Monday, April 02, 2012

Reasonable Doubt on Non-Contingency

Continuing to work together with Kairosfocus to build a worldview.

Previously, I paused at the principle of cause and effect because of the way it was worded:
[d] if something has a beginning or may cease from being -- i.e. it is contingent -- it has a cause.
The wording implies we can systematically distinguish between contingent and non-contingent things. Later, as he explained the system for making the distinction, Kairosfocus introduced a mathematical equation as an illustration:
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.)
His point is the truth of the equation, the fact it is correct, is an example of a non-contingent thing. That is, the specific truth is itself a necessary being.

I was confused by this reasoning and shared my thinking about contingency and the mathematical equation:
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.
In response to this post, Kairosfocus commented:
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.
I don't think this response resolves my issue. First, I had said--or meant--that the truth of 2 + 3 = 5 is contingent upon not only materiality but also principles of reason/intelligence and something to interpret the operation of the principles with respect to materiality. Another way of expressing my thinking is to ask:
Is it still true that 2 + 3 = 5 if (i) the universe is wholly uniform and immaterial, if (ii) there are no principles of reason that differentiate or manipulate information, and if (iii) there are no living or non-living entities in the universe that act or are acted upon by anything?
I don't see how the equation can have any meaning if a combination of at least two factors listed above is true. In a uniformly immaterial universe without principles of reason, how can 2 + 3 = 5 mean anything? There would be no 2, no 3, no 5, no operators, no vehicle or target for any action or conception, and no method governing difference between things (as there is no difference between things).

So, while I trust it's correct that the truth of 2 + 3 = 5 does not depend on materiality, as Kairosfocus says, my point is that the truth of the equation does depend on other factors. Indeed, the fact we need to conceive of numbers as sets demonstrates my point. If, contra factor (ii), there are principles of reason--such as conceptions of sets and operators--and if, contra factor (iii), there are entities such as ourselves that reconcile existence with (ii), then the equation can be true without reference to materiality. Yet, if either (ii) or (iii) holds along with (i), then the equation has no status whatsoever.

The proverbial jury remains out on non-contingency, as far as I can tell. It doesn't seem as though we have any way to verify whether something or some state exists with no contingencies whatsoever. If we have no way to verify, I'd like to use my version of the principle of cause and effect:
[d-1] A thing, A, has a beginning or may cease from being. A thing, A, therefore, has a cause.
This cause and effect principle flows seamlessly to the principle of sufficient reason:
[e] “Of everything that is, it can be found why it is.” (Principle of Sufficient Reason, per Schopenhauer.)
I think [d-1] and [e] work beautifully together, the first asserting the pervasive contingency of the universe and the second promising that everything in the universe can be traced to preceding factors.

Once we jettison non-contingency, we make quick progress. Everything falls together so nicely I wonder whether we should invoke another of Kairosfocus's principles:
[f] “to say that what is is, and what is not is not, is true.” (Aristotle, Basic Definition of Truth, i.e. “telling it like it is.”)
Perhaps contingency is and non-contingency is not: this may be the truth.

Now, non-contingency may very well be true, even if we cannot verify it. I do not deny the possibility that non-contingency exists. But I have no special judgment to make about non-contingency until I know how we systematically and independently verify it. In other words, the empty set may be a mental model of non-contingent reality, or it may be a mental model of human reasoning.

In this post, I've tried to isolate that part of the principle of cause and effect that seems to me unassailable. For the assailable part--that is, for non-contingency--I have explained why I see things like mathematical truths to be contingent, as these truths cannot exist or continue apart from a combination of factors. My explanation will falter or fail if it can be demonstrated that the truth of 2 + 3 = 5 does not in fact rely on any factor, that it is true regardless of a physical universe, principles of reason, and an interpreting entity that mediates the previous two items. I have argued that at least two factors must be present for the equation to have any meaning at all, let alone truth. Finally, by reducing the scope of the principle of cause and effect, I have sketched how it integrates flawlessly both with the principle of sufficient reason and the "telling it like it is" principle.

Clearly, the principle of cause and effect is a critical point of divergence. Things that are caused and create effects are commonplace and not a philosophical problem. Things that have no causes and no effects--well, I don't know that we spend time wisely dwelling too much on these, if they are at all. But things that have no causes yet create effects: these are the things that concern us. Our specific concern is how we prove that a thing is un-caused. Yet what seems unquestionable is that reasonable people can doubt the existence of un-caused things.


  1. LT:

    I now see a box, when I was going to close the tab.

    Here is the substantial text of an emailed comment:

    >> I am seeing only a black zone in your comments section. I can find no way to submit a comment in the usual way, and have no time to troubleshoot. I think Blogger is doing some changes.

    My remarks, since I have little time, are brief:

    1 --> I spoke before, to the question of your earlier suggestion on material dependence, as that is where causal "switches" are a most evident issue.

    2 --> Z-F set theory shows how this is abstract and establishes numbers thence number relationships independent of materiality. (Isn't it amazing to see numbers built up from nothing by the power of logic! Give the empty set, then the set that has it, then the set that has the empty set AND the one with the empty set, and assign: 0, 1, 2, . . . )

    3 --> Next, observe a key distinction: logical relationships of ground and consequent are distinct from causal ones that affect beginnings, sustenance in existence, and termination thereof.

    4 --> Specifically, we cannot switch off the relevant laws of logic, they logically constrain what is possible in all possible worlds. Always have, always will, cannot be otherwise. The laws of logic are not on/off switches.

    5 --> So, the truth in "2 + 3 = 5" is not contingent on such laws of logic, this and other necessarily true propositions are constrained by them, as is anything else that may exist. >>


    Hope that distinction helps.

    GEM of TKI

  2. Thanks, KF. I generally agree with your point in #3, at least the first part of it. Point #4 is interesting. Every possible universe operates under physical constraints and limitations. No possible universe is unconstrained.

    I agree that the truth of 2 + 3 = 5 is constrained by logic. Yet I still see that truth as depending on a combination of factors, as in logic and materiality or as in logic and intelligence--or as in logic, materiality, and intelligence. As I've reasoned truth seems impossible without a combination of factors. Thus, we really do agree and we are both pretty happy with worldviews built on the six principles we've discussed.

  3. LT: Passed back for a moment. There is no reason to believe physical laws and parameters considered as constitutive of particular [sub]universes are necessary. And possible worlds are not just physical or material. That is a part of the ideas context of Z-F set theory. KF

  4. KF,

    "There is no reason to believe physical laws and parameters considered as constitutive of particular [sub]universes are necessary."

    I'm not sure how to parse the above:
    * "There is no reason to believe physical laws and parameters are necessary."
    * "There is no reason to believe physical laws (and parameters considered as constitutive of particular [sub]universes) are necessary."
    * "There is no reason to believe physical laws and parameters (considered as constitutive of particular [sub]universes) are necessary."

    In any event, I think it's a stretch to say there's no reason to see the necessity of physical laws and parameters. I've argued there's pretty good reason to hold this view. And of course, I have not reduced everything to only physical laws and parameters.

    I cannot comment with any intelligence on Z-F set theory. But if this is a mathematical theory, which it appears to be, then I don't see how it applies to what we are talking about, which is ontology (the study of what there is). We are talking about non-contingent things and how to identify/verify them.

    I don't want to move on before you agree, but perhaps it will be more fruitful to suspend our cause-effect disagreement and think about what's next. We have the six principles more or less settled. How do we move on (or up) to larger worldview concerns?

  5. LT:

    I was not aware until just now that you had continued.

    Physics, per modern discussions of cosmology, is seen as contingent. The laws are not laws of logical necessity or physical necessity, and many parameters and observed quantities patently could have been otherwise. That is where the whole discussion of how finely balanced the set of physics behind our observed cosmos is, relative tot he requisites of C-chemistry, aqueous medium, cell based life. Just one example, the physics is set up so that the first four elements are H, He, C, O [in a balanced abundance pivoting on a resonance that led Hoyle to his monkeying with the physics of the cosmos remark] and close to this is N. The first two get you to stars, the next two to water and organic chemistry. The fifth, to proteins. And the properties of both C and H2O are astonishing relative to the requisites of life. Hoyle's overall comment is about put-up jobs.

    In short, there is no reason to imagine that there is a super-law that forces the ratio of protons to electrons to be as finely balanced as it is [ 1 part in 10^37 IIRC]. A little this way or that, and our universe would not be possible as the electromagnetic force is long range and dozens of orders of magnitude stronger than gravity.

    The next issue is that Z-F set theory addressed the paradoxes of naive set theory, by starting with the empty set, then constructing a cascade of further sets {}, then the set of the empty set, then the set of the empty set and that containing the empty set, and so forth, which have cardinality 0, 1, 2, 3 . . .

    So, we construct the set of numbers and set the context for mathematical operations and relationships, without locking ourselves into the paradoxes that stem from another approach that speaks of say a definable collection of objects. Russell's barber shop paradox saw that off: the men in the village shave themselves or are shaved by the barber, so who shaves the barber?

    1. KF,

      Well, enough time has passed so that I forget some of the immediate context of what we had been discussing.

      IIRC, we had talked about the truth of 2 + 3 = 5 depending on several factors, though not all factors at once. Physical laws comprised one factor. Relative to the truth of the equation, physical laws are necessary.

      Are specific physical laws necessary in themselves? You say "no," and that is consistent with my view which continues to wonder where the case for purely necessary beings comes from.

  6. Thus, mathematics exists in an abstract space that nonetheless impinges on and constrains the material world. Hence, we see hints of why there is sch a thing as the astonishing power of mathematics to predict and guide control of material reality. In short, mathematics is a case where we do have logical, propositional constraints that are rooted in logical necessity and which determine what is possible. As opposed to physics as such, which as we just saw is contingent and empirical: we have to actually try and see it for ourselves, with all the provisionality that the resulting inductions have.

    We also see through mathematics, where the mental can impinge on and constrain the physical. For if the empirical is X and X mathematically entails Y then we can firmly expect Y. My favourite case was when Young, post the double slit exercise, put forth the wave theory of light. It was objected that then a small ball would have a dot of light in the middle of its shadow, per the mathematics of superposition. Ludicrous!


    Someone went out and tested.

    The little dot of light is there, just as implied by the wave effect established as credibly real through the double slit experiment and interference fringes.

    So, the issue of the reality of necessary propositional truths and the way this can constrain material phenomena is pivotal and decisive.

    But then, this is so on much less august cases too.

    If we have three pennies in one pocket and two more in another, then put the two together, we expect to find five. And if we find less, we go looking for a hole.

    Beyond, lies the world of the complex frequency domain to which the gateway is the derivation that among other things tosses out the famous Euler relation: 1 + e^i*pi = 0. I used to set my students to work pole spotting as they observed all sorts of objects. In short, we find ourselves in a world where decidedly mental and logical relationships have real world force.

    Powerful real world force that we use routinely in our designs and technologies.

    (And maybe that is part of why I do not find it particularly hard to see how mind and body can be linked!)



Feel free to comment if you have something substantial and substantiated to say.