one of the great arguments of philosophy and reduced it to a problem in computer science once we've introduced an axiom here or there.
Our ``self-aware self'' of Atman, then, is a peculiar system of information, as is Brahman. Where that information is, how it is directly or indirectly represented, whether it is a map made out of territory that it maps or territory that contains metastructure that is a map of the territory itself or any other horrendous ``ordered'' statement seeking to give priority to one aspect or the other - none of these are necessary truths, but they can all be viewed as information, as a territory that is its own map, a map that is the territory. This will permit us to develop a ``materialistic'' view of the universe that is at the same time ``idealistic'', and eliminate one of the oldest, silliest, most bullshit-fraught debates of all philosophy.
If one examines the list above, one can see that the meta-axioms break up into several general self-consistent categories. Some of them appear to relate to the mechanical process of reasoning, its mathematics as it were, and establish the fact that our knowledge will be based on a formal reason of uncertainty that we must axiomatically derive as our first order of business.
Three of them pretty much say that we are only going to reason about the real Universe, in the decomposition of Self as that which is doing the knowing (whatever that might be) and Other which is one part of that which we seek to know. Our real goal is the knowledge of Brahman - of everything, both our selves and everything else. Hopefully we are wise enough to ``know'' that we cannot realize this goal as Self per se, and if we aren't it will be (provisionally) proven mathematically later in a cute twist of information theory.
One of them more or less defines the permissibility of the process of knowledge and reason, the creation of a symbolic projective reduction ``in our thoughts'' with a successful correspondance with our ongoing sensory stream.
Several of them appear to be very odd, at least by the standards of the usual boolean formulation of axioms, ``such and such is true''. They require us to use judgement to ordinally rank axiom sets themselves. We know that we cannot say ``this set of axioms is correct''; these meta-axioms give us a way of saying some axiom sets are better than others.
There are two more meta-axioms we will list separately. These are ``completely optional'' in the sense that neither of them can fundamentally alter any system of knowledge of the real world that we might build up using the system above. Axiom sets that are based on them clearly lose rank via the meta-axiom of parsimony - in both cases they add something that isn't strictly necessary from the point of view of ``simple'' correspondance of our axiom-based map and the actual terrain being contemplated through our reason and senses. In both cases they arguably gain rank from the application the meta-axiom of esthetics, which, we recall, intuitive and subjective. In both cases there are practical aspects associated with their use both pro and con.
Finally, I'm adding them as meta-axioms (where they will really function as axioms in any developed theory) because they can conceivably alter the selection of axiom sets through an ``esthetic''-like criterion and because they might lead to incompleteness or inconsistency in addition to a lack of parsimony, so their inclusion might well require alteration or removal of the meta-axiom of consistency (which opens up a can of worms, sure, but so be it). They are:
This meta-axiom is simple: Axioms associated with the concept we name ``God'' can be included in our worldview. Such axioms will generally, but not always, make the resulting worldview less parsimonious. They may, however, make it more esthetic or intuitive. Either way, a significant fraction of humanity explicitly includes God axioms in their worldview - sometimes highly elaborate and non-parsimonious ones that are egregiously inconsistent with the parsimonious axioms that otherwise lead to a highly consistent and satisfactory state of knowledge about Brahman. It would therefore be foolish to attempt a meta-axiomatic formulation that we can use to create a worldview that does not at least have room for it.
This meta-axiom leaves one with a broad range of choices for specific axioms of deity in your worldview. All of the following are possible ``prime axioms'' of deity that might be included before being dressed up (or not) with further axioms:
Note that atheism is an axiom of deity as much as rational deism or scriptural theism. Worldviews without the meta-axiom of deity are de facto agnostic - they neither affirm nor deny the possibility of God's existence, they merely omit any explicit consideration of the issue from their axiom set and leave it parsimonious, since it is always possible to explain any sensory experience either with or without an aware God as all or part of the explanation.
Also note that an equivalently parsimonious way of implementing the agnosticism that I would argue is everyboy's true state of knowledge with respect to God, is by asserting rational deism not in the form of an axiom but in the form of a proposition to which one initially ascribes a neutral degree of belief. One can then alter one's degree of belief on the basis of axiomatic reasoning, personal experience, intuition, esthetics, or humanistic considerations, just as one would if presented with any other proposition. Ultimately, you can believe what you want, just don't call the beliefs themselves ``reason''.
We will discuss axioms of religion in a later chapter in some detail. Here I wish to make only one or two remarks. First, it has commonly been asserted that belief in God is irrational or in some way contradicts scientific knowledge. Entire books are written promoting atheism and citing the success of a worldview without God as evidence of nonexistence, accusing believers in God of ``delusion'' and indulgence in other emotionally loaded forms of nonreason.
As we can hopefully see at this point, individuals who advance such an argument are simply using implicit meta-axioms - statements that cannot be proven and might not be true - to select the axioms they use to develop their ``reason''. We have also seen how a simple outer product of ``God exists'' with an otherwise untouched set of axioms establishing an e.g. scientific worldview is absolutely consistent. This means that no scientific evidence can bear on whether or not God exists! Note well - it can neither prove it nor disprove it. As was observed long ago by Hume, even the observation of apparently supernatural events may well mean that we don't have the right axioms for natural ones, we may be misinterpreting the perfectly natural shadows, while nonobservation of supernatural events is equally ambiguous in the other direction.
This sort of ad hominem ``science'' based attack abuses the meaning of the word ``reason'' and incorrectly draws conclusions from science that science is not axiomatically equipped to provide. Science, too, is based on ``faith'' (as I'm hoping this book has made abundantly clear) - reason itself follows from sets of arbitrary choices that cannot themselves be justified by reason.
Correct reasoning proceeds from axioms, and axioms are not
self-evident truth, they are provisional truth, a working set of
assumptions that cannot be proven. One can freely assert that God
exists and reason on that basis, or assert that God does not exist and
reason on that basis, or not assert anything either way and reason on
that basis. Reason does not, and in some sense cannot, apply to
process of selecting your axioms, at least not until you have
meta-axioms some of which will themselves be arbitrary. It is as
senseless as asserting that all triangles must have angles that
sum to
radians simply because you can prove it from the axioms of
plane geometry. The correct statement is not a necessary truth, it is a
contingent one: ``If I make the following definitions and assumptions,
then I can rationally prove that the sum of the angles of any triangle
is
radians.'' If this is true for a subject with the rational
purity of mathematics, how much more true for the statement ``If I make
the following, far larger set of definitions and assumptions, then I can
prove that God does not exist.''
Here is why I think that humans are esthetically inclined towards at the very least agnostic deism, and why even many scientists (myself included) often ascribe a positive degree of belief - that it is likely true that the Universe itself is in some sense self-aware - even as the Deity thus revealed is nothing at all like the athropomorphic gods axiomatized in most established scripture-based religions. I can easily imagine there being nothing at all by a process of subtraction applied systematically to everything that I am aware of (not unlike our earlier exercise in doubt, but one that goes further and removes from ``existence'' everything we doubt as we encounter it). We can doubt that all the objects of our senses exist. Throw them out. This leaves us with our sensations - but I can imagine not experiencing sensations, so out they go. Memory is a simulated sensation, so I forget. Even my thoughts I can imagine quieting, one at a time, until I no longer think about ``things''.
The one thing ``I'' cannot do, is imagine my self not existing, because I am not a thing, I am a process and imagination is a contingent process. There is an irreducible difference between Self and Nothing that is not ``sensory'' in the usual sense - even when I think no thoughts about Other I don't cease to be as there is a thought I cannot unthink. As long as that ``I'' remains, the existence of nothingness is literally unthinkable.
Is that ``I-ness'' the process of information processing, the core loop at the heart of self-awareness the irreducible property of existence? It is difficult to say, and certainly cannot be proven. However, most of the people I know who do believe in God do so not because of a causal argument (which requires that we assume all sorts of silly things about time and what the word ``cause'' really means, most of them inconsistent with their use in physics) but because of a time-independent, an intuitive and existential argument connected to ``I-ness'', one based on association, not logic, one that appears to be the very heart of certain Upanishads and the very axiomatically minimal pronouncements of Buddhism on the subject.
``Nothing'' is a logical possibility and we can come close to imagining it, we can build semantic symbol sets and define a mathematical view of nothing such as the empty set, or throw out even the empty set so we are left with no set theory at all, the Void, the converse of existence itself, the complete lack of information, not even a single 0.
Yet there is something - minimally my Self - that contradicts nothingness. There is reason to believe (as we build axioms for worldviews that seem to work) that my existence is contingent on the existence of something else and not the other way around, something that is ``conserved'' and persistent over time and space as I axiomatize everything consistently with my experience. Yet my own existence seems to have an irreducible information component associated with awareness itself. My information content is already self-aware. That information content is only a part of the information content of Brahman. It seems reasonable, then, that the awareness of Brahman is not less than my own, and indeed includes mine as a very small part of its greater whole. Let me, then, identify Brahman as the self-aware Universe, an entity whose information content is equivalent to, and sustained by, its own awareness of that information content.
Note well that adding this axiom alone alters none of the conclusions of a parsimonious worldview to which it is attached. Nor is it (with no further axioms) any less parsimonious. It simply postulates a self-aware Brahman whose ``thoughts'' are all the information there is whether or not that information is itself viewed as being sustained by ``mind'' or by ``matter''. The two become indistinguishable - mind is matter is mind - a point of view that consistently echos our own, fully axiomatized, understanding of our own Atman and hence has a compelling esthetic.
In any event, you hopefully can see clearly why it is important to have a meta-axiom of deity. Propositions concerning God are bound to come up. One of our favorite human pasttimes is to argue about the metaphorical equivalent of whether Charlie prefers us to self-flagellate with a codfish or a trout, or whether it is to be done on alternate Tuesdays or every Tuesday, and then run out and scream ``Jihad!'' and slaughter those that disagree with us lest they seduce our children away from the axiomatically true faith, corrupt our women5.11, and ultimately cause Charlie in a vain moment to Unmake our world. If nothing else, we must understand to role and limitations of reason, in particular where and why the fundamental argument is over the axioms, the unprovable beliefs, and has nothing to do with ``reason'' at all.
The meta-axiom of deity arises very naturally from our attempts to imagine the Universe as it might exist without awareness. Any such attempt ends up with neither more nor less ``reality'' than ``the set of all functions on the real line''. We can write down the words, we can imagine a few examples drawn from the set, yet when we really try to imagine it, we simply cannot. It is so large, so absolutely homogeneous, it is absolutely devoid of any meaning
Again, this isn't really a metaphor. A perfectly mechanical Universe seems like it can be broken down into a set of information that can be reduced to a very large (possibly infinite) convolution of over all of the similarly infinite set of all possible functions on the real line, with some mapping made between the symbols in those functions and the mechanism. Ultimately, we arrive at a very large (possibly infinite) integrodifferential form that represents the stationary union of all world lines for all the components that aggregate to form the Universe. ``We'' may be encoded into all that information somewhere, but the result is as cold and sterile as an unplayed DVD sitting in the exact center of infinite, empty, space for all of eternity, unmoving. Even if we knew the precise contents of every bit of ``information'' on the DVD, there might well be higher order information encoded on it that can never be interpreted without a decoder.
For us to make sense of it with our own personal Atmans, the information has to be decoded within our Selves. We, in turn, do not live in a vaccuum; we live in a complex structure of society and family and economy and ecology with rules that are an amazing, dynamic, self-modifying, evolving, self-organizing, chaotic, violent, ethical, loving, braiding of intermediary sets of structured rules and meta-rules that sure - are ultimately supported by the elementary particle dynamics of their most microscopic constituent parts.
However, the 37th electron from the left in some protein in some cell of the skin near the tip of one of my fingers, whose coordinates form one tiny bit of the information content of the whole Universe, follows rules that in no way resemble the real rules that are causing me to internally articulate these rather abstruse metaphysical speculations and volitionally push that finger down against the ``l'' key on my laptop (yes, the one I hit in between the preceding quotes, not the one in laptop) so that this particular electron exerted one tiny part of the total force that activated the key that triggered a most unlikely chain of electronic events in an object that I truly believe can be said to have ``co-evolved'' with humanity to record a piece of self-referential truth that somehow - through a chain of highly ordered intermediary steps that any sane person would have bet against at several trillion to one only a few instants earlier - are causing electrons in your brain to perform a peculiar little dance that trigger biomechanical gates to open and close, affinities and proteins to ``permanently alter'', as ``you'' start understand and henceforth remember the point that I'm making in this absurdly long sentence (that is also a paragraph).
In other words, a strictly mechanistic interpretation of ``knowledge'' is certain to end up woefully incomplete even if one precisely knows the underlying mechanism. Any five year old child knows what a clock does and how to interpret its state. Most full grown adults couldn't begin to build one, or take one apart and fix it again. One could imagine assembling robots that built absolutely perfect clocks. One cannot imagine the robots knowing ``how to tell time''.
This is the more is different rule of complex systems5.12 In physics we often discover that when you take a bunch of very simple things and put them together, structure spontaneously self-organizes and new composite things emerge with their own rules. You put those things together in turn and new things with still more different rules appear. Quarks and gluons build into nucleon. Nucleons build into nuclei. Electrons and nuclei build into atoms. Atoms build into molecules. Molecules build into crystals, into solids, into gases and liquids, into stars and planets and seas and land. Molecules build into proteins, and proteins start to self-replicate. Self-replication turns out to be self-sustaining and because of the mathematics of a stochastic selection process rapidly diversifies until self-replicated bags of protein stand up and walk. Along the way, they develop a marvelous ability to carry within themselves an image of the world they walk through, a semantic map. They develop volition, society, philosophy.
In the end, they create whole languages of complex rules that formalize the semantic reconstruction and transmission of self-similar structures that are doing their best to capture the essential information content of the original Unverse of elementary particles and project that information back to the ``truth'' that casts all of the shadows, information echoing information. This high level knowledge, rules organized on an entire tower of structural re-orderings, bears no resemblance whatsoever to the simple, parsimonious ruleset that physicists seek today to consistently describe that original truth and allow us to chart forward at each level and at least in principle understand the emergence of new structural truths.
Those high level rules are important. They absolutely constitute a part of the ``knowledge'' we hope to recover. Ultimately, they affect actual matter as my pounding fingers and the future course of your own life (after you've read all of these words) well attest. It is entirely possible that the ``ideas'' that are being conveyed will convince people to stop fighting about religion, agree on a common set of axioms to govern human knowledge and affairs at all levels of human endeavor and that twenty years from now we will all live in peace ``because'' (in very small part) of my 37th electron, somewhere in the tip of a finger's continuing efforts. It is possible that if ``I'' don't keep typing, and in the future take many highly complex steps far beyond the realm of physics to ensure their publication and dissemination, that twenty years from now the planet will be a radioactive wasteland with a vastly reduced human population.
I therefore introduce the meta-axiom of humanism. It basically states that it is permissible to judge axiom sets and worldviews from a human point of view, and to introduce propositions that lead to theorems that can create systems of ethics on top of our systems of more microscopic knowledge. We can judge propositions romantically. In fact, we can use human judgement with its strange mix of reason and intuition and wishful thinking to define high order structures and thereby create them. Our only constraint is that the overall result has to be rationally consistent, or at least as rationally consistent as we each (one at a time) can make it within our own worldview. That is the path of reason, where we know that unreason is in fact the only real devil that plagues human existence5.13.
In one sense the meta-axiom of humanism alone is sufficient to admit axioms of deity. Deity is, in its purest essence, the assertion that there is non-accidental high order meaning in the sea of information that makes up Brahman, higher order meaning than I can personally discover, meaning that embraces everything I know and think and feel, every event everywhere in the Universe, so that a symphony is not just a pattern of 0's and 1's burned with a moduluated coherent light beam onto a shiny disk, not even just a peculiar dynamical state that persisted in a body of air and described (approximately) by various well-understood physical laws, but is music, is (among many other things) the way the music makes me feel, is my memory of the music long, long after it was played, is the experiences I had that before listening that made me receptive to the music and capable of being moved by it.
It admits still more. Humanism allows us to formulate absurd axioms such as a ``right to life, liberty, and the pursuit of happiness''. Are such things ever ``true'' or ``false''? Of course not. They are propositions being offered up as a provisional basis for certain kinds of reason. We will judge them in part by how well they work. In that sense, they are no different from the axiom we call ``the law of gravitation''. This too, is an axiom, an assumption, a provisional fiction. It too is judged, ultimately, on how well it ``works'' in an entire system of axioms making up a worldview. It too is (as we now best understand it) simultaneously true and false within our most functional systems - it is an excellent approximation of the truth, but is not yet consistent with other things we believe to be true and hence likely false. So, too, a ``right to life'' is inconsistent with the theory of evolution, our very strong belief that all things die, our occasional need to kill both animals and humans - and yet is still in the right context a beautiful, romantic truth, one that can be used with other similar axioms as the basis for contextual ethical reasoning.
More amazing, the assertion of the axioms created the context. Humans can create new structures of knowledge with reason. Or rather, with a mix of reason, experience, and an entirely unreasonable process of intuition and romantic endeavor. A reasonable man would never have written down these words, but rather would have stuck with the far more bleak (and honest) assessment of Hobbes that life in a state of nature is ``ugly, nasty, brutish and short''.
Any theory of knowledge that denied us this creative impulse, that seeks a purely mechanical, meaning-free knowledge of Brahman, is one that may well be wildly popular with linear thinkers but that ultimately will be sterile and unproductive. The meta-axiom of humanism must be used with great caution in our construction of a common worldview because it can be used for evil ends as easily as for the good. Not an easy thing to do since to do either one it must first define the good, and must do so in a way that minimizes internal inconsistency and conflict with science and mechanism and remains reasonably parsimonious (simple and easy to understand). It also opens the way for axioms of politics, and of course there are many groups with many dogs in many fights in all of human affairs.
All in all, this meta-axiom will work best when it is teamed with many axiomatic expressions of the meta-axiom of doubt and open-mindedness, particularly as expressed in (for example) the Bill of Rights in the United States Constitution.
We will therefore begin our list of meta-axioms with the so-called Laws of Thought5.14 These laws have been around for over 2500 years, being formulated in the West (it is believed) originally by Parmenides5.15 who likely influenced both Plato and Aristotle. Aristotle wrote down a triplet of laws that were accepted as the basis of logical reasoning for thousands of years.
In the East they were also writing down theories of knowledge with rules for inference5.16 5.17 that were roughly contemporary with those of the Greeks but that differed somewhat in their purpose in directing thought.
For now, however, we'll stick with the Greeks. Aristotle's version of the Laws of Thought are:
These laws need to come with a ``warning label'' to read before using them, one that is somewhat contrary to accepted beliefs in the theory of logic. We want to use these laws strictly as meta-axioms. This is how they are used, practically speaking, in mathematics - one develops mathematical theories from axioms in accordance with these rules, rejecting theories that contain contradictions or holes as being ``worse'' than theories that are consistent and complete. We will not use them directly on any thing, though, because the only thing we have to use them on (so far) is our Self!
Even after we find reason to believe in things again, though, we will avoid applying these laws directly to them. This is out of sheer intellectual conservatism. We will do so because it is not, actually, perfectly clear that these laws should apply to ``things''. Sure, many people find might it ``inconceivable'' that a thing can be and not be at the same time, or that a thing might not be itself. On the other hand, philosophy has been plagued from the beginning by the very arrogance and surety of philosophers, and excessive belief in anything places one in intellectual chains.
For example, it is quite possible to find circumstances in quantum theory that can be interpreted as a thing ``being and not being'' at the same time, and to this day overcoming this ``classical'' mental prejudice is a major hurdle for student of physics who seek to learn it. Quantum theory similarly blurs our concept of identity in, for example, a Bose condensate where identical particles lose their individual identity and form a collective one. In recent years the Law of Excluded Middle has come under attack even from logicians, and an entire (intuitionist) theory of logic that omits it has been developed. These examples suggest, if nothing else, that it is easy to find real systems where the naive application of the Laws of Thought is misleading and actually hinders one's search for understanding.
Another major problem with ``inconceivability'' being the criterion for falsehood is that it is hardly a constant or universal condition - it is essentially subjective. Students begin thinking of quantum theory as being inconceivable because they cannot yet conceive it and it is inconsistent with their classical experience, but end with at least some sort of conception based on new experience. Similarly, what I find inconceivable might not agree with what you (or nature) find inconceivable.
Our search for knowledge seems unlikely to end up giving us truth, but it might give us wisdom. Wisdom can often be found outside of the dry pages of philosophy and logic. Let us recall just two little bits of wisdom, first that ``There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy'', second the scene in The Princess Bride where Vizzini announces that thing after thing done by Westley (in the person of the Dread Pirate Roberts) is ``inconceivable'', finally leading Inigo Montoya to observe ``You keep using that word. I do not think it means what you think it means.''
These neatly summarize objections to the idea that anything (but our own non-existence while we are experiencing our existence) is true or inconceivable. It seems sensible to consider conceivability not as a necessary condition for truth, but rather a practical one, and further agree that the Laws of Thought are more a recipe for logical consistency in symbolic reasoning (and hence meta-axioms) than axioms that can or should be applied directly to our sensory stream, where we will require both meta-axioms and axioms galore to identify any sensory experience with a presumptive ``thing'' that can be represented by a symbol.
Since we plan to use logic to evaluate our entire system of axiom-based beliefs as we develop it, and since this system will necessarily involve a lot of semantics and manipulation of symbols, we would do well to make the evaluation process as mechanical as possible.
This can most conveniently be accomplished by using the formulation of logic written down by George Boole, who transformed Aristotle's laws into a formal algebra of two-valued symbols. This algebra reduces tedious proofs in logic to a form of numerical computation. Boolean logic is the basis of modern binary computing, where it governs both the logic of program flow and arithmetic (where things like integer arithmetic are reduced to a complex axiomatic schema of boolean operations on ``bits'' - binary digits - that are elementary boolean variables).
We will find the following boolean algebraic form5.18 of the laws of thought to be the most useful. In it,
stand for propositions that can take on the two ``truth values''
T
and
F
, 
means logical and, 
means logical or, 
means negation (a unary process that switches
T
to
F
and vice versa as shown), and 
means logically
equivalent or ``has the same truth value, whatever that might be'' for
the expressions on either side.
F
and
F T
and
T
and
F
T
T
and
and
and
and
and
Of these, identity, contradiction, excluded middle, together with commutativity, associativity and distributivity are considered axioms and the rest (and of course far, far more statements named and otherwise) can then be derived as theorems, but this partitioning is somewhat arbitrary and not unique.
Note that these laws, plus suitable axiomatic extensions specific to each special case, are the meta-axiomatic foundation of pretty much all set theory and mathematics. I say meta-axiomatic because (as noted) they apply to propositions, not things - they are used, among other things, to reject possible algebraic (abstract) axiom sets when it is shown that they lead to internal contradictions and inconsistencies. We are about to embark on a fairly complicated process of bootstrapping the derivation of a set of algebraic laws that generalize to the (suitably axiomatized) ``real world'' and which will supercede these rules. These new laws will in fact end up including the old ones as the special limiting case of certain knowledge of unconditional truth or falsehood, a limit that will never be realized in the real world for anything but our direct, ongoing empirical knowledge of our Atman.
Things are looking good. We are now armed with two things with which to fight the PED: the certain knowledge of our own Atman, and a framework that can be used to determine logical consistency of symbols that can take on two values that we can interpret as true or false (or, just as easily, as 1 and 0 in a system of plain old arithmetic, so don't get carried away with any implicit semantics). In order to use the latter to reason, though, we have to have something to reason about.
So far, the only thing we have to reason about is the dizzying array of shadows being percieved by our Atman on the walls of our own personal cave. By the time we are old enough to remember anything at all, these shadows have self-organized into certain fairly consistent categories (where I make no claims to exhaustivity or exclusivity):
A huge portion of our experiencing of awareness, as one can see, is associated with the abstraction of patterns. Our experiencing a mix of direct sensation and memory and thought and dream and insight generates propositional relationships that gradually organize into the appearance of an outside world that is associated with all of these sensations and experiences, an outside world with a very distinct structure.
This possible illusion of an outside world is so strong and compelling to our senses that we are usually at least teen-agers before it occurs to us to doubt it, to wonder what (if anything) lies behind its existence, to wonder if ``reality'' is indeed there at all or if only our thoughts and apparent sensory experiences of it exist, for that is all that filters inward to our Atman.
So, before there is any point in choosing any axioms about the real world, we need a meta-axiom to answer the question: Is there a world outside that corresponds to our perception?
This is a tough one. We cannot use any sort of evidence to answer it using pure reason; trust me - no matter what is advanced as an ``argument'' begs the question because if there is no outside world, our evidence of one is all illusory. Basically, this decision is a completely free choice that each of us must make for him or herself.
That doesn't mean that we can't have reasons for our choice; only that those reasons are at best a form of consistency, not deduction from known truth. For example, a common reason given for believing in an outside world is that things appear to happen there that ``we'' do not control with our own volition. It causes us pain, it surprises us. It fails to grant us all of our wishes. If there is truly nothing outside of ourselves, then we somehow are intrinsically dual, with one part telling a detailed story of an apparent world and the other forced to endure the story as a player.
Even so, there are those who (seriously or not) propose this as a philosophy. It is metaphysical solipsism5.19 , and of course it cannot be refuted by means of logic alone. This, in turn, logically implies that no other theory that contradicts it in any way can be proven by logic alone (because if it could, it would suffice to disprove solipsism see the law of excluded middle), which pretty much takes care of all the bullshit theories proposed by philosophers from the beginning of time right there (and is actually a fairly succinct statement of the entire book to this point, come to think of it).
Fortunately, we are dealing here with meta-logic, not logic; wisdom, not knowledge. I don't know about your dream, but mine has been more than a bit painful. Furthermore, it has behaved as if it would if there was a real world out there. If you choose to believe that your Atman is all that exists, that you are alone, that all that you appear to observe in the Universe is a dream that you alone are dreaming and control, permit me to suggest that you dream up this book, or a table-thumping Samuel Johnson5.20 , or a Zen Master who is impatient with that sort of bullshit and who can give you an entirely imaginary whomp upside the head that creates enough of the sensation of make-believe excruciating pain that you convince yourself to - just for the heck of it - believe that reality is real before you inflict on yourself something that really hurts when you fail to treat your own dream with the respect that you apparently demand of yourself5.21.
Solipsism of any flavor is a dead end. So even though in reality we are technically epistemological solipsists when we make the observation that our sensory impressions do not have a logically necessary connection to an outside world, to make sense of those sensory impressions we'll either end up picking axioms that create an understanding of them just like the one we'd arrive at by assuming that reality is real or we'll end up imagining ourselves behaving as if we were psychotics in the outside world we are imagining, complete with imaginary kind men in white jackets with their imaginary thorazine. Or we'll end up imagining ourselves walking in front of rapidly moving cars because the cars aren't really there and stop imagining things at all, quite abruptly. Imagine that!
I'm going to strongly suggest that you adopt as a meta-axiom that there is, in fact, a real world external to your Self that is the object of your senses. There is nothing ``transcendental'' about this suggestion; it's just that if you don't there is little point in continuing to imagine yourself reading this book for reasons that are left as an exercise to the reader's imagination.
Ah, I see that you're continuing. I'll take that as acquiescence. All our future axioms will revolve around an external reality that exists in some association with our sensory experience, and will subsequently limit our imaginings to be about that fundamentally presumed real world. Only thus can the chains that bind us within our personal cave be removed.
Here is a list of additional meta-axioms that we will use at various points in future chapters to help us ordinally rank axioms proposed to help us understand the real world, in no particular order. They don't require justification, of course - you are welcome to reject them or consider alternatives and I actually recommend that you do so if only as an exercise - but I'll still say a few words about why I choose them. Some will have quite profound consequences.
For example, consider our most basic argument in ordinary logic:
The latter statement is true enough, but
have equally well stood for
any assumped truth, include ``God does not exist''. If we
admit unneeded but logically permissible conjunctions such as this, two
things will happen. First, people will be able to create a special
class of axioms that lead to what I call ``fairy theories''. This kind
of theory claims that anything you wish to explain is explained by the
mechanism of invisible fairies that make it happen. Fairy theories,
like solipsism, are always logically defensible, but we will
reject them anyway if we have a non-fairy theory that gets by with fewer
assumptions. The second is that sure as God made little green apples,
somebody will add the word only and reason backwards: Only
if God exists can we understand how pigs can fly, pigs fly, so God
exists. We'll just put a stop to all this right here. If we can
squeeze axioms out of a theory and get to the same conclusions, we will
consider the ``gooped up'' theory to be worse than the smaller, cleaner
one.
This is a modern statement of Ockham's Razor5.23 the idea that a simpler theory that leads to the same conclusions is better, all other things being equal. Note well that this meta-axiom isn't to be narrowly or foolishly applied - we should fully realize that there will be grey areas where competing theories cannot be resolved by our current state of knowledge even as we judge simpler ones to be a bit better, all things being equal.
There is one other very important set of meta-axioms we want very much to ``import'' so that we can use them without restriction. Those are the axioms of mathematics. In particular, we want to build a quantitative system of beliefs about the real world. We already knew that we were going to need this when we decided to use plausibility, or believability to rank-order potential axioms to use as a base of knowledge about the real world. If we use (for example) real numbers to describe plausibility and attempt to build an algebra of plausible reasoning, we then inherit the concepts of greater than and less than that we can use to sort out propositions in some sort of order.
This algebra will have a structure that is very similar to the boolean algebra above. Very very similar. For our next little burst of meta-axiom and axiom selection, we turn to work done by Richard Cox5.24 , E. T. Jaynes5.25 , and Claude Shannon5.26 and the algebra of inference.
Socrates - And surely this instinct of the dog is very charming;-your dog is a true philosopher.
Glaucon - Why?
Socrates - Why, because he distinguishes the face of a friend and of an enemy only by the criterion of knowing and not knowing. And must not an animal be a lover of learning who determines what he likes and dislikes by the test of knowledge and ignorance?
We are now ready to begin our quest to recover knowledge in earnest. We have decided that we are going to use the meta-axioms of logic and mathematics as the basis for reason itself. We have decided to begin with the fundamental prior meta-axiom that the Universe we appear to perceive has an external objective existence - it ``is not us'' even though our perceptions of it per se ``are us''. That is not to say that our final conclusions need to be dualistic - nothing will prevent us from forming the unions of our knowledge of Self and our knowledge of Other and forming the knowledge of All, so to speak. It just gives the reasoning process a direction - we will reason from our certain knowledge of Self to an axiomatically supported knowledge of Other.
The way we must proceed is to introduce axioms concerning
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Copyright © 1993, 1994, 1995, 1996,
Nikos Drakos,
Computer Based Learning Unit, University of Leeds.
Copyright © 1997, 1998, 1999, 2000
Ross Moore,
Mathematics, Sydney.
is true as well'', where
is any proposition
that does not ``couple'' to the assertions and proofs in what one
already knew.
