To keep an eye on (10) we invoke certain assumptions about experiments that have been conducted so far to confirm the physical theory. We assume that the obtained experimental results do not falsify the differential equation given the limitations that observers who con-duct the experiments face. The basic idea is to take into account some of these limitations in the construction of our mathematical representation. If certain limitations characterize the observation of experiments then it would be prudent if our mathematical construction that represents the experiments took these limitations into account. Building in observational limitations to the mathematical representation may protect us from surplus mathematical structure that reflect unobservable features or properties of the world.
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Table 2: An argument for the Desired Solution Thesis.
(1) Assume that solutions of the differential equation are representationally successful and empirically ade-quate in experimental cases we have tested so far.
(2) Introduce another mathematical construction which meaningfully satisfies the differential equation and thus is another candidate for representing of a physically possible world. For the time being let us refer to this alternative mathematical construction as a ‘wolution.’
(3) Formulate thewolution counterpart of an initial value, an initial value problem, continuous dependence of solutions on initial values, and well posed initial value problem: ‘winitial value,’ ‘winitial value prob-lem,’ ‘continuous dependence ofwolutions on winitial values,’ and ‘well posedwinitial value problem.’
(4) Show that by construction a well posed winitial value problem have a unique wolution that depends continuously onwinitial values.
(5) Show that by construction there are either no non well posed winitial value problems or that a non well posedwinitial value problem do not havewolutions (that awolution only exists if it is unique and depends continuously on winitial values).
(C1) If wolutions represent physically possible worlds then only wolutions of a well posed winitial value problem represent physically possible worlds.
(6) Argue that wolutions share the representational success and empirical adequacy of solutions in the experimental cases we have tested so far.
(7) Show that by construction a wolution, a winitial value, and a winitial value problem reduces, in an appropriate sense, to a solution, an initial value, and an initial value problem, respectively. Thus if wolutions represent physically possible worlds then a solutionsthat has awolution reducing tos can be regarded as anindirect representationof a physically possible world. If however there is nowolution that reduces tosthensdoes not represent a physical possible world neither directly nor indirectly.
(8) Show that by construction if an initial value problem is well posed then the correspondingwinitial value problem is also well posed.
(9) Show that by construction if an initial value problem is not well posed then there is either no corre-spondingwinitial value problem or the correspondingwinitial value problem has nowolution.
(C2) If wolutions represent physically possible worlds then a solution of a well posed initial value problem does (indirectly) represent a physically possible world.
(C3) Ifwolutions represent physically possible worlds then a solution of a non well posed initial value problem does not represent a physically possible world neither directly nor indirectly.
(10) Argue that wolutions are better motivated as mathematical representations of the experimental cases we have tested so far than solutions.
(11) Argue that (6), (7), and (10) provides sufficient reason to preferwolutions as representations of physically possible worlds.
(C4) A solution of a well posed initial value problem does (indirectly) represent a physically possible world.
(C5) A solution of a non well posed initial value problem does not represent a physically possible world.
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On the other hand we should also strive to disengage a representation of the world from the particular viewpoint of the potential observers since we want the representation to be a representation of the world and not a representation of the world-as-seen-by-a-particular-observer. A way to achieve a balance may be to construct our representation on the basis of the following recipe: in the first step we create a mathematical representation that respects the limitations of particular observers and in the second step we disengage particular observers from the representation by removing the limitations we imposed in the first step.
What are the limitations faced by observers of the experiments that have been conducted so far? Recall from Chapter 2 that the main motivation cited for the requirement of con-tinuous dependence is the assumption that an observation is limited to finite measurement precision; another such limitation is that an observation only lasts for a finite time. Accord-ingly we are going to assume that any potential observer (any observer who is or who could potentially be in the position to confirm the physical theory) faces two kinds of limitations:
that her observation lasts for a finite time and that her measurements have a finite precision.
Our mathematical construction then implements the two-step recipe sketched above for local dynamical theories of classical physics. First we define the notion of a bolution-chunk that takes finite time and finite precision limitations into account; then in two steps we remove these limitations to arrive at the notion of a bolution.
5.4.1 Bolution-chunks
Assume that all observations of experiments share a common feature, namely that they last for some finite time and that data is available with finite measurement accuracy. Since local dynamical theories of classical physics rely on continuous representation of at least some physical quantities it is typically the case that measurement data is compatible with
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more than one solution of the governing differential equation – that measurement data is compatible with all solutions which are so close to each other that we can’t discern them within a given measurement precision. Hence it seems that any particular experiment should be represented by not one solution of a differential equation, but by what we are going to call abolution-chunk: a maximal set of solutions of a differential equation defined on a finite time interval which stay within some finite distance from each other throughout this time interval2. The greater the measurement precision the narrower the bolution-chunk’s width;
some bolution-chunks are proper subsets of other bolution-chunks, some pairs of bolution-chunks have non-zero intersections, and yet some other pairs are disjoint or can have different time intervals with different lengths.
Figure 2: Bolution-chunk: a maximal set of close solutions throughout time intervalT.
A bolution-chunk thus represents the viewpoint of a particular observer who conducts her observation for a given finite time and with a given finite precision. On the other hand a bolution-chunk is laden with the limitations of the particular observer; the observation could have lasted for a longer period of time and that the measurement precision could have
2We carry out the construction using the framework developed for abstract differential equations in AppendixA. AppendixB also recalls the basic details. Thus the differential equation is assumed to take the formB.0.1. For definition of a bolution-chunk and length and width of bolution-chunks see Definition 13. We assume that the distance between solutions is given by the norm k.k of the Banach space E. See discussion later.
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been more refined3. We now thus proceed to remove these limitations.
5.4.2 Bolution-path
Assume that our observer extended the length of her observation. As she would still face observational limitations her longer observation would still be compatible with a set of solu-tions that stayed close to each other. Thus her longer observation would still be adequately represented by a bolution-chunk of a particular width. This new, longer bolution-chunk needs to be consistent with the older, shorter bolution-chunk in the sense that the longer bolution-chunk can only consist of solutions whose segments were already contained in the shorter bolution-chunk.
The observation could be continued even further, potentially up to any given point in time; the observation could also have been conducted for a shorter time. Hence the adequate representation for a potential observer who has a limited measurement capability but who lives long enough to see how the world unfolds is an assignment, to any given length of time, of abolution-chunk of this length, with the restriction that a longerbolution-chunk can only consist of continuation of solutions which are already present in a shorter bolution-chunk.
Such an assignment of bolution-chunks to time-lengths will be called a bolution-path4. Given mild technical conditions bolution-chunks exist and any bolution-chunk can be extended to a bolution-path5. Whether this extension is unique depends on the differential equation. If the time evolution preserves distance between the solutions, e.g. when the time evolution is unitary, then extension of a bolution-chunk to a bolution-path is unique.
If the time evolution does not preserve distance between solutions, then extension of a
3Chapter 2 calls attention to the dangers of this assumption. Here we go along with intuitions from classical physics and assume that an ideal observer’s measurement precision could be arbitrarily refined;
adding a lower boundary to our construction is feasible but it would change the results we end up with esp.
regarding conclusion (C4).
4See Definition14.
5See Proposition6and Proposition7.
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Figure 3: Bolution-path: an assignment of bolution-chunks to all points in time.
bolution-chunk to a bolution-path may not be unique (if solutions diverge it is not). In case a bolution-chunk does not determine the bolution-path then having represented the experiment only up to a certain point in time puts limits on how the observed system can behave in the future but it could still behave in discernibly different ways.
5.4.3 Bolution
The bolution-path removes the limitation that particular observations are only conducted for a finite length of time. Observations could have been made with better precision as well.
Had our observer made her observations with a better precision her adequate representation of the experiment would still be a bolution-path but it would be a bolution-path with a narrower width. This new, narrower bolution-path needs to be consistent with the older, wider bolution-path in the sense that the narrower can only contain solutions which were contained in the wider.
We may assume that measurement precision could have been even better, potentially reaching any finite accuracy. Hence a representation that removes the assumption that the measurement precision is a given finite value is an assignment, to any given positive
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Figure 4: Bolution: an assignment of bolution-paths to all levels of precision.
measurement precision, of a bolution-path with a width defined by this precision, with the restriction that a bolution-path with better precision is contained in a bolution-path with worse precision. Such an assignment ofbolution-paths to levels of measurement precision is a bolution6.
Again, given some mild technical conditions, any bolution-path can be extended to a bolution7. The extension is not unique; a series of bolution-paths can “zoom on” many solutions in the bolution-path, leading to different bolutions8.
What is a bolution, then? We can think of abolutions as a more elaborate mathematical representation of a possible world than a solution; abolution also tells, for any measurement precision and for any length of time, how observations would be represented if we took into account the limitations of the observer who conducts her measurement with said precision for the said length of time.
We offer the considerations leading to the definition of the bolution as evidence that (6) bolutions share the representational success and empirical adequacy of solutions and (10) bolutions are better motivated as representations of the experiments through which we tested
6See Definition15.
7See Proposition8.
8For a more precise statement see Proposition9.
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the theory. If we then generalize and (11) take bolutions to represent physically possible worlds then we should not let ourselves be led astray by the fact thatbolutions are defined via solutions. Logical primacy should not be conflated with representational primacy; if bolutions are the mathematical constructions that represent physical possibilities then the representational role some solutions might have is only derivative from the representational role of bolutions. As the next section shows some solutions do indeed have such indirect representational role as in certain circumstances a bolution reduces to a solution.