Physics Collectivism vs Mathematics Individualism

It is sometimes claimed that physics is a collective effort, while mathematics is rather a collection of individual efforts. Let us test this hypothesis by analysing the use of the first person plural pronoun "we".  

Let us start with physics noting how Andrew Strominger as a modern physicist of reputation invited by Lex Fridman to an interview entitled Black Holes, Quantum Gravity, and Theoretical Physics responds the following question by Lex:

• What is the effort of quantum gravity (to reconcile the standard model and general theory of relativity as the main open problem of modern physics)?

Andrew tells Lex:

• The one fully consistent model we have is string theory.
• We do not know if in any sense string theory describes the world.

We see that Andrew uses the pronoun "we" to send a message that modern physics is a collective effort, or more precisely a selective collective effort performed by an inner circle of leading modern physicists. For someone outside this group (which is not well defined) to say "we" in this context would be pretentious. By using an exclusive "we" Andrew signals that he belongs to the inner circle. 

There is another inclusive "we" often used in mathematical reasoning starting with a phrase like "We assume that x is a positive real number". Here "we" includes the reader or just anybody without asking for any qualification.

We know that the mathematics of Calculus was invented (in competition) by Leibniz and Newton, who both could say with confidence that "I invented Calculus", but never "We invented Calculus". 

More generally, a mathematician would rarely use "we" in the exclusive inner circle meaning, since the inner circle typically would have only one member. Mathematical theorems or theories mostly carry just one name like Brouwer's Fixed Point Theorem or Galois Theory. 

Maybe this indicates a difference between (modern) physics and mathematics, with physics being shared by physicists as some form of common reality or fiction, while mathematics is a human construct created by individuals like pieces of literature or art as individual efforts. 

But it may be tempting for both a physicist and mathematician to use "we" to signal membership in some group of prestige behind some theorem/theory. You can make your own observations...

On Wolfram's Concept of Computational Irreducibility

This is a continuation of the previous post.

Computational Irreducibility CI is used by Stephen Wolfram to describe that prediction of the state of a system requires full resolution of the evolution of the system by computation; to predict the outcome of tossing a coin by computation requires precise resolution of the motion of the coin from toss to landing. Wolfram seeks an explanation of the 2nd Law in terms of CI as an expression of Computational Impossibility. 

Wolfram's agenda (initiated when he was 12) connects to mine (since 90s), both serving to give an explanation of the 2nd Law or irreversibility or direction of time, based on computation, although fundamentally different in details. 

As a key example I consider turbulent fluid flow which is (mean-value) computable forward in time but not backward in time, thus computationally reducible forward in time but computationally irreducible backward in time, which gives time a direction. More precisely, consider an airplane wing moving through air from left to right meeting still air and leaving a wake of turbulent air. Drag and lift of the wing as force mean values are computable without resolving the flow to physical scale, thus expressing computational reducibility as shown in the book Computational Turbulent Incompressible Flow. 

However, reversing the flow with thus incoming turbulent flow from the right, will not give back the still air to the left, but instead a turbulent wake. To return to still air at the left after reversing the flow, would require solving an exponentially unstable problem without any cancellation (in associated linearised dual problem), thus suffering from CI.

The flow of air around a wing is thus an example of irreversible flow, with the irreversibility resulting from lack of cancellation in laminar flow, as carefully explained in the book. The forward problem is computationally reducible, while the backward problem is computationally irreducible = Irreversibility = 2nd Law. 

Turbulent dissipation of large scale kinetic energy into small scale heat energy can serve as a prime example of an irreversible process, which is open to study by computational solution of the Euler/Navier-Stokes equations, see post on Euler CFD and Euler's Dream. Construction of such a model from scratch in the spirit of Wolfram's New Kind of Science has not been made. More generally, processes involving some form of friction or other losses are irreversible for the same reasons as presented. 

A key fact is that these losses cannot be avoided, because forward evolution without losses requires infinite precision/infinite computational yet is necessary, while the show must go on and the show is not just heat diffusion. 

Recall that heat diffusion is easy forward but difficult backward and so is an example of irreversibility, but a trivial example. Turbulent flow is a non-trivial example. 

Wolfram's computational physics has been met with skepticism by the physics community, and so his version of the 2nd Law may not be applauded. My version stays closer to real physics in the form of real turbulent flow... 

It is remarkable the the simple code of Euler CFD (as an extended version of the 5-point scheme for the 2d Laplacian) generates the whole spectrum of turbulent flow. Remarkable.

Recall the essence of the 2nd Law: Physical systems with nearby particles moving with different velocities, typically arising in slightly viscous flow turning turbulent or moving parts mechanics under friction, develop increasing velocity gradients controlled by turbulent/friction dissipation into heat energy as small scale unordered kinetic energy. This is an irreversible process which cannot be avoided. 

In other words: Perpetuum mobile in the form of a system with moving parts is impossible: There will always be some turbulence/friction transforming motion into heat energy in an irreversible process. 

On the other hand an atom in ground state is a form of perpetuum mobile, but then in the form of a system without moving parts, as explained in Real Quantum Mechanics: Electrons in ground state do not move around the kernel, but rather turn around like someone unable to get to sleep, while the electron density is stationary. On the other hand, in a radiating atom, electron densities do shift between energy levels under energy exchange.   

Elaboration of Wolfram's Explanation of the Second Law

da Vinci's explanation of the 2nd Law = Mine.

Stephen Wolfram is now presenting work on the 2nd Law of Thermodynamics based on the notion of computation, which connects to a view I have been advocating. The role the 2nd Law is to give an explanation in mathematical terms of the observed fact that certain macroscopical physical processes are irreversible even though the underlying microscopics based on Newton's Laws appears to be reversible. 

Boltzmann tried to give an explanation in terms of statistics, which has not convinced Wolfram nor me, since we have both tried to replace statistics by computation as a process taking a system from one configuration at a certain time $t$ to a next configuration at $t+dt$ with $dt$ a small time step.  My theory is presented in the books Computational Turbulent Incompressible Flow and Computational Thermodynamics with Johan Hoffman and blog posts.

I will not here seek to summarise Wolfram's theory, which he admits is not easy to fathom, but just briefly recall the elements of my theory which is different by qualifying the general idea of computation in terms of the concepts of (i) finite precision computation and (ii) stability/well-posedness (connecting to Wolfram's computational irreducibility).

The phenomenon of turbulence of fluid motion demonstrates the role of (i) and (ii) in explaining why certain processes are not reversible. The basic feature of turbulent flow is that it contains a range of scales from large to small with large carrying kinetic energy in ordered coherent form and small carrying heat energy as unordered incoherent form. 

Fluid flow may be seen as initiated in large scale coherent form like the laminar flow in a river before a water fall in which it transforms into small scale turbulent flow by the fall developing strong velocity gradients, see picture above. In the fall coherent laminar flow develops into turbulent flow thus transforming large scale coherent kinetic energy into small scale incoherent heat energy. 

This is a stable process in the sense that the large scales and amount of heat energy will change little under finite precision of the computational process over a time. In other words, the large scale forward process is stable or well-posed. It is possible to break down large scale kinetic motion into small scale heat energy with only finite precision = Easy. You can easily break a glass by a hammer without much precision.

On the other, reversing the flow through the water fall is impossible, because it would require infinite precision to coordinate incoherent small scale turbulent flow/heat energy into large scale laminar flow /kinetic energy = Difficult. To assemble a glass smashed into pieces is difficult/impossible. 

Digital computation has finite precision. We do not have to ask physical processes to be carried out by some form of computation with infinite precision; finite precision is enough.  

Summary of Irreversibity/2nd Law: 

• Forward motion in time is possible/easy under finite precision: physics. 
• Backward motion in time is impossible/difficult under finite precision: not physics.          

PS1 Wolfram gives a (human) observer with certain perception capabilities a key role, like in the Copenhagen interpretation of quantum mechanics. But a Universe without observer is also a Universe maybe of more fundamental interest. 

PS2  There is a connection to both previous and next post describing turbulent flow as predictable chaotic and irreversible flow.   

Predictive Chaos

A common argument in the current debate about global climate is that the Earth climate system is chaotic and so cannot be computationally modeled and predicted. The argument is used to question alarmistic predictions of climate change, but also open to unrestricted alarmism in the sense that anything can happen from virtually nothing. 

Basic examples of chaotic systems are (i) coin tossing and (ii) turbulent fluid motion. In both cases pointwise prediction in space and time is impossible, but mean-values can be predicted with high precision. In coin tossing with a perfect coin the quotient of number of heads and tails quickly approach 1 as the number of tosses increases. The drag and lift as mean-values in space and time of the forces on an airplane wing subject to turbulent flow of air, can be accurately computationally predicted by solving the Navier-Stokes equations. This is the subject of the monumental treatise Computational Turbulent Incompressible Flow. See also previous posts under label chaotic system.  

How is it possible that mean-values can be accurately computed but point-values not? The reason is that turbulent flow is (like the weather) fluctuating with alternating ups and downs (like the motion of a tossed coin before landing). Therefore mean-values can be predicted by computational simulation of the fluctuations without asking for point-wise accuracy. 

We understand that chaotic motion as fluctuating turbulent motion in a certain sense is more predictive than non-fluctuating motion like that of a pen left in upright position.

While the weather as pointwise chaotic motion is not predictable over more than a week, climate as mean value of weather changes only slowly and so can be predictive, more or less. For example, we can well expect to enter into a new ice age within hundreds of years as an effect of major factors like the orbit of the Earth. 

Stock markets appear chaotic, yet can be predicted over time using models including major relevant factors. 

In any case, reference to chaotic systems as being unpredictable can be misleading and so has to be qualified.  

PS This post connects to the previous post on the unfortunate present formulation of the Navier-Stokes Clay Millennium Problem forgetting the completely foundational aspect of well-posedness in the sense of small output effects of small perturbations, which is a property of turbulent solutions of Navier-Stokes equations: Drag and lift of a wing remain the same under small perturbations of incoming flow and geometry. The present formulation appears to confuse smoothness with well-posedness.