Godel and Atheism
I was going through a saved home directory from a HD that died a couple years back, and found this piece, which deserves to be saved. The file it came from is dated Mar 8 2004. (if it looks like crap it’s because I didn’t want to reformat it for the web)
I’ve been dwelling on the contents of this e-mail for more than an hour, couldn’t sleep because of it. So, for any e-mail that I write, this’ll be epic. Also, this is what I really learned over last summer break, while I took, and was supposedly learning, History of Science, and Life Science 4: Genetics.
Statements of Personal Philosophy, and the books I got it from, in order of importance.
BOOKS I’VE READ
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G�del’s Proof — Ernest Nagel
delightfully short and exciting work, captures the philosophical essence of G�del’s work without the cumbersome set-theoretic notation.
G�del, Escher, Bach: An Eternal Golden Braid — Douglas R. Hofstadter
A wonderful book that has taught me more about math, reason, and consciousness than Zen and the Brain, even though I’ve only read half of both.
The Fountainhead, Atlas Shrugged, Anthem — Ayn Rand
nice fictional pieces that tell a good tale. though I think she holds logic as an absolute, which, in my opinion, is a mistake. Her philosophy is pretty sound, and is nicely expounded upon in some of her other works. I particularly enjoyed Anthem for its narrative style. Atlas Shrugged is a culmination of all her thought and effort, if you’re only going to read one.
Zen and the Art of Motorcycle Maintenance — Robert Pirsig
a lovely frame story of a philosophical journey in search of a platonic Quality (which can probably be substituted for just about any platonic Ideal) during a physical trip through the midwest and down memory lane. Gets even better when you read about the history of it’s author (Guidebook to Zen and the Art of Motorcycle Maintenance — Ron Di Santo)
Heart of Darkness — Joseph Conrad
a nice read about the depths of the human psyche, gotta love the depiction of the British.
Zen and the Brain: Toward an Understanding of Meditation and Consciousness — James H. Austin
big thick volume that reads like a scientific journal article that won’t stop. Much can be learned about the physical structure of the brain itself
I feel that I should take a brief walk through G�del’s Proof….
THINGS YOU SHOULD KNOW
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Paradox by Bertrand Russel.
Let S be the set of sets which DO NOT contain themselves as an element, most sets are like this.
Some sets that DO contain themselves as elements are very strange.
Example1: the set of all thinkable things, is, itself, a thinkable thing, and therefore contains itself as an element.
Example2: the set of all possible combinations of numbers, is, itself, a possible combination of numbers, and therefore contains itself as an element.
So far we’re good. But wait: is S an element of S ?
well, logically, by the way we defined S: S is an element of S if and only f S is not an element of S.
What a paradox!!!
Self-Referential Statements are a common source of paradox, and my special categories BOTH and NEITHER.
Ex1: This statement is True.
well, if you assume so, then it is. if you assume not then it is not.
so, really, since it is consistent both ways, the statement is BOTH True and False.
Ex2: This statement is False.
well, if you assume it’s true, then it says it’s false. if you assume it’s false, it says it’s true.
so, really, since it is inconsistent both ways, the statement is NEITHER True nor False.
Historically, Ex2 was taken to be a paradox and to be avoided.
Definitions: Mathematical, Meta-mathematical, axiom.
Mathematical statements are those describing relationships between numbers.
ie. 1+1=2
Meta-Mathematical statements are those statements which are about numbers.
ie. 2 is the first and only even prime
Most mathematical work is accomplished through meta-mathematical arguments.
Axiom is some statement to be assumed true, usually self-evidently. It’s what you build your Theorms on.
HISTORY AND MOTIVATION
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The mindset of mathematicians at the turn of the 1900’s was to axiomize all of mathematical work up to that point, and remove any and all self-referential statements that might breed inconsistencies, and thereby cleanse the entire field of doubt, because, of course, doubt is the bane of logic and truth. It was with this mindset that Alfred North Whitehead and Bertrand Russel, both [mathematical philosophers/philosophical mathematicians/philosopher mathematician/mathematician philosopher] (best type of person, in my opinion) devoted 20 years to compiling a veritable tome of mathematics known as Principia Mathematica (yes, Newton also published a different book under this title). The idea was to define logic as mechanical notation shuffling according to preset rules, such that a computer, with no intelligence or knowledge of the underlying ‘meaning’ of the symbols involved, could proove stuff, if you supplied it with a starting position, the axioms. So detailed was the language of Principia that they never got around to 1+1=2 until somewhere around page 200 of the 3 volume set. I don’t think that anyone has actually ever read the whole work.
It was in response to this work that G�del published his _On_Formally_Undecidable_Propositions_of_Principia Mathematica_and_Related_Systems_. A work that was humorously devoted to Whitehead and Russel, and which in 20 pages completely underminded what they had done in 20 years and 3 hefty volumes. Naturally his argument rests on something they were trying to get rid of: the self-referential statement.
GEOMETRY REFERRED TO IN RESULTS (best to skip and come back)
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Axioms of Euclidean Geometry:
1. It is possible to draw one and only one straight line from any point to any point.
2. From each end of a finite straight line it is possible to produce it continuously in a straight line by an amount greater than any assigned length.
3. It is possible to describe one and only one circle with any centre and radius.
4. All right angles are equal to one another.
5. (Euclid’s fifth axiom). Through a given point not on a given straight line, and not on that straight line produced, no more than one parallel straight line can be drawn.
Ax5 is important because it was for some 100’s of years thought to be derivable from the other 4, but nobody could find out how. Eventually someone got smart and figured out that if you had instead:
5. (Hyperbolic axiom). If P is any point and AB is any straight line not passing through P (even if produced), then through P there are straight lines YPZ and WPX such that: (1) YPX is not a single straight line. (2) YPZ and WPX are each parallels to AB. (3) no straight line through P entering ^ZPX is parallel to AB.
You get Hyperbolic Geometry. Then there’s also Reimannian Geometry, and lots others, none of which are True in the absolute sense of the word.
THE PROOF
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1. assume that mathematics is complete, that we know all axioms and can, from them, derive all things (given the time)
2. It is now, in principle, possible to have some algorithm test all possible statements in our symbolic language, namely Principia Mathematica, to see wether or not that statement is derivable from the axioms.
3. At present no such algorithm is known to exist, and if we can show it to be impossible, then 1 is null and void.
4. To do this you start out making some statements about numbers (meta-mathematical statements), you organize all these into a table and number them.
5. Now you start asking things like, is statement X true ?
I sorta forget what happens next, but eventually you get some statement somewhere that can be looped back to talk about itself, and the whole system comes crashing down. ie. your mathematical statements have some mapping to your meta-mathematical statements, so the meta-math is really talking about itself not math.
6. because this inevitable happens somewhere, it must be that not such test algorithm can exist, and therefore that mathematical systems, all of them, are incomplete.
RESULTS
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Take, the historical example about geometry. You have some axioms 1-4 and you have a statement 5 that you want to derive from 1-4. well someone showed it couldn’t be done, that 5 had to be taken as another axiom. Then, if you assume it, you get Euclidean Geometry, if you assume something else, you get a Noneuclidean geometry. There will always be some statement out there that you can find, which must be assumed as another axiom in your system. In this way the field of mathematics will branch out into some sorta intricate tree.
The results of all this can be taken two ways:
1. mathematicians have prooven themselves into good job security, ’cause there shall always be some statement out there waiting to become an axiom
2. it’s a lost cause, because you can no longer find absolute Truth, nor will your search ever end if you try.
I’m gonna go with the 1st interpretation myself.
Mathworld states G�del’s work a little differently, but it’s really all the same.
Beware ye seekers of truth, those that claim to have it:
If you have a system that claims to be able to prove itself true(consistent), then that system is inconsistent.
PERSONAL PLUG
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I could ramble on more about consciousness and its manifestations (G�del, Escher, Bach) or about subjectivism (Zen and the Art of Motorcycle Maintenance) or the logic of atheism (Ayn Rand, with support from Godel and Hofstadter). But, though I be atheist, all is not lost. If you show me somewhere that I might be wrong, I’ll readily go back and rethink everything, atheism doesn’t make me particularly happy. (Assuming they are mutually exclusive, given the choice between happiness and knowledge, I choose knowledge.) You’ll have a tough time, though, ’cause G�del ruined my belief in the absolute.
Is the maxim ‘There are no absolutes.’ itself an absolute? is probably the best place to start, ’cause I don’t really have that answered.
Also, during the summer, inspired by G�del, i came up with a new way of looking at the manner in which the field of mathematics can be derived, you have but to ask, and I’ll write another installment. In retrospect, that brief walk wasn’t so brief.
“…nice fictional pieces that tell a good tale. though I think she holds logic as an absolute…” Ayn Rand holds the virtue of reason as an absolute for man’s survival. Where is her error?
That was a good summer.
Re: Favela
I apologize for this response being so delayed. But I think the primary error with Rand is holding an absolute of anything at all. So far, the sciences (being the most rigorous application of logic) have remained remarkably quiet about both ethics and morality. With the rise of the ‘New Atheism’ I believe that in a few years we will begin to see some good results, coming from the more social sciences, such as psychology, that will address this historic lack.
But honestly, how many purely rational people have you met? I haven’t met a single one, not even myself, yet I, and everyone else seem to be surviving just fine. The first troglodyte had severely handicapped mental faculties, yet they too survived. Bacteria don’t have any mind to speak of and they’re everywhere. So the capacity for logical, rational thought is not necessary for survival, man’s or any other creatures.
Things get much more interesting when you probe more deeply into how much rationality we really possess. We are certainly psychologically compelled toward the minimization of cognitive dissonance. In many cases this is a basic compulsion that leads us towards more rational behavior, yet a psychotic splitting of the mind or the ‘donning of hats’ is also an effective, though arguably irrational, coping mechanism.
Furthermore I think that holding any principle as an absolute is a mistaken approach. Culturally, we’ve always had exceptions for every ‘prohibited’ action. For example Murder is Wrong. We know this because not only is it logically inconsistent, it’s also asymmetric, and you simply can’t build a large stable society if it’s allowed. Yet, we’ve reserved exceptions: you can kill in self-defense, death penalty is still acceptable, you’re allowed abortion. (I don’t want to debate the correctness of these exceptions, I’m just pointing out that they exist) If you examine all of our culture’s moral prescriptions, I’m confident that you’ll find exceptions to all of the rules; absolutes simply don’t exist.
Also, I’m sure a mathematical logician somewhere has had to have shown that absolutes can’t exist because they’d invariably lead to inconsistency. Russel’s Paradox comes to mind.
Besides, the real beauty of the social systems we’ve created is that some of them (freemarket capitalism) work exceptionally well despite the fact that humans are known to behave irrationally. This really is our greatest achievement.
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The only absolute is that there are no absolutes.