Rather interestingly, D. R. Hofstadter seems to be working with Emmanuel Sander on a book.

Stergos D. Afantenos

It is great to see a blogger with some common sense about Comment Policy and I congratulate you on your conciseness.

Ka kite
Ken Allan
from Middle-earth

It appears that the fundamental laws of quantum mechanics are discontinuous, according to our current best physical theories. Space is divided into Planck lengths and time is divided into chronons. Therefore stock prices are likely discontinuous at both the macroscopic and microscopic scales.

Fractals seem to be good at modeling many phenomena. A fractal may have a regular, predictable structure, yet be nonlinear, non-smooth, and discontinuous.

About your counter-example, I have to say that discontinuous phenomena do not not imply discontinuous causal laws. For example, even the most turbulent liquid flow is produced by simple linear hydrodynamic laws. This also applies to the weather.

In your example, linear, continuous laws, applied to complex systems, that make decisions, like brains and computers, produce a final result that sometimes is discontinuous or chaotic, but this says nothing about the basic laws that govern brains and computers.

Other example, an asteroid in direct path to the heart follows linear and continuous physical laws of inertial movement, gravitation, conservation of energy etc. But the macroscopic effect are destructive, abrupt discontinuous and chaotic.

But the other way around is not true: chaotic or discontinuous laws does not produce continuous or non chaotic phenomena.

-complicated laws produce complicated phenomena

-in some cases, underlying linear laws produce linear environments (with smooth fitness landscapes) where life can evolve.

These are the strong point in my argumentation.

Anyway, I take your advice.

I agree that some inductive biases work better in our universe than others, and that this is due to the structure of our universe. Where I disagree with you and Günther is concerning exactly what inductive biases work best. I believe that we have not yet done the required experiments that would support any well-founded claims about the inductive biases that work best in our universe. You claim the biases are smoothness, continuity, simplicity, and parsimony. These are certainly popular biases, but what evidence do we really have for them? It seems to me that almost all of the arguments in favour of these biases rely on appeal to intuition and introspection. Where are the experiments? Where is the science?

Furthermore, these terms, smoothness, continuity, simplicity, and parsimony, are vague. There are formal definitions for all of these terms. Now we need to show that the formal definitions really work, on a wide variety of real-world problems. Instead, researchers tend to justify the formal definitions by claiming that they capture our intuitions. This is not science.

You are trying to support your favourite inductive biases (smoothness, continuity, simplicity, and parsimony) by pure argument, without experimentation. This is not science. If you want to convince me, stop arguing and start experimenting.

By the way, I would call parsimony and simplicity synonymous, but smoothness and continuity are distinct from each other and distinct from simplicity. So you are advocating three distinct inductive biases (smoothness, continuity, and simplicity), yet you talk about them as if they were all the same. They are not.

Here is a real-world example of a situation in which the continuity bias leads to false predictions. Suppose that shares in the XYZ company are selling for $100 each at 1:00 PM. Then there is a news report about fraud and scandal in the XYZ company, and the share prices drop to $5 each at 2:00 PM. It is wrong to assume that there is a time between 1:00 PM and 2:00 PM at which the share price is $50. Are you going to tell me that the share price must pass through $50 at some time between 1:00 PM and 2:00 PM, because of the anthropic principle?

I don´t think that I say anything similar to what Günther Greindl says. I say that, considering

- the weak anthropic principle

- the multiverse hypotesis, where each universe can have it´s own mathematical formulation (see Max tegmark Which mathematical structure is isomorphic to our Universe?)

- the fact that Darwinian evolution need a relatively simple, continuous , smooth fitness landscape, not plagued with random peaks

it follows that we must live in a certain universe where macroscopic phenomena must obey smooth, continuous, and parsimonious laws for the fitness landscape to be that way; that is, to permit life.

It is our universe, the thing that has been selected for life to exist inside just because it is simple, so we succeed when we try to explain real phenomena through the most smooth, simple, continuous, and parsimonious laws (some of them, that does not work always, I agree with you). Our universe has the bias. It is not by chance.