Sci Fi TV science or fiction?

The fourth hypothesis, building on formal results by

Kolmogorov, Solomonov and Chaitin, claims that mathematics is so useful in

describing the natural world because it is the science of the abbreviation of

sequences, and mathematically formulated laws of nature enable us to

compress the information contained in the sequence of numbers in which we

code our observations. In this tradition, laws are equivalent to the shortest

algorithms capable of generating the lists of zeros and ones representing the

empirical data.

THE ALGORITHMIC VIEW OF THE LAWS OF NATURE

Suppose that the temporal evolution of any physical system is describable

by finite strings of real numbers, corresponding to operational measures of

physical magnitudes (temperature, pressure etc.). We can have two cases:

such strings can be ordered

(111000111000111000…)

or truly random

(0100110101100110…).

In the first case, the string can be generated by a simple instruction (“print

111000 n times”), which is much shorter than the list itself. In the second

case, the string appears as truly random, where “appear” is meant to stress

that while we can show that a finite string is not random by giving the

generating law (algorithm), we can never prove that a string is random (this

is a version of the halting theorem).

At this point we can give two definitions, based on algorithmic

complexity theory, which will be relevant for our purpose:

Definition1: the complexity of a string is the length of the shortest

algorithm capable of generating it

Definition2: a string is said to be algorithmically compressible when

there is an algorithm capable of generating it, such that its information

content (number of bits) is much less than that of the string.

As an illustration of these definitions, consider that a string like

{1, 4, 9, 16, 25, 36, …} (1)

is obviously not random, since it can be trivially obtained by squaring the

positive integers in the list

{1, 2, 3, 4, 5, 6, …} (2)

If the numbers in (2) correspond to measured magnitudes in such a way

that, say in a temporal interval of 1, 2, 3 seconds (the input data), a body

travels 1, 4, 9 meters (the output), then the existence of a rule generating (1)

from (2) shows that (1) is algorithmically compressible. The above algorithm

is – modulo the constant ½g. – Galileo’s law of free fall, generating the

spatial intervals (1) from the square of the temporal intervals (2).

In a word, by following the metaphor of scientific laws regarded as the

software of a physical system, we discover that searching for laws is

generating the string of numbers expressing the experimental measures.

Such a length – the complexity of the string – will be equal to that of the

original string only if the latter is composed by apparently random numbers,

and does not obey any known law.

The idea that scientific laws are an economic synthesis of all the

information contained in our observations is certainly not new, and in this

algorithmic approach it finds a new, rigorous and precise formulation. It was

especially Ernst Mach who regarded science and its theories and laws as an

economic “summary” of our observations. As he wrote (Mach, 1896):

Science is a form of business. Its purpose is to find the maximum amount

of the infinite eternal truth with the minimum amount of work, in the

minimum expenditure of time and with the minimum amount of thought

effort. After having made explicit the philosophical consequences that seem to

follow from the software metaphor for scientific laws, we can now finally

discuss a possible explanation of the applicability of mathematics, due to the

physicist John Barrow (Barrow, 1992):

science exists because the natural world seems algorithmically

compressible. The mathematical formulae that we call laws of nature are

economical reductions of enormous sequences of data expressing

changes of state of the world: here is what we mean by intelligibility of

the world…Since the physical world is algorithmically compressible,

mathematics is useful to describe it because it is the language of the

abbreviation of sequences. The human mind enables us to make contact

with that world because our brain has the ability of compressing complex

sequences of sense data in shorter form. Such abbreviations make

thought and memory possible. The natural limits that nature poses to our

senses prevent us from overloading our brains with information about the

world. Such limits are security gates for our minds.

 

 

Just for the record, I will remind my quote which caused all this discussion

Dean wrote: In a sense, the laws of physics themselves are a wonderful example of compression.