# Determinism, Indeterminism, Probability, and the nature of the laws of physics—a second take…

After I wrote the last post [^], several points struck me. Some of the points that were mostly implicit needed to be addressed systematically. So, I began writing a small document containing these after-thoughts, focusing more on the structural side of the argument.

However, I don’t find time to convert these points + statements into a proper write-up. At the same time, I want to get done with this topic, at least for now, so that I can better focus on some other tasks related to data science. So, let me share the write-up in whatever form it is in, currently. Sorry for its uneven tone and all (compared to even my other writing, that is!)

Causality as a concept is very poorly understood by present-day physicists. They typically understand only one sense of the term: evolution in time. But causality is a far broader concept. Here I agree with Ayn Rand / Leonard Peikoff (OPAR). See the Ayn Rand Lexicon entry, here [^]. (However, I wrote the points below without re-reading it, and instead, relying on whatever understanding I have already come to develop starting from my studies of the same material.)

Physical universe consists of objects. Objects have identity. Identity is the sum total of all characteristics, attributes, properties, etc., of an object. Objects act in accordance with their identity; they cannot act otherwise. Interactions are not primary; they do not come into being without there being objects that undergo the interactions. Objects do not change their respective identities when they take actions—not even during interactions with other objects. The law of causality is a higher-level view taken of this fact.

In the cause-effect relationship, the cause refers to the nature (identity) of an object, and the effect refers to an action that the object takes (or undergoes). Both refer to one and the same object. TBD: Trace the example of one moving billiard ball undergoing a perfectly elastic collision with another billiard ball. Bring out how the interaction—here, the pair of the contact forces—is a name for each ball undergoing an action in accordance with its nature. An interaction is a pair of actions.

A physical law as a mapping (e.g., a function, or even a functional) from inputs to outputs.

The quantitative laws of physics often use the real number system, i.e., quantification with infinite precision. An infinite precision is a mathematical concept, not physical. (Expect physicists to eternally keep on confusing between the two kinds of concepts.)

Application of a physical law traces the same conceptual linkages as are involved in the formulation of law, but in the reverse direction.

In both formulation of a physical law and in its application, there is always some regime of applicability which is at least implicitly understood for both inputs and outputs. A pertinent idea here is: range of variations. A further idea is the response of the output to small variations in the input.

Example: Prediction by software whether a cricket ball would have hit the stumps or not, in an LBW situation.

The input position being used by the software in a certain LBW decision could be off from reality by millimeters, or at least, by a fraction of a millimeter. Still, the law (the mapping) is such that it produces predictions that are within small limits, so that it can be relied on.

Two input values, each theoretically infinitely precise, but differing by a small magnitude from each other, may be taken to define an interval or zone of input variations. As to the zone of the corresponding output, it may be thought of as an oval produced in the plane of the stumps, using the deterministic method used in making predictions.

The nature of the law governing the motion of the ball (even after factoring in aspects like effects of interaction with air and turbulence, etc.) itself is such that the size of the O/P zone remains small enough. (It does not grow exponentially.) Hence, we can use the software confidently.

That is to say, the software can be confidently used for predicting—-i.e., determining—the zone of possible landing of the ball in the plane of the stumps.

Overall, here are three elements that must be noted: (i) Each of the input positions lying at the extreme ends of the input zone of variations itself does have an infinite precision. (ii) Further, the mapping (the law) has theoretically infinite precision. (iii) Each of the outputs lying at extreme ends of the output zone also itself has theoretically infinite precision.

Existence of such infinite precision is a given. But it is not at all the relevant issue.

What matters in applications is something more than these three. It is the fact that applications always involve zones of variations in the inputs and outputs.

Such zones are then used in error estimates. (Also for engineering control purposes, say as in automation or robotic applications.) But the fact that quantities being fed to the program as inputs themselves may be in error is not the crux of the issue. If you focus too much on errors, you will simply get into an infinite regress of error bounds for error bounds for error bounds…

Focus, instead, on the infinity of precision of the three kinds mentioned above, and focus on the fact that in addition to those infinitely precise quantities, application procedure does involve having zones of possible variations in the input, and it also involves the problem estimating how large the corresponding zone of variations in the output is—whether it is sufficiently small for the law and a particular application procedure or situation.

In physics, such details of application procedures are kept merely understood. They are hardly, if ever, mentioned and discussed explicitly. Physicists again show their poor epistemology. They discuss such things in terms not of the zones but of “error” bounds. This already inserts the wedge of dichotomy: infinitely precise laws vs. errors in applications. This dichotomy is entirely uncalled for. But, physicists simply aren’t that smart, that’s all.

“Indeterministic mapping,” for the above example (LBW decisions) would the one in which the ball can be mapped as going anywhere over, and perhaps even beyond, the stadium.

Such a law and the application method (including the software) would be useless as an aid in the LBW decisions.

However, phenomenologically, the very dynamics of the cricket ball’s motion itself is simple enough that it leads to a causal law whose nature is such that for a small variation in the input conditions (a small input variations zone), the predicted zone of the O/P also is small enough. It is for this reason that we say that predictions are possible in this situation. That is to say, this is not an indeterministic situation or law.

Not all physical situations are exactly like the example of the predicting the motion of the cricket ball. There are physical situations which show a certain common—and confusing—characteristic.

They involve interactions that are deterministic when occurring between two (or few) bodies. Thus, the laws governing a simple interaction between one or two bodies are deterministic—in the above sense of the term (i.e., in terms of infinite precision for mapping, and an existence of the zones of variations in the inputs and outputs).

But these physical situations also involve: (i) a nonlinear mapping, (ii) a sufficiently large number of interacting bodies, and further, (iii) coupling of all the interactions.

It is these physical situations which produce such an overall system behaviour that it can produce an exponentially diverging output zone even for a small zone of input variations.

So, a small change in I/P is sufficient to produce a huge change in O/P.

However, note the confusing part. Even if the system behaviour for a large number of bodies does show an exponential increase in the output zone, the mapping itself is such that when it is applied to only one pair of bodies in isolation of all the others, then the output zone does remain non-exponential.

It is this characteristic which tricks people into forming two camps that go on arguing eternally. One side says that it is deterministic (making reference to a single-pair interaction), the other side says it is indeterministic (making reference to a large number of interactions, based on the same law).

The fallacy arises out of confusing a characteristic of the application method or model (variations in input and output zones) with the precision of the law or the mapping.

Example: N-body problem.

Example: NS equations as capturing a continuum description (a nonlinear one) of a very large number of bodies.

Example: Several other physical laws entering the coupled description, apart from the NS equations, in the bubbles collapse problem.

Example: Quantum mechanics

The Law vs. the System distinction: What is indeterministic is not a law governing a simple interaction taken abstractly (in which context the law was formed), but the behaviour of the system. A law (a governing equation) can be deterministic, but still, the system behavior can become indeterministic.

Even indeterministic models or system designs, when they are described using a different kind of maths (the one which is formulated at a higher level of abstractions, and, relying on the limiting values of relative frequencies i.e. probabilities), still do show causality.

Yes, probability is a notion which itself is based on causality—after all, it uses limiting values for the relative frequencies. The ability to use the limiting processes squarely rests on there being some definite features which, by being definite, do help reveal the existence of the identity. If such features (enduring, causal) were not to be part of the identity of the objects that are abstractly seen to act probabilistically, then no application of a limiting process would be possible, and so not even a definition probability or randomness would be possible.

The notion of probability is more fundamental than that of randomness. Randomness is an abstract notion that idealizes the notion of absence of every form of order. … You can use the axioms of probability even when sequences are known to be not random, can’t you? Also, hierarchically, order comes before does randomness. Randomness is defined as the absence of (all applicable forms of) orderliness; orderliness is not defined as absence of randomness—it is defined via the some but any principle, in reference to various more concrete instances that show some or the other definable form of order.

But expect not just physicists but also mathematicians, computer scientists, and philosophers, to eternally keep on confusing the issues involved here, too. They all are dumb.

Summary:

Let me now mention a few important take-aways (though some new points not discussed above also crept in, sorry!):

• Physical laws are always causal.
• Physical laws often use the infinite precision of the real number system, and hence, they do show the mathematical character of infinite precision.
• The solution paradigm used in physics requires specifying some input numbers and calculating the corresponding output numbers. If the physical law is based on real number system, than all the numbers used too are supposed to have infinite precision.
• Applications always involve a consideration of the zone of variations in the input conditions and the corresponding zone of variations in the output predictions. The relation between the sizes of the two zones is determined by the nature of the physical law itself. If for a small variation in the input zone the law predicts a sufficiently small output zone, people call the law itself deterministic.
• Complex systems are not always composed from parts that are in themselves complex. Complex systems can be built by arranging essentially very simpler parts that are put together in complex configurations.
• Each of the simpler part may be governed by a deterministic law. However, when the input-output zones are considered for the complex system taken as a whole, the system behaviour may show exponential increase in the size of the output zone. In such a case, the system must be described as indeterministic.
• Indeterministic systems still are based on causal laws. Hence, with appropriate methods and abstractions (including mathematical ones), they can be made to reveal the underlying causality. One useful theory is that of probability. The theory turns the supposed disadvantage (a large number of interacting bodies) on its head, and uses limiting values of relative frequencies, i.e., probability. The probability theory itself is based on causality, and so are indeterministic systems.
• Systems may be deterministic or indeterministic, and in the latter case, they may be described using the maths of probability theory. Physical laws are always causal. However, if they have to be described using the terms of determinism or indeterminism, then we will have to say that they are always deterministic. After all, if the physical laws showed exponentially large output zone even when simpler systems were considered, they could not be formulated or regarded as laws.

In conclusion: Physical laws are always causal. They may also always be regarded as being deterministic. However, if systems are complex, then even if the laws governing their simpler parts were all deterministic, the system behavior itself may turn out to be indeterministic. Some indeterministic systems can be well described using the theory of probability. The theory of probability itself is based on the idea of causality albeit measures defined over large number of instances are taken, thereby exploiting the fact that there are far too many objects interacting in a complex manner.

A song I like:

(Hindi) “ho re ghungaroo kaa bole…”
Singer: Lata Mangeshkar
Music: R. D. Burman
Lyrics: Anand Bakshi

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# Determinism, Indeterminism, and the nature of the laws of physics…

The laws of physics are causal, but this fact does not imply that they can be used to determine each and everything that you feel should be determinable using them, in each and every context in which they apply. What matters is the nature of the laws themselves. The laws of physics are not literally boundless; nothing in the universe is. They are logically bounded by the kind of abstractions they are.

Let’s take a concrete example.

Take a bottle, pour a little water and detergent in it, shake well, and have fun watching the Technicolor wonder which results. Bubbles form; they show resplendent colors. Then, some of them shrink, others grow, one or two of them eventually collapse, and the rest of the network of the similar bubbles adjusts itself. The process continues.

Looking at it in an idle way can be fun: those colorful tendrils of water sliding over those thin little surfaces, those fascinating hues and geometric patterns… That dynamics which unfolds at such a leisurely pace. … Just watching it all can make for a neat time-sink—at least for a while.

But merely having fun watching bubbles collapse is not physics. Physics proper begins with a lawful description of the many different aspects of the visually evident spectacle—be it the explanation as to how those unreal-looking colors come about, or be it an explanation of the mechanisms involved in their shrinkage or growth, and eventual collapse, … Or, a prediction of exactly which bubble is going to collapse next.

For now, consider the problem of determining, given a configuration of some bubbles at a certain time $t_0$, predicting exactly which bubble is going to collapse next, and why… To solve this problem, we have to study many different processes involved in the bubbles dynamics…

Theories do exist to predict various aspects of the bubble collapse process taken individually. Further it should also be possible to combine them together. The explanation involves such theories as: the Navier-Stokes equations, which govern the flow of soap water in the thin films, and of the motion of the air entrapped within each bubble; the phenomenon of film-breakage, which can involves either the particles-based approaches to modeling of fluids, or, if you insist on a continuum theory, then theories of crack initiatiation and growth in thin lamella/shells; the propagation of a film-breakage, and the propagation of the stress-strain waves associated with the process; and also, theories concerning how the collapse process gets preferentially localized to only one (or at most few) bubbles, which involves again, nonlinear theories from mechanics of materials, and material science.

All these are causal theories. It should also be possible to “throw them together” in a multi-physics simulation.

But even then, they still are not very useful in predicting which bubble in your particular setup is going to collapse next, and when, because not the combination of these theories, but even each theory involved is too complex.

The fact of the matter is, we cannot in practice predict precisely which bubble is going to collapse next.

The reason for our inability to predict, in this context, does not have to do just with the precision of the initial conditions. It’s also their vastness.

And, the known, causal, physical laws which tell us how a sensitive dependence on the smallest changes in the initial conditions deterministically leads to such huge changes in the outcomes, that using these laws to actually make a prediction squarely lies outside of our capacity to calculate.

Even simple (first- or second-order) variations to the initial conditions specified over a very small part of the network can have repercussions for the entire evolution, which is ultimately responsible for predicting which bubble is going to collapse next.

I mention this situation because it is amply illustrative of a special kind of problems which we encounter in physics today. The laws governing the system evolution are known. Yet, in practice, they cannot be applied for performing calculations in every given situation which falls under their purview. The reason for this circumstance is that the very paradigm of formulating physical laws falls short. Let me explain what I mean very briefly here.

All physical laws are essentially quantitative in nature, and can be thought of as “functions,” i.e., as mappings from a specific set of inputs to a specific set of outputs. Since the universe is lawful, given a certain set of values for the inputs, and the specific function (the law) which does the mapping, the output is  uniquely determined. Such a nature of the physical laws has come to be known as determinism. (At least that’s what the working physicist understands by the term “determinism.”) The initial conditions together with the governing equation completely determine the final outcome.

However, there are situations in which even if the laws themselves are deterministic, they still cannot practically be put to use in order to determine the outcomes. One such a situation is what we discussed above: the problem of predicting the next bubble which will collapse.

Where is the catch? It is in here:

When you say that a physical law performs a mapping from a set of input to the set of outputs, this description is actually vastly more general than what appears on the first sight.

Consider another example, the law of Newtonian gravity.

If you have only two bodies interacting gravitationally, i.e., if all other bodies in the universe can be ignored (because their influence on the two bodies is negligibly small in the problem as posed), then the set of the required input data is indeed very small. The system itself is simple because there is only one interaction going on—that between two bodies. The simplicity of the problem design lends a certain simplicity to the system behaviour: If you vary the set of input conditions slightly, then the output changes proportionately. In other words, the change in the output is proportionately small. The system configuration itself is simple enough to ensure that such a linear relation exists between the variations in the input, and the variations in the output. Therefore, in practice, even if you specify the input conditions somewhat loosely, your prediction does err, but not too much. Its error too remains bounded well enough that we can say that the description is deterministic. In other words, we can say that the system is deterministic, only because the input–output mapping is robust under minor changes to the input.

However, if you consider the N-body problem in all its generality, then the very size of the input set itself becomes big. Any two bodies from the N-bodies form a simple interacting pair. But the number of pairs is large, and worse, they all are coupled to each other through the positions of the bodies. Further, the nonlinearities involved in such a problem statement work to take away the robustness in the solution procedure. Not only is the size of the input set big, the end-solution too varies wildly with even a small variation in the input set. If you failed to specify even a single part of the input set to an adequate precision, then the predicted end-state can deterministically become very wildly different. The input–output mapping is deterministic—but it is not robust under minor changes to the input. A small change in the initial angle can lead to an object ending up either on this side of the Sun or that. Small changes produce big variations in predictions.

So, even if the mapping is known and is known to work (deterministically), you still cannot use this “knowledge” to actually perform the mapping from the input to the output, because the mapping is not robust to small variations in the input.

Ditto, for the soap bubbles collapse problem. If you change the initial configuration ever so slightly—e.g., if there was just a small air current in one setup and a more perfect stillness in another setup, it can lead to wildly different predictions as to which bubble will collapse next.

What holds for the N-body problem also holds for the bubble collapse process. The similarity is that these are complex systems. Their parts may be simple, and the physical laws governing such simple parts may be completely deterministic. Yet, there are a great many parts, and they all are coupled together such that a small change in one part—one interaction—gets multiplied and felt in all other parts, making the overall system fragile to small changes in the input specifications.

Let me add: What holds for the N-body problem or the bubble-collapse problems also holds for quantum-mechanical measurement processes. The latter too involves a large number of parts that are nonlinearly coupled to each other, and hence, forms a complex system. It is as futile to expect that you would be able to predict the exact time of the next atomic decay as it is to expect that you will be able to predict which bubble collapses next.

But all the above still does not mean that the laws themselves are indeterministic, or that, therefore, physical theories must be regarded as indeterministic. The complex systems may not be robust. But they still are composed from deterministically operating parts. It’s just that the configuration of these parts is far too complex.

It would be far too naive to think that it should be possible to make exact (non-probabilistic) predictions even in the context of systems that are nonlinear, and whose parts are coupled together in complex manner. It smacks of harboring irresponsible attitudes to take this naive expectation as the standard by which to judge physical theories, and since they don’t come up to your expectations, to jump to the conclusion that physical theories are indeterministic in nature. That’s what has happened to QM.

It should have been clear to the critic of the science that the truth-hood of an assertion (or a law, or a theory) is not subject to whether every complex manner in which it can be recombined with other theoretical elements leads to robust formulations or not. The truth-hood of an assertion is subject only to whether it by itself and in its own context corresponds to reality or not.

The error involved here is similar, in many ways, to expecting that if a substance is good for your health in a certain quantity, then it must be good in every quantity, or that if two medicines are without side-effects when taken individually, they must remain without any harmful effects even when taken in any combination—that there should be no interaction effects. It’s the same error, albeit couched in physicists’ and philosopher’s terms, that’s all.

… Too much emphasis on “math,” and too little an appreciation of the qualitative features, only helps in compounding the error.

A preliminary version of this post appeared as a comment on Roger Schlafly’s blog, here [^]. Schlafly has often wondered about the determinism vs. indeterminism issue on his blog, and often, seems to have taken positions similar to what I expressed here in this post.

The posting of this entry was motivated out of noticing certain remarks in Lee Smolin’s response to The Edge Question, 2013 edition [^], which I recently mentioned at my own blog, here [^].

A song I like:
(Marathi) “kaa re duraavaa, kaa re abolaa…”
Singer: Asha Bhosale

[In the interests of providing better clarity, this post shall undergo further unannounced changes/updates over the due course of time.

Revision history:
2019.04.24 23:05: First published
2019.04.25 14:41: Posted a fully revised and enlarged version.
]

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Here are a few interesting links I browsed recently, listed in no particular order:

“Mathematicians Tame Turbulence in Flattened Fluids” [^].

The operative word here, of course, is: “flattened.” But even then, it’s an interesting read. Another thing: though the essay is pop-sci, the author gives the Navier-Stokes equations, complete with fairly OK explanatory remarks about each term in the equation.

(But I don’t understand why every pop-sci write-up gives the NS equations only in the Lagrangian form, never Eulerian.)

“A Twisted Path to Equation-Free Prediction” [^]. …

“Empirical dynamic modeling.” Hmmm….

“Machine Learning’s `Amazing’ Ability to Predict Chaos” [^].

Click-bait: They use data science ideas to predict chaos!

8 Lyapunov times is impressive. But ignore the other, usual kind of hype: “…the computer tunes its own formulas in response to data until the formulas replicate the system’s dynamics. ” [italics added.]

“Your Simple (Yes, Simple) Guide to Quantum Entanglement” [^].

Click-bait: “Entanglement is often regarded as a uniquely quantum-mechanical phenomenon, but it is not. In fact, it is enlightening, though somewhat unconventional, to consider a simple non-quantum (or “classical”) version of entanglement first. This enables us to pry the subtlety of entanglement itself apart from the general oddity of quantum theory.”

Don’t dismiss the description in the essay as being too simplistic; the author is Frank Wilczek.

“A theoretical physics FAQ” [^].

Click-bait: Check your answers with those given by an expert! … Do spend some time here…

Tensor product versus Cartesian product.

If you are engineer and if you get interested in quantum entanglement, beware of the easily confusing terms: The tensor product and the Cartesian product.

The tensor product, you might think, is like the Cartesian product. But it is not. See mathematicians’ explanations. Essentially, the basis sets (and the operations) are different. [^] [^].

But what the mathematicians don’t do is to take some simple but non-trivial examples, and actually work everything out in detail. Instead, they just jump from this definition to that definition. For example, see: “How to conquer tensorphobia” [^] and “Tensorphobia and the outer product”[^]. Read any of these last two articles. Any one is sufficient to give you tensorphobia even if you never had it!

You will never run into a mathematician who explains the difference between the two concepts by first directly giving you a vague feel: by directly giving you a good worked out example in the context of finite sets (including enumeration of all the set elements) that illustrates the key difference, i.e. the addition vs. the multiplication of the unit vectors (aka members of basis sets).

A third-class epistemology when it comes to explaining, mathematicians typically have.

A Song I Like:

(Marathi) “he gard niLe megha…”