... one1
Wikipedia: http://www.wikipedia.org/wiki/Wikipedia. This is a Wikipedia reference. In an online or active version of this book, this footnote becomes a working ``hot link'' to a useful Wikipedia article. For people reading a paper copy, I can only hope that either you are already amazingly literate, well read, and know offhand all about that of which I speak, or that you read this book somewhere near a web browser.

By the way, it should become clear from my frequent use of this as a Universal Resource that in my opinion Wikipedia is well on its way toward becoming the crowning achievement of human civilization - literally an online, free repository for all non-encumbered human knowledge, such as it is. 

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... ``42''2
Wikipedia: http://www.wikipedia.org/wiki/42. See? All human knowledge. Try looking up ``42'' on Encyclopedia Brittanica and it will laugh at you and return all sorts of irrelevant facts from World War II. 
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... Pratchett3
Wikipedia: http://www.wikipedia.org/wiki/Discworld. Author of the Discworld novels and a perfect master of all that is in this work and then some. In fact, Terry Pratchett could be the world's greatest living philosopher. Scary, that. 
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... ``Enlightenment'',4
Wikipedia: http://www.wikipedia.org/wiki/Satori. Or ``satori'' - where in proper Zen Buddhism satori is a word that perhaps better translates as ``transient epiphany'', an experience of deep understanding that may well last but a moment before life reaches out and drags you away to deal with kids and TV remotes. We'll shoot for something that lasts a bit longer, but be satisfied with what we can get. 
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... Zen5
Wikipedia: http://www.wikipedia.org/wiki/Zen. This is one place to learn a bit about Zen, if you like. Alan Watt's lovely book The Way of Zen is another.

Please understand, however, that however much we will talk about some of the ``philosophy'' underlying Zen practice, this is not a work on True Zen. Or even Fake Zen. It is a work on philosophical existential metaphysics, the fundamental basis of knowledge itself. Pretty serious satori, as far as that goes. 

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... actually6
Especially the student whacking part, which, as a professional educator with students who often blow off doing the massive amounts of homework I assign, I cannot help but admire. 
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... Enlightenment7
Or even understand that they should try. Or not try, since another precept is that if you try you won't succeed - Buddha succeeded only after he stopped trying. The key thing is to not try just right after spending years of your life trying to not do the things that I cannot tell you don't work. Or is that one too many negatives? Damn... 
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... exercise8
And not a terribly easy one at that, at least if you try to maintain this state of outer awareness for more than a few seconds at a time. 
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... cream9
OK, fine. Put down the book and get yourself a bowl of ice cream and then come back. After all, your brain consumes 1/3 of your total calorie input every day, and you're about to use yours a lot so a little extra sugar, chocolate, and fat can't hurt. 
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... process10
Sorry, but there will be a rather lot of this sort of thing in this book. Can't be helped. This book is about what you know, and cannot be read without thinking about what you are thinking about. Computers (as purely logical entities) tend to get trapped instantly into infinite loops by this sort of recursion; humans don't. Something to think about... 
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...actions11
Such as getting up and swatting your children on the bottom - with a banana - and removing the TV controller from their greedy little hands so you can actually concentrate on reading this book and avoid all possibility of emergency room trips and stitches. There. Isn't that better? Nothing like a Zen ``clearing blow'' to guide young and chaotic minds... and help them learn to act in ways that don't have a significant risk of injury. 
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... fitting12
At least it is meet and fitting according to an axiom-based, unprovable, value system that we have yet to overtly select or describe. You may disagree. Your brain may also be (mis)wired in such a way that it doesn't do terribly well on the ordinal sorting and emotional thing so that for you there isn't much difference between being hungry or full, having healthy safe children or children dripping with their own blood. If so, evolution will eventually sort this sort of thing out... 
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... channel13
Or even hard core. I actually couldn't tell you what kind of pornography is on the Playboy channel because naturally I've never visited it. Maybe it just shows tastefully done short videos involving puppies and butterflies and fully clothed Amish farmers. 
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... water''14
Wikipedia: http://www.wikipedia.org/wiki/Koan. This link (at the bottom) has a whole lot of links to online Koans, or ``Zen Puzzles''. Most of them are really pretty silly, but there they are. The ``best'' collection of Koans and commentary that I've thus far found is ``Zen Flesh, Zen Bone'', a collection of four primary Zen and Pre-Zen collections with commentary, by Paul Reps and Nyogen Senzaki. 
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... special''15
Yes, this too is a quote from an ancient Zen Master. 
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... fruit16
See: http://www.phy.duke.edu/$\sim$rgb/Poetry/hot_tea/hot_tea/node9.html. 
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... rest17
For a variety of excellent reasons that I'll go into later, computer metaphors abound in this work. If this is a problem for you, well, it's too late to take the book back. You paid for it and have read this far. Might as well finish it anyway. Right? 
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... axioms18
Wikipedia: http://www.wikipedia.org/wiki/Free will and determinism. Of course, because this is itself an axiom and therefore a free choice you can always choose to believe that you are forced by biology, physics, or invisible fairies to believe that your choice of beliefs is determined and not free.

It's entirely up to you. 

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... bazaar''19
Or perhaps, if you prefer, ``axiom bizarre'', because there is something strange and wonderful about choosing what we believe. 
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... wisdom20
Yes this is all a bit sentimental, and some of you are probably making gagging motions as you read this. Don't worry though - later I put down very precisely what I consider wisdom to be, quoting an actual luminary or two. At that point you're supposed to smack yourself in the forehead and go `So that's what he meant by knowledge and freedom, yeah.' Probably won't happen, but hey, I try. 
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... brain1.1
GIYF: reptile brain cortex In all GIYF links, you'll have to look for some likely links (ones that aren't obviously fiction or sexual solicitations) and click-n-read. 
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... 1.2
Wikipedia: http://www.wikipedia.org/wiki/Brain. In this specific case, between Google and Wikipedia you should learn that your reptile brain is one of the oldest (in evolutionary terms) parts of your brain, the part that controls your physical anatomy and very basic survival functions like eating, sleeping, fear of death, aggression, and sex. Your reptile brain is where a lot of your basic cold animal hungers reside - the ones you share with snakes and lizards. It is very, very selfish. Your limbic (emotional and judgement brain) and neocortex (language and logic and higher abstact thought) are layered on top of the reptile brain quite literally like layers of icing on a jello cake. 
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... inherited1.3
Wikipedia: http://www.wikipedia.org/wiki/Phylogenetics. I'm linking this article early because we will frequently have good reason to talk about phylogenetics, especially the notion that ontogeny recapitulates phylogeny. This nifty-sounding sound bite basically refers to the observation that for the most part, the development of individual organisms from a single (fertilized egg) cell to the finished product recapitulates the stages the same organism went through in the process of evolving. This makes sense - the theoretical mechanism of evolution enables a species to add something that enhances survival, but there is no real mechanism for it to take something away unless it actively and negatively affects survival. This is directly visible in our DNA, which contains many inactive segments that are basically ``fossils'' that once, perhaps, performed important functions. 
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...cortex1.4
GIYF: mammal brain cortex Also check out the previous Wikipedia article on the brain. Basically mammal cortex is the next set of cortical layers out from reptile cortex - the neocortex. Note that describing cortical layers and brain structures as ``mammal'' or ``reptile'' is a somewhat simplistic view of brain evolution and function, especially in humans, but is nevertheless a useful one for my purpose here. 
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... cerebrum1.5
Wikipedia: http://www.wikipedia.org/wiki/Telencephalon. The cerebrum is more properly called the Telencephalon, and contains the various cortical layers and regions. As one ascends the phylogenetic/evolutionary scale from reptiles to humans, the most striking change is the systematic addition of layers of cortex-based processing systems with neuronal connections to and from the phylogenetically older structures within. 
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...memes1.6
Wikipedia: http://www.wikipedia.org/wiki/Meme. This is not a terribly original argument on my part, of course... 
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... algorithm1.7
Wikipedia: http://www.wikipedia.org/wiki/Genetic Algorithm. 
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... ones1.8
A process that led, step by small step, to the discovery that mounting multiple nuclear warheads on rocket-engine driven spears worked really, really well and could kill off whole species of animals or entire continents worth of competing tribes. 
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... mythopoeic1.9
A nifty word that means ``giving rise to mythical narratives'' in case you didn't know. 
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... SquarePants1.10
Wikipedia: http://www.wikipedia.org/wiki/SpongeBob SquarePants. 
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... unstructured1.11
This point of view is advanced in a number of contemporary speculative works, notably in The Lucifer Principle by Harold Bloom. I strongly recommend it. 
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... cares1.12
OK, OK, yeah, sure, evolution doesn't care about anything because it isn't a sentient causal agency, it is a process. This is an anthropomorphizing metaphor, because evolution, in spite of having absolutely no ``intelligence'', is perfectly capable of bringing about changes as if it cared about them. It really is a hell of a metaphorical watchmaker, for those teleologists out there, and indeed is directly responsible for every single watch that has ever been discovered in the middle of a desert... 
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... role1.13
As opposed to ``proof by induction'', which is a common methodology for stepping over an unbounded set to prove an assertion. This sort of ``successor'' induction plays a key role in the axiomatic development of arithmetic, as we shall see. 
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... causality1.14
Occult in both the sense of hidden and magical, more often than not. 
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... processing1.15
Wikipedia: http://www.wikipedia.org/wiki/Lateralization of brain function. Here you will learn that your brain has two distinct halves, separated by a membrane called the corpus callosum that mediates communications between the two. The left brain is predominantly analytic; symbolic language and associated sequential symbolic reasoning seems to be dominant. The right brain is instead associated with emotion, visualization, imagination, and the formation of long term memory. 
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... chain1.16
For example, in the branch of physics called statistical mechanics. 
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... biosphere1.17
Wikipedia: http://www.wikipedia.org/wiki/Gamma ray burst. 
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... paradigm1.18
Just in case you cared. 
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... wrong1.19
Although I've been tempted on a few occasions to ``argue'' with a two-by-four upside the head when some sanctimonious scoundrel has added a healthy does of religious guilt to the burden of someone with cancer. Neither does it explain why God would do such a horrible thing as to inflict a hurricane-driven drowning or death by cancer on an innocent little child, although religious dogma does its best by transferring the blame, somehow, back to the rest of us. 
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... Earth1.20
Unless cats and dogs really are superior beings decended from space aliens who genetically engineered us to take care of them and provide them with a near-idyllic life, but then lost their awesome psychic powers and found themselves at our mercy. Or is that just a movie plot? 
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... survive,1.21
``Survival is good'' is one of those built in axioms, or instincts, that is hardwired into the brains of pretty much anything that has a brain. It isn't irresistable - suicide and altruistic sacrifice both serving as evidence that this is so - but it is pretty darn strong. 
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... one1.22
I can hear you thinking ``Aha! But what about $X$! They have a true language and use tools...'' for some value of $X$, say, bees, or maybe chimpanzees. To which I can only reply no they don't and if they do, who cares. You get my point. 
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... Dilemma1.23
Wikipedia: http://www.wikipedia.org/wiki/Prisoners Dilemma. 
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... selfish1.24
Er, I think. How did that go, again? 
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... generations1.25
I personally would argue further that for most humans the human brain is still only borderline capable of engaging in true analytical reasoning without the amplification of intelligence inherent to using symbolic reasoning on external media. With the possible - and note well that I say possible - exception of transcendent supergeniuses such as Ramanujan who did not apparently require paper and pen or clay and stylus or sand and stick to work out complex symbolic proofs, even the brightest physicists and mathematicians are crippled without the ability to do algebra on paper. 
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...$^,$1.26
Wikipedia: http://www.wikipedia.org/wiki/Ramanujan. A lovely article that otherwise has nothing to do with our current discussion. 
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... cause1.27
The SUW question and question chains are quite familiar to any parent of small children, as is the actual, rationally unprovable answer: ``Because!'' 
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... question2.1
Wikipedia: http://www.wikipedia.org/wiki/Galileo. We'll have more to say about Galileo's prosecution by the Inquisition for having the temerity to challenge the geocentric view of cosmology that appears repeatedly in the Bible. The current article in Wikipedia (in my opinion) presents a view that makes the Catholic church appear far more progressive than any reading of the primary documents justifies. It is important to view the event in the context of the ongoing and contemporary ``revolt'' of Martin Luther. Many thought - correctly - that Galileo's work would further inflame this rebellion against the authority of the church, and Galileo's work was indeed a factor in the Enlightenment. Individuals within the church may well have been progressive, but the superorganism itself quite rightly sought to defend its collective soul at all cost. See the Appendix on Galileo in this book. 
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...$^,$2.2
Wikipedia: http://www.wikipedia.org/wiki/Trial of Socrates. We'll have a lot to say - much of it good clean fun - about Socrates. Socrates managed to run afoul of both the secular and the religious authority, making fun of the former and and challenging the Gods that supposedly protected Athens against its many enemies. Socrates also claimed (metaphorically, of course) that he had a daemon in his head that told him what the ``good'' was and how to avoid mistakes and that generally guided his reason. We will have more to say about this daemon, which has a very Zen feel to it, the watcher that watches the watcher watching the world. For example, one is very tempted to assert that Socrates was groping towards a concept of freely chosen personal meta-axioms that form the substance of this daemon and are the basis for its ``whisperings'', that ought to preceed the choice of axioms of religion, political view, ethical view and not follow it. 
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... language.2.3
In addition to providing one with symbols that could be manipulated using (say) clay or sand or papyrus as an extension of the human brain, writing extended the social lifetime of ideas. Abstract discoveries developed before writing became common had a much harder time being ``remembered'' unless they were adopted as a critical part of an oral tradition within some superorganism, usually a religion. Thus the Vedas and the great Hindu Epics survived until written Sanskrit could capture them, but how many stories, tales, myths, were lost before they were recorded simply by being forgotten? Without the critical support of a written language, even if there was a super-genius of logic and mathematics born in, say, 1200 B.C.E. in India, or in Greece in 800 B.C.E. their work did not survive long enough to be written down. 
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... times2.4
Wikipedia: http://www.wikipedia.org/wiki/Parmenides. 
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... 2.5
Wikipedia: http://www.wikipedia.org/wiki/Socrates. 
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... 2.6
Wikipedia: http://www.wikipedia.org/wiki/Plato. 
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... 2.7
Wikipedia: http://www.wikipedia.org/wiki/Aristotle. 
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... 2.8
Wikipedia: http://www.wikipedia.org/wiki/Heraclitus. 
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... 2.9
Wikipedia: http://www.wikipedia.org/wiki/Pythagoras. 
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... India2.10
Wikipedia: http://www.wikipedia.org/wiki/Indian Logic. 
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... China2.11
Wikipedia: http://www.wikipedia.org/wiki/Logic in China. 
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... hand2.12
The Nyaya school, in particular, introduced the notion of four sources of knowledge: Perception, inference, comparison and testimony. These are not at all incompatible with the structure of knowledge as presented in this book. The Aristotelian syllogism (and its formal descendants) is a useful tool after one chooses axioms but provides no guidance on how to make those choices and little insight on how knowledge arises in the first place. The Indian school of logic was carried into China along with Buddhism itself, where it mixed with e.g. the social rituals and legalisms of Confucianism and the mystical dynamics of Tao to eventually form ``Zen logic'' - which isn't, at least by the standards of the West.

Zen emphasizes direct perception of the Real, unclothed by the (to it) distractions of language, formal syllogism and inference, very tightly connected to schools and heuristic scriptural tradition. If anything, Zen views syllogism and formal reason as an insidious trap, where your answers are preconditioned by your deepest and most unquestioned beliefs (your personal axioms, as it were) - a point of view with which this work very much agrees. The Koans of Tao and Zen are hence in some sense equivalent to syllogism as a foundation for ``the rational'' in a culture but their purpose is very different. In a sense they attempt to bring one to a transcendant realization of the right axioms for Enlightened living, from which point ordinary commonsensical reason in pretty much any language or culture permits one to arrive at right conclusions. 

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... logicians2.13
And individuals who can be counted as all three at once. 
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... breakfast!2.14
From Alice in Wonderland, in case you forgot... 
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...abstract.2.15
Ho, ho, ho. Like you aren't reading this book... 
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... knowledge.2.16
Sorry, but I'm a metaphor fanatic, largely because human language is so marvelously nonlinear and compressive. You will just have to live with this, or go buy a book on Logic and Set Theory and work out about umpty zillion empty theorems that, when they are done, tell you nothing about the real world you live in beyond what you already knew... 
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... Hopi).2.17
Wikipedia: http://www.wikipedia.org/wiki/Sapir-Whorf hypothesis. An important question in the Philosophy of Mind or Philosophy of Language is to what extent our conceptualization of the Universe is shaped by our language (and other ``learned'' filters). The famous ``example'' of this is Whorf's analysis of the Hopi language (although there are many other related analyses that have now been performed for many other languages) where he asserted that an individual raised to think in Hopi might have an easier time understanding, say, Relativity Theory because linguistically time is treated in exactly the same way as space, where in English the concept of time is built into verbs and sentences as tense. 
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... landscape.2.18
Wikipedia: http://www.wikipedia.org/wiki/Optimization (mathematics). Yes folks, you heard it here first. Your language, your personal axioms, your beliefs all give you an understanding of the Universe that is a solution to a generalized optimization problem in a very high dimensional, very abstract space. 
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...truthiness2.19
Wikipedia: http://www.wikipedia.org/wiki/Truthiness. The link in case you've never watched The Colbert Report on television. We discuss the notion of truthiness itself below, so you don't really need to go read this before proceeding. 
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... case.2.20
The Politically Correct way of gently saying wrong, wrong, wrong. 
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... revolutions.2.21
Wikipedia: http://www.wikipedia.org/wiki/Einstein. Which is not really lazy at all, hmmmm. Einstein was, in fact, considered both lazy and rebellious by his teachers who couldn't understand why he didn't give a rodent's furry behind for most of what they ``insisted'' that he be drilled in in school. Really, one can go down the list of philosophical revolutions and find a lazy heretic iconoclast behind nearly every one. This is the tragedy of Aristotle and Euclid and Newton - their contributions, however awesome and majestic, didn't come with a warning label that they were just one small step on a path that we are making up as we go, don't take this too seriously, let the buyer beware! 
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... understood2.22
Or not. Don't worry about it yet. Hopefully you will, eventually. 
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... sense2.23
Really, of course, one has to use a great deal of both to figure out the various factoids I pitch around in this chapter, but I couldn't resist the line. 
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... box2.24
Wikipedia: http://www.wikipedia.org/wiki/Schrodinger's Cat. I tell you, cats and philosophers have a sorry history together. Schrödinger's ``infernal device'' is just one example. An ancient Zen Master told his surrounding students that if they couldn't answer one of his silly Zen Koans he would cut in half a perfectly inoffensive (but handy) feline. Naturally, they failed, and so did the cat.

I promise, the only cats injured in the writing of this book were metaphorical ones. 

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... states.2.25
The difference is profound - later we will get a taste for this by considering quantum versus classical computing, where there are very strong connections between computatability and logic, and where being/not-being is replaced by a more prosaic truth/falsity as represented by 1/0. 
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... like2.26
Well, possibly not. But the failures either aren't germane to the argument or if anything make them still more cogent, as we'll shortly see. In any event, infinite resolution requires an infinite amount of energy. Our current belief that the electron is a truly point-like particle stems from the extrapolation of finite measurements to a presumed infinite limit and the problem of needing to figure out what holds an electron (or any ``elementary particle'') together if it has finite extent. 
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... effect2.27
Wikipedia: http://www.wikipedia.org/wiki/Aharonov-Bohm effect. 
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... pairs2.28
Wikipedia: http://www.wikipedia.org/wiki/Heisenberg Uncertainty Principle. Conjugate variables are things like $x$ (position) and $p$ (momentum) which cannot both be known (measured) simultaneously to arbitrary precision. 
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... logic2.29
Wikipedia: http://www.wikipedia.org/wiki/Spock. If you've never seen any of the original Star Trek series and are clueless about Spock, you have my deepest sympathies, but you can still follow this link to get an idea of what I'm talking about. 
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... you2.30
I will refrain from asserting that I ``prove'' anything at all beyond any doubt for ``reasons'' that are hopefully already self-consistently clear. It's so difficult to be rational about reason, after all. 
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... English2.31
Yeah, yeah, yeah, Parmenides wrote in Greek, things get translated to Latin, eventually folks write about them in English, who knows what was originally meant. Who cares? We happen to be reading a book that was written in English, don't we, so let's just smooth down those ruffled feathers and move on. 
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... terminology2.32
Wikipedia: http://www.wikipedia.org/wiki/Semantics. This is a decent place to learn exactly what ``semantics'' really is, if you care. Or you can take my word for it that semantics is all about the true meaning of the symbols used in reason, the map that is not, in fact, the terrain. Ooo. More on this later. 
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...$^,$2.33
Wikipedia: http://www.wikipedia.org/wiki/Semiotics. You might also want to read a bit about semiotics, although it is more concerned with ``human'' communication and reason than set theory or information theory, alas. Still, you'll often hear both of these terms bandied about as if they are important to whatever ``thought'' turns out to be. 
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... objects2.34
Without getting into any sort of debate over what a ``thing'' or ``object'' is, mind you. For the purpose of this discussion, it is essentially a unique label or algebraic symbol that can be assigned any semantic meaning we like if we are asserting that the laws of thought are to be truly universal or at least a constraint on the statements concerning relationships between objects in the Universal Set. 
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... tricky2.35
Wikipedia: http://www.wikipedia.org/wiki/Universal set. 
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... irrelevant2.36
The original Greek actually was closer to ``being is'' anyway, not that we care as we analyze the English forms. 
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... used2.37
Again I'll treat you to a bit of insight from physics. Time is not what you classically and human-experientially think it is, at least not if you accept the extremely rational and well reasoned and in fact mathematically precise conclusions of the theory of relativity, for which there is a wealth of empirical evidence and which also is such a beautiful theory that it is difficult to imagine it not being at least one part of what is true. 
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... Semantics''2.38
Wikipedia: http://www.wikipedia.org/wiki/General Semantics. Gawds, you are doubtless saying by this point, is there anything that cannot be referenced at the introductory level, with lovely links through to more advanced stuff, through the Wikipedia? The answer is asymptotically approaching ``no'' in the limit as fast as some of the world's brightest and most altruistic people can make it so... 
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... it3.1
Here and now, at any rate. 
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...paradoxes3.2
Wikipedia: http://www.wikipedia.org/wiki/Paradox. Yes, you know the word, but the word has a fairly specific meaning in the context of mathematics and set theory. 
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...antimonies3.3
Wikipedia: http://www.wikipedia.org/wiki/Antimony. Points where two laws - things that we cannot easily imagine being different - lead to contradictions within a theory that uses those laws in its development. 
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...axioms3.4
Where we deliberately defer discussing just what an axiom is for several more chapters, sorry. You can always look ahead and come back... 
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... theory3.5
Wikipedia: http://www.wikipedia.org/wiki/Set Theory. 
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... theory''3.6
Wikipedia: http://www.wikipedia.org/wiki/Naive Set Theory. This of a shy, blushing, Set Theory, one that isn't scarred enough by bitter experience to be trusted out on the street. How charming! 
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... Cantor3.7
Wikipedia: http://www.wikipedia.org/wiki/Cantor. A Very Smart Guy. He pretty much defined infinity as we know and love it today. 
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... theory3.8
Wikipedia: http://www.wikipedia.org/wiki/Axiomatic Set Theory. 
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... theory3.9
Wikipedia: http://www.wikipedia.org/wiki/ZFC. This is real mathematics and not for the faint of heart. In fact, this can be made into a foundation of real mathematics as a constraint that strongly influenced its choice of axioms, not to mention its axiom of choice. 
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... Neumann3.10
Wikipedia: http://www.wikipedia.org/wiki/Von Neumann. A man who was a giant in both mathematics and computation. 
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... set3.11
Wikipedia: http://www.wikipedia.org/wiki/Power Set. This is the set of all permutations of objects drawn from a a given set, the set of all subsets of a set. 
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... Universe3.12
Wikipedia: http://www.wikipedia.org/wiki/Von Neumann universe. This is a class of sets, a hierarchy of sets that are generated by transfinite recursion of the power set from the empty set. It is extremely useful in the development of arithmetic from set theory. 
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... members3.13
We will ignore for the moment the issue of how to deal with ``continuous'' (non-denumerable) sets. After all, in any interval of the real line there are an infinite number of points, and we surely cannot list them all. In fact, we cannot list a nonzero fraction of them, and we can think of infinitely many ways to generate infinite lists of points that cannot be listed. Hmmmm. 
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...$S$3.14
Sorry about that. I'm pretty sure that this works out to make sense, if you work at it... 
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... times3.15
Wikipedia: http://www.wikipedia.org/wiki/Natural number. A very similar construction, starting with the empty set $\{\}$ being equated with 0, is one way of relating natural numbers to sets, where every natural number is recursively linked to a set consisting of all the sets that preceded it. 
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... breakfast3.16
A quote from Alice in Wonderland, in case you forgot. 
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... theory3.17
Wikipedia: http://www.wikipedia.org/wiki/Positive set theory. Note that parts of the above resemble positive set theory - which is also quite existential - as much as anything. However it has axioms related to the need for axiomatic set theories to be able to resolve concepts such as equality of two predicate descriptions in terms of subsets pulled from $\Pi(S)$ (only, as far as I can tell) - it does not explicitly address the $\Pi^n(S)$ hierarchy in general. 
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... live3.18
I can just hear the Mind vs Matter enthusiasts dragging out their siege equipment and donning their metaphysical armor. The actual existential physical Universe is mind. No! (clang, bash) It is matter! (thump) Owww, getting whonked with that rock that hurt. But did it hurt in your mind, or in your matter? Was the rock mind, or was it matter? If you don't mind, let me assert that it doesn't matter... 
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... equality3.19
Wikipedia: http://www.wikipedia.org/wiki/Axiom of equality. Also look at the axiomatic set theories, as they almost invariably have one. 
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... does3.20
Which again sounds odd, as we blithely talk about nonexistence existing, so to speak. 
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... ago3.21
Wikipedia: http://www.wikipedia.org/wiki/Descartes. We'll spend a lot of time talking about Descartes later, and you've probably already heard of his I think, therefore I am. 
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... unique3.22
Except that it isn't unique. The empty set of $P$, the set of all pennies, is no pennies. Since this is a finite, Univeral set, we can insist that the complement of the empty set is in fact $P$. On the other hand, the Universal set of all quarters $Q$ also has an empty set, with complement $Q$. The empty sets of these two Universal sets are distinguished by insisting on closure within the respective set Universes and permitting me to say things like ``I'm out of quarters'' when in fact I am not out of pennies! This difficulty is usually eliminated in axiomatic set theories by not permitting the action of forming the complement of the empty set (and frowning strongly on Universal sets in general) because you might get something really crazy, like the Mother of All Set Theories, the Universal Universal Set. Or God. Or something like that. Hard to say, really - nobody's ever tried it. 
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... theory3.23
Null sets in measure theory are even more concrete - they can contain objects from the universal set in question, e.g. points on the real line - as long as those points have zero measure. There can be infinite numbers of these objects and the entire set can still have zero measure. For example, the set of all rational numbers has measure zero on the real number line and hence belongs to the null set of measure theory, which is not the one I propose here. 
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... Universe3.24
Wikipedia: http://www.wikipedia.org/wiki/Milne Model. I remember with a certain wry joy learning about something called ``Milne's Empty Universe'' when studying astrophysics and general relativity. Kind of a boring place to live, of course... 
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... set3.25
There are all sorts of interesting things in this paragraph - enough that it is worth pointing them out. For one thing, a truly empty Universe - coordinates with nothing anywhere at any of the coordinates - is a moderately creepy concept, a mathematical concept. To a Logical Positivist (a philosophical school made mock of somewhat later in this work), the entire idea of an empty Universe (or mathematics in general) is probably meaningless, although a good mathematician or theoretical physicist has no problem whatsoever with it and it is thus to normal human beings not meaningless.

For another, I use terms like ``putting things into'' to describe set-theoretic statements such as $S = S \bigcup \O$. The language is that of operators that act on one thing to produce another, which is a valid construct in mathematics and in fact we could (I promise) build an algebra of creation and anihillation operators that act on a suitably defined ``empty set'' to create non-empty sets. This is, in fact, the algebra of quantum field theory (which has more bells and whistles, of course).

The equals sign in mathematics, however, is a symmetric entity that has no implicit ``time'' or notion of ``action''. It describes a static true relationship. You can write an equation forwards or backwards and it says the same thing. It is a pure abstraction of the notion of identity itself. Two algebraic representations that are equal are the same thing. Even in algebra, however, algebraic derivation retains a sense of order and operation in the steps performed on one equation to transform it into another, and certain operations are only conditionally permitted.

In computer science, the equals sign really stands for logical assignment, for operational equality. A = B + C; in the C programming language stands for ``take the contents of the memory locations labelled B and C and add them, and place the result into the memory location labelled A''. Any of these work to describe the way the empty set is algebraically manipulated, but they mean very different things and just by using language cleverly I can predispose a discussion about them to proceed along very different lines. But you probably knew that. 

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... Paradox3.26
As you should have learned by following my previous Wikipedia link, this ``set'' was discovered/invented by Bertrand Russell in 1901 while working on his Principia Mathematica. Russell observed that the set of all sets that do not contain themselves is a bit ``odd''. In particular, does this set contain itself? Hmmm, the answer appears to be a bit cloudy. Try again later. 
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... knots3.27
We'll get there, don't worry. 
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... theory3.28
If you aren't familiar with the Barber paradox now, soon you will be. You will be. 
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... Poof3.29
Poof is somehow too modest a term for ``it vanishes in a blaze of hard radiation releasing immense amounts of energy into the Universe on the way down'' but it will have to do for now. 
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... theory3.30
Only more so, since real black holes preserve a few of the coordinates of the stuff you dump into them - charge, mass, and at least some bounds on location. These set theoretic black holes preserve nothing - they suck any set in the Universe into a (non) state of total nonexistence, not just the ``emptiness'' of vacuum, of the empty box. 
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... logic3.31
A concept that would openly offend any Zen master, especially ones who are good at it. 
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... Mu3.32
Wikipedia: http://www.wikipedia.org/wiki/Mu (negative). Ain't Wikipedia wonderful! 
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... clarity3.33
Or, in the case of Musashi, the ability to take a crude ``sword'' fashioned out of a spare oar from a boat and with a single blow slaughter a top-gun wannabe named Sasaki Kojiro armed with a razor-sharp katana and a highly developed ``strategic'' technique but still in a state of mental confusion regarding the Void. Individuals who have truly mastered the conceptualization of the null set are often portrayed as having considerable power over the non-null Universe, perhaps because they know how to take its complement in different ways and hence select their reality... 
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... legitimately3.34
To be formally precise here, ``legitimately'' is not easy to come by. It is worth noting that it took Russell and Whitehead close to a full page - more if you embed the referenced theorems - of some of the nastiest algebra on the planet to ``prove'' that 1+1 = 2 in a formalism that excluded this kind of contradiction. Real Mathematics Is Not Easy. Fortunately this is not a work on real mathematics. I keep saying that, so it must be true, right? 
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... themselves3.35
Note well that I am very careful to specify that our barber is a man where this is frequently omitted in statements of the paradox in both books and online. Obviously if we refer only to a barber, that barber that might in fact be a woman or a blob-shaped hermaphroditic space alien and there is no essential paradox. 
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... set3.36
And you, dear reader, are going to have to try to keep this in mind as you read, as I'm not going to keep pointing it out and if you forget you'll start asking yourself if null isn't really the same as empty. It isn't, as it isn't a set. Calling it a set is merely a convenience of (and trap of) the language - referring constantly to the null no-set would be tedious. There is something delightfully self-inconsistent by referring to it as a set anyway, sort of a two-word embodiment of the Russell paradox. 
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... ``imaginary''3.37
Two terms that are often confused in ordinary language or rhetoric. I recall with great fondness 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.'' 
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... aliens3.38
Armed with razors and threatening to shave all men that do not shave themselves, perhaps? 
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... unlikely3.39
At least I think that I do. Noting well that this is the George W. Bush era as I write this... 
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... Mu3.40
You are following those links, aren't you? Of course you don't have to, but if you don't you're pretty likely to miss some of what I'm trying to say as the Wikipedia articles are rich with connections. Of course it might take you a year to read the book if you really follow all this. Tell you what, go ahead and read the book one time just straight through, then try it again following the links. That way you'll get the point much better anyway. 
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... symbols3.41
As these words should make perfectly clear, of course. 
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... well-defined3.42
Whether or not you care about what happens at the point $x = 0$ depends upon what you are going to do with the expression, which could easily lead us to a discussion of measure theory and the other definition of null set, to which the single point $x = 0$ belongs in cases where one is using the expression under an integral sign. Who cares if one leaves a single mathematical point out of an otherwise well-behaved integral? But this particular digression is once again way too close to real mathematics and hence is only offered parenthetically for those that already understand it. 
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... world3.43
This is nearly a quote, in fact, from one of his earliest works referenced elsewhere in this book. 
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... dictionaries4.1
By assumption here a dictionary written in the language it defines. 
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... represented4.2
Remember, Burroughs fans, Tarzan managed the miracle of learning written language from just having a pile of books and a children's primer dictionary handy in the jungle hut where his parents died. In my opinion this makes Tarzan a transcendental genius that makes Newton, Einstein, Goethe, Ramanujan look like dullards... 
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... them4.3
Sure, WIYF, no doubt. 
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... unique4.4
A fact that, recall, really bothers General Semanticians. 
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... structure4.5
The author is deeply indebted to Richard Palmer for teaching him this general idea as part of an introduction to the general topic of Complex Systems in the sense studied at the Santa Fe Institute - GIYF and WIYF, naturally. 
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... derived)5.1
Note also that one example they give of an axiom is the Law of Contradiction from the previous chapter! The same dictionary gives a mathematical or physical definition of ``law'' that is very definitely not that of an axiom. Clearly there is a great deal of semantic confusion that underlies the epistemological confusion being addressed by this work. That this mistake absolutely pervades even the language is evident from the fact that this commonly accepted dictionary definition is technically incorrect in its supposed logic and mathematics definition. 
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... level5.2
Where, for the sake of argument, University level introductory Calculus will still be considered ``elementary'' for the most part. Hey, nobody said this would be easy - otherwise somebody would have already done it! 
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... time5.3
There are some lovely books for the lay reader that manage the same thing in prose fiction, such as Flatland. 
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... reality5.4
You think that I joke here, but for the world the philosopher Hegel served as a sort of ``Antichrist'' or ``straight man'' of rational philosophy. No conclusion was too absurd for this master logician, including a ``proof'' that there could be no planet of any kind in the orbital interval between Mars and Jupiter, published months after the discovery of the Asteroid belt. Ba-da-bum. 
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... again5.5
Or if we are careful. This process precisely defines what happens at the leading edge of physics, where each newer, bigger theory has to completely swallow its predecessors and not explain any less but which precedes from different axioms. 
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... students5.6
Mathematically competent students, that is. 
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... Dynamics5.7
Which is alas out of print and no you can't borrow my copy. Amazon sometimes can locate a used copy; there are also some in physics libraries. And somewhere one of my ex-students is walking around with my original copy and if/when they read these words I'm certain they'll put it into a mailer and send it back... 
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... observer5.8
And this turns out to be a time-ordered, non-relativistically-covariant kind of statement that is just plain wrong. From the inside of the Universe you were always entangled with the system and no actual change occurred in your combined state. There is no such thing as the ``collapse of a wavefunction'' in a properly relativistic reversible quantum system, only ``ignorance of the initial state of all of the entangled Universe outside that small part you call ``the system''. If you are a physicist or mathematician, you can see the truth of this by reviewing the derivation of the generalized Master equation in the quantum theory of open systems, but it is not for the mathematically faint of heart. I may have to write a wikipedia article for the GME just to put something out there for lay people to understand, as google reveals nothing immediately useful to anyone but theoretical physicists.

The GME is generally not taught to physics graduate students, alas, unless they are in one or two subspeciality fields. This means that most physicists are not terrible aware of its existence or how it works in quantum statistics. Consequently, damn few non-physicists have ever heard of it, including the ones that teach or write in philosophy departments. This in turn leads to silliness that often goes under the name of ``quantum philosophy'' where it is asserted e.g. ``wavefunction collapse'' as a truly random process opens the door for free will (as if randomness is any more free than determinism) or the even sillier ``new age quantum'' stuff that appears in the movie ``What the (bleep) is it All About''.

Sigh. 

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... this5.9
What is a ``space'' of axioms, anyway? We clearly need some axioms to describe it... 
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... theory5.10
For example, the theory of classical chaos arises in part because for any sufficiently complex interacting system, the set of closed orbits is of measure zero; there are usually a countable infinity of them while in between there are an infinitely bigger uncountable infinity of open orbits that never quite repeat. This, in turn, is one of several reasons why the Universe cannot be classical and also structured and persistent (given our classical empirical understanding of the forces of nature). 
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... Russell5.11
A Really Smart Guy(tm) in the annals of modern philosophy, who thoughtfully refrained from contributing lots of Hegelian Bullshit while still writing with great insight upon the basic problems of philosophy. He was helped by being, actually, a decent logician/mathematician who was working on the formal limits on what can be known from logic. 
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... G\"odel5.12
An archetypical mathematical genius who was so brilliant that he couldn't manage his own personal life and accidentally starved himself to death when his wife became too sick to cook for him. History is full of examples of super-brilliant people who are somehow, incredibly, simultaneously too stupid to come in out of the rain. Or (for example) to put down a stylus and address an invading soldier armed with a very sharp piece of metal and dressed in bloody armor respectfully by (for example) grovelling a bit while declaiming ``Slay me not, oh master, I am the Great Archimedes and a spoil of war and your king will Have Your Head if you take mine...'' 
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... later5.13
Yes, as a kid I had an Eight Ball, and still think that as a tool for prophecy and logical analysis it is unparalleled. It beats the hell out of Tarot cards, and is cheaper too! 
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... chapters6.1
Not just ``the Church'' of course, but rather all religions in all primitive cultures, especially the really successful and scary religions with a harsh memetic defense system. Recall from the introduction that religions preferred to have a monopoly on logical and philosophical discourse so that they could make people believe seven impossible things before breakfast. 
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... ASCII6.2
A mapping between characters and binary numbers used in pretty much all computers for handling character data. 
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... ``pseudoquestion''6.3
Just in case it isn't obvious, I'm making a formal definition of the term pseudoquestion (or its cousin, the pseudostatement) here. We need to be formal, because this is related to logic and mathematics and there is no room for sloppiness. This usage is common enough, if you look for it, although it is usually used in polemic discourse to discredit some question or statement - ``That's not a real question, it's a pseudoquestion.'' Here it is not used in anything like the polemic sense - a pseudostatement will be defined to be one whose unambiguous truth value cannot fundamentally be demonstrated within an axiomatic system of reasoning. Axioms are therefore all pseudostatements. So are the Laws of Thought. 
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... ordered6.4
This is the famous dartboard problem in measure theory. If you throw and infinitely sharp dart at a 2-dimensional dartboard surface, it will strike some point. Which point? Well, it could be a random point, in which case every point has equal probability of being struck, including the point in the exact center of the board. That probability is (paradoxically) zero because a point has measure zero with respect to the finite area of the dartboard, but some point is nevertheless struck. Alternatively the point struck could be completely non-random - the dart might be designed and directed to hit precisely in the center. One cannot infer anything about the mechanism (or lack of mechanism) that produced the hit by examining the point itself - it lives in a different space altogether. This metaphor can be iterated indefinitely, just like truth/false loops in the syllogistic example above. 
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... Hume6.5
See, for example, his Enquiry concerning Human Understanding. Or not, it isn't too important and the basic idea is presented in spades below. 
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... Russell6.6
There is a lovely book called Problems of Philosophy by Russell you might want to look over. Now I personally have a hardback copy of the original Home University Library text from 1912 sitting on my chest as I write this, but fortunately it is also available on the web for free at http://www.ditext.com/russell/russell.html. Russell was a Very Smart Man, and this book - which I read only when the first draft for Axioms was long since completed and had been on the web for a year or more - is a lovely sort of pre-Gödelian and not sufficiently mathematical version of what I'm attempting here. Its mistakes are sufficiently illuminating that I'll have occasion to refer to it very specifically from time to time, so be prepared. 
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... Descartes6.7
Meditations. 
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... Berkeley6.8
A Dialogue Between Hylus and Philonous. 
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... Kant6.9
Metaphysics. Of course. 
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... optimization6.10
No good reference, sorry. This is just one of the things I do on computers though, so I'll tell you what you need to know in a self-contained way. 
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... systems6.11
This is a fairly broad branch of fairly new mathematics. Y'know, phase transitions, earthquakes, economics, statistical mechanics, and all that. Simple stuff. Check out http://www.santafe.edu/ 
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... geneto-memetics6.12
You might try reading The Lucifer Principle by Bloom at some point if this appeals to you. 
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... contradiction7.1
My computer science roots ring through here, as I'm using `!' as the symbol for negation instead of logic's $\sim$. Another good reason for this choice is that I'm using LATEX, and typing ! requires a single keystroke while I have to enter $$\backslash$sim$ to get a $\sim$. Always nice when personal inclination meets efficiency... 
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...7.17.2
As we introduce the Aristotelian concepts of $A$ and its dual, $!A$, I supposed that it is time to give a web redirect - visit The Institute of General Semantics, http://www.general-semantics.org This institute (founded by Alfred Korzybski) pushes a somewhat touchy-feely version of some of the ideas advanced in this work, and is addressed later on in a section all its own. 
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...symbolically7.3
See e.g. http://en.wikipedia.org/wiki/Contradiction 
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... Penguin7.4
Given that this is true, I expect all religious computer users to immediately stop using Microsoft products and convert over to Linux... 
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... derivation7.5
If nothing else, this may keep religious individuals who are not computer users from accusing me of being a ``wise fool'', as has been so often done in sermons attacking people who are convinced by silly little things such as physical evidence that, e.g. - much of the Bible, the Koran, and other religious scripture with creation myths incorporated therein are wrong. Their argument would be something like: God is not a penguin! If God is not a penguin, then Robert Brown is not a wise fool.

Hmmm, I like this argument too! 

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... proud...7.6
Probably not, actually... 
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... time7.7
Not my wife, my kids, or anybody who actually knows me, that's for sure... 
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... themselves7.8
Hmmm, I think that would be all lions. I foresee real trouble ahead if we are ever to assert that Socrates was a ``lion'' among philosophers... 
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... different7.9
Nor have we exhausted the possibilities. Really we have something like True (existentially correct, whatever that means) and Provable (deducible from axioms and appropriate laws of thought), True and Unprovable, False (existentially incorrect) and Provable (to be incorrect), False and Unprovable, True and Inconsistent, False and Consistent, Unknowable (period) and more... 
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... absurd7.10
In fact, in a beautiful Gödelilan twist, this statement cannot be proven by reason. It is also incorrect, but that's beside the point. 
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... reasoning7.11
It is also castigated for being mostly unoriginal, egocentrically presented, and both bigoted and sexist, with examples of ``reason'' or the lack thereof that are highly demeaning to women. In spite of this there are some decent ideas mixed into the bullshit (and we have to expect there to be bullshit to the extent that it is a philosophical work). I will therefore personally throw no stones from my glass house as this work is mostly unoriginal, and it is likely that at least some readers will find it egocentrically presented although I really do try to make it clear that its theme is far from original but all too often forgotten. It very definitely is not sexist and can think of no reason it should be demeaning to women, however hard it is on lions and penguins. Perhaps I'm a closet speciesist. 
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... aconite7.12
Oops. Aconite, or wolfsbane, or monkshood, is a pretty garden flower that contains one of the more deadly and dangerous plant poisons, one that regularly kills people or pets or livestock that eat it or even get it onto their skin). 
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... apple''7.13
However true that might be for apples, it doesn't work as well in physics where an object might not either ``be at $\vec{x} = (1,1,1)$'' or ``not be at $\vec{x} = (1,1,1)$''. 
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... Universes7.14
Hopefully it has occurred to you, dear reader, that I have just spent a great deal of time trying to communicate to you the concept of the inconceivable. Furthermore, unless you are a complete dodo-brain, you probably understood the concept, even though the ``understanding'' of it is much like trying not to think of the word ``rhinoceros'' for the next thirty seconds. It would have been really easy if I just hadn't told you what not to think of.

Now you can begin to appreciate the difficulty of teaching (and learning) Zen. In fact you could have been Enlightened, if only someone had smacked you in the head with a banana while you were reading this footnote, but now it is too late because you're still thinking about rhinoceroses, aren't you. The banana thing only works if you aren't thinking about large grey endangered species with nasty tempers, hunted for their horns. The bulk of Zen is about how to stop thinking about animals with heavy plate-like skin that live in dry parts of Africa mixed in with Wildebeests and such, usually by means of meditating using the word ``rhinoceros'' as a mantra for a decade or so until it loses all possible association in your mind with animals whose horns are often ground into potency-enhancing elixirs in Oriental medicine.

I'll stop now. 

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... is8.1
No joking now. This PED is real and lies far too near, beneath the facade of the ``reality'' in which we live. It is to combat this despair that humans create religions and elaborate systems of belief in an answer to the SUW-level questions, for only an answer here can provide a foundation for all the more mundane answers that follow. Even with such an artificial foundation (or perhaps because of its manifest artificiality) the PED looms in every human life, to yawn agape when one is depressed, sad, or chemically imbalanced. In a fundamental sense it is the PED that is a prime factor in most suicides. We require faith to live, and we require a faith that isn't obviously inconsistent with the reality in which we live to not lose it when things get tough. 
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... will8.2
And the mentally healthy don't have much more. We just do a better job of ignoring that fact and revelling in the joy of a Universe in which, for the most part, it feels like we do. 
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... contradictions8.3
An exercise, of course, that is doomed to failure in so many ways. 
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... example8.4
On the other hand religions, for the most part, have yet to actually act on this axiom and attempt to reconcile e.g. science with scripture because the reconciliation would absolutely require the rejection of much scripture, which would violate the Prime Axioms of religions in general, see below. 
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... questions9.1
Google up ``loaded questions'' in the wikipedia, of course, if you don't recognize the term, or just wait a second. 
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... fairy''9.2
What, your mother never told you that to teach you to mind your manners at the table? Good. 
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... reader9.3
And I'm sure that at least some of you who are gnashing your teeth as I disrespect one or more of your favorite philosophers past and telling all your philosopher friends what a silly fellow I am. 
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... Germans9.4
It isn't just Hegel. Well, Hegel is such a perfect foil, I mean fool. I mean look, greater Germany contributed plenty of truly excellent natural philosophers and my own geneology has a significant fraction of German ancestry in it so I'm hardly disrespecting German Intellect - but what can you say that is good about Hegel, Schopenhauer, Nietzsche (now there's a guy didn't just look down into PED but rather jumped right in and set up housekeeping, although viewed as dark poetry some of his stuff doesn't read too badly). Kant and Wittgenstein, especially both should have known better - Wittgenstein was Russell's student and Russell definitely understood the inevitability of Hume's conclusions and made important contributions in the development of Gödel's theorem. It isn't possible to ``transcend'' rational thought and it isn't about language. It's about pure logic. In particular, it's all about the basis of logic, axiomatic reasoning. 
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... did9.5
If not, you can probably learn at least a wee bit by looking at Russell's little book online and googling for a wikipedia reference such as: http://en.wikipedia.org/wiki/David_Hume. This is actually a lovely article and summarizes quite a lot of Hume's basic contributions to philosophy beautifully. 
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... choose9.6
Yes, I really mean the italicized part. If you've ever watched, for example, Monte Python's Life of Brian you should deeply understand my point. 
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... color(s)9.7
All the best examples in probability theory involve urns. God knows why. 
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... Urn''9.8
Polya was a rather famous Hungarian mathematician, who did a number of things with probability and is also famous for his work on problem solving methodology. Unfortunately he's not Greek, so I couldn't frame this as ``What's a Grecian Urn''. 
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... world9.9
Loosely speaking, of course. 
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...Axioms9.10
Ta-da-BUM! 
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... race10.1
I believe that it was Arthur C. Clarke who pointed out that any sufficiently advanced technology is indistinguishable from magic or Godlike powers. Historical examples abound, as do Science Fiction stories such as A Connecticut Yankee in King Arthur's Court. If I went back a mere three hundred and fifty years and flicked my Bic in the wrong crowd - that is, nearly any crowd on Earth at the time - I could end up being burned alive or revered as a God or Shaman, depending on whether or not I was packing a few hand grenades and an Uzi with several spare clips along with my Bic. 
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... Man10.2
Reference to another moderately famous Science Fiction short story by Damon Knight that got made into an episode of The Twilight Zone. Told as a cookbook... 
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... societies10.3
Technically, to be fit enough to survive it only had to be relatively beneficial, which historically has often meant that outsiders got their hearts cut out at the altar first. 
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...socio-memetic10.4
I could, of course, write a whole section of this book on memes alone. Or I could plaigerize (even more than I am already) from other books, like The Lucifer Principle by Bloom. Or I could just send you to the Wikipedia for a quick lookup and a damn fine article. Instead, I'm only giving this important concept this one lousy footnote, aside from whatever you get out of context as you read, so listen up. For all practical purposes ``memes'' are to social/human groups (superorganisms) what genes are to ordinary organisms. The encode the organization - get it? Organize? Organism? Common root? Another metaphor is that they are the ``program'' of a social-neural network computer whose components are humans. There are still other metaphors. Relax. A meme is, fundamentally, pretty much a social axiom or a proposition of sorts that follows from social axioms. All clear? 
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... excommunication10.5
Or, in olden days, execution in a variety of usually very public, very painful ways. To prevent other ``cancer'' cells from forming, of course, should they be actually thinking for themselves. Maybe modern medicine should learn from this - not just cut out cancer cells, but impale them on pins, sear them with fire, force them to recant their cancer down on their knees before crushing them beneath heavy stones, all where the other cells can see them. 
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...mistaken10.6
Certainly not about something as elementary and important as (for example) where the world came from and how we came to be here upon it. Or about the future history of Damascus. 
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... again10.7
I leave it to your own judgement as to whether or not physicists or computer scientists, who in general have the coolest toys on the planet, actually work for a living as it is. I am both. On the other hand I'm a teacher, and teaching is damn hard work... 
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... integument10.8
That's ``skin'', y'all. What point is there in having a vocabulary if you never use it? 
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... prose10.9
Words like ``integument'' aside, of course. Look, I told you what it meant. My kids have to build up their vocabulary for their eventual SATs so we use big words around the house. 
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... apostatical10.10
Sigh. Apostasy is ``the act of renouncing your religious or political beliefs''. 
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... anywhere10.11
Potentiate I say because most people are well enough defended in the beliefs they inherit from their parents and native culture as children, which after all fit them ``well enough'', like an old pair of jeans or a particularly comfortable burka. Even after they come to understand intellectually how infirm our basis for understanding things intellectually or otherwise really is, it takes a little something, a spark, a word, having a soggy banana squished right onto your head unexpectedly, to actually bring about the Zen experience of Enlightenment. Which is not rational, per se, it is experiential. This experience is common to all disciplines - scientific, religious, political, sports, work - and it tends to happen to the prepared mind, paradoxically at a time when it is unexpected. 
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... again10.12
Assuming that you ever. 
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... right10.13
It should be carefully noted that the answer to this rhetorical question is that humans have the ``right'' to do anything they want and can get away with, literally. This statement is thus emotional (and practical) persuasion, not logic. 
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... book10.14
With the deepest, most profound apologies to any high school or university students where in fact, somebody is making you read this book. Tell you what. You can quit here. Tell your professor or teacher that I said so. Chances are that they'll be so impressed that you were actually doing your assignment so carefully that you got to this footnote and read it that they'll send you out to play frisbee in the bright afternoon sun with tears in their eyes. I do, however, invite you to finish the book off on your own just on the off chance that you might find it interesting or informative or useful or because your teacher is still going to hold you responsible for knowing what is in it whether or not they ``force'' you to read it... 
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... Human10.15
The ``elect'' on the other hand, sometimes get a bit of freedom to speculate on this, but when they do they always tread a fine line between orthodoxy and heresy. Some really great heresies have been invented in Christianity, for example, in just this way. Not surprising given the Gödelian traps and axiomatic contradictions, given the socio-memetic requirement of keeping it simple for the masses which supercede a silly thing like logical consistency. 
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... it10.16
Every human on the planet should read the letter of Saint Bellarmine to the Carmelite provincial Paolo Foscarini (a public supporter of Galileo and the Copernican model of the solar system). To make it easy, I include a copy in an Appendix 
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... se10.17
On the other hand, ``every point is in the middle of an infinite line'', right? As I like to tell my students about the time we cover Galileo and Copernicus, the Church and Aristotle and most teenagers are right after all and the entire Universe does revolve around your own, personal navel, just the way that you see it. 
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... genitalia10.18
No, seriously. Don't think about that. Pull yourself together, man! Elephants have really big ones too, it doesn't mean that yours isn't big enough! 
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... obsequious10.19
I'm going to let you look this one up. It'll be good for you. 
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... presence10.20
Right before asking His Help in averting disaster, providing enough to eat, avoiding pain and death, and so on 
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... best10.21
Just like Voltaire's Candide. 
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... exists10.22
Although one way out of the quandry is that it doesn't exist, and that all that we see of Evil in the world is ethically equivalent to the scripted out acting in a gangbusters widescreen Horror movie, all full of rape, robbery, murder, disembowellings, beatings, eye-gouging, genital electrocution, genocide, and topped off with a healthy dose of sickness and old age, earthquake and hurricane and above all death, untimely death. All acting. The ones getting hurt, they're not real, they're just extras, constructs, they have no souls. One day we (those of us with real souls) will all sit around some metaphorical bar in heaven drinking heavily of Perfect Beer and laugh about it. 
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... Evil10.23
Given that God is Good, it really helps for Him to have a Fallen Guy to take all the blame for Evil, even though any bruised four year old can tell you that if Johnny (18) is beating up on Tommy (4) with Dad (40, and supposedly strong and in charge) sitting there just watching it all happen, Dad is probably drunk. Besides, Dad could have decided to have just one kid - Tommy - and not bothered siring Johnny at all, or maybe sent Johnny off to military academy where he can get picked on by upperclassmen and taught to Be a Man. The devil is at best a straw man, and this is morally All Dad's Fault, unless there really is a higher Good being served by things being the way that they are, pain and suffering and Evil and all. 
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... Devil10.24
Let's give the concept its own capital letter as this being is often portrayed as being gifted with God-like powers only somewhat less than those of God, at least as far as screwing around with human affairs is concerned. 
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... it10.25
What the hell does this mean? Don't ask me. 
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... drivel10.26
Which is obviously an anagram for ``r devil'', suggesting some sort of cosmic message that some of the axioms inserted r (from the) devil, or maybe ``r d evil'', suggesting that religions r d(oing) evil. In which case they should be revil'd. I'll go take my medicine now, OK? You don't have to quit reading. 
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... human10.27
Works for my dogs, anyway. Well, ok, it doesn't actually work with my dogs, it works with some of my dogs, the ones that do not Sin and stay in the yard God gave them as delimited by the action of their zap collars. Dogs too stupid to learn from being autozapped often end up buried, as one of mine unfortunately is, out beneath a peach tree somewhere, after being hit by a truck on the highway of life while chasing deer and having an otherwise great old time. I'd say it was evolution in action if it weren't for the fact that they are all fixed anyway. There's a moral in there somewhere if you can find it. 
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... Ways10.28
I've tried this, too, with my dogs. Doesn't work as well. Some sort of language barrier, however much they like the attention. A cat, of course, spits on either method. Literally. 
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... expect10.29
The dog metaphor, for example, is now getting really strained. Only in Gary Larson cartoons would Ginger come up to Sally and tell her ``The Master has revealed to me in a mystical revelation that you are to give me all of your Chunks and then lick the fleas out of my fur.'' Riiiight. 
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... devil10.30
I am not making this up. Google on Orgone Blasters if you don't believe me. Visa card in hand. 
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... bit10.31
Let's see, ``is the world I see really there?'' Decisions, decisions. Yep, it is. Guess it wasn't that hard after all. 
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... desert10.32
Actually I don't know about the bald part. Or the fat part, which is actually pretty unlikely come to think about it. Really, who had cameras back then? 
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... faith10.33
Nostradamus, anywone? 
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... Be10.34
While, on the other hand, having no visible problem with war, genocide, hurricane, earthquake, sickness, disease, death, or any of the other equally self-certain, self-serving religions on the planet. But I digress. 
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... mind10.35
Debateably, in the case of playing WoW. 
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... debate10.36
Also known as ``open intellectual war''. Nobody said that philosophers were sissies. Well, actually, that probably has been said from time to time, but only because Socrates gave the rest of us a bad reputation by meekly drinking hemlock instead of rousing the disgruntled youth and taking over in a revolution against the establishment. 
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... heavens13.1
In case you haven't figured it out yet, I'm quite fond of the metaphor, yes I am. Shaken, not stirred. 
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... argument13.2
Which anybody even then could see was Bullshit, but just in case there are any holdouts - and there are - I beat this dead horse a bit later on myself. Of course it never was very politic to criticize the conclusions of a Saint unless you were a bigger better Saint. 
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... ways13.3
Hibert found some 23 axioms were required to fill in all the gaps left by or presumed by Euclid and his successors. 
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... man13.4
Hard on cats though. Even very bright people can be so silly. In the East, the question to the master is ``Do dogs have Buddha Nature?'' (souls, loosely speaking) and the answer is to whack your student upside the head with a dog, or make an inscrutable remark about the wind whistling through holes in heads. In the West, it was Descartes announcing that ``Cats have no souls'' (Buddha nature, loosely speaking) and - apocryphally, at least - throwing his cat out of his upper story window to demonstrate the lack of moral sin inherent in killing a cat. Mmmrrrrreooow-splat!

I prefer the more interesting questions of whether or not Buddha had Dog nature and whether Descartes' cat landed on its feet, shook its head a couple of times, and moseyed off to philosophize to the extent its spirit permitted on a warm sunny wall belonging to a cat-lover far away from Descartes. Perhaps Voltaire's, if it's nine lives stretched out so long... 

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... events13.5
Presuming of course, the reality of the other people and their memories of an objective past. Both of which are, of course open to doubt. Individuals who are married will no precisely what I mean, here. 
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... Sum13.6
I think therefore I am. 
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... myself13.7
I smell the remnants of his Jesuit education and probable influence by St. Anselm's Ontological argument in spite of himself, since this smacks of imagining something so great that existence must be one of its great characteristics. 
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... Guardian13.8
Sorry, obscure literary/cinematic reference. I'm not telling which. 
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... real13.9
Except for the ones that don't agree with those of my wife... 
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... straightforward13.10
For physics and math groupies, contemplate either outgoing wave and incoming wave Green's Functions, or (my favorite) advanced and retarded Green's Functions and Dirac's truly marvelous paper on Radiation Reaction. Or write me for some even better references, e.g. Barut, McManus, Wheeler and Feynman. Or think about the Generalized Master Equation, whereby any closed quantum system is in a stationary state until it is split into a system plus a (stochastically described) bath. 
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... Universe13.11
Prove that it isn't. You can't, can you. I mean that you really can't - Green's theorem in four dimensions says so, because $\int_{S} = \int_{S-\Omega} + \int_\Omega = \oint_{\partial (S-\Omega)} + \int_Omega$ where the latter results from integrating by parts to convert the exterior volume integral of some closed four-dimensional domain $\Omega$ into a surface integral over a consisten four-dimensional boundary condition. This is a fancy mathematical way of stating the same kind of thing I was saying in English when I asserted that I might have come into being (memories intact) a second ago and might disappear (memories disappearing with me) in a second or two where from the inside of my thread of existence I Could Not Tell. 
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... wrong13.12
Just in case this is correct, I'm sitting here trying as hard as I can to think up a huge number of people who are going to appear to buy this book so I can convince myself that I'm getting filthy rich on the proceeds and can buy that imaginary yacht I've always been pretending to dream of. Righto. Works for me... 
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... ``Bullshit''13.13
Or ``Thus I refute you'' if you want to emulate Johnson. 
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... believe13.14
Where I mean that literally, as disbelief in e.g. Descartes axioms above is a form of practical insanity that will (I firmly believe) leave you dead, first time you walk off the top of a building thinking that this time gravity might not work. You believe whatever you like, but try not to splatter all over my shoes... 
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... it13.15
Sorry, but Real Science is often done using things like pure thought, dumb luck, random experimentation (the Monkey approach), serendipity, admixed with all kinds of social-interactive scientific-memetic exchanges and a certain amount of intellectual theft in the down-home rastiest genetic optimization process the world has ever seen, except maybe the one that evolved us and that is - marginally - more fun on a Saturday night. It goes way beyond just formulating hypotheses and trying to experimentally test them. 
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... cake13.16
This is often done in the unfortunate context of the Science Fair, also known as ``the place where I had to come up with a truly mindless experiment about feeding chipmunks different brands of popcorn and learned that a) chipmunks hate popcorn, all brands, and b) science is really boring and irrelevant and strictly for nerds''. If secondary school students built just one Tesla coil capable of arcing lightning three feet out into the air of the room (frying computers for ten classrooms around and just looking as powerful and dangerous as it in fact is), you'd have them winning the Nobel prize in physics within a decade. Science is fun, and it should be taught as if it were fun. But I digress. Again. 
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... Theory13.17
As I will describe below, a Fairy Hypothesis says something like ``Invisible fairies whisper in the ears of chipmunks in a secret language and cause one chipmunk in three to love butter-soaked popcorn enough to leave their chipmunk-families and try to hop on the front fenders of popcorn trucks as they make their delivery rounds every morning, resulting in the observed carnage of chipmunks on the streets.'' Even if you design a perfectly beautiful experiment that catches on film movies of fat little popcorn-fed chipmunks jumping at the fenders of popcorn trucks and being squished, is it evidence of invisible fairies? Is it really evidence about the inner psychic state of the chipmunks at the instant they jump? Maybe they just suffer from self-esteem problems due to their un-chipmunkly portliness and want to End it All and choose the instrument of their unhappiness as the means. 
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... experiments13.18
The word validated in Science has a funny meaning that has led to much confusion. To give you a hint, nothing is ever proven in science, and no good scientist is ever certain that any given theory is true, no matter how good the evidence. The proper way to view scientific ``proof'' is a state of conditional belief. That is, when I say ``I believe that Newton's Theory of Gravitation is a proven fact'' and is in fact a Law of Nature, what I mean is that ``I believe that no reliable experiments have ever discovered Newton's Theory of Gravitation to be inconsistent with their results in at least the non-relativistic, non-quantum mechanical arena, where many such experiments with macroscopic gravitationally attracting bodies have been performed any one of which might be expected to egregiously fail were the Theory egregiously wrong, and with the understanding that I might have been lied to about those experiments, that experiments to determine whether or not antimatter gravitationally repels matter have not yet been done and that they (or the discovery with sufficiently accurate probes of a deviation from $1/r^2$ at short or intermediate or very long distances, or anything else one might thing of) might prove it one day to be wrong.'' Which we don't usually write all the way out because it is pretty long, and everybody who paid attention in their science classes instead of spending the time reading their favorite piece of scripture and praying for their teachers' souls already knows it.

Next thing you know, some numb-nut who obviously was such a student in their youth is claiming that the ``Theory of Evolution'' or the ``Theory of Gravitation'' are ``only theories and not proven facts'' so that other theories like the ``God designed and build the human eye out of plasticine using his own strangely anthropomorphic eyeball as a model'' theory should get equal time in the schools. Right... 

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... example13.19
Einstein's famous ``God does not play dice with the Universe'', for example. 
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... ``likely''13.20
In quotes to emphasize the fact that this word is basically meaningless without axioms that tell you how to compute probabilities and more axioms to tell you how to compute probabilities relevant to anything concrete. Russell made this mistake in Problems in Philosophy and I'd prefer not to repeat it, so once and for all, quotes or not, the word ``likely'' is always used in the same way I use ``reasonable'' - utterly unprovable and irrational, but somehow sensible for all of that. That's ``probably'' - eep - the way Russell meant it as well, but he should have said so, somewhere. 
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... is...13.21
One of my favorite mathematical paradoxes - imagine a dartboard and a dart with an infinitely sharp tip - a tip consisting of a single mathematical point. There are an infinite number of points on a dartboard, and when the dart is randomly thrown at the board the probability of hitting any particular one is therefore zero. Yet some point is hit - the probability of hitting the dartboard itself is unity. Hmmm, possibly the purest Zen koan imaginable. 
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... experience13.22
The Internet Encylopedia of Philosophy, http://www.utm.edu/research/iep/l/logpos.htm 
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... sputter13.23
Or should be sputtering, that is. C'mon, now, give it a try. 
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... meaning13.24
I might just as easily formulate as an alternative axiom that the only way to determine whether or not a statement is meaningful is to see if I understand it, if it has any meaning to me, whether or not I can prove it by means of experience. Or that a statement only has meaning if it can be proven by a passage in the Bible. Or that nothing has meaning. Or that everything, all propositions, have meaning, if only as themselves. This is an axiom in the ``unprovable assumption'' sense because it is not the only possible criterion of meaning and indeed is a little bit of an odd one by the standards of the dictionary and the cognitive process itself. 
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... proof.13.25
Is a question ``meaningful'' if it could in principle be answered by a suitable experiment or when it is answered by a suitable experiment? What about classes of experiment? Science is all about beliefs of what will happen in some future time based on observations of the past, yet one cannot use LP as a mechanism to prove that the future will be like the past even in principle. Without a lot more axioms, almost any nontrivial question becomes ``meaningful'' only after it is answered. This is simply nonsense - nobody uses the term ``meaning'' in that way. I can perfectly well understand questions such as ``was Frodo a virgin when he left on the Quest of the Ring'' that cannot, actually, be answered by experience even in principle. LP confuses the meaning of the word ``debateable'' with the meaning of the word ``meaningful''' and gets them wrong either way. 
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... perception''13.26
...and equally likely have a very hard time understanding why one cannot ask where an electron is and how fast is it going at the same instant in time to arbitrary precision. As I believe Feynman once is alleged to have said, ``Nobody understands quantum mechanics...'' 
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... space-time13.27
For the non-physics-groupies out there, an ``event'' isn't something like a black-tie soiree; rather it is a single four-dimensional space-time point $(\vec{x},t)$. 
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... Jaynes13.28
Jaynes was a master of both quantum electrodynamics and statistical mechanics and in fact developed a model I studied and used myself in application to resonant optical systems. His ``maximum entropy'' approach to the generation of probabilities is, as you should recall from earlier chapters, in my opinion the best way to axiomatize the process of inference in all of science in part because it works to allow one to derive statistical mechanics and thermodynamics that (self-consistently) empirically works.

I am indeed fortunate to have a copy of his 1994 draft, and hope to see it published one day. Jaynes did not derive his results from a consideration of sets, which I think may have been a mistake in his analysis of logic itself, but otherwise his reasoning was transparent and quite lovely - he certainly clearly exposed his axioms (and pointed out the fallacies in many alternative sets of fallacies) as he went along. Jaynes makes precise what Russell rather sloppily referred to as ``probable correctness'' or ``plausibility'', which guides real human reasoning far more than the actual rules of logic. 

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... question13.29
Which it does, of course. Duh... 
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... infinity13.30
Except, of course, that not only are there at least two kinds of infinity, with one ``greater than'' the other and both infinitly great, but through many quirks of mathematics many of the ``fool's'' intuitions about infinity are just plain wrong. For example, one might think that there are ``more'' (a larger infinity of) rational numbers than there are of integers (think about it). One would be wrong. Think about it some more. 
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... isn't13.31
If we're being sticklers for logic and laws of thought here, and if we're not why are we playing with syllogisms in the first place? 
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... knot13.32
Sorry. I'm a bad, bad man. 
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... other13.33
Or as my mother liked to say, ``If wishes were horses then beggers would ride...'' 
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... logic13.34
This isn't to pick on the West. The East does a better job of looking there, but they do just as bad a job at getting hung up on scripture and politics and human beings and other bullshit. This is with the notable exception of Zen, which would attempt to teach you precisely the same conclusions as this entire book by reciting just the right Gödelian koan to you as a perfect haiku and then, while you are still in a logically stunned state, whacking you with a banana. Unfortunately, I couldn't get the publishers of this book to distribute it with a banana so you could whack yourself. Fortunately, they are readily available at any supermarket and are even cheap. So at your convenience, please get a slightly overripe banana, and while puzzling over Gödel's theorem and improveable axioms and the Pit of Existential Despair, whack yourself on the head with it (or get a loved one to do it for you from behind your back - it is better if you don't know exactly when you're going to get whacked on the head by a banana in Life) hard enough that the pulp squishes and oozes down onto your face. It won't hurt, and the exercise isn't at all silly. I promise, you will then be Enlightened. As soon as you stop laughing. 
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... sun13.35
You can see that I'm a repressed writer of fictional prose, right? 
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... dumped13.36
See? 
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... ``watch''13.37
Sorry, I just can't help it. 
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... miracle13.38
Note well that I'm not asserting that these miracles ``must have been performed by God''. Personally, I think that evolution, physics and the Universe is God, and that our hands and eyes are God's hands and eyes. Nope, can't prove it. Can't prove the Law of Gravitation or that the scientific method is ``correct'', either, but I tend to believe in them too... 
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... table14.1
And by my own Zen contribution to mankind's wisdom, the squashed banana dribbling down from your scalp to your mouth, leaving behind a kind of slimy trail. 
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... FoscariniA.1
See http://www.fordham.edu/halsall/mod/1615bellarmine-letter.html 
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... CouncilA.2
of Trent 
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... 1633A.3
http://www.fordham.edu/halsall/mod/1630galileo.html 
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... glozingA.4
``To give a misleading or false interpretation.'' Sorry, this is a word that isn't terribly common any more and you probably don't have a dictionary handy. 
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... news''A.5
By divine providence I happened to be teaching gravitation in an introductory physics class the very day that the announcement was made to the world. I had great fun announcing to my class with a very serious demeanor, that it was at last all right to believe in Newton's theory of gravitation, the Copernican model of the solar system, and that it turned out that Galileo was right after all. The pope Himself had said so!

Sarcasm, sure, but gentle sarcasm, with a point. The commonly accepted methodology for determining what is and isn't knowledge is directly contrary to that required by any superorganismic religion, as therein knowledge must be based on scripture, not sense or science. In the subsequent discussion that followed, nobody in my class was fooled into thinking that Galileo's exoneration meant anything at all to the contrary. 

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... truthA.6
Except this one, of course, which is not proveable but is true nonetheless. 
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