A theory of incomplete measurements: Towards a unified vision of the laws of physics

Foreword

Albert Einstein formulated his theory of relativity between 1905 and 1915, explaining large-scale phenomena like gravitation. In parallel, quantum physics was being developed by a large community of physicists, explaining many mysteries in the microscopic world. Both theories made great advances since then. However, physicists today are still stuck on the inability to reconcile our best theories for the infinitely large and the infinitely small. The interpretation of the solutions of the equations, when trying to unify the quantum and relativist viewpoints, still leads to major inconsistencies with what we observe in nature.

Both theories are also seen as strange, paradoxical even. After more than a century, quantum physics and relativity are still not taught in school. Students are not even exposed to any XXth-century physics before reaching a scientific university. We continue to format our human brains from a very young age to the Newtonian mechanistic and deterministic paradigm. I think this is a major obstacle to the global understanding of our universe: we become incapable of conceiving another way of thinking. To overcome this blockage, it is necessary to take a step aside, to look at things from another angle, to devise original methods, in order to teach modern physics to future generations.

Christophe de Dinechin is a French computer scientist, with major contributions in video games, programming languages and operating systems. His experiences in programming allowed him to apprehend physics from another point of view: that of discontinuity and pixelation. By using this approach, and after realizing that changing scale inevitably introduces measurements errors, he has revisited physics with new concepts and tools, focusing primarily on measurements and probabilities. He explains his "Theory of incomplete measurements" in a very didactic way in the book that you are holding. To make his thoughts more accessible to many, Christophe begins each chapter with an intuitive and popular level of explanation, followed by a separate section containing the mathematical equations, program source code and demonstrations for specialists.

As a result, this is not just some arid book about abstract theoretical physics. It is also a book about teaching, and about thinking, which will benefit any curious reader, and answer many questions for anyone who has been puzzled about what modern physics is really trying to tell us. The subtitle, "Toward a unified vision of the laws of physics gives away this intent. In that respect, this book reminded me of Douglas Hofstadter’s excellent Gödel, Escher, Bach: An Eternal Golden Braid", which had such an impact on a whole generation of scientists. Both authors introduce their readers to really complicated concepts in a way that anyone can appreciate and remember.

It is no coincidence that I first met Christophe in a wonderful trans-disciplinary "Art-Science-Pensée" (Art-Science-Thought) colloquium, organized by Dr. Paul Charbit in Mouans-Sartoux, on the French Riviera. As a regular participant in this one-of-a-kind gathering of French scientists, philosophers, artists and thinkers, Christophe was presenting his point of view on some well-known problems in physics, the kind that are still misunderstood and misinterpreted by a majority of people, including educated scientists. I am not a physicist, and I was quite naive on the subject, but I appreciated the clarity of his talk. His presentation gave me a glimpse of a new way of presenting and understanding modern physics, improving our collective vision of the world. I had the intuition that his approach could truly bring a new vision for scientific research, so I encouraged him to continue popularizing his ideas. We kept exchanging for a few years, and Christophe allowed me review his book as he was writing it. His original and sometimes provocative approach was for me a revelation, allowing me to make conceptual leaps and to open many doors in my understanding of several mysterious physical phenomena.

I am filled with gratitude that this book is finally being published. I hope that a large public, especially young students, will read it, and that it will be welcomed with curiosity and openness by physicists, perhaps provoking constructive reactions and discussions. I have no doubt that it will allow great advances in our way of apprehending and understanding the world around us. Who knows, it might even awaken us to a true paradigm shift, to a vision of the universe where, as Christophe puts it, there is no space-time continuum, first because there is no continuum, second because there is no space and no time.

Alice Guyon, PhD in Neurosciences

Research Director at CNRS – University Côte d’Azur – IPMC

Introduction

In the beginning was the word

Beginning of the Gospel according to Saint John

For over a century, physics has been in a strange situation, so uncomfortable as to be unbearable. In order to correctly describe the universe, we need not one, but two theories, known as quantum mechanics and general relativity.

These two theories, which emerged almost simultaneously at the turn of the XXth century, were true conceptual revolutions, upturning our understanding of time, space and fundamental bricks that make up our universe. Since their inception, both were experimentally tested time and time again. We know for certain today that both are remarkably accurate.

However, there is only one universe. It seems superfluous to need two theories to describe a single universe. This is the reason why a number of highly regarded physicists dedicated so much effort into building a unified theory of the universe, meaning a theory that combines the best aspects of relativity and quantum mechanics. These efforts began quite early. As an example, Albert Einstein himself spent a number of his later years working on an unsuccessful unified field theory.

Unfortunately, and that’s the root problem, quantum mechanics and general relativity proved to be mathematically incompatible. Reasons are given further in this book, but to make it short, attempts at integrating tools and methods from the two approaches lead to physically absurd results, such as infinite values where experimentation only gives finite measurements.

Theories produced that way therefore do not work. In some specific cases, cruelly inelegant techniques such as renormalization can be used to produce so-called effective theories, meaning theories that, at least experimentally, work in some particular domain of validity. But their theoretical basis are not satisfactory, and the existence of such solutions has not yet been demonstrated in the more general case.

Consensus is therefore that the problem is very difficult. However, I believe that I solved it. Such a statement is extraordinary, and can seem presumptuous, all the more because I am more of a computer person than a physics researcher. Such an extraordinary statement requires a demonstration to match. I hope that this book will provide it.

The solution that I am proposing is both extremely simple and frighteningly complicated. Conceptually, the base idea to extend the principle of relativity to all forms of measurement is almost obvious, although the details definitely are not. That extended principle simply requires that we must be able to do physics regardless of the physical measurement apparatus being used, including one that is imperfect, defective or did not complete its measurement. The proposed approach removes a number of hard-to-explain axioms to replace them with common sense choices about measurements. Finally, a number of troublesome issues with earlier theories disappear. This is how a summary of the theory can be given in a few short pages, see next section.

From a practical standpoint, on the other hand, the simplest equations from the classical theories have to be replaced by mathematical monstrosities that are extremely hard to manipulate. These concepts are well understood in the field of computer science, notably in the context of image or digital signal processing, but the required tools are not necessarily suited for theoretical analysis, and often leave us no choice but numerical processing.

In short, I reluctantly suggest a new way of doing physics that is notably more difficult, at least regarding fundamental concepts such as equations, variables, or changes of reference frame.

First, it becomes necessary to deeply redefine the very notion of equation or variable in physics, specifically to clarify the transition from the physical universe to the mathematical model. Whereas  Equation in an equation could previously be read as "the speed", it can at best be interpreted in the proposed approach as a particular measurement of speed performed using a specific measurement device. This idea was already somewhat emergent, at least as far as space and time measurements are concerned, in the theories of special and general relativity.

Next, we need to invent a new kind of transform to deal with changes of reference frame, that generalizes to all kind of measurements, including imperfect, noisy or incomplete ones. This new transform is conceptually simple, and gives a good theoretical foundation for well known measurement units, but it introduces a real technical difficulty that can only be resolved using algorithm and similar operations with mathematically problematic properties. For instance, where classical physics would represent a rotation using matrix algebra, and where general relativity uses tensors, the new theory resorts to an algorithm similar to what computers use to rotate an image. Among other undesirable properties, the algorithm is not mathematically reversible nor linear, except possibly as an approximation.

Physicists are used to taking advantage of properties such as linearity, continuity, differentiability, or symmetries, to the point where such properties are sometimes treated as a given in quantum mechanics or relativistic theories. Regrettably, they cannot reasonably be considered as generally valid in the proposed approach. At best, some of them can be recovered under special simplifying conditions intended to enable some specific reasonings. This always comes at the cost of severe restrictions regarding the domain of validity for said reasonings.

If this was not sufficient to spread a smell of sulphur, the most serious limitation of the new theory proposed here, and therefore, of my entire demonstration, is that it is somewhat difficult to make it predict new and unknown things. This theory may not immediately give rise to strange and spectacular predictions the way relativity and quantum mechanics did in their time. This does not mean that the theory does not make its own predictions, but rather, as we shall see, that it differs from older theories primarily in unexpectedly mundane and obvious everyday cases.

In other words, the theory presented in this book does contradict quantum mechanics or relativity, but instead of doing so in exotic realms like black holes, this happens in verifiable daily experiments like digital camera sensors or watching the stars through a foggy telescope. One can only hope that more spectacular forms of experimental validation will emerge, since this is how physics progresses.

In any case, I invite you to discover the "Theory of incomplete measurements", which I hope will let physics move forward again with renewed momentum, resuming this kind of continuous forward progress that always made science so interesting. Even if that does not happen, at the very least criticism of the ideas presented here will allow me to personally progress.

That is sufficient for me to have the audacity to write this book

Organization of this book

If you cannot explain it simply,

you do not understand it well enough.

Albert Einstein

Explaining a fundamental theory of physics is a very perilous exercise. I wish to keep this publication easy to understand for the layman, while offering enough information for specialists to dig deep and verify my statements, hypotheses or proofs.

This is the reason why each chapter is split in two parts, namely an introductory section, in which I try as much as possible to focus on the meaning and intent of a topic, followed by another more technical section in which I will put all equations, whether classical or new, as well as algorithms or computer source code that are necessary for in-depth understanding. In doing so, I tried to make it so that a non-specialist reader can get a good understanding of the theory being presented without having to read the second section.

The book begins with a summary of the theory in a few pages¹. The goal of that brief exposé is to give a broad overview sufficient to convince a doubtful reader that the essential ideas are pertinent. This summary also sets the limits of the discussion and of the theory being presented, clarifying that it absolutely makes no claim to be a theory of everything, as such unifying efforts are sometimes described. I also hope that this will help specialists quickly find their bearing in my presentation of the subject, to precisely understand what I believe to have demonstrated as well as the work that remains to be done.

The main body of this book is made primarily of four large sections:

An even more general relativity sets the foundations for the approach of physics that I am proposing. These foundations are naturally anchored in a long history of physics, what Isaac Newton once called "the shoulders of giants". A large fraction of the ideas being presented are nothing but a reformulation of the knowledge accumulated over many centuries. I will try to highlight what I believe to be new or specific in my statements.

Representing the laws of nature discusses the emergence of laws and patterns. Where classical physics uses algebraic laws to express relationships that are observed in nature, the theory presented here uses a statistical and numerical approach that enables an interesting link with well-known notions such as entropy. This approach also unifies previously distinct concepts, such as "laws of nature and change of reference frame". In short, this section explains how to transpose known laws into the new framework.

Towards unification reinterprets the two major theories of the XXth century, general relativity and quantum mechanics, in the context of ideas that were presented previously. Tools and methods will be presented to almost automatically convert classical models using continuous equations to a discrete model that predicts the same thing. This construction will eliminate the main obstacles to a theory of physics that can simultaneously be compatible with both quantum mechanics and general relativity, while going even further.

Something new will highlight the essential differences between the theory being presented in this book and the models that were elaborated earlier, regarding either concepts, interpretation, tools or reasoning. This will be done by exploring a number of cases that are difficult to model with earlier theories. In that last section, I will be engaging in more speculative exploration of interesting corners of physics. I will also explain the need to limit, sometimes drastically, the domain of validity of known experimental procedures.

Summary of the theory

Whatever we conceive well we express clearly

and the words to state it will flow easily

Nicolas Boileau

Putting measurements at the center or physics reasonings

Since Isaac Newton, we have gotten used to representing laws of physics using equations such as  Equation , which in this particular case describes the experimental observation that the force, written  Equation is equal to the product of the mass, written  Equation , by the acceleration, written  Equation . The commonly accepted meaning of such an equation is that  Equation is a real number, and that  Equation and  Equation are vectors, in other words mathematical entities with a behavior that has been quite well known for a very long time.

This use of mathematics in physics has been so successful that nobody today ever questions it. Of course, the kind of mathematics being used became increasingly sophisticated and complicated, over time, but the idea to associate a mathematical quantity to a physical observation is unavoidable today. Similarly, measurement instruments were perfected over and over again, but without ever putting into question the idea that we measure quantities, and therefore numbers.

In reality, such an equation presents a real puzzle. Does the validity of this equality depend on the way  Equation ,  Equation or  Equation are being measured? Laws of physics are never written with a specification of how to measure force, mass or acceleration. Therefore, the classical response to this question seems to be that this really does not matter, and that the problem is correctly handled as long as one can evaluate the measurement error.

However, in practice, the fact that two measurement instruments are not equivalent is well known. It is necessary to calibrate instruments, and to remain within their range, or else the measurement becomes invalid. Consequently, one cannot too naively assimilate a physical measurement to a number. Strangely enough, quantum mechanics as well as general relativity entirely ignore this aspect of things, including the mundane notion of measurement unit, which does not emerge naturally in either theory.

The proposed approach was called Theory of Incomplete Measurements precisely because it takes the opposite approach of putting the act of measuring at the heart of the reasoning. One will therefore focus first on the crucial question of knowing what the variables that physicists put in their equation truly represent, in other words, to understand the precise meaning that  Equation ,  Equation or  Equation might have in the equation. We must then take into account that they are the result of physical measurements, and not purely abstract mathematical entities.

Extending the principle of relativity to all measurements

This approach could just as well have been called the Theory of extended relativity, because the starting hypothesis requires a generalization of the approach Albert Einstein followed to elaborate first the special theory of relativity, and then the general theory of relativity. A possible formulation of the philosophy leading to the relativity principle that Einstein made so popular is that one should be able to do physics correctly irrespective of the frame of reference being chosen. In special relativity, physics must state the same thing irrespective of the speed of the observer; in general relativity, physics must remain valid even in the presence of acceleration or gravitation.

The new principle of extended relativity that the theory of incomplete measurements requires states that we must similarly be able to do physics irrespective of the measurement instruments being chosen. Physics must remain invariant even with measurements of space and time that are blurred by fog, with a voltmeter that still oscillates, or with an oscilloscope equipped with noisy probes. In short, one should be able to perform physics experiments with measurements that one can consider as incomplete, in the sense that we take it as a fact that they do not represent a complete knowledge of the system being measured.

This formulation is clearly an extension of the relativity principle to all measurements, which justifies the phrasing extended relativity principle. In the theory of incomplete measurement, we will no longer consider any measurement as perfect or ideal, but only take into consideration the seriously imperfect physical measurements we actually have access to. Even time and space as a universal background that exists irrespective of its contents disappear, and must be replaced by measurements of duration and distance.

Of course, the introduction of the principle of general relativity by Einstein had, in a sense, simplified things, because it made it possible to precisely formulate physics in a uniform way, irrespective of the frame of reference. On the other hand, it became necessary to specify the reference frame being used, and switching from one reference frame to another now requires a more complex mathematical set of tools than for earlier theories. In particular, in order to deal with acceleration, the theory had to take the curvature of space-time into account, and to use tensors to represent physical quantities.

In the same way, the theory presented here simplifies things insofar as it makes it possible to reason about physics using any measurement instrument, including those that are imprecise or badly calibrated. On the other hand, it becomes necessary to specify which measurement instruments are being discussed, and new kinds of mathematical transforms are needed to go from one measurement instrument to another.

These transforms are noticeably more complicated than general relativity’s tensors. For instance, a physical rotation that only required matrix algebra with comfortable properties such as linearity, reversibility and continuity will, in the case of the theory of incomplete measurements, have to be replaced with an algorithm similar to what computers use to rotate pictures. Such algorithms have none of the nice properties physicists are used to, except as an approximation in specific cases. For example, we will cover a number of really common cases, including simple ones like digital zooms in smartphones, where the changes of referential in the new theory are not reversible.

General relativity can then be understood as a special case of the theory of incomplete measurement, where we only take into consideration very particular kinds of measurements and very specific kinds of conversions between similar measurements of space and time, and under a simplifying condition called "Gaussian Co-Ordinates" by Albert Einstein, postulating that all the measurements being considered can be treated as continuous and twice differentiable.

This Gaussian coordinates hypothesis, which Einstein had carefully delineated in the formulation of his theories, is therefore no longer interpreted as an intrinsic property of the universe itself, or at least of space-time, but rather at most like an observable property of specific measurements. The most striking consequence is that we no longer can legitimately talk about space and time anymore, but only about measurements of space and time, some of which are anything but compatible with the Gaussian coordinates hypothesis. As we shall see, examples are not hard to find.

We chose the processes we call measurements

Focusing on the measurement operation itself makes it possible to take interest in what makes a physical measurement. The theory of incomplete measurements defines a measurement as being a choice amongst all the possible physical processes, more precisely:

a physical process

connecting an input and an output chosen ahead of time,

that can be repeated with an identical result,

that provides information solely about its input,

this information being given by an observable change in the output,

observation that can be given a symbolic interpretation for example using a scale graduation that can results from a calibration process.

Experimentally, there are such physical processes. There are also physical processes that do not have these properties, demonstrating that we are making a choice when we select something as a measurement.

Measurement results are discrete

A key observation is that, as far as we know, all physical processes that can provide a symbolic interpretation do so in a discrete manner. In other words, physical processes do not naturally produce real numbers. Quite to the contrary, our measurement instruments all feature discrete scales and not continuous ones, and such scales give a finite number of observable results.

It turns out that the mathematical incompatibilities between quantum mechanics and general relativity derive precisely from the use of real numbers, for example in the computation of integrals that diverge. This specific kind of divergence cannot emerge with finite sums on finite quantities. The primary obstacle blocking the unification of quantum mechanics and general relativity simply disappears if one never uses real numbers.

A probabilistic knowledge of physical systems

In order to represent the knowledge about a physical system provided by measurements, one can consider the probability of getting each of the possible measurement values the apparatus is susceptible of producing. For example, knowledge about a measurement that gives two possible results, such as "presence or absence", can be represented by two probabilities, the probability of presence and the probability of absence. If these are the only two possible results, the sum of the two probabilities must be one.

This kind of representation transparently corresponds to the description of a state in quantum mechanics. In particular, if one prepares a system so that we have a certainty to get a particular result, then the probability of getting that result is one, and the probability of getting any other result is zero. That matches so-called eigenstates that immediately follow a measurement in quantum mechanics. Conversely, any uncertainty regarding the result of the measurement corresponds to an "entangled" state with more than one non-zero probability for measurement results.

The theory of incomplete measurements derives its name from its ability to enable quantitative evaluation of measurements that do not give a complete information about the system being studied. That makes it possible to handle the case where the measurement process has not completed yet. A measurement that is progressing leads to a set of probabilities that shrinks as the measurement converges and as uncertainties about the result diminishes. The theory can then apply to measurements that are not instantaneous.

The precise correspondence between this representation and the state as envisioned in quantum mechanics exhibits a number of similarities, but also a few important differences. In particular, the measurement result in this context is always discrete and not continuous. This eliminates a few difficulties, as pointed out previously, but also introduces some new ones, in particular the correct way to adapt techniques and mathematical theorems that were initially demonstrated in the continuous domain.

The theory of incomplete measurements makes it possible to explain and justify, relatively simply, most of the axioms and properties associated to quantum mechanics. For example, the collapse of the wave function emerges as a necessary consequence of the requirement that measurements must give the same result when repeated. We refuse to call measurement any physical process that does not have this property. Similarly, the traditional shape of the wave function as a complex-valued function on all of space can be deduced from the fact that we measure the position of a particle by determining its presence or absence in every point of space.

Quantum mechanics can therefore be seen as a special case of the theory of incomplete measurement under specific hypotheses about linearity and continuity of the measurements. It is no longer restricted to the microscopic world, and macroscopic quantum experiments can be devised easily. We will provide a number of examples in this book, including large-scale analogs to many signature quantum mechanics experiments, without shunning the most mysterious ones, like delayed choice experiments and the so-called quantum eraser.

Non-relativistic and non-quantum systems

Exploring what happens when the hypotheses required from quantum mechanics or relativity are no longer valid opens a very interesting field of study. The Gaussian coordinates hypothesis explains why it is so difficult to apply general relativity at a small scale, while linearity and continuity explain why quantum mechanics breaks down quickly with scale.

Fortunately, many reasonings held in the framework of traditional theories apply identically in the theory of incomplete measurements, under specific conditions. For example, a reasoning that uses symmetries as the basis for conservation laws, like Noether’s theorem, remains valid as long as differentiability conditions apply.

There exist physical experiments that do no fall within either the boundaries of general relativity or quantum mechanics. We already mentioned the case of imprecise or imperfect measurements, but this has only a limited practical interest, except insofar as it shows how we have long solved these problems in practice with techniques such as repeating measurements and averaging them out to reduce noise or uncertainty.

The theory being presented also makes it possible to consider the evolution of systems along arbitrary measurement axes. The theory of incomplete measurements is not restricted to time-based evolution. A simple example is repeating the same experiment in space, but this can be generalized to other kinds of measurement. This leads to a statistical formulation of the evolution of physical systems, which plays the role of a fundamental equation in the theory, and establishes a strong link with thermodynamics. This formulation can be done without any reference to time.

Once the specific conditions that define general relativity or quantum mechanics are well understood, it makes sense to explore other categories of constraints. We can for example study what happens in systems where multiple axes of time and space are necessary to faithfully model the system, like a swarm of bees or drivers on a highway; discuss systems where one property of measurements does not hold, like rolling a dice; model modern technologies such as speculation in modern microprocessors, where numerous possible and uncertain outcomes are deliberately held in flight simultaneously as an optimization.

The results we get studying such systems may not be particularly novel, but it is satisfying to get them in an enlarged theoretical framework of physics that also naturally includes both quantum mechanics and general relativity.

A new step in physics, but no quantum gravity

The ambition of this book and the theory presented within is clearly set: proposing a credible way to finally unify general relativity and quantum mechanics in a single theoretical framework of physics.

The proposed unification is achieved by adopting a new extended principle of relativity that applies to all types of physical measurements. Doing so requires a precise definition of physical measurements, and new tools to study them.

Let us state from the get go that this idea, while it truly offers a form of unification, will not achieve what many expected from any grand unified theory of physics, notably a precise quantum model of gravity. Quite to the contrary, it will highlight previously ignored problems in building models ranging all the way from the atomic to the galactic scale.

Furthermore, the cost of the new approach is very high, at least on three points:

First, we must give up, at least in the general case, many hypotheses so far considered as solid, such as continuity or differentiability, and consequently, to find alternatives to the very useful mathematics that require such hypotheses.

Second, w need to learn how to convert between arbitrary measurements. The required mathematical tools are particularly unwieldy, lacking any desirable properties such as reversibility, although they are relatively well understood in the field of computer science, notably in the context of digital image processing.

Finally, the system evolution as predicted by the theory is not very specific, often being driven by raw data rather than purely theoretical considerations. This makes it unlikely that the new theory will ever predict the kind of spectacular effects, such as curvature of light by gravity or antimatter, that experimentally confirmed general relativity or quantum mechanics in their time.

Nonetheless, many retroactive results, where the proposed theory gives better answers than historical ones, seemingly provide sufficient validation to submit it to collective scrutiny. A simple example is that the new theory only predicts particle positions that can actually be measured, whereas quantum mechanics predicts probabilities of presence at places where there is no sensor to detect the particle, because the wave function is defined over all of space.



Grand Unified Operating System (https://septheory.wordpress.com/2018/06/03/grand-unification)

Part I - An even more general relativity

How mathematics are being used in physics

If I have seen further than others,

it is by standing upon the shoulders of giants

Isaac Newton

A big step forward in our understanding of the universe has been the transition from the qualitative to the quantitative. Knowing that there are many days in a year is a useful piece of information for a human society, but being able to state without a doubt that each year contains approximately 365 days (and a quarter) is a significant progress relative to the previous statement, since it has much better predictive power.

The two statements express a relationship between the year and the day. We are in both cases comparing physical quantities of the same kind, time. However, the quantitative relation can be expressed with increased precision using an equation such as  Equation , where  Equation is the number of days, and  Equation is the number of years. Such an equation is mathematically perfectly defined. It can be true for arbitrarily large or precise values of  Equation or  Equation . For instance, it is true for the number of years below:



as long as the number of days is precisely:



From a physical standpoint, on the other hand, stating that the equality remains valid with these particular values is highly suspect, for at least three reasons.

First, the known age of our universe is roughly 13.78 billion years, a number that can be written with ten digits. What does a number of years with 42 digits mean in that context, as far a physics is concerned?

Second, the equation relates specific physical phenomena, one measuring the rotation of the Earth around the Sun, the very definition of a year, and the second one measuring the spin of the Earth around its axis, defining the day. Over the span of a few years, we can observe a proportionality relationship. However, modern science tells us without a doubt that this relation does not hold at the scale of a few billion years.

Finally, the mathematical equation is only exact if all decimals match. How can we talk, over a duration that far exceeds the age of the universe, of an identity regarding infinitesimal fractions of a second?

Physical equations therefore must be understood and handled in a very different way than mathematical equations. That is the purpose of this chapter.

A conditional equality

We can first note that the example of an absurdly large or precise number demonstrates that the physical equality is only valid under some conditions. Unfortunately, these hypotheses are all too often entirely implicit. We need to find a way to make them explicit.

A first relatively simple idea is to add a precise set of hypotheses. In our example, this can include "while the universe exists, or for a given number of years, or regarding precision, a fuzz factor like within 2%". This is obviously only moderately satisfying, but it is still better than not mentioning any of the hypotheses at all.

Such working hypotheses can, most often, be guessed or deduced from the experimental context. A few readers may think that this means the observation has very little value. We can naturally tolerate that we don’t need to belabor obvious hypotheses, as long as they are correctly established one way or another. In particular, many hypotheses are implicit as being necessary for the observed system to simply exist or make sense.

Yet, the reason for paying attention to working hypotheses is to be able to analyze them in order to guarantee, as much as possible, that they are known and documented wherever necessary.

What numbers are we talking about?

The example we gave of proportionality between the number of days and the number of years illustrates quite well that the quantities in a physical equation cannot be too naively assimilated to mere real numbers, since such numbers can be arbitrarily large, small or precise. This begs the question of the numerical representation we should consider as a valid alternative.

Are we talking about whole numbers? A number of days or years can certainly be considered as a whole number. This is the information we have in a date. For instance, when we talk about February 17, 1969, we identify a specific day by counting the 17th day of the second month of the 1969th year in our calendar. We are indeed talking about whole numbers, even if for historical and practical reasons, we count them in a special way, for example by giving names to months or days in the week. We can still write a date using only numbers, like 02/17/1969.

However, it is unlikely that this kind of numbering will be used by someone who wants to use the equation. This is made apparent by the constant 365.25 that shows up in the formula: we apparently talk about decimal numbers. We are also used to talking about fractions of days ("half a day) or years (three years and a half").

Does that mean that we are talking about rational numbers, the set that mathematicians call  Equation ? Probably not: fractions suffer the same problems as real numbers, namely they can be arbitrarily large or precise. As a matter of fact, the absurd values given earlier in this chapter can be represented as fractions. It seems impossible to consider the number in physics as being any fraction.

The answer is actually quite well known if one thinks about it in terms of physical measurements. If we measure the number of days, for example by counting the alternation between day and night, we naturally get a whole number. We can talk about half a day, but only by using another measurement, such as the position of the Sun in the sky, or by counting hours, minutes or seconds, and then applying a similar proportionality equation to convert between hours and (fractions of) days. These equations are simple and well known. There are 24 hours in a day, 60 minutes in an hour, or 60 seconds in a minute. Consequently, we can state that there are 86400 seconds in a day.

By applying these unit conversions, we can easily talk about small fractions of a day such as a 86400th of a day. It is, indeed, a fraction, but not using an arbitrary denominator. Without another device specifically measuring the 1029299th of a day, it makes little sense to talk about such fractions of a day.

A good representation of numerical values in physics appears to be similar to fractions, but with a numerator that results from a counting operation, and a denominator that derives from a choice of measurement device, something we often call a measurement unit. As a result, the numerator has a limited range of possible values, those that correspond to a physical counting, for example how many times the Sun rises. The denominator has a constant value that indicates the precision and unit of the measurement².

This is only the beginning of an answer. Reality is more complicated.

Equalities are inexact

Another difficulty to consider is that equalities in physics are always inexact, at least if we treat them as strict mathematical equalities. They are only true within errors and approximations. The relationship between the number of days and years depends largely on the precise definition given to the number of days and the number of years.

The figure below³ illustrates the problem using a very similar relationship, namely the approximate proportionality between the duration of a day as measured using astronomical observations and a high-precision measurement of time, such as an atomic clock, that makes it possible to count very small fractions of a second.



Deviation of day length from SI day (Wikimedia, under public domain)

The figure makes it clear that even on such a simple example, the precise relation expressed by the equation is not a simple proportionality. The deviation from the idealized multiplication is a somewhat arbitrary and noisy curve that can only be inferred from observations. Presumably, the variations result from a large number of factors, ranging from fluid movements in our planet to interaction with various celestial bodies.

Naturally, we can as a first order of approximation consider that the proportionality remains valid, since the deviation from this model is only a few milliseconds on a duration of 86400s. Still, from a strictly mathematical standpoint, the equality is not verified.

A physics equation must therefore be regarded as being true with some error. This is not just an hypothesis about the equation, such as the "within 2%" precision specification we mentioned earlier. Such a specification would describe a chronometer that measures the 50th of a second, and therefore has a built-in precision of 2% However, what we are talking about here is not related to the limited precision of the measurement device. Quite the opposite, measuring with increased precision is what allows us to detect the deviation between the simplified model and reality.

We must therefore talk about a "physical error", something that captures all aspects of the system we cannot directly account for in a simplified model.

We will later have to return to the problem of converting between different units. We shall see that such conversions are the fundamental brick used to build any physical law in the theory of incomplete measurements.

As indicated in the introduction, the second part of this chapter on the next page contains informations that are slightly more technical. A reader who simply wants to understand what the theory of incomplete measurements states but does not feel a need to know how it says it can completely ignore that second section of each chapter, and start reading again at the beginning of the next chapter.

Going further

There are good reasons to consider that a variable in a physical equation does not belong to the set  Equation of real numbers, nor does it belong to the set  Equation of rational numbers, nor even to the set  Equation of natural numbers. From that standpoint, the theory of incomplete measurements departs radically from earlier physical theories.

Finite cardinality of the set of measurement results

The theory of incomplete measurements generally considers that the results of physical measurements belong to a set of finite cardinality, which may not always be known precisely. For experiments such as checking the value of a dice, cardinality is easy to determine (six in that specific example). In other experiments such as measuring distance by counting steps, the cardinality is much higher, but is exact value is not known and largely irrelevant.

We can determine an upper bound to this cardinality. Nobody can reasonably have walked more than ten steps per second continuously for a duration of a thousand years, so that we can state with certainty that nobody ever measured a distance by counting more than 316,000 billion steps⁴. We can even state that this cardinality has some exact value, meaning that there once was an illustrious anonymous individual who happens to have counted the largest number of steps ever counted. Ultimately, it does not matter, since the precise value of this cardinality is of no consequence for most physical reasonings we can make about distance measurements.

Equality within conversion noise

As the graph shown above illustrates, the relationships between two distinct physical measurements are almost never an exact proportionality. We can still consider the quantities as practically proportional, and any deviation can be treated as a noise.

Let us call  Equation the standard duration of a day, as measured using a very precise clock, and expressed as a number of seconds. Let us call  Equation the astronomical duration of a day, derived from an observation of the apparent position of the Sun, and expressed as a number of days.

The simplified equation relating the two dimensions is a simple proportionality that we can translate in English as "The duration of a solar day is 86400 seconds":

 Equation

This simplified relationship is mathematically inexact.

A more accurate version must introduce a deviation relative to the simplified model which, unless we have more information, can only be considered as some random noise with values that change from one measurement to the next:



In some cases, it might be possible to model the noise itself, but this is not the most general scenario. Usually, we know nothing about this noise. The base measurement gives us incomplete information about the system. This is another meaning we can give to incomplete measurement.

The real difficulty in the following chapters will be to understand how to correctly reason about measurements that we know from experience to be incomplete in the sense given above.

What is a physical measurement?

Does the tree falling in the forest make noise if nobody hears it?

Koan Zen

Physicists are used to talking about physical quantities. In an experimental setup, these quantities are the result of measurements. We saw in the previous chapter a small fraction of what this implies regarding differences between physical and mathematical quantities. We still need to answer the question what is a physical measurement?

As indicated in the introduction of this book, we propose here an axiomatic definition of measurements in six bullet points. A physical measurement is:

a physical process

connecting an input and an output chosen ahead of time,

that can be repeated with an identical result,

that provides information solely about its input,

this information being given by an observable change in the output,

observation that can be given a symbolic interpretation.

This does not mean that we cannot find physical processes that do not have these properties, only that such processes are not considered as usable physical measurements.

Our species selects, ahead of time, through experimentation, which ones of the physical processes can be used as measurements, and which ones cannot. This is a choice we make, not a property of the universe.

A measurement is a physical process

Physics is not some kind of magic. If someone uses a voltmeter, that must be done within the framework of the normal laws of physics. No one needs to go outside the matrix or to look at the universe from outside.

A physical measurement is therefore, first and foremost, a perfectly normal physical process. When we measure a length using a pocket ruler, we do not perform any particularly exotic physical operation. We simply align the ruler with the object to measure, and then we count the tick marks between the two points we are interested in.

A physical measurement can easily be performed by a fully automated system, even one as simple as a bimetallic strip, meaning two pieces of metal that dilate differently. The bending of such strips is used to measure and regulate temperature in old-style thermostats.

Such examples demonstrate that, in my opinion, it is entirely superfluous to put any consciousness into the picture in order to explain measurements, unless we extend the notion of consciousness to two small chunks of metal, which I consider as unreasonable from a philosophical standpoint.

It might sound strange to talk about bringing consciousness into the picture simply to define how a physical measurement operates. However, this is a very common idea, in particular in the domain of quantum mechanics. For example, one of the widely accepted interpretations of quantum mechanics is called "Consciousness causes collapse", defined as follows on Wikipedia⁵:

In his treatise The Mathematical Foundations of Quantum Mechanics, John von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the Schrödinger equation (the universal wave function). He also described how measurement could cause a collapse of the wave function. This point of view was prominently expanded on by Eugene Wigner, who argued that human experimenter consciousness (or maybe even dog consciousness) was critical for the collapse, but he later abandoned this interpretation.

Television shows and books have actually been written on this basis:





Consciousness in quantum mechanics: Youtube thumbnail for a video from PBS

Consciousness and quantum mechanics: Cover of a book by Michel B. Mensky

The theory of incomplete measurement deals with entangled states almost like quantum mechanics from a mathematical standpoint. However, it does so without resorting to consciousness at all. This is a Good Thing™. Consciousness is extremely hard to define, in particular outside the context of biology, and arguably should not have any place in the foundational principles of any theory.

More generally, the theory presented here attempts to do away with the hard-to-understand and hotly debated postulates of quantum mechanics. It appears possible to precisely define physics without having to resort to obscure and often counter-intuitive first principles.

A measurement connects preselected inputs and outputs

For the measurement to be of any use, it must apply to an object that we can identify ahead of time, the input of the measurement, and it must provide a result through a change in its output, that also needs to be known ahead of time, such as a display device. In other words, one must be able to tell, prior to the measurement, what is being measured and how the result of the measurement will manifest.

For example, a voltmeter must measure potential differences between its input probes, and show the value of that potential difference on a screen, which may be an analogue galvanometer with a scale graduation, or some kind of numerical display activated by an analogue-to-digital converter. A pocket ruler measures a distance along its edge, and that distance can be measured by counting the tick marks: the tick marks are the output. A bimetallic strip measures temperature by bending, and we can roughly estimate the temperature by looking at how far the strip bends. The output does not need to be something we can see: some fire alarms measure carbon dioxide levels and manifest a high level using a loud sound.





A voltmeter or a carbon monoxide alarm are examples of measurement devices

Some physical processes do not have this property, but we cannot use them to perform a measurement. For example, burning a wooden log takes a specific input, the log, but the result, namely the heat and smoke, is not easily readable as giving information about the log. Consequently, we generally do not consider this process as a measurement of the log.

Conversely, throwing a dice has a well identified output, the number shown by counting dots on the face of the dice showing up after it stops spinning. However, a desirable property of dices is that the output number shows a very low correlation with anything. It is intended to be random.

These are some of the reasons why fire is not a good measurement device, and why we do not traditionally use dices to measure the position of the hand of the person throwing them, the temperature of the room, or anything else for that matter.





Dices and fires are examples of physical processes that do not measure their input

A measurement can be repeated

In order for a measurement to be reliable, it needs to give the same results in a predictable and repeatable manner.

This property is often understood as a repetition in time. Right after we measure 230V in an outlet using a voltmeter, and assuming we do not change anything, we expect to measure something close if we perform the measurement again. If we measure the width of a page as being 21 centimeters, and double check that measurement, we expect the same result.

In practice, other kinds of repetition matter, notably repetitions in space. For example, if the picture sensor in a digital camera is said to have a resolution of 20 megapixels, that means that same measurement device is repeated roughly 20 million times identically, 60 million times if you count the sensors for the three primary colors separately. This makes it possible to measure the position of a very large number of photons simultaneously, and therefore to reconstruct a very precise picture of the light field landing on the sensor. If the measurement was not almost identical from a photosite to the next, we wouldn’t be able to get the whole picture.



A digital picture sensor on a modern digital camera repeats the same measurement millions of times in space

Experiments can also be repeated along some other arbitrary measurement axis, in order to establish laws or correlations. In this case, the axis along which the measurement is repeated can be arbitrary. Pollsters, for example, ask the same question to a large group of people. It is well known that the measurement quality is better if the sample of individuals fairly represents the whole population. In other words, the basic measurement process, asking a question, must be repeated with a large number of people who must be sufficiently widespread across the spectrum of age or socio-economic characteristics. We are repeating the measurement along a mathematical axis chosen to make the measurement as significant as possible.

These remarks explain how the more general formulation of the theory of incomplete measurements cannot possibly be written as time-dependent, but instead using arbitrary variables, which can be used to elaborate the observed laws of physics.

A measurement only depends on its input

Another key feature of good measurements is that their result only depend on the measurement’s input, on what is being measured. Obviously, when you measure the length of an object, it is undesirable for the result to depend on its color or on air pressure.



A ruler should not depend on your age, on air pressure or on temperature

This is the reason why all measurement instruments are built to focus on a specific object. In the case of a voltmeter, the manufacturer will put extra work to minimize any undesired noise that would affect the input signal. The probes, in particular, will be carefully insulated, so as to only pick potential from their tip.

Even without taking operator security into account, if the voltmeter probes were two uninsulated wires, the chances of picking up electrical potential from the wrong place would increase, and this would make such voltmeters less useful.

Like the previous choices, this condition can be valid in some restricted physical domain. For example, measurements made with a tape ruler normally only depend on the length of the object being measured. However, if temperature is high enough, the ruler will dilate or even melt. Strictly speaking, the measurement result depends on ambient temperature. Nevertheless, under normal conditions, the temperature factor can be entirely ignored.

The example of a melting tape measure shows how a same physical process can continuously transition between a state where it is an acceptable measurement and another where it becomes a very bad one. Yet, we are talking about the same universe and the same physics, so it seems reasonable to be able to deal with both cases in the same theory.

One ambition of the theory of incomplete measurements is precisely to treat the two cases in the same conceptual framework. Naturally, the tape measure at low temperature will yield measurements that generate a rather classical Newtonian space. On the other hand, if the tape measure changes shape, becomes elastic or soft, then the measurement becomes partially or even totally inoperative. Information about the system becomes probabilist, and the description will gradually morph into something that might diverge a lot from classical physics.

The output of a measurement is directly observable

It is not sufficient for the measurement to be sensitive to its input. The measurement result must also manifest as a directly observable variation on the output. Observation here does not necessarily refer to a conscious observer, as we demonstrated above with the example of a bimetallic strip thermostat. it only means that there is a change in the output that can directly be detectable, including by another mechanism.

For measurement instruments intended for human use, the change in output can be visual, as is the case for a voltmeter, audible like for a fire alarm, or use other senses like the touch-based haptic feedback of modern smartwatches and smartphones. A measurement device can also generate a mechanical, electrical or digital signal intended for some other device, like the email your printer may send when it runs out of ink. Biological senses in living organisms communicate their measurement results using complex biochemical mechanisms such as variations of electrical potential along nerve paths. All these variations fit the same directly observable category as far as the theory of incomplete measurements is concerned, since in all cases that is true for the intended target of the signal.

We can easily build counterexamples. Consider a voltmeter with a broken display:



A voltmeter with a broken display is no longer a valid measurement device

Possibly, the rest of the voltmeter works perfectly. However, as long as the screen cannot display the result, the device is useless. This scenario can also happen gradually, for instance as you walk farther away from the device. This movement does not change the physics of the measurement process, but it may make the output information unreadable. It should be possible to express how the system evolves in the same way, irrespective of whether the display is readable or not.

There are more subtle counterexamples. If one projects an Earth-based laser beam on the Lunar surface, the Earth-Moon distance will amplify the tiniest movements of the laser. A displacement of one meter on the Lunar surface would correspond to a fraction of a degree, roughly one part in ten millions. Therefore, we built a very sensitive angle measurement device.



A very sensitive angle measurement that becomes unusable if you move it too quickly

However, it is relatively easy to move the laser with movements that are wide enough to move the spot on the Lunar surface faster than the speed of light. All you need is a rotation that is faster than about 200 degrees per second, in other words less than a turn per second. There is no contradiction with the theory of relativity, because no energy is transmitted between the successive spots illuminated by the laser.

However, as soon as the apparent movement on the surface of the moon is faster than light, it becomes impossible for anyone to follow it, irrespective of the physical device they use. Since you can no longer follow the beam, you cannot use that process to measure the original angle. We have another case of a continuous transition between valid and invalid measurements.

The output change has a symbolic interpretation

A last essential property of physical measurements is that they must generate information, in other words provide a symbolic interpretation for the changes that are observed on the output.

The symbolic value can be as simple as a binary signal, like the output of a fire alarm signaling a dangerously high temperature, or the bimetallic strip in a thermostat breaking an electrical circuit to signal a comfortably high temperature in a room.

Naturally, this can be more sophisticated, like a digital display or a computer screen, or one of these extraordinary measurement devices that you only find in laboratories:



Complex measurement instruments can have a sophisticated symbolic output

The most frequent case is that the symbolic interpretation leaves room to some uncertainty. For example, if a tape ruler has tick marks every millimeter, it may still be possible to visually estimate a length as being "between 15 and 16 millimeters, or alternatively, precisely 23 millimeters".

Scientific articles often summarize a set of results using "error bars", which would make it possible to visually estimate the probability distribution of the measured values. Another technique is to display the data using point clouds that make possible correlations visually apparent.

In other words, the existence of