Frequentism in a Laplacian Universe

Frequentism in a Laplacian Universe Fedde Benedictus July 3, 2015

Abstract This short note serves to give a brief introduction to the topic to be discussed in my dissertation in the chapter I am currently working on. The view on frequentism that is presented here is a development of the views of Hume [1] and Laplace [2]. I expect that it is similar to the frequentist view of von Mises [3], but further research is necessary to support this conjecture. I aim to conduct this further research in the coming weeks.

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Degrees of Global Probability are Global Relative frequencies

We consider the Laplacian universe as an infinitely extended 3D container with only matter in it. The behaviour of this matter is wholly determined by causal relations (described by Laplace’s principle of sufficient reason); Newton’s three laws of motion, and the law of universal gravitation. Laplace’s demon (LD), knowing all positions and velocities of all particles at one instant (the exact constitution of the space-container at a certain point in time), could calculate precisely the state of the universe at some earlier or later instant. We can picture Laplace’s universe as a newtonian spacetime hypercube. The exact constitution of the 3D space-container at a certain point in time is a timeslice in such a 4D hypercube. We will call this 4D universe UL . For an agent with perfect knowledge the matter in UL can be considered as consisting of perfectly spherical particles whose centres of gravity describe infinitely smooth paths. Any true statement about these particles that this agent can make will be true at all times, so every statement that is true in UL is a universal statement. Laplace held that for his omniscient demon probabilistic considerations would be of no value. We can see this 1

when we realise that in UL there are no separate events (because everything is causally related1 ). Therefore there are no sequences of events and also no relative frequencies. There are no degrees of probability. Probability is therefore a subjective notion, as it depends on an observer’s subjective (necessarily incomplete) degree of knowledge. We will see below that degrees of probability arise when observers with imperfect knowledge define separate events by applying a measure on UL . From this measure relative frequencies arise and thus appears probability. For our further arguments we define the following: • we call a measure on the whole of UL a global measure • such a global measure may serve to define global sequences of events • we call the relative frequencies of events within global sequences global relative frequencies We remark that global sequences can have any orientation within UL . They are not necessarily diachronic, but may be (partly) synchronic2 .

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Degrees of Observed Probability are Global Correlations

Now consider a 4D UL with a measure on it that denotes the events a, b, c, d... We investigate whether a causes b. We are therefore not interested in the global relative frequency of either a or b, but rather in how often a and b succeed each other: we want to know the global correlation between a and b. To investigate this correlation we take random events from the 4D UL . We find sequences of events of the form ‘abadde...’ or ‘eecbb...’ etc. Our understanding of what is a cause determines which subset of all such possible sequences we are interested in. We must, for example, decide whether we are investigating cartesian or newtonian causation (only the 1

This argument of course depends on how you define ‘event’. This issue will be revisited in my dissertation in my discussion of the views of Mill. There is also a link here to the contemporary debate on propensities. One could say that some physical propensity actually provides a measure. But then LD would no longer be omniscient, so we are no longer discussing UL ) 2 In our discussion of induction we will see that Reichenbach’s straight rule of induction amounts to looking at only global sequences represented by straight lines in a UL described in cartesian coordinates.

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latter allows for non-local action). Are we only interested in sequences in which b immediately succeeds a? For an investigation of local causation we must look for sequences in which a and b 1. always succeed each other 2. there are no events between a and b (the absence of an event also counts as an event) 3. do not appear alone in the sequence (depending on whether we are looking for sufficient or necessary causes3 ) The choice of what observed relative frequencies to regard as belonging to the same global sequence thus is informed by the notion of causation we entertain. Because these relative frequencies in turn determine degrees of probability, we might be tempted to conclude that our notion of causality defines probability. But such a conclusion overlooks the possibility that there is some physical propensity to any causal structure that motivates observers to apply a certain measure to it. We therefore arrive at a somewhat more subtle conclusion. The definition of a causal relation between events as implying the co-occurrence of these events guarantees that regarding observed relative frequencies as degrees of probability is a rational choice4 .

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Example & Methodological Rules

Suppose we formulate the hypothesis h: “p causes f”. We may think of p as representing a situation in which some object is pushed from a table and f a situation in which this same object falls to the floor. We test h by investigating many events. In this situation there are two methodological demands that any experimenter would make. To properly test h 1. we must vary the situation in which the experiment is conducted; and 2. we must repeat the experiment often in every one of those situations Say we test h in two ways: 3 If we want to know whether A is a sufficient cause of B, then instances of A alone should not appear in the sequence. If, on the other hand, we wish to know whether A is a necessary cause of B, then both A and B should not appear alone 4 A choice is rational if it serves to protect agents from dutch bookkeepers

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1. by investigating 5 differently shaped objects which are pushed from a table (this plausibly results in observing af & bf & cf & df & ef) 2. by investigating 5 instances of pushing a particular book from the same table (this plausibly results in observing pfpfpfpfpf) How should the two methodological demands be understood when applied to this case? Varying the experimental situation can be regarded in our example as taking samples from all events in UL as randomly as possible. We investigate many different situations in which an object is pushed from a table (many a’s) [demand 1]. But a occurring by itself is not what we are interested in. We therefore investigate the 4D neighbourhood of a to see whether it contains many instances of ab [demand 2]. The first methodological demand therefore leads to an investigation of many global sequences whereas the second demand implies that the investigated sequences become as long as possible. That the relative frequencies both within samples as well as in a large number of samples converge towards a stable value is guaranteed by the strong and weak law of large numbers, respectively5

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Acknowledgments

Professor Dennis Dieks and David Hume.

References [1] Hume, D., An Enquiry Concerning Human Understanding, originally published in 1748 [The version I used was published by Hacketing Publishing Company in Indianapolis in 1993] [2] Laplace, P. S. de Philosophical Essay on Probabilities, originally published in French in 1814 [The version I used was published in 1902 by Wiley & Sons in London] [3] von Mises, R. Probability, Statistics and Truth, originally published in German in 1921. [The version I used was published in 1981 in Dover] 5

Inherent to the application of these laws is the assumption that all events in a global sequence are equiprobable.

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