William M. Briggs

Statistician to the Stars!

Under-determination, Quus, And Why It’s Cause That Counts (And With A Taste Of Grue)

The Calculator, a.k.a. the device.

The Calculator, a.k.a. the device.

Turns out Arthur C. Clarke was on to something (who told him?). See, scientists, via one of the NASA Rovers, found this device on Mars. It is roundish, the color of the Martian soil and occasionally displays, or is thought to display, what appear to be numbers. In order to keep listeners of NPR calm—a jittery crowd at best—news of the discovery was kept quiet. But I learned of it, and since no NPR listeners venture to this site, it’s safe for me to tell you about it. Just don’t tell anybody else.

Scientists have decided the device has two “inputs”, which are thought to be two protuberances in the “back”. Here’s where it gets interesting. Through a series of inferences, it has been decided that the displayed numbers are correlated (I mean this word in its plain English sense) to the “inputs”, which have been discovered to be “activated” (the flash different colors) in the same base of numbers as the display.

Put plainly, (and I’ll convert the numbers to base 10 for ease of understanding), the display is the sum of activations of input “A” and of input “B”. Mathematically, A + B = D(isplay). So far, since the Rovers have not had much chance of observing the object, dubbed The Calculator, the activations of A and B have never been greater than 56 individually. I mean A,B < 57, which necessarily implies D < 113.

Naturally, since it is plain this is a device, scientists want to know its purpose. Theories are flying around NASA thick and fast. And, oh boy, have scientists become exercised over these theories. There have even been fist fights! (That’s another thing you mustn’t tell the NPR crowd, who view scientists as priestly; we don’t want to disillusion them.)

Though there are more theories than there are scientists, three rough camps have coalesced. Camp 1 says its coincidental that so far A,B < 57, thus always A + B = D no matter the number of activations of A and B. Camp 2 thinks the folks of Camp 1 have been standing too close to the jet fuel. Camp 2 theorizes that the activations are “obviously” caused by two types of cosmic rays, which if they were to exceed some tolerance, they would cause D = 5. This, they say, is a derivation of string theory. I mean, if either A,B > 56, then the function is no longer a straight plus, but is instead a “quus”; i.e.

     A + B = D, if A,B < 57,
     A + B = 5, if A,B > 56.

Camp 3 thinks…well, what Camp 3 thinks of the others can’t be printed on a family blog. But I can summarize and say Camp 3 has called for the folks in Camps 1 and 2 to be gently sedated and trucked to some tenure-granting institution for less strenuous employment. Anyway, Camp 3 says A + B = D for any number of activiations, but that after some period of time the device must start to degrade and that, because of various technical reasons,

     A + B = D, before Date,
     A + B = C < D, after Date,

and where the inequality is strict. We have plus and quus already, so call this (and this is my suggestion, not the scientists’) “cuus”.

Funny thing. The observations of the device are consistent with each of these three theories—and with many more theories, too. Recall I’ve only given you the three most popular. Obviously, none of the theories put forward by any of the scientists are inconsistent with the observations. We conclude from this that the physical observations are indeterminate; I mean, the state of the device, or the world plus the device, do not fully determine the device’s purpose.

Still, even though the facts are indeterminate, we’d still like to know which of the plus, quus, cuus theories is right. I have no idea. We’ve already learned that no theory “has” a probability, so there is no joy to be found in searching which of theories is more “likely”. Of course, we could use each theory to make predictions and see which is better in some decisionable sense, which is what I recommend in the interim. But we’ll never know which of plus, quus, cuus, or even some other theory, is true until we understand the purpose of the device. And we’ve seen that the facts alone do not and can not determine what this purpose is.

To understand the purpose is to understand, in part, the cause of the device. Cause is of four aspects: the formal or form, the material, the efficient, and the purpose, final, or end. In this case, the form is obvious enough: the device is “disguised” or made to look like a rock. The material is unknown at this point, but it’s thought to be at least a rocky covering, or something which simulates rock. The efficient cause is, all agree, some kind of intelligence and whatever comprises the internal workings. Whether the designer is Martian or some clever human is unknown.

But what about the purpose of the device? Well, that’s what the real unknown is. If it turns out that some Martian (or whomever) designed the device to count activations, however these are brought about, then plus is the right theory. If instead the final goal of the device was to count cosmic rays, then we’re on to quus. Now it could be that quus and cuus are right, on the guess that the harsh cosmic rays are causing the degradation. That means the quus-purpose is right, but the cosmic rays efficiently cause a degradation which leads to cuus. So we have to be careful to keep in mind what part of the cause we’re examining.

If somehow we discover the user’s manual or tech specs for the device (and could translate them), then we’d know the cause of D—we’d know all aspects of the cause, and then we’d know the theory. And, as should now be obvious, what holds for this Martian device holds for all devices, whether made by Martians or via “natural processes.” It is only after we have knowledge of cause that under-determination ceases to be a problem.

Knowledge of cause is above, or rather beyond or deeper than, knowing what happens. Even beasts can know what happens, but they don’t and can’t understand why. Knowledge of cause is the grasping of essence, of the natures and substantial forms of the objects under consideration. None of these things are material in themselves, but are universals above and beyond the material world. Thus to come to knowledge of cause is to understand universals, which we get through a form of induction. Induction is the immaterial “movement” from finite particularities to an infinite generality and is such that only rational creatures can accomplish it. (I still owe readers an explanation of the different types of induction; but see below for the links on grue for some of this.)

The “quus” example is from Saul Kripke, as many will recognize. If not, head over to Ed Feser’s place for more information; particularly see his “Kripke, Ross, and the Immaterial Aspects of Thought” in the American Catholic Philosophical Quarterly. Quus isn’t usually presented with respect to functioning devices of unknown cause, but of language and thought. How can we be sure that the fellow who is speaking to us (about some mathematical operation, say) really meant plus instead of quus or even cuus? Or how can we know that we ourselves mean plus and not quus?

I have concentrated on the epistemology, because uncertainty is my main interest, but the point about quus is broader and is mainly metaphysical. But thinking along those lines has marvelous implications for the immateriality of our rational thought. I ignore all that here since it takes us too far afield. I merely want to demonstrate the induction is present when making abstractions about universals.

Incidentally, when people get on to quus they often bring up grue. Regular readers will know grue is a non-problem for induction, too which is necessary for understanding cause, and the grasping of cause is the solution to grue. See this paper or this post. The power of induction goes unrecognized.

14 Comments

  1. “The power of induction is constantly abused” seems a more likely interpretation.

  2. Deduction:

    1 + 2 = X, calculate X given an operation (addition) and inputs (1, 2).

    Induction:

    A, B = 3. Calculate A, B when the operation might not be known but you have the result of the inputs and operation.

    OR

    A, B, C … Z with many operations and plenty of time = EARTH. Calculate the operations and factors and timeline to produce Earth.

    Induction involves making intelligent guesses. For instance,

    1 + 2 = 3. But 1 * 3 = 3; so given “3” you have two equally plausible paths to it (rather more than two, it’s actually infinite).

    Occam’s Razor, often taken to be “proof”, is simply a way to reduce the plausibilities to a manageable, but possibly wrong, number of possibilities.

    Science is largely inductive. We see the consequence or result of unknown operations operating over unknown time to produce the observation.

    Initially the possibilities are infinite, including that we were created yesterday or maybe we haven’t actually been created yet. So you create possibilities and then try to disprove your own guesses.

    As you disprove your guesses you end up with fewer and fewer plausible paths to whatever has been observed. I assume in the case of natural processes that nobody is deliberately trying to conceal the operations. In case someone is concealing is presence or locations, it is trivially easy to defeat induction-with-razor by simply creating a simple, but wrong, explanation.

  3. So that’s where that thing went! I’ve been looking for it. I’ll spare you the grue-some Amazing how dropped thing manage to find the most unlikely and least accessible places to hide.

    Knowledge of cause is above, or rather beyond or deeper than, knowing what happens. Even beasts can know what happens, but they don’t and can’t understand why.

    How do you know this?
    Even humans can’t answer “Why?” The answer inevitably is to insert yet another cause. If given X causes Z, the answer is little more than saying the chain is X — B — Z which doesn’t really answer the question. Is this what you mean by “understand”?

    How do you know beasts don’t do this as well? Because they don’t write papers on X causing Y? Or maybe they have a different/simpler (and therefore inferior) B-cause?

  4. Induction is a process by which one guesses at the inputs and process that lead to whatever you observe; but to disprove bad guesses one then uses deduction. It is like “checking your work” in mathematics.

    Carbon dating is a good example. Induction allows you to guess that carbon 14 decaying to Nitrogen 14 takes a predictable amount of time using a fraction-of-the-remnant operation. But it was a guess, possibly “inspired.” Good guesses do not arise ex nihilo.

    This is tested deductively, that is to say experimentally, by taking samples of known age and using the process that one has guessed exists (by induction). Thus the method is validated to a certain degree of accuracy but not more than that. One of the assumptions is that the proportion of C14 in the atmosphere is constant. Experiments have shown C14/C12 ratio to be somewhat variable over time and so dating isn’t exactly a simple mathematical function although the function underlies the method.

  5. Sheri, I’m concerned how you get to that.
    About the grue post:
    I’ve obviously missed a lot of interesting debates about language and information. Given that I wasn’t reading around then why do I feel like I’ve read that before? More chicken soup, obviously or more sleep.
    I liked the comment about Shakespeare’s dismantlement.
    About breaking down verse to where it loses it’s meaning.
    A flower can be broken up to discover it’s form and function and cellular mechanics but it’s essence or meaning will still be a mystery. I won’t be looking to physics for an answer.
    “Full many a flower was born to blush unseen and waste it’s sweetness on the desert air.”
    On this post:
    I’s all chinese to me!

  6. Briggs

    December 29, 2015 at 6:02 pm

    DAV,

    “How do you know beasts don’t do this as well? Because they don’t write papers on X causing Y?”

    In brief, yes.

  7. The solution for grue has been known since the early days of computing.

  8. Briggs

    December 29, 2015 at 11:02 pm

    Spetzer86,

    I’d be delighted to learn of some references here.

  9. Device to meet you! 😉

    JMJ

  10. JMJ has read Clarke

  11. Briggs,

    The grue is a legendary? monster from the very old text-based game called Zwork. Anytime you moved into a dark room, the grue could (and usually would) get you. http://zork.wikia.com/wiki/Grue

  12. Briggs

    December 30, 2015 at 6:57 pm

    Spetzer86,

    I can see that there was some assembly language programming deep in your background.

  13. JH, does that symbol mean JH? It’s a new one.
    I remember when I thought your name was John spelt Jhon! (with help from my mischievous screen reader) I just thought you’d played with the letters.
    So you were John. All Johns I know are male.
    Once I realised you were JH I still thought you were male! The secondary assumption stuck along with the first mistake.
    It all made sense when I realised that mistake. A LONG time after i’d worked out that you were sweet, sensitive and not John-like and only when you told me, actually.

    When I first read this post I thought it was clear enough. The more I thought about it the more confused I became!
    “Shall I keep reading to understand what the confusion’s about?” Now I’m thinking I’ve missed something really important.
    In other words my impression is that I don’t think there’s a problem, I have to second think this to make it a problem. It’s about perspective, without which we’re all at sea without a compass.
    At least I’m clearer about induction. This is like the grammar of thinking.
    That’s about it and don’t fall out you two!

  14. Oopse,
    This comment belongs on the previous post!
    As for this post,
    Huston’s got it sorted.

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