Norman Wildberger: Insights into Mathematics

Mathematician Norman Wildberger presents an educational program for a wide audience as Insights into Mathematics connecting to the principles I have followed in Body and Soul and Leibniz World of Math.

A basic concern of Wildberger is how to cope with real numbers underlying analysis or calculus, geometry, algebra and topology, since they appear to require working with aspects of infinities coming with difficulties, which have never been properly resolved, like computing with decimal expansions with infinitely many decimals and no last decimal to start a multiplication. Or the idea of an infinity of real numbers beyond countability.

I share the critique of Wildberger but I take a step further towards a resolution in terms of finite precision computation, which can be seen to be the view of an applied mathematician or engineer. In practice decimal expansions with a finite number of decimals are enough to represent the world and every representation can be supplied with a measure of quality as a certain number of decimals as a certain finite precision. This offers a foundation of mathematics without infinities in the spirit of Aristotle with infinities as never attained  potentials representing "modes of speaking" rather than effective realities. 

In particular the central concept of "continuum" takes the form of a computational mesh of certain mesh size or finite precision. With this view a "continuum" has no smallest scale yet is finite and there is a hierarchy of continua with variable mesh size.    

The difficulty of infinities comes from an idea of exact physical laws and exact solutions to mathematical equations like $x^2=2$ expressed in terms of symbols like $\sqrt{2}$ and $\pi$. But this can be asking for too much, even if it is tempting, and so lead to complications which have to be hidden under the rug creating confusion for students.

A more down-to-earth approach is then to give up exactness and be happy with finite precision not asking for infinities.  

Philosophy of Statistical Mechanics?

Collapsed pillars of modern building.

Let us continue with the post Three Elephants of Modern Physics taking a closer look at one of them. 

We learn from Stanford Encyclopedia of Philosophy the following about Statistical Mechanics SM: 

• Statistical Mechanics is the third pillar of modern physics, next to quantum theory and relativity theory. 
• Its aim is to account for the macroscopic behaviour of physical systems in terms of dynamical laws governing the microscopic constituents of these systems and probabilistic assumptions.
• Philosophical discussions in statistical mechanics face an immediate difficulty because unlike other theories, statistical mechanics has not yet found a generally accepted theoretical framework or a canonical formalism. 
• For this reason, a review of the philosophy of SM cannot simply start with a statement of the theory’s basic principles and then move on to different interpretations of the theory.

This is not a very good start, but we continue learning: 

• Three broad theoretical umbrellas: “Boltzmannian SM” (BSM), “Boltzmann Equation” (BE), and “Gibbsian SM” (GSM).
• BSM enjoys great popularity in foundational debates due to its clear and intuitive theoretical structure. Nevertheless, BSM faces a number of problems and limitations
• There is no way around recognising that BSM is mostly used in foundational debates, but it is GSM that is the practitioner’s workhorse.
• So what we’re facing is a schism whereby the day-to-day work of physicists is in one framework and foundational accounts and explanations are given in another framework.
• This would not be worrisome if the frameworks were equivalent, or at least inter-translatable in relatively clear way...this is not the case.
• The crucial conceptual questions (concerning BE) at this point are: what exactly did Boltzmann prove with the H-theorem?

This is the status today of the third pillar of modern physics formed by Boltzmann 1866-1906 and Gibbs 1902 as still being without a generally accepted theoretical framework, despite 120 years of deep thinking by the sharpest brains of modern physics. 

Is this something to worry about? If one the pillars apparently is shaky, what about the remaining two pillars? Who cares?

Recall that SM was introduced to rationalise the 2nd Law of Thermodynamics stating irreversibility of macroscopic systems based on deterministic reversible exact microscopics. This challenge was taken up by Boltzmann facing the question: If all components of a system are reversible, how can it be that the system is irreversible? From where does the irreversibility come? The only way forward Boltzmann could find was to replace exact determinism of microscopics by randomness/statistics as a form of inexactness. 

 

In the modern digital world the inexactness can take the form of finite precision computation performed with a certain number of digits (e g single or double precision). Here the microscopics is deterministic up the point of keeping only a finite number of digits, which can have more or less severe consequences on  macroscopic reversibility. This idea is explored in Computational Thermodynamics offering a 2nd Law expressed in the physical quantities of kinetic energy, internal energy, work and turbulent dissipation without need to introduce any concept of entropy.

Replacing SM by precisely defined finite precision computation gives a more solid third pillar. But this is new and not easily embraced by analytical/theoretical mathematicians/physicists not used to think in terms of computation, with Stephen Wolfram as notable exception.  

PS1 To meet criticism that the Stosszahlansatz underlying the H-theorem stating that particles before collision are uncorrelated, simply assumes what has to proved (irreversibility), Boltzmann argued:

• But since this consideration has, apart from its tediousness, not the slightest difficulty, nor any special interest, and because the result is so simple that one might almost say it is self-evident I will only state this result.

Convincing?

PS2 Connecting to the previous post, recall that the era of quantum mechanics was initiated in 1900 by Planck introducing statistics of "energy quanta" inspired by Boltzmann's statistical mechanics, to explain observed atomic radiation spectra, opening the door to Born's statistical interpretation in 1927 of the Schrödinger wave function as the "probability of finding an electron" at some specific location in space and time, which is the text book wisdom still today. Thus the pillar of quantum mechanics is also weakened by statistics. The third pillar of relativity is free of statistics, but also of physics, and so altogether the three pillars offer a shaky foundation of modern physics.  Convinced? 

Challenges to the 2nd Law

The book Challenges to the Second Law of Thermodynamics by Capek and Sheenan starts out describing the status of this most fundamental law of physics as of 2005:

• For more than a century this field has lain fallow and beyond the pale of legitimate scientific inquiry due both to a dearth of scientific results and to a surfeit of peer pressure against such inquiry. 
• It is remarkable that 20th century physics, which embraced several radical paradigm shifts, was unwilling to wrestle with this remnant of 19th century physics, whose foundations were admittedly suspect and largely unmodified by the discoveries of the succeeding century. 
• This failure is due in part to the many strong imprimaturs placed on it by prominent scientists like Planck, Eddington, and Einstein. There grew around the second law a nearly inpenetrable mystique which only now is being pierced.

The book then continues to present 21 formulations of the 2nd Law followed by 20 versions of entropy and then proceeds to a large collection of challenges, which are all refuted, starting with this background:

• The 2nd Law has no general theoretical proof.
• Except perhaps for a dilute gas (Boltzmann's statistical mechanics), its absolute status rests squarely on empirical evidence.  

We learn that modern physics when confronted with the main unresolved problem of classical physics reacted by denial and oppression as cover up of a failure of monumental dimension. The roots of the present crisis of modern physics may hide here. 

Computational Thermodynamics seeks to demystify the 2nd Law as a result of finite precise computation  meeting systems developing increasing complexity like turbulence in slightly viscous flow.  

Physicists confronted with proving the 2nd Law. 

Is Radiative Heat Transfer a Resonance Phenomenon Between Bodies?

Computational BlackBody Radiation offers a new proof of the Planck-Stefan-Boltzmann Law PSB based on electromagnetic wave resonance under deterministic finite precision computation, taking the form

• $Q = \sigma (T_A^4 - T_B^4),$       (1)

where $Q$ is (normalised) radiative transfer of heat energy between two blackbodies A and B with temperatures $T_A$ and $T_B$ Kelvin, and $\sigma$ is the SB constant. If $T_A>T_B$ then the heat transfer is from A to B.

This is to be compared with the 1900 proof by Planck based on particle/quanta statistics typically expressed on the following form involving only one blackbody of temperature $T$:

• $Q = \sigma T^4.$       (2)

Comparing (1) and (2) we see that (1) expresses the radiative heat transfer between two bodies in resonance, while (2) is supposed to express the radiative heat transfer from one body independent of surrounding bodies and thus without resonance. 

In particular, (1) expresses that heat transfer from A requires the presence of a receptor B with lower temperature. On the other hand (2) appears to express that a body can radiate (spit out quanta/photons) without receptor, or assuming the presence of "empty space" at 0 Kelvin acting as receptor. In this case the body at higher temperature will spit most and so win the combat. 

This leads to the following questions: 

• Does radiative heat transfer from one body need a receptor at lower temperature?
• Does radiative heat transfer involve a resonance phenomenon between bodies? 

The new proof of PSB suggests that the answer is YES, while the standard proof suggests NO. What does physics and observation say? YES or No? Is radiative heat transfer carried by electromagnetic waves or particles/photons? An answer that it is both is no good. The questions concern basic physics and must be answered.

Compare with resonance between two tuning forks:

 To be compared with a particle model with both forks spitting out particles/phonons?

PS Read about Planck's struggle to prove (2) in Quantum Mechanics at the Crossroads starting with Schrödinger Against Particles and Quantum Jumps by M. Bitbol and continuing with Max Planck's Compromises on the Way to and from the Absolute by J. L. Heilbron.

Yes, it is not a good idea to resort to compromises in science, which is the essence of politics. Planck was not happy with his particle/quanta statistics and neither was Schrödinger, yet it has come to serve as a fundamental part of quantum mechanics following Born-Bohr. Real Quantum Mechanics in the spirit of Schrödinger presents a new realist deterministic approach based on waves instead of particle statistics.  

 

Similarity Between Fluid Turbulence and Radiative Energy Transfer

In the TNT Radio interview in the previous post I suggest a similarity between fluid turbulence and radiative heat transfer connecting to a phenomenon of high frequency "cut-off".  

Fluid motion transfers large scale ordered kinetic motion/energy over a cascade of successively smaller and smaller scales into a smallest scale depending on viscosity of unordered kinetic motion/energy as turbulence perceived as heat energy with the smallest scale representing the cut-off. The total energy transfer from large to smallest scale shows to depend mainly on the largest scales with thus little dependence on the smallest scales set by viscosity, thus with little dependence on viscosity once small enough. This is Kolmogorovs law of finite rate of turbulent dissipation.

Computational BlackBody Radiation describes radiative heat transfer between two bodies/oscillators $B1$ and $B2$ of temperature $T1$ and $T2$ with $T1>T2$ as an ordered resonance phenomenon with a frequency "cut off" increasing with temperature with energy balance for frequencies below the lower cut-off for $B2$, while frequencies above and below cut-off for $B1$ are absorbed by $B2$ in the form of unordered high-frequency oscillations perceived as heat energy by $B2$. The result is transfer of energy from the warmer body to the colder body. 

In both cases there is thus a split between ordered large scale motion and unordered small scale motion beyond cut-off perceived as heat energy. In both cases the small scale unordered motion perceived as heat energy is the consequence of an impossibility of sustained ordered motion: In fluid motion an impossibility of transferring energy to smaller scales in ordered fashion, and in radiative transfer an impossibility to balance frequencies above cut-off for the colder body, as an effect of finite precision. 

There is a connection to the 2nd law of thermodynamics with the transformation of large scale motion into heat energy as small scale unordered motion, is irreversible. In radiation it means that heat energy transfer is one-way from warm to cold.

Computational vs Statistical Physics

Statistical physics was created by Boltzmann (1844-1906) in an attempt to explain observed irreversibility of thermodynamic processes as a necessary evolution from more ordered/less probable to less ordered/more probable states in a microscopic particle-collision model of a gas. This was captured in Boltzmann's macroscopic equations derived from an assumption of molecular chaos (StossAnzahlAnsatz) stating that particle velocities prior to collision are uncorrelated. Boltzmann's H-theorem states that a gas left alone will approach a uniform rest state with a Maxwellian velocity distribution. 

Statistical physics is based on some assumption of statistical nature, such as molecular chaos, to be compared with computational physics where the evolution of a gas as a collection of colliding particles is simulated simply by computing the trajectories of all particles subject to collision with chaos/unordered motion as an emergent phenomenon without any assumption. 

One can argue that computational physics is real physics because particle trajectories subject to Newton's laws of motion = real physics,  are computed. On the other hand, statistical physics is not real physics in the sense that real physics cannot do statistics and decide to evolve according to an assumption of molecular chaos.  

On the other hand it is possible for human beings to do statistics by computing mean values and standard deviations in particle-collision models. 

Statistical physics was developed before the computer when computational physics could not deliver. Today with the computer computational physics can answer the questions posed in statistical physics, see Euler Right! showing physics emerging in a discrete finite element model by computation. 

PS The classical approach is to derive a continuum model in the form of a partial differential equation from a particle model. A computational model can then be derived by discretising the differential equation using the finite element method, which can be viewed as a form of particle method, in a way closing the circle with the particle model as the real model and the continuum model as a fictional model.  

The World as Analog Computation?!

                                     Augmented reality by digital simulation of analog reality.

Sabine Hossenfelder expresses on Backreaction:

• No, we probably don’t live in a computer simulation!

as a reaction to the Simulation Hypothesis:

• According to Nick Bostrom of the Future of Humanity Institute, it is likely that we live in a computer simulation. And one of our biggest existential risks is that the superintelligence running our simulation shuts it down. 

Sabine starts her discussion with

• First, to get it out of the way, there’s a trivial way in which the simulation hypothesis is correct: You could just interpret the presently accepted theories to mean that our universe computes the laws of nature. Then it’s tautologically true that we live in a computer simulation. It’s also a meaningless statement.

And she gets support from Lubos Motl stating:

• Hossenfelder sensibly critical of our "simulated" world.

Is it then meaningless to view the World as the result of analog computation? I don't think so, with arguments presented at The World as Computation.

The main idea is that if the world is an analog computation, then it may well be possible to simulate the world by digital computation, and if that is possible we may perhaps better understand and control the world to our benefit. 

And the other way: If the world is not analog computation, then chances for simulation by digital computation are slim, and then what? 

Recall that the basic principle of classical rational deterministic physics is to view the evolution of the world as the result of sequential analog computation as transformations of inputs into outputs according to laws of physics. In short: The World as a Clock according to Laplace. Or more precisely, The World as a Clock of Infinite Precision, since the laws of physics are supposed to be satisfied exactly.

Sabine's standpoint is logical as an expression of the complete collapse of classical rational deterministic physics in the spirit of Laplace into the irrational quantum world of modern physics without determinism, for which the idea of input-output computation no longer is valid. The non-computational aspect of quantum physics comes out in the multi-dimensional form of Schrödinger's equation, which makes it impossible to solve by digital computation. 

But the complete collapse of rationality/determinism in modern physics is a serious blow to physics as science and I have sought a way to avoid collapse by modifying Laplace's dictum into The World as a Clock of Finite Precision and by giving Schrödinger's equation an alternative three-dimensional form as realQM, both inviting to simulation by digital computation.  

Sabine's post expresses the paralysis created by the Copenhagen Interpretation of quantum mechanics presenting a world which is not understandable and therefore not computable and therefore not understandable...a world view which we do not have to accept because there are alternatives to explore...

There is no evidence that we live in a computer simulation (because the world is not digital), but there is much evidence that an analog world can be simulated by digital computation, and that opens endless possibilities of enhancing the analog world by simulated worlds as augmented reality...

Is the Quantum World Really Inexplicable in Classical Terms?

Peter Holland describes in the opening statement of The Quantum Theory of Motion the state of the art of modern physics in the form of quantum mechanics, as follows:

• The quantum world is inexplicable in classical terms.
• The predictions pertaining to the interaction of matter and light embodied in Newton's laws of motion  and Maxwell's equations governing the propagation of electromagnetic fields, are in flat contradiction with the experimental facts at the microscopic scale.
• A key feature of quantum effects is their apparent indeterminism, that individual atomic events are unpredictable, uncontrollable and literally seem to have no cause.
• Regularities emerge onlywhen one considers a large ensemble of such events.
• This indeed is generally considered to constitute the heart of the conceptual problems posed by quantum phenomena, necessitating a fundamental revision of the deterministic classical world view.

No doubt this describes the predicament of modern physics and it is a sad story: It is nothing but a total collapse of rationality, and as far as I can understand, there are no compelling reasons to give up the core principles of classical continuum physics so well expressed in Maxwell's equations. 

If classical continuum physics is modified just a little by adding a new element of finite precision computation, then the apparent contradiction of the ultra-violet catastrophe of black-body radiation as the root of "quantization", can be circled and rationality maintained.  You can find these my arguments by browsing the labels to this post and the web sites Computational Black Body Radiation and The World as Computation with further development in the book Real Quantum Mechanics.

And so No, it may not be necessary to give up the deterministic classical world view when doing atom physics, the view which gave us Maxwell's equations and opened a new world of electro-magnetics connecting to atoms. It may suffice to modify the deterministic classical view just a little bit without losing anything to make it work also for atom physics.

After all, what can be more deterministic than the ground state of a Hydrogen atom?

Of course, this is not a message that is welcomed by physicists, who have been locked since 90 years into finding evidence that quantum mechanics is inexplicable, by inventing contradictions of concepts without physical reality. The root to such contradictions (like wave-particle duality) is the linear multi-d Schrödinger equation which is picked from the air as a formality without physics content, but just because of that being inexplicable. To advance, it seems that a new Schrödinger equation with physical meaning should be derived...

The question is how to generalise Schrödinger's equation for the Hydrogen atom with one electron, which works fine and can be understood, to Helium with two electrons and so on...The question is then how the two electrons of Helium find co-existence around the kernel. In Real Quantum Mechanics they split 3d space without overlap....like East and West of global politics or Germany...

Physics as Massive Repeated Failure

                                             All human beings are different, and the same.

The World as Computation presents an approach to physics based on a concept of finite precision computation, which opens to digital simulation of physical processes viewed as analog computation, in general in the form of time-stepping relaxation towards satisfaction of certain equations expressing conservation of mass, momentum, energy et cet.

In this world of finite precision computation, turbulence as a main mystery of classical mechanics, can be understood as the result of a computational relaxation process aimed at producing an exact solution of Navier-Stokes equations of fluid mechanics, which however misses this goal because every exact solution of Navier-Stokes with small viscosity is unstable and thus is not reachable by relaxation.

Turbulent flow as fluctuating, complex and locally unpredictable flow, or physical reality in all its complexity, thus can be seen as the result of repeated failure to compute an exact solution:

• The ideal (exact solution) is unstable and thus cannot take real form. 
• What can take real form is an approximate solution resulting from a finite precision computational process aimed at producing the exact solution.  

The fact that all snow flakes, leaves and human beings are different can thus be explained by the fact that there are so many ways to fail (more or less), while there is only one road to complete success, but a road that cannot be followed because it is too narrow.  

The complexity of the world can thus can (maybe) be viewed as a result of massive repeated failure, rather than the creation of a creator of infinite fantasy.

It is maybe illuminating to realise that (according to Kevin Kelly in What Technology Wants): The closer a face hews to an ideal average human face, the more attractive we find it.

It is natural to compare with machine made copies of a mechanical part of a car or a microprocessor, which are constructed with the explicit goal to make them all identical and interchangeable, a goal that is less obvious when nature is constructing (although ans to us seem pretty identical and interchangeable).   

Multiscale Finite Element Method as Informational Structural Realism

Luciano Floridis concludes in The Philosophy of Information that the world on microscopic scales cannot be neither discrete nor continuous, since each of these standpoints is contradictory, but can maybe instead be described by Informational Structural Realism (ISR) as scale dependent structural description leaving out the true nature of the elements forming the structure. ISR can be viewed as synthesis of the discrete and continuous without internal contradiction and thus potentially as a useful world view.

This is nothing but the finite element method in multi-scale form, originally developed in structural mechanics, as scale dependent discretization of continuous differential equations, where a true physical realisation of the finite elements is not possible nor necessary.

This connects to my attempt to describe a complex world including turbulence and quantum mechanics as analog finite precision computation simulated by digital finite precision computation, where the flow of information under computation represents ISR and the true physical nature of the analog computation is unknowable, but irrelevant.

Why is then both discrete and continuous physics impossible? Because both requires infinite resolution: a discrete point particle or discontinuity has zero size and a continuum has no smallest size. Thus both discrete and continuous physics requires infinitely small resolution and thus an infinite amount of information on any scale. If you don´t think this is asking for too much, then you should reconsider your notion of the infinite.

Quantum Information Can Be Lost

Stephen Hawking claimed in lecture at KTH in Stockholm last week (watch the lecture here and check this announcement) that he had solved the "black hole information problem":

• The information is not stored in the interior of the black hole as one might expect, but in its boundary — the event horizon,” he said. Working with Cambridge Professor Malcolm Perry (who spoke afterward) and Harvard Professor Andrew Stromberg, Hawking formulated the idea hat information is stored in the form of what are known as super translations.

The problem arises because quantum mechanics is viewed to be reversible, because the mathematical equations supposedly describing atomic physics formally are time reversible: a solution proceeding in forward time from an initial to a final state, can also be viewed as a solution in backward time from the earlier final state to the initial state. The information encoded in the initial state can thus, according to this formal argument, be recovered and thus is never lost. On the other hand a black hole is supposed to swallow and completely destroy anything it reaches and thus it appears that a black hole violates the postulated time reversibility of quantum mechanics and non-destruction of information.

Hawking's solution to this apparent paradox, is to claim that after all a black hole does not destroy information completely but "stores it on the boundary of the event horizon". Hawking thus "solves" the paradox by maintaining non-destruction of information and giving up complete black hole destruction of information.

The question Hawking seeks to answer is the same as the fundamental problem of classical physics which triggered the development of modern physics in the late 19th century with Boltzmann's "proof" of the 2nd law of thermodynamics: Newton's equations describing thermodynamics are formally reversible, but the 2nd law of thermodynamics states that real physics is not always reversible: Information can be inevitably lost as a system evolves towards thermodynamical equilibrium and then cannot be recovered. Time has a direction forward and cannot be reversed. 

Boltzmann's "proof" was based an argument that things that do happen do that because they are "more probable" than things which do not happen. This deep insight opened the new physics of statistical mechanics from which quantum borrowed its statistical interpretation.

I have presented a different new resolution of the apparent paradox of irrreversible macrophysics based on reversible microphysics by viewing physics as analog computation with finite precision, on both macro- and microscales. A spin-off of this idea is a new resolution of d'Alemberts's paradox and a new theory of flight to be published shortly.

The basic idea here is thus to replace the formal infinite precision of both classical and quantum mechanics, which leads to paradoxes without satisfactory solution, with realistic finite precision which allows the paradoxes to be resolved in a natural way without resort to unphysical statistics. See the listed categories for lots of information about this novel idea.

The result is that reversible infinite precision quantum mechanics is fiction without physical realization, and that irreversible finite precision quantum mechanics can be real physics and in this world of real physics information is irreversibly lost all the time even in the atomic world. Hawking's resolution is not convincing.

Here is the key observation explaining the occurrence of irreversibility in formally reversible systems modeled by formally non-dissipative partial differential equations such as the Euler equations for inviscid macroscopic fluid flow and the Schrödinger equations for atomic physics:

Smooth solutions are strong solutions in the sense of satisfying the equations pointwise with vanishing residual and as such are non-dissipative and reversible.  But smooth solutions make break down into weak turbulent solutions, which are only solutions in weak approximate sense with pointwise large residuals and these solutions are dissipative and thus irreversible.

An atom can thus remain in a stable ground state over time corresponding to a smooth reversible non-dissipative solution, while an atom in an excited state may return to the ground state as a non-smooth solution under dissipation of energy in an irreversible process.      

Modern Physics: Meaningless Sacrifice of Causality, Rationality and Reality?

Hermann von Helmholtz in Conservation of Force (1862-63): Reason we call that faculty innate in us of discovering laws and applying them with thought...there is a kind, I might almost say, of artistic satisfaction,when we are able to survey the enormous wealth of Nature as a regular-ordered whole--a cosmos, an image of the logical thought of our mind.

Modern physics in the form of relativity theory and quantum mechanics was born from a perceived impossibility of solving the following "problems" using methods of classical deterministic continuum physics:

1. Second law of thermodynamics (irreversibility in formally reversible systems).
2. Blackbody radiation (including avoidance of an ultraviolet catastrophe).
3. Existence of a unique aether medium for propagation of electromagnetic waves. 

Boltzmann "solved" 1. by introducing statistical physics, thus giving up classical determinism or causality.

Planck "solved" 2. introducing a smallest quantum of energy, thus giving up the classical continuum of rational mechanics.

Einstein "solved" 3. by freeing electromagnetics from an aether, thus giving up classical coordinates of space and time describing reality. 

In each case the sacrifice of pillars classical physics was monumental and the grandness of the sacrifice was taken as a sign that it was inevitable and thus justified: No physicist would be willing the give up so much, unless it was absolutely necessary, as expressed by Planck excusing his introducing of the quantum:

• ...the whole procedure was an act of despair because a theoretical interpretation had to be found at any price, no matter how high that might be...

But if one day it shows that 1-3 in fact can be handled using a mild extension of classical deterministic continuum physics, then the monumental sacrifices would be unnecessary and then without rationale.

And yes, it may be that such a mild extension is possible in the form of finite precision computation exposed on The World as Computation.

This connects to Helmholtz' approach to 1. with heat as partly "incalculable" or "disordered" energy as energy with limited capability of being transformed to other forms of ("calculable") energy. This brings us back to the peak of classical physics represented by the mechanism of Helmholtz:

• Natural phenomena should be traced back to the movements of material objects which possess inalterable motive forces that are dependent only on spatial relations.
• Science, the goal of which is the comprehension of nature, must begin with the presupposition of its comprehensibility  and proceed in accordance with this assumption until, perhaps, it is forced by irrefutable facts to recognise limits beyond it may not go.

It thus appears to be possible to handle 1. and 2. by classical mechanism modified by finite precision computation. Further, 3. may be handled as suggested by the British physicist Ebenezer Cunningham (1881-1977) by viewing an aether is an immaterial space-time coordinate systems with the observed non-existence of a unique aether medium simply as an expression of the possibility of choosing many immaterial aethers/coordinate systems.

It thus may be that the monumental sacrifices made by modern physicists in order to cope with 1-3, are not necessary, and as such represent human stupidity, rather than heroic victory of the power of the human mind as official truth of modern physics propagated by modern physicists. 

Physics as Analog Finite Precision Computation vs Physics as Statistics

I am exploring an approach to physics as "analog finite precision computation" to be be compared with classical physics as "analog infinite precision physics" and modern physics as "physics of dice games" or "statistical physics".

The step from classical to modern physics was forced upon physicists starting in the mid 19th century when it became clear that the 2nd law of thermodynamics could not be found in classical infinite precision physics of irreversible systems. The way to achieve irreversibility was to assume that atoms play dice games with the outcome of a throw of a dice inherently irreversible: To "unthrow" a dice was (correctly) understood to be impossible and thus irreversibility was introduced and the paralysis of reversible classical physics was broken. So far so good.

But the fix came with severe side effects as real physics independent of human observation was replaced by statistical physics representing "human understanding", as if the world goes around just because some physicist  is making observations and claim them to be understandable. Einstein and Schrödinger could never be convinced that atoms play dice, despite major pressure from the physics community.

The unfortunate result of this collapse of rationality of deterministic physics, has led modern physics into wildly speculative physics of strings and multiverse, which nobody can understand.

But there is a milder way of introducing irreversibility into classical reversible physics, and that is to view physics as analog computation with finite precision instead of infinite precision.

This connects directly to a computer operating with finite decimal expansion of real numbers as a necessary restriction of infinite decimal expansion, in order to allow computations to be performed in finite time: In order to make the world go around, and it does go around, and thus not come into a halt, physical processes cannot be realised with infinite precision and thus finite precision computation is a must in a world that goes around. It is thus necessary, but it is also sufficient to introduce irreversibility into classical reversible physics.

Finite precision computation thus solves the main problem which motivated the introduction of statistical physics, but in a much more gentle way and without the severe side effects of full-blown statistics based on dice games.

Finite precision computational physics is represented by the modern computer, while statistical physics would correspond to a "dice computer" throwing a dice in every step of decision, just like the "dice man" created by the pseudonym Luke Rhinehart. The life of the "dice man" turned into misery, which can be compared with (reasonably) successful ordinary (reasonably controlled) life under finite precision, without a dice but with constant pressure to go onto the next day.

So if you want to compare finite precision analog physics to modern statistical physics, make the thought experiment of comparing your usual finite precision computer, which you use to your advantage, to a "dice computer" which would be completely unpredicatable. This is the comparison between an experienced computer wiz often getting reliable results, to a totally inexperienced user pushing the keys randomely and getting garbage.

Or make the comparison of getting married to a person which follows a principle of "finite precision" to a person like the "dice man" who is completely unpredictable. What would you prefer?

Few ideas can change your view in the same way as "physics as analog finite precision computation". Try it!

Crisis in Physics vs Computational Physics

The May14 issue of Scientific American asks the following questions:

• Does Physics Have a Problem?
• Crisis in Physics?

These questions naturally present themselves because modern theoretical physicists have driven themselves to search for the truth on scales which are either too small (string theory) or too big (cosmology) to be assessed experimentally. But theory without experiment may well be empty theory and that may be the meaning of the crisis. Of course, advocates of string theory like Lubos, forcefully denies that there is a crisis in physics. But there are other blog voices…and leading physicists show little hope..

But modern physicists have a new tool to use and that is computational physics, which offers an experimental laboratory without the scale limits of a physical laboratory. 

Computational physics needs computable models, but both quantum mechanics and general relativity are based on models which are not computable, and so there is a lot of work to be done. The question is if modern theoretical physicists have the right training to do this work.     

Quantum Theory: Flight from Realism

The book Quantum Theory and the Flight from Realism by Christopher Norris is introduced by:

• Norris examines the premises of orthodox quantum theory as formulated most influentially by Bohr and Heisenberg….as requiring a drastic revision of principles which had hitherto defined the very nature of scientific method, casual explanation and rational enquiry.
• Putting the case for a realist approach which adheres to well-tried scientific principles of casual reasoning and interference to the best explanation, Norris clarifies the debate…

Norris continues:  

• In this book I examine various aspects of the near century-lonh debate concerning the conceptual foundation of quantum mechanics (QM) and the problems it has posed for physicists and philosophers from Einstein to the present. They include the issue of wave-particle dualism; the uncertainty attaching to measurements of particle location or momentum, the (supposedly) observer-induced "collapse of the wave-function"; and the evidence of remote superluminal interaction between widely separated particles.
• It is important to grasp exactly how the problems arose and exactly why - on what scientific or philosophical grounds - any alternative (realist) contrual should have been so often and routinely ruled out as a matter of orthodox QM wisdom. 

This is an important book with the important mission of bringing realism back to physics after a century  of anti-realist confusion ultimately corrupting all of science and with the adoption of climate alarmism by the American Physical Society as the tragic anti-realist irrational expression. 

Today digital computation opens new roads to bring realism back to physics viewed as analog computation.  See The World as Computation.

Schrödinger's Equation: Smoothed Particle Dynamics

Eigenfunctions of the Hamiltonian for the Hydrogen atom with eigenvalues representing the sum of kinetic and potential energies, with Schrödinger's equation as a smoothed version of the particle dynamics of a harmonic oscillator.  

This is continuation of the previous post How to Make Schrödinger's Equation Physically Meaningful + Computable. Consider the basic case of the Hydrogen atom with one electron (normalized to unit mass and charge):

• $ih\dot\psi + H\psi =0$,
• $H\psi =\frac{h^2}{2}\Delta\psi +\frac{1}{\vert x\vert}\psi$,

where $\psi (x,t)$ the complex-valued wave function depending on coordinates of space $x$ and time $t$ with the dot denoting differentiation with respect to time, $H$ is the Hamiltonian operator and $h$ Planck's constant.

In terms of the real part $\phi$ and imaginary part $\chi$ of $\psi =\phi +i\chi$, Schrödinger's equation takes the system form

1. $h\dot\phi +H\chi =0$,
2. $h\dot\chi - H\phi =0$.

If $\phi_E(x)$ is an eigenfunction of the Hamiltonian satisfying $H\phi_E =E\phi_E$ with $E$ the corresponding eigenvalue, then the solution can be represented as

• $\phi (x,t)=\cos(\omega t)\phi_E(x)$,     $\chi (x,t)=\sin(\omega t)\phi_E(x)$, 

with $h\omega =E$, which expresses a periodic exchange between the two real-valued wave functions $\phi$ and $\chi$ mediated by the Hamiltonian $H$.

We can see 1- 2 as an analog of the equation for a harmonic oscillator $\ddot u+\omega^2u=0$ written in system form (with $h=1$)

• $\dot\phi  + \omega\chi =0$
• $\dot\chi  - \omega \phi = 0$,

where $\phi =\dot u$ and $\chi =\omega u$, with solution

• $\phi (x,t)=\cos(\omega t)$,     $\chi (x,t)=\sin(\omega t)$.  

Here the velocity $\phi =\dot u$ connects to kinetic energy $\phi^2 =\dot u^2$ and $\chi =\omega u$ to potential energy $\chi^2 =\omega^2u^2$ and the dynamics of the harmonic oscillation consists of periodic transfer back and forth between kinetic and potential energy with their sum being constant.

Returning now to the Hydrogen atom, we obtain multiplying 1 by $\phi$ and 2 by $\chi$ and integrating in space the following the energy balance

• $h\frac{d}{2dt}\int\phi^2\, dx + \int \phi H\chi \, dx =0$
• $h\frac{d}{2dt}\int\chi^2\, dx - \int \chi H\phi\, dx =0$,    

where 

• $ \int \phi H\chi \, dx = \int \chi H\phi\, dx =\frac{h^2}{2}\int\nabla\phi\cdot\nabla\chi\, dx +\int\frac{\phi\chi}{\vert x\vert}\, dx$,

which shows upon summation (by the symmetry of $H$) that

• $\frac{d}{2dt}\int\phi^2\, dx =\frac{d}{2dt}\int\chi^2\, dx =0$, 

which allows normalization to  

• $\int\phi^2\, dx = \int\chi^2\, dx = \frac{1}{2}$,
• $\int\vert\psi\vert^2\, dx = 1$, for all time. 

Further, multiplying 1 by $\dot\chi$ and 2 by $\dot\phi$ and subtracting the resulting equations shows that

• $\int (\phi H\phi + \chi H\chi)\, dx$ is constant in time. 

We can now summarize as follows:

A. We see that the solution pair $(\phi ,\chi )$ of 1 - 2 as the real and imaginary part of Schrödinger's wave function $\phi$, represents a periodic exchange mediated by the Hamiltonian $H$ with balancing associated total energies 

• $\int \phi H\phi (x,t)\, dx = \frac{h^2}{2}\int\vert\nabla\phi (x,t)\vert^2dx +\int\frac{\phi^2(x,t)}{\vert x\vert}\, dx$,
• $\int \chi H\chi (x,t)\, dx = \frac{h^2}{2}\int\vert\nabla\chi (x,t)\vert^2dx +\int\frac{\chi^2(x,t)}{\vert x\vert}\, dx$    

as the sum of kinetic and potential energies.

B. We see that Schrödinger's equation for the Hydrogen atom can be viewed as a smoothed version of a harmonic oscillator with the smoothing effectuated by the Laplacian and with $h$ acting as a smoothing parameter.

C. We see that the system form 1- 2 combines the spatial eigenfunction $\phi_E$ with a periodic time dependence without introducing energy beyond the kinetic and potential energies defined by the Hamiltonian, thus associating these energies to frequency as the essence of quantum mechanics.

D. We see that quantum mechanics and Schrödinger's equation can be given an interpretation which closely connects to classical mechanics, as smoothed particle mechanics, which avoids the common mystifications of particle-wave duality, complementarity, wave function collapse and statistics forced by insistence to use a multidimensional wave function defying a direct physical meaning.

Extension to several electrons can be naturally be made following the idea of smoothed particle dynamics. For details see Many-Minds Quantum Mechanics.

New Theory of Flight Presented to the World

Simulation movie of airflow around a jumbojet in landing configuration at large angle of attack.

The revised version of New Theory of Flight has now been submitted to Journal of Mathematical Fluid Mechanics for expected swift publication.

This article together with my former students Johan Hoffman and Johan Jansson represents the summit of my scientific career as a combination of mathematical analysis and computation. The article asks for a major revision of text book aerodynamics and opens new roads to aerodynamic design. And it is not a joke…Finally, The Secret of Flight can be revealed to humanity.

Once the article has appeared in JMFM the new theory will be launched in a press release to media. Stay tuned….

Here is the Summary of article:

• The new theory shows that the miracle of flight is made possible by the combined effects of (i) incompressibility, (ii) slip boundary condition and (iii) 3d rotational slip separation, creating a flow around a wing which can be described as (iv) potential flow modified by 3d rotational separation. 
• The basic novelty of the theory is expressed in (iii) as a fundamental 3d flow phenomenon only recently discovered by advanced computation and analyzed mathematically, and thus is not present in the classical theory. 
• Finally, (iv) can be viewed as a realization in our computer age of Euler’s original dream to in his equations capture an unified theory of fluid flow. 
• The crucial conditions of (ii) a slip boundary condition and (iii) 3d rotational slip separation show to be safely satisfied by incompressible flow if the Reynolds number is larger than 106. For lower Reynolds numbers the new theory suggests analysis and design with focus on maintaining (ii) and (iii).

Fluid Turbulence vs Quantum Electrodynamics

Horace Lamb (1849 - 1934) author of the classic text Hydrodynamics: It is asserted that the velocity of a body not acted on by any force will be constant in magnitude and direction, whereas the only means of ascertaining whether a body is, or is not, free from the action of force is by observing whether its velocity is constant.

There is famous quote by the British applied mathematician Horace Lamb summarizing the state of classical fluid mechanics and the new quantum mechanics in 1932 as follows:

• I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic.

Concerning the turbulent motion of fluids I am happy to report that this matter is now largely resolved by computation, as made clear in the article New Theory of Flight soon to be delivered for publication in Journal of Mathematical Fluid Mechanics, with lots of supplementary material on The Secret of Flight. This gives good hope that the other problem of quantum electrodynamics can likewise be unlocked by viewing  The World  as Computation:

• In a time of turbulence and change, it is more true than ever that knowledge is power. (JFK)

Blackbody as Linear High Gain Amplifier

                                      A blackbody acts as a high gain linear (black) amplifier.

The analysis on Computational Blackbody Radiation (with book) shows that a radiating body can be seen as a linear high gain amplifier with a high-frequency cut-off scaling with noise temperature, modeled by a wave equation with small damping, which after Fourier decomposition in space takes the form of a damped linear oscillator for each wave frequency $\nu$:

• $\ddot u_\nu +\nu^2u_\nu - \gamma\dddot u_\nu = f_\nu$,

where $u_\nu(t)$ is oscillator amplitude and $f_\nu (t)$ signal amplitude of wave frequency $\nu$ with $t$ time, the dot indicates differentiation with respect to $t$, and $\gamma$ is a small constant satisfying $\gamma\nu^2 << 1$ and the frequency is subject to a cut-off of the form $\nu < \frac{T_\nu}{h}$, where 

• $T_\nu =\overline{\dot u_\nu^2}\equiv\int_I \dot u_\nu^2(t)\, dt$, 

is the (noise) temperature of frequency of $\nu$, $I$ a unit time interval and $h$ is a constant representing a level of finite precision.  

The analysis shows under an assumption of near resonance, the following basic relation in stationary state:

•   $\gamma\overline{\ddot u_\nu^2} \approx \overline{f_\nu^2}$,

as a consequence of small damping guiding $u_\nu (t)$ so that  $\dot u_\nu(t)$ is out of phase with $f_\nu(t)$ and thus "pumps" the system little. The result is that the signal $f_\nu (t)$ is balanced to major part by the oscillator

• $\ddot u_\nu +\nu^2u_\nu$, 

 and to minor part by the damping

• $ - \gamma\dddot u_\nu$,

because 

• $\gamma^2\overline{\dddot u_\nu^2} \approx \gamma\nu^2 \gamma\overline{\ddot u_\nu^2}\approx\gamma\nu^2\overline{f_\nu^2} <<\overline{f_\nu^2}$. 

This means that the blackbody can be viewed to act as an amplifier radiating the signal $f_\nu$ under the small input $-\gamma \dddot u_\nu$, thus with a high gain. The high frequency cut-off then gives a requirement on the temperature $T_\nu$, referred to as noise temperature, to achieve high gain.   

Konrad Zuse on The World as Clock with Finite Precision

                                      Konrad Zuse pondering if physics is digital? If so, time has a direction.

I have recently discovered that the idea which I have pursued in different pieces of work, the idea to view physics as a Clock with Finite Precision, was expressed in 1969 by the German computer pioneer (and civil engineer) Konrad Zuse (who constructed the first working computer named Z3 in 1941) in the remarkable article Calculating Space on Digital Physics, starting out with the following experience which I share with Zuse:

• The work which follows stands somewhat outside the presently accepted method of approach, and it was for this reason rather difficult to find a publisher ready to undertake publication of such a work. 

Here are some highlights from Calculating Space, connecting in particular to my book The Clock and the Arrow: A Brief Theory of Time about the 2nd law of thermodynamics and the direction of time:

• It is obvious to us today that numerical calculations can be successfully employed in order to illuminate physical relationships. 
• To what extent are the realizations gained from study of calculable solutions useful when applied directly to the physical models? Is nature digital, analog or hybrid? And is there essentially any justification for asking such a question? 
• The examples of digitalization of fields and particles which have been preented are in their present unfinished form still far removed from being able to serve in the formulation of physical rules. Nevertheless, they give a rough impression of the possibilities for using the tools of the automaton theory to answer physical questions.
• The question to what extent it is possible to consider the entire universe as a finite automaton depends on the assumption which we make in relation to its dimensions. 
• An infinite information content is required for an unlimited spacetime element. It is practically impossible to simulate such a model with computers because of the necessity of infinite number of places required. 
• The sources of error are correspondingly great in the extremely large number of collisions between gas molecules, and these errors quickly lead to deviations from theoretical processes.
• This means that the better the causality rule is approximated in the reverse time direction, the more calculations we must be prepared to carry out in our model. This leads to the result that simulations of universal systems with causality functioning in both time directions belong to the category of “unsolvable” problems.
• Of course, it can be said that this is true only for calculating simulative models. But this result should encourage us to reconsider the matter. Are we justified in assuming a model of nature for which no calculable simulation is possible? 
• From this point of view, it appears that the frequently advanced argument of determination in both time directions should be fundamentally reexamined.
• But if Zuse didn’t hit upon the concept of universal computation (as Turing did), he was interested in another very deep question, the question of the nature of nature: “Is nature digital?” He tended toward an affirmative answer. (from Afterword)