Friday, March 23, 2012

Contingent Propositions

Math: Sometimes the truth is contingent.
I have been a bad blogger lately--if not posting is bad. I have been monumentally busy at work, leading sometimes five, sometimes two different proposal teams. I also have the introduction to literature class I teach, and that demands time and focus. Unfortunately, the class is not going as well as I would have liked. More than previous classes, the current group seems especially apathetic toward poetry. I'm having a hell of a time helping them connect. I remain optimistic, though, because poetry has a way of sticking with a person. My dissertation progresses, albeit slowly.

In the interest of keeping up, I want to post a small update for consideration. I have been conversing with Kairosfocus on worldviews and first principles. One of the paths we took recently involved distinguishing contingent and necessary beings. To illustrate a necessary being, Kairosfocus offered this:
Any entity that is dependent for its ability to function on the particular co-ordinated physical arrangement of parts is contingent, as if the parts are moved around or separated such a composite entity will cease to be or will break down.

Auto parts shops have a surprisingly deep philosophical significance, never mind that chilling, long low whistle from under your car when the mechanic is looking at it.

By way of contrast, the truth asserted in the structured set of symbols: 2 + 3 = 5 always was, will always be, cannot be denied on pain of absurdity, etc. It cannot break down and does not need to be repaired.

It is a necessary being.

(We need not trouble ourselves for the moment on the 2400 year old debate on whether such may only be instantiated in physical entities. Suffice to say that such mathematical or more broadly propositional truths capture assertions about reality that may or may not be true, but if true can have very powerful implications. Thence, the “unreasonable” effectiveness of mathematics in science: If X then Y, holds, once X is found.)
I have been turning over this bit in my mind. Something about Kairosfocus's statements about 2 + 3 = 5 has seemed problematic to me.

The problem, as I have tried to work it out, is that the truth of 2 + 3 = 5 is contingent. Let me explain:
For the truth of 2 + 3 = 5 to begin or continue, we need (1) a material universe; (2) principles of rationality, such as the law of identity; and (3) a computer, that is, a being to arbitrate between the universe and rationality so as to determine the truth of expressions. There may be additional needs, but these three factors seem essential at the least.

We note also that the falsity of 2 + 3 = 4 depends on (1) to (3).

The truth or falsity of these expressions is an effect of the three factors of universe.
In my estimation, then, it is incorrect to say “the truth asserted in the structured set of symbols: 2 + 3 = 5 always was, will always be, cannot be denied on pain of absurdity, etc. It cannot break down and does not need to be repaired.” Specifically, the incorrect parts are “always was” and “will always be.” The expression 2 + 3 = 5 is true only as long as we have a material universe where things are identical only to themselves and interact with one another in regular ways. As long as we have, in other words, all three factors in play: materiality, regular constraints, and a translating/computing intelligence.

I should point out that my comments do not deny or reject necessary beings, per se. I am only saying that the specific truth of 2 + 3 = 5 is actually contingent and not necessary. I have not yet thought far and deeply enough to say what this conclusion means for the principle of cause and effect and the principle of sufficient reason.

1 comment:

  1. LT:

    I too am busy, with a budget season and related matters. I understand, having neglected my main blog, since I am also busy writing a course unit or two -- having broken through a 6-week writer's block.

    To help you in brief, 2 + 3 = 5 does NOT depend on materiality, as sets can contain ANY entities. In fact, to address the Russell paradox, Zermelo-Fraenkel set theory joined to the von Neumann cumulative hierarchy -- yup THAT JvN from "Mars" again [there is a suspected nest of "Martians" in and around the old Austro-Hungarian empire's universities) -- builds in effect the set of natural numbers out of a chain of sets based on emptiness!

    A simple look (let's duck the power sets issue) is to take the empty set O-slash. Take the set that contains this. Then put the two together to form a 3rd set. The fourth set collects the first three. Repeat ad infinitum. Ordinal numbers appear.

    We have just created a successive collection that sets up the natural numbers, with cardinality at each step bound to that of the sets. These may be assigned labels, numerals 0, 1, 2, 3 . . .

    Then, identify operations such as add and equal. The join operation add, on a set of cardinality 2 and one of cardinality 3 will yield one of cardinality 5.

    At no stage have we depended on materiality, just logic. Though it is helpful to use our background as corporeal beings experiencing a physical world.

    Numbers and operations on them APPLY to the physical world we experience, but do not depend on it to have validity, by extension, the same holds for logic.



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