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Thought and the Joy of Sets

Here is at least one way that the Laws of Thought are commonly stated:

Where I deliberately use various versions of the verb ``to be'' conjoined with ``negation'', the version of these laws given by modern logicians might use ``be true'' and ``be false'', although these are not precisely the same thing. Truth and Non-truth are much subtler concepts than Being and Non-Being, and the latter concept pair is subtle enough to cause us no little amount of difficulty.

Note well that I say ``modern logicians'' in the paragraph above. In some very deep sense mathematicians and philosophers and logicians2.13 have advanced formal logic itself more in the last hundred or so years than they did in the previous indefinite interval of time preceding the late 1800's or early 1900's. Yet historically, systematic analysis of our understanding of things began with these simple Laws of Thought as expressed by the Greeks. That's an excellent reason to postpone until a later chapter the discussion of formal logic per se and concern ourselves with the existential analysis of a Real Universe using the Laws of Thought themselves. After all, our ultimate concern is with knowledge of a Universe that actually exists, and the rules of logical inference are perfectly happy establishing abstract relationships between abstract symbols, often relationships and symbols that ``exist'' only in our imagination, that shadowy realm of looped experiential reality where we can hallucinate, dream, and think of six impossible things before breakfast!2.14We will therefore begin our examination of Thought with the concrete instead of the abstract.2.15

We will find that the concrete (as abstractly expressed by the Laws above) is a bit mushy and hasn't quite set yet, but that it contains a few hard nuggets that somehow must form a foundation for all knowledge.2.16 Let us begin like proper scientists, even though we really don't know what it means to be a proper scientist yet, by being very, very skeptical. I mean, like, cynical. Mistrusting. Doubtful. We will try to understand the Laws of Thought in some sense that doesn't require words or other symbols by chopping the words themselves apart and attempting to realize the concepts in a way that is more existential than analytical.

To put us into the proper frame of mind, let us begin by doubting all of these Laws. The fact that they are usually presented as self-evident truths should not deter us. First of all, they are not self-evident truths - we haven't even figured out what they mean as they are expressed in English (or for that matter in Greek, or Hindi, or Hopi).2.17 Second, a point that will be relentlessly hammered in this text is that historically, a tremendous number of advances in e.g. mathematics, physics, psychology, sociology, and really pretty much all of human intellectual endeavor have come about when people rejected ``obvious truths'' and explored the consequences of alternative ``truths''. Being a physicist and knowing way more than is good for me, I can even express why this is so in terms of optimization theory on a rough (non-convex) landscape.2.18 ``Truth'' is often analyzed in some convex region of this ``logical landscape'' in such a way as to effectively ``lock'' analytic conclusions in some reasonably self-consistent way to some optimum, where making small changes in some logical argument yields immediate change away from optimal truthiness2.19 , allowing proponents of the accepted optimum to assert its obvious and unquestionable correctness.

However, then along comes an iconoclast who makes a ``long jump'' in the underlying propositional space on the basis of some heretical intuition or another, and the next thing you know all of human understanding is reorganized around a new optimum - at least until the next brilliant iconoclast comes along. Hence the Copernican/Galilean revolution, hence the Newtonian (classical) revolution, hence the non-Euclidean and non-Aristotelian revolutions, hence the ongoing Gödelian revolution, hence the Relativity revolution, hence the Quantum revolution, hence.... you get the idea.

Interestingly, as a teacher and researcher, I have long observed that it is dangerous to know too much about any subject if one wishes to advance that subject, just as (paradoxically enough) it is dangerous to know too little. For example, physics (say, classical Newtonian non-relativistic physics) is generally presented to impressionable young students in an overwhelmingly logical way. A logical calculus of being is developed and carefully connected to everyday observations. Objects are assigned coordinates and causality is formalized in terms of differential equations of motion that are so overwhelmingly perfect and internally consistent that the observation of an incredibly strong correspondance between this system and everyday experience seduces the mind into saying ``Aha! This is it! Truth!''

If this isn't enough, we enlist the Brain's own biochemistry and hormonal system, designed to help us survive in a hostile world where fight or flight can become necessary at any moment by punishing and rewarding students to a greater or lesser extent in the various versions of Academe around the world according to how well they master this system via conditioned learning as if it really were Truth.

But of course it is not, only an approximation that appears to work because of the relative scale of a single constant, because of our silly insistence on viewing time as an independent variable indexing causal time evolution instead of a spatial coordinate of essentially static events, because of our insistence on the idea that ``objects'' exist at all at specific points in space and time with certain ``obvious'' requirements of continuity on the underlying functional description of that ``existence''. All of which, alas, turns out to not be the case.2.20 Small wonder, then, that we have to wait for students to come along who are a bit lazy, who are rebellious, who are too stubborn or stupid to be properly conditioned, who are unafraid to go all the way back to the beginning and start over making different basic assumptions in order to precipitate one of these conceptual revolutions.2.21 That which is imagined to be ``known'' is a trap to our imagination from which we can escape only by dint of immense effort and a certain amount of pain.

When it comes right down to it, Truth is pretty much Truthiness. Truthiness is a quality of knowledge assigned on the basis of intuition or instinct - ``from the gut'' - instead of on the basis of rigorous logical analysis and connection to things like facts. Very shortly we'll have you hanging out over the existential Pit of Despair, where we will make the true but unprovable assertion that logical analysis and what we consider to be ``facts'' are inventions of our intuition, possibly supported by our instincts, and hence logical analysis itself is illogical. With that understood2.22 we will now proceed to analyze the truthiness of the Laws of Thought.

What kind of beast, we might start by asking, are these Laws? No matter how self-evidently correct you might imagine them to be, ultimately they are semantic assertions in a language, and (like thought itself) exist only in the realm of our imaginations. To ``do'' anything with them we must imagine a set of definitions and a formal reasoning process in a symbolic language in which the Laws are assumed to be constraints on the symbolic objects about which we wish to reason. Ultimately, such an imaginary reasoning process will take a given set of imagined (and hence conditionally true) premises and reason ``correctly'' according to the rules to some given (necessarily conditional) conclusion, where we defer until later just what this process is and how it works.

Since we are making the whole thing up in our heads, we can therefore perfectly reasonably ask whether or not we can equally well think up other definitions and rules for a a symbolic reasoning process constituting Laws of Thought in which the imagined objects of the reasoning process are not required to strictly either ``be'' exclusively or ``not be'', with no middle ground possible. Maybe we can, maybe we can't, but there is no harm in trying.

Although we won't discuss the details until later, we already have what we need to develop the primary ideas of this book, neatly captured in the observation that the process is imaginary and truths are at best conditional. This means that there is a fundamental rational disconnect between abstract rational systems and the real world that can only be bridged by means of a system of axioms that permits us to relate our symbolic conclusions to our instantaneous experience.

Subject to this whole raft of assumptions, if you are reading this at all you are old enough and wise enough to know that the real reason we adopt the Laws of Thought is that they seem to work to describe our experiences trapped as memories of that which we call the ``real world''. It seems worthwhile then, to look at some of the details of that real world as revealed by our experience in the form of experiments, as well different sorts of experiments with alternative forms of reasoning systems. Our goal will be to see whether the Laws above are really assumptions or whether they are instead necessary truth - not assumptions at all, but rather things that cannot be conceived of as being any different.

Let us begin, then, by looking at a wee bit of physics.

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