The Probability Of A Bottle Broken Into N Pieces When Struck By A Hammer

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Logic is the study of relationships between propositions, and probability, which is the same, is thus the completion of logic to instances where the relationships are not certain.

From this it follows (by a long path which we’ll skip over today) that not all probabilities are calculable. And anyway, it is proved easily by example. Here’s is one which I heisted from Henry Kyburg Jr (Epistemology and Inference; chapter on “Chance”). The proposition of interest is:

    q = “This (really quite tasty) bottle of 2009 Muga Rioja Reserve will break into N pieces when struck by a hammer”,

and we want to know the probability of q. The answer is that there is no answer: there is no “probability of q.” This non-answer answer holds for all q which are not self-referential, and the reason is that, as said, probability is a measure between propositions. Here there is only q; there is no between. This is the end of the lesson. If you’re in a hurry, there’s no need to read further.

Because the question makes some kind of sense, though, it encourages (what we can call) the Subjective heresy, which is the belief that probabilities are subjective “feelings.” The most virulent form of this heresy is the diversity-loving academic who spouts “all truth is relative” or “there is no ‘truth’.” Skip that. Subjective probabilists presented with q and asked its probability are tempted beyond resistance to provide an answer. The move requires them to provide unstateable evidence so that q can be put in a relation. Such evidence is in the form of a complex proposition, which might look like this:

    e = “I feel this and that.”

Given e, the Subjectivist can say “the probability of q given e is X”, where X is usually stated with astonishing precision.1 The danger is e is likely to vary based on the Subjectivist’s last meal (fast-food Chinese produces very small Xs). Of course, if e is rigorously specified, then it is quite possible for “Pr(q|e) = X” to exist. The only problem is that this is not the answer to the question; instead, it is the answer to “Given e, what is the probability of q?” And nobody asked that.

Relative frequency is the number of times a proposition is true (given a set of observations) divided by the number of times a proposition is true or false (given the same set of observations). Relative frequency is not probability; further, relative frequencies often do not exist. Counterfactuals do not have relative frequencies, nor do hypotheticals (Martians wearing hats). Worse, followers of the Frequentist heresy (which is the claim that probabilities are RFs) are also prone to the Subjective heresy.

Notice that q does not specify how and under what circumstances the bottle is to be struck. If we are to keep q, unadulterated as logic requires,2 then these informational tidbits must remain forever unknown. The Frequentist, like the Subjectivist, is tempted beyond all ability to withstand and invents this information. “Suppose,” he might say, “we set the bottle on a concrete pavilion and have a drunken one-eyed orangutan belt it with a 20 OZ Solid Steel Nail Eagle Claw Hammer (my father’s and therefore my choice) whilst I try to feed it bananas.” This—and it must be something like this, with the experiment precisely defined—becomes the e. Or, rather, the e’, because it is still impossible to compute Pr(q|e).

The Subjectivist (as above) agglomerates emotions to e’ but the Frequentist must insist that the experiment he imagined be carried out. The results from these experiments (call them r) are tacked onto e’ with the result that, after a very long time, just after the last trump sounds to be exact, he finally can state Pr(q|e’r). Never wait on a Frequentist.

We’re back at the deeply unsatisfactory result that q has no probability, not in isolation, not without respect to some other proposition. Since we do not know what this other proposition is, we are unable to compute. Computing is scientific; science is numbers. Perhaps the “real life” quality of q is what causes people to invent information. They see the question (reminder: “What is the probability of q?”) as a scientific one, which it is not. It is logical and logic is not a branch of any empirical science. The tendency to invent is yet another symptom of the rabid scientism which permeates our culture.


1There are times when the Subjectivist heresy isn’t quite: these are when you have actual bottles and hammers in front on you and an experiment will be carried out. “What is the chance?” is then tangible and a tacit invitation to supply missing information. People may use their knowledge of hammers and bottles to hone in on a probability, though unless a person’s knowledge of the physics of breaking bottles is magisterial, this probability should be no more than a wide interval (yes, interval). It only becomes a fixed number when this person makes a bet (perhaps only a mental wager with himself), which is a decision to act in a certain way. A decision or bet is not a probability, but an act. And anyway, probabilities are immaterial, like logical statements, and are not acts.

2Believing you have God-like control over specified propositions is like telling your calculus teacher that you decided not to answer the questions he gave you, but ones you invented based on his questions.

Technical notes: Suppose we agree to amend our question with an e which describes an experiment, like with the orangutan. We are still ignorant of N. A secondary temptation arises to let N be any number from 0 (the bottle might not break) to infinity. Bottles cannot break into infinitely many pieces, but supposing they might allows mathematics to enter, and allowing mathematics to enter is what scientism demands. This in fine if the real experiment was of definite interest, but the approximation necessarily implies the calculations will be too certain.

19 Thoughts

  1. I sorry, but the first sentence reads strangely to me. Is it a statement or a question? Could you rephrase it?

  2. “The tendency to invent is yet another symptom of the rabid scientism which permeates our culture.”

    Are you foaming at the mouth Briggs?

  3. Ahhh!

    Great post. This fits in a current discussion I’m having with unrepentant hypothesis testers with a taste for P-values.

  4. Actually the non-breakable space was kind of appropriate. It makes the probability calculation a lot easier.

  5. There’s a related issue not touched upon yet. What constitutes a piece?

    Strike a real-world glass bottle with a hard object such as a hammer, or drop it on hard ground, and the number of major pieces might be predictable at least to some extent; but there will also be anywhere from dozens to thousands of smaller pieces, ranging from half-centimetre slivers to pieces the size of sand grains or below. Do these count as pieces? And if not, where is the line drawn?

  6. Fletcher Christian,

    What kind of answer would you expect after hitting the bottle? And what then after hitting the bottle repeatedly as in footnote one?

    Sounds like all of this is alcohol abuse, but then, maybe these are empty bottles?

  7. The original question as such is untestable. The answer to that one is: “don’t know”

    However, in safety test procedures you could design a bottle fragmentation test, in a similar fashion as the Mythbusters do work. You set up a mechanism that can blow a given force on a bottle and you can set up a binary pass/fail. You then can construct a distribution curve at which force how many bottles wil break. Using this probability density curve you can make an estimate when the bottle will break.

    Crash test dummmies work like this all the time.

  8. ummm…

    I think it is possible to make a bottle that will, when subject to quite a wide range of external shocks, break into a fixed, and thus predictable, number of pieces.

    In general, e can be rigged to make Pr(q|e) predictable – given that e includes some feelies, this is, after all, what Las Vegas lives on.

  9. Las Vegas works with well-known distributions except for sports betting which they are just in it for the juice and don’t care what the outcomes are.

  10. As suggested by Hans Erren a study of fragmentation can be made. It has been done many times. There are many studies of how various materials fragment with shock. There are algorithms that are capable of predicting the particle size distribution of shocked materials of various shapes with shocks of various magnitudes. The predicted sizes range down to sub micrometer. The military is one interested party in such research as are those in the mining industry.

    As an example of the ability of the algorithms and knowledge of fragmentation, I showed pictures of a test device I was using to some colleagues at another laboratory. They asked that I let them make a prediction before I told the the results. The prediction was extra ordinarily accurate. Nevertheless, They would never attempt to say exactly N particles.

    If I roll a six sided object with sides numbered one through six, what is the probability of a three? How is this a measure between propositions?

  11. Joe,

    “If I roll a e = ‘six sided object with sides numbered one through six’, what is the probability of q = ‘a three’?”

    Pr(q|e) = 1/6.

    “How is this a measure between propositions?”

    Just that way.

  12. Between means two. Does “Just that way” mean between 3 and not 3? I do not understand between. Is it because I have no way of knowing how many possible pieces glass as I know 6 sides of a die?

    You have a way of being mysterious. There is something missing in “Just that way.”

  13. This thing about the subjective-objective dichotomy reminds me of George Berkeley’s question, “Does a tree fall in the forest when nobody can hear it?”

    Here is the objective answer from above:

    There once was a man who said ‘God
    Must think it exceedingly odd
    If he finds that this tree
    Continues to be
    When there’s no one about in the Quad.’

    Dear Sir: Your astonishment’s odd:
    I am always about in the Quad.
    And that’s why the tree
    Will continue to be
    Since observed by yours faithfully…God.

    (Nigel Warburton, A Little History of Philosophy)

  14. Is the bottle empty or full?

    The optimist says it is empty. (Since you are about to smash it and who would do that to a full bottle of wine.)

    We used to break glass bottles all the time, though we were seldom concerned with the number of resulting pieces. In the impact test, a hammer of a specified weight and configuration was hauled back on a fulcrum a specified number of degrees and released. The bottle would either crack or not and, depending on how representative of the lot, would indicate the likelihood that the lot would survive shipping. A variant test would increase the angle of deflection to increase the impact force.

    The burst test filled the bottle with water and increased the pressure either to failure or to the “safe” pressure. The results were treated as truncated reliability data.

    There was a similar test for thermal ware, but I was not engaged in that.

    Shards became interesting only when looking for failure points. The bottles were wrapped snugly in transparent tape and stressed to failure. The pattern of cracks could be analyzed by a glass engineer to identify the point on the bottle where failure commenced.

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