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Questioning the world's deterministic behaviour, I shall present an example which seems to defy any certainty about the recurrence of events and is (obviously) a result of faulty logic, but I would like to know why.

Suppose we conduct an experiment of a free falling object which we consider to be a dimensionless point dropped from a given height h.

According to Hume's problem of induction we don't know for sure that the object will fall to the ground only because it has done so every time we repeated the same experiment in the past.

So, some possible outcomes would be: A1 = falling on the ground, A2 = stopping at h/2 and then float there, A3 = stopping at h/4, A4 = stopping at h/8, and so on dividing the height by 2 repeatedly since all these outcomes are not logically impossible.

Since the possible outcomes of the experiment are countably infinite, the probability of getting A1, which we would consider as the event most likely to happen, is actually zero.

What's wrong about this example?

  • "According to Hume's problem of induction we don't know for sure that the object will fall to the ground only because it has done so every time we repeated the same experiment in the past." WRONG: according to Newton's law of universal gravitation we know for sure that the object will fall to the ground. – Mauro ALLEGRANZA Jun 24 '18 at 10:06
  • As an aside, you've missed (uncountably) many other possible outcomes, such as stopping at πh/4 and floating. – Hurkyl Jun 24 '18 at 10:40
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    Aren't laws of physics based on induction? This is circular logic... – Jevaut Jun 24 '18 at 12:47
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    I think you are making an assumption that all of the possible outcomes are equally likely. Considering that A1 happens most often, it would have a greater likelihood of happening. – Frank Hubeny Jun 24 '18 at 13:51
  • Just because you don't know what's going to happen does not refute determinism. How the world works and what we know about how the world works are two different things. – Bram28 Jun 24 '18 at 18:41
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If we conduct the experiment of an object falling to the ground 1 million times and the object falls to the ground (A1) 1 million times, we can conclude that the probability of A1 is 1 for this 1 million-time experiment and that the probability of A2, A3, A4, and other As are 0 in this experiment.

Although we do not know whether the object will do so in the next million or the next billion times or not, we do know that the probability of A1 is certainly not 0 because if it were 0, it would never ever have fallen to the ground. But A1 can still be 0.01 or less with the very very unusual fluke that the object did fall to the ground in all these million falls. The exact probability of A1 for the next million or next billion experiments cannot be known for certain until after those experiments have been done.

Now, even if the probabilities of A2, A3, A4, and other As in the next million or next billion experiments is not 0, it does not mean that they will sum up to infinity or 1 and make the probability of A1 become 0. This is because not all infinite series are divergent, and the sum of them are infinite; convergent infinite series do exist, and the sum of them converge to a certain number that is not always 1. For example, if the probabilities of A2, A3, A4, and other As are the series of 1/2^2, 1/3^2, 1/4^2, and so on (=1/4, 1/9, 1/16, and so on), the sum of this infinite series will be about 0.6449. That leaves the probability of A1 to be about 0.3551.

In conclusion, the argument of this example and the assertion that A1 is actually 0 are wrong because

  1. if the object does fall to the ground once, A1 will not be 0 no matter what A2, A3, A4, and other As are, and
  2. the summation of infinite series of A2, A3, A4, and so on is not always infinite or always 1 (as implied in the argument).
  • So, we can not determine A1 for the future, but it oughts to be a number greater than zero and less or equal to one? – Jevaut Jun 24 '18 at 18:00
  • That’s right. Even if A1 was 1 in the past million falls, we cannot use that to make an induction that it will always be 1 in the future. We only know that its’ greater than 0 and less than or equal to 1. But, in reality, if A1 was 1 in the past million falls, the probability that it will not be 1 in the future is extremely small and practically negligible. That’s why, practically or in our daily living, we take that A1 as 1 as the basis for whatever we will do that is related to the fall of that object. – user287279 Jun 24 '18 at 18:16
  • Physical laws are like that. Physicists never claim that their physical laws that are derived from the evidence they have are absolutely correct. The laws are just the best explanations for now (for the existing evidence). If new definite evidence comes out that contradict the laws, they have to adjust or change the laws. This always happens, the most notable one may be that Newton’s gravitation equations had to be changed to Einstein’s equation. – user287279 Jun 24 '18 at 18:25
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I think there are a few problems with your scenario:

First, you're treating lack of knowledge about the possible outcomes to mean that there are infinite possible outcomes. But you don't know that. For all you know--and this is compatible with Hume's view--there could be exactly two possible outcomes: the point-mass falls, or the point-mass turns into a chicken (I include that bizarre image to underscore that Hume's point can be generalized far beyond just the movement of objects in spacetime). Or there could be only 37 possible outcomes (which is itself assuming some sort of non-determinism, because if the situation were deterministic, there would always be only one possible outcome). We just don't know how many there really are. We don't know what "the real rules" behind causality are. That's Hume's point, that all we have are collections of observation of seeming causation (from which we have induced the laws of physics) but there is never any necessary connection between event A and event B.

But let's say for the sake of argument that we somehow magically knew that there were indeed an infinite number of possible outcomes, all with the same probability, for the dropping of the point-mass. You take that to mean that therefore every one has a probability of 0. But that cannot be the case, because there has to be some outcome, every time, even if that outcome is "nothing happens" (that is, the point-mass hovers where it was released). Therefore there is a guarantee of an outcome, and therefore its probability cannot be 0, and therefore the probability of all other possible outcomes (since they have equal probabilities) cannot be 0. In fact, this makes sense when you consider that 1/infinity does not equal 0. It's just that the limit of 1/n as n goes to infinity = 0. And what that means is that each outcome has an infinitely small probability, but a non-zero one.

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Since the possible outcomes of the experiment are countably infinite, the probability of getting A1 is actually zero.

This reasoning is simply invalid; the conclusion doesn't follow from the premise.

You've skipped over why you think this argument form to be true; I'm guessing you've made the usual fallacy of presuming all outcomes are equally likely, but there's absolutely no reason why that should be true.

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    If the possible events were equally likely would the conclusion follow from the premise? – Frank Hubeny Jun 24 '18 at 13:55
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    @FrankHubeny - I'd say it would. – PeterJ Jun 24 '18 at 14:49
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Theoretical infinity, and pragmatic limits

Quantum mechanics shows at the very small, you get to a point where things exist as both a wave and a particle. This stops the infinite splitting of space into ever smaller bits, and says here infinity stops.

So the ability of infinite stopping places as something falls relies on the idea things can become infinitely small, but quantum mechanics stops this. As there is not infinity here, then the finite stops the answer being zero, but a small number and therefore a definable outcome.

This is what I call the pragmatic outcome, though theoretically infinity could exist here, it does not. I suspect this is essential to the basis of our existence, so time as a limit to how small you can go, though we have not yet found it.

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I take this question to be, does the problem of induction contradict determinism?

The problem of induction essentially means we have to modify our understanding of what is true to account for the fact the universe has the final 'say' https://en.m.wikipedia.org/wiki/Problem_of_induction#Notable_interpretations

This view is incompatible with determinism as a principle, because it says however true it seems to be it is limited in proportion to the strength of evidence and critique it has withstood, and that new evidence could contradict it.

We can derive necessary true consequences from the nature of mathematical abstractions, but those abstractions will always be subject to corrections, amendments, in light of new observations. In this sense we can understand how all our ideas are not strictly true, because they are special cases or limited examples of larger truths we think they will fit in to. This is very different, and incompatible with, the idea of permanent truths about the relatiinship of definitely defined mathematical entities.

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