thoughts on politics

May 14, 2010

Axioms of existence

Filed under: Uncategorized — Matt @ 5:49 am

Recently, I have come across three proposed axioms that govern mathematics and human life/existence in general, and such things as these provide fertile ground for thinking.

  1. The Axiom of Choice — I first encountered this tonight after surfing through a few layers of web pages on abstract algebra.  (I was reading about that to try to find some good concepts and proof ideas to show to one of the students I work with.)  This axiom (as far as I can tell, it is axiomatic) is limited (as far as I can tell) to mathematical use only, although inasmuch as that application has to do with taking our rules of logic to their furthest extent, its use may pour over into other areas of inquiry.
    The axiom of choice states that it is always possible  to pick one object from each set of a collection of non-empty sets.  It was surprising to me when I first read this that such a statement even needed to be made, yet after reading some relevant examples, I was surprised to find that this was an essential fact that I (not to mention mathematicians throughout history) had taken for granted.  A vital stipulation of the axiom, as stated where I found it, is that a rule for how to pick the objects does not have to be given, or even to be exist, for it to still be possible to choose objects.
    Why would a rule on how the objects are picked matter?  Imagine that each of our sets is just a random group of integers, such as [2, 6, 235], [-78, 0], and [490, 23, 293875928371].  With just three sets, we can perform the choosing for ourselves in less than a second.
    But what if our collection is all possible subsets of the integers, a collection which is infinite and consists of sets which are finite as well as infinite (for example, odd numbers)?   To do work with these sets or objects from them, we must, or so we think, at least know what we are dealing with, and the only way to do that systematically with an infinite number of sets is to define the rule for taking an element from each.  A choice function for this collection (I think) could be as follows: For a given set S in our collection, f(S)=0 if 0 is in S, the least positive number in S if 0 is not in S, or the greatest negative number in S if S contains no positive numbers nor 0.
    However, there are collections and sets where we could not find a rule.  An example (thought of off the top of my head) would be sets composed of any number of integers and/or objects from my desk.  This could include of course numers, but  also a rubber band, a book, a box of tissues, a guitar capo, a cell phone, a bunch of change, an Amazon credit card statement, and so on.  Here, it is less obvious how to specify a rule to pick objects for sets without integers.  (This may fail, I realize, because there is not an infinite amount of objects on my desk, but I hope it illustrates that problems can arise with more complexity and arbitrariness in the makeup of our sets)
    What the axiom of choice says is that it is possible (and therefore allowable in formal proofs) to choose objects from sets, without specifying a rule for how to do so, and even in cases where no such rule may even be possible.  This is extremely powerful, and the wikipedia article on the subject points out numerous equivalent results throughout mathematics, and results that rest on the axiom.  In probably every math class I took at UGA, at some point the phrase “Pick an object from each set” was uttered (or something equivalent), but never did I question whether that action was doable.
    Don’t expect a complete logical justification or defense of this principle from me right now, though.  It will be food for thought for a while.
  2. The Action Axiom — Ludwig von Mises specifies this axiom at the opening of Human Action.  The axiom simply states that “Man acts.”  From this (to my understanding of Austrian economics) all other economic knowledge is derived.  Whether Mises was the first to enunciate this axiom, I do not know.
    His usage of the word “act” is defined in the first sentence of the first chapter: “Human action is purposeful behavior.”  Put other ways, this is “will put into operation and transformed into an agency, is aiming at ends and goals, is the ego’s meaningful response to stimuli and to the conditions of its environment, is a person’s conscious adjustment to the state of the universe that determines his life.”
    In the first part of Human Action, Mises goes through the ins and outs of this definition, problems that could be raised as objections to it, and some first consequences of it.  One consequence is the scale of value, which determines the action a man takes at any one time.  Since action is purposeful behavior to remove unease, and since man always has at least one source of unease which he is capable of attempting to mediate or eliminate, and since he can only perform one action at a time, a scale of values arises whereby men act on the dissatisfaction that is most urgent to them at the time.  (This also assumes scarcity, which ought to be a self-evident fact of the universe, and at any rate is assumed by other schools of economics; Mises addresses the performance of one action at a time.)  Another result that closely follows, well known in other schools of economics, is the law of diminishing returns, which Mises just calls the “Law of Returns.”
    Here we have (moreso in Human Action than in this blog post) a firm, rigorous, deductive derivation of commonly known economic laws, whose proofs are usually given with graphs of smooth curves and calculus.  Mises terms the study of human action “praxeology,” and says that economics is only the most well-developed area of this science.  I suppose this is because production and trade are obviously important as subjects of study, and more easily parsed because of pricing elements, goods produced, and such.
    The amazing thing about the action axiom, similar to the axiom of choice, is  the recognition of a fundamental element of not only all economic activity, but of all conscious endeavors undertaken by men.  It is so self-obvious that it was overlooked for thousands of years of inquiry by men, and so its terrible implications were also not understood.  Unfortunately, today, the action axiom is really only widely known on one side of the campus of Auburn University today, as well as at the Manhattan barbecue restaurant Hill Country once a month and a few other pockets around the world, but at colleges it has been shunted aside and forgotten.  This is how economics was studied, in a logical and careful manner, in the School of Salamanca, Spain, long before Adam Smith, and certainly Carl Menger and Ludwig von Mises ever wrote.  If any of its conclusions are to be proved wrong, the error in the reasoning must be pointed out.  Later chains of reasoning notwithstanding — I’m not so fresh on them right now — the action axiom seems to be solid.
    (Sadly, I cannot find the simple statement of the axiom in Human Action right now, but all the other citations are from page 11 of the Scholar’s Edition.)
  3. The Existence Axiom — I am nearing the end of Atlas Shrugged, but I just reached the chapter called “This is John Galt Speaking,” which is probably the roughest rough patch of all in the book.  In truth, this 60-page unbroken monologue (!) is less a piece of dialog in a book than a full exposition of Ayn Rand’s personal philosophy, just stuck right in there in the middle of a novel.  I got 8 pages in last night; I fully expect reading the whole thing to take several days, whereas I’ve had other days where I got through 60 full pages of normal parts of the book.
    In the dialog, John Galt takes over the airwaves of the whole country to explain the philosophy and moral system of the producers who have been victimized by society.  He begins at the start of the start, positing the mind as man’s one tool of survival, as an animal has strength and speed and plants have, I dunno, the sun.  Man must use his mind to attain the things he values for his life.  Actions he takes toward self-preservation are natural and moral; thought therefore is the basic moral action, while having the things he values brings him happiness.  However, unlike plants and animals for which attaining valued things is an automatic instinct, for man pursuing his goal is choice; he must decide to think, and in this way, morality is also therefore a choice.
    Proceeding from here, Galt gives the kernel of his system of reason as the “Existence Axiom:”  “existence exists.”  That’s it.  From this follow two immediate corollaries: the existence of something, and that consciousness exists.  It seems so bare a statement, but it is understood and indeed necessary for all else we know and do.  Why should it even need to be stated that existence exists?
    In the world of Atlas Shrugged, most of society has unknowingly arrived at agreement on what John Galt calls the “Morality of Death,” where the mind of man is turned against himself, or rather disabled so that he cannot fend for himself and ultimately not survive.  To preach this ideology, the academics and public leaders partially rely on demonstrating a number of inherent supposed contradictions of existence and of morality, from which it is concluded that morality is non-real, truth is unknowable or impossible, and the mind is useless at best and man’s greatest weakness at worst.  Galt unmasks the fallacy of this ideology by reducing it to a contradiction with the existence axiom.
    In a universe of absolutes where the existence axiom rules, contradiction is impossible, and of the many consequences of this recognition is the idea that man must use his mind to improve his life, attain the things he values while adhering to the moral code, and ultimately find happiness.  All this from the simple sentence “existence exists.”

What is it that makes these axioms, with all of their vital ramifications, so hard to identify?  Partly, it is that they are so deeply and universally embedded in our experience that we do not notice them until we reach much deeper problems which seem baffle all the prior assumptions we had held.  It seems that their discovery must follow concrete experience lacking understanding.  Without having observed a multitude of economic transactions, Austrian economists could not have discovered the action axiom operating behind them all.  Not until contemplating and parsing many more derivative mathematical questions for thousands of years could mathematicians notice the function of the axiom of choice in all of them.  Yet these truths are the most basic and fundamental to all of our systems of thought.

I feel some trepidation in asserting that an axiom is true; I rather prefer “seems to be true.”  I do not know whether an axiom can be positively asserted by anyone to be true, for axioms are defined to be self-evidential and requiring no proof.  However, if no proof can be given for the truth of an axiom, does that mean it is incontrovertible?  How could an axiom be debated, if it is ultimately simply supposed to be accepted as a given fact of the universe?  It seems that an argument could turn on whether someone simply refused to accept some axiom, or some specific part of one, because it was not self-evidential to them.  This is important, because while the axiom of choice is only important/understandable to those with some mathematical training and I can’t imagine anyone attempting to deny the existence axiom, the action axiom on the other would probably be extremely controversial to discuss.  Perhaps the difficulty is that the definition of axiom is a little subjective.  Something may be self-evident to one, but not obvious to another.  Again, it’s hard to imagine someone denying that existence exists, but the axiom of choice in mathematics was not accepted for quite a while after it was first proposed.  (Not to say that its widespread acceptance makes it true, either.)

One can see, each of these three axioms are basic in concept yet primitive in importance.  The question comes up, at least to me, is there an infinite number of such axioms?  I can see no reason for there to be a limit on the number of axioms to be formulated by humans, and it seems silly to attempt to arrive at some final list of 10 axioms or whatever.  However, it just doesn’t seem like there’s room for too many axioms as deep as this.  Each one refines in a strong way our understanding of the world around us; how many new subterranean logical structures are there for us to discover?

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