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February 15th, 2013:

Axioms, Postulates, Maps, Territory ….

It began as a response to an interesting comment by Philip Scott Thomas at

http://www.countingcats.com/?p=13915&cpage=1#comment-307104 ,

but I’m afraid it got out of hand…. :(

PST,

Gödel! Indeed. I’m still trying to find a layman’s restatement of The Theorem. Although I believe the theorem says far more, for now I go with “No closed system can explain itself,” by which I mean precisely that every non-trivial logical system must be built on postulates: starting propositions about the system that we assume without proof. (A given person may or may not consider such a postulate self-evident. More on that below.)

You wrote, “the starting point of a system may also be a self-evident statement. I took you to be referring to statements “without which it is impossible to think rationally.” Which, on reflection, I think isn’t what you meant at all. However, that’s going to be my starting point just the same.

However, the description in boldface is vitally important. Such statements are the ones properly called AXIOMS.

Axioms do not in themselves constitute the starting point of any system except the null system–the one having NO system-specific propositions (be they postulates or theorems).

Axioms are about how human logic must proceed. Postulates are starting, hence unprovable, PRE-assumptions about relationships within some particular system.

The axioms are the same whenever we reason logically, regardless of what we’re reasoning about. The postulates are the foundational premises we make about the SPECIFIC system under discussion. It is a POSTULATE that “there is a real world out there which exists whether anyone thinks so or not.”

Here is another foundational postulate: “There is a God, whose properties are ….”
And another: “There exists no ‘God’ whose properties are ….”

We then posit (postulate) a bunch of attributes of the object “God” and see what else we must include in, or exclude from, our set of postulates in order to keep the system logically valid.

Unfortunately there is plenty of precedent for confusing axioms and postulates, as the logicians seem to have adopted a notation and a system of thought where they call all fundamental rules and premises “axioms.” They distinguish between the REAL axioms, which are the rules of thought without which reasoning is simply impossible — such as “A = A” — and postulates, which are the presumptions we’re going to make when building a logical system, by calling the first something like “L-axioms” and the second something like “P-axioms”–I forget the exact labels, but they do refer to both as “axioms” and then add a designator to distinguish between them.

As for “pulling [postulates] out of your arse,” yes, you CAN do that, though you may not end up with anything much–or with anything that seems applicable to the real world. You can start with some collection of elements–numbers, geometric shapes, barbers–and posit, arbitrarily, one or more statements about them (“propositions”). You can then work out the logic. You may start with the set of all barbers, and then postulate that there is a subset of the barbers who shave all those, and only those, who don’t shave themselves. Now that IS a valid logical system. Unfortunately, following out the logic leads to the conclusion that said subset of barbers is the Empty Set. (Russell chose to say that therefore the concept of the subset so-defined is meaningless. I agree with him, but that’s just me. *g*)

With this postulate you have constructed a logical system, albeit one which is inherently trivial. (The system’s elements are “all barbers who shave only those who do not shave themselves. The starting proposition is, “The set of such barbers is not empty.” But the postulate is self-contradictory. This is why Russell called the concept of such barbers meaningless, and it’s why I say we have here a system containing no propositions which are true–thus it is an example of the trivial logical system.)

But some sets of postulates–starting assumptions, givens of the system–which although not intuitively obvious, let alone obviously relating to something in the Real World, yield a rich system chock-full of non-trivial theorems (provable statements about the relationships among various elements of the systems, and about relationships among subsets of the elements of the system).

Such as the starting postulates of Euclidean plane geometry. (E.P.G. may have elements and propositions inspired by the real world, and it may yield results which help us to understand the real world, but its elements, being pure abstractions such as lines with no width, points with no extent at all, etc., have no actual physical referents in the real world. E.P.G. is the map, not the territory.)

When we reason about the Real World, we are trying to develop rules or “maps” that will let us navigate our way to a better understanding of it. So somewhere, there are and must be foundational postulates about that world, one of which is going to be along the lines of “There exists an object X in the real world, which has the property that….”

As a matter of fact, although I’ve never worked out such a system as the very idea bores me, I don’t see why you can’t posit a solipsistic Universe, where all that is exists only in your imagination. I think you could construct a valid logical system around that postulate. I think in fact it would be easy, depending on how nit-picky you are about making unassailable statements–which is a language, or a meta-language, issue.

As for “certain postulates [about the Real World] are self-evident”–well, yes, they are, but the problem is that some people find one postulate self-evident [say the existence of god(s)] and others find the exact opposite self-evident. Now, that does not mean that both those people are correct. Self-evidence may be in the eye of the beholder, but Reality is not: and that is one of MY foundational postulates ABOUT THE REAL WORLD. (It could be fun to write a fantasy novel where that postulate is false. And of course Solipsism is one example of a system that denies the existence of reality external to some particular person’s consciousness.)

Also your own example. It’s self-evident to you and me that all races, say, are human. However, there are two terms in there that need definition: “races” and “human.” The simple fact is that throughout history there have been groups of humanoids who thought that other groups of humanoids were, at best, only somewhat human: subhuman in fact. Famously, a certain large subset of ethnic Europeans held that opinion of Negroes to be self-evident; and those who are rigorously Muslim learn that self-evidently non-Muslims are subhuman. (For instance, only Muslims can, properly speaking, be “innocent.” The concept is simply not applicable to non-Muslims, who can only enjoy full Personhood, at least in current Islamic doctrine, if they convert to Islam. This boils down to denying the full humanity of non-Muslims.)

Nevertheless, it’s self-evident to me personally that all your groups are groups of humans, and also that their members share the same fundamental nature as all other humans…and that therefore they are subject to the same laws of nature, wherefore it is wrong for anyone to assume authority over any other fully-competent (to exclude children and the mentally dysfunctional) human. Which is precisely to say it’s wrong to interfere (unjustly) with another’s self-determination. Which is to say that we all have the same “rights” to life, liberty, and property, by nature.

I’m afraid it seems to me that your penultimate paragraph is circular. :>((

I’m glad if my posting interested you. I’m delighted to see your thoughts on the matter, and I’d look forward to reading more of them.

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