In this paper I shall discuss the problem of cognition of\ud
laws of nature on the following different levels of understanding:\ud
First level of understanding of laws of nature: the Greek\ud
Ideal of Science\ud
Second level: Space Time Invariance\ud
Third level: Dynamical Laws\ud
Fourth level: Statistical Laws\ud
Fifth level: Laws and Causality\ud
Sixth level: Chaotic Motion\ud
Seventh level: Initial conditions and Constants of Natur

Weingartner, Paul

English

Elektronisch archivierte Theorie - Sammelpunkt

  367
On the Cognition of Laws of Nature 
Paul Weingartner, Salzburg 
In this paper I shall discuss the problem of cognition of 
laws of nature on the following different levels of under-
standing: 
First level of understanding of laws of nature: the Greek 
Ideal of Science 
Second level: Space Time Invariance 
Third level: Dynamical Laws 
Fourth level: Statistical Laws 
Fifth level: Laws and Causality 
Sixth level: Chaotic Motion 
Seventh level: Initial conditions and Constants of Nature 
Before I shall begin with the first level of cognition or 
understanding a short clarification of different meanings of 
the expression ‘law of nature’ will be given: 
L1  the “law” as it “is” in the thought of the inventor or 
discoverer 
L2  the “law” as it “is” in the things which are ordered 
or described by it 
L3  the “law” as a law statement formulated in some 
scientific language 
L4  the “law” as an ideal true law, w.r.t. which laws 
known at present in the sense of L3 are ap-
proximations 
L5  the “law” as ideal conceptual entity more or less 
independent and separated from law statements. 
First Level: the Greek Ideal of Science 
In order to be able to describe and explain movement we 
need to distinguish something which changes relative to 
something which does not change. This important 
distinction is pointed out by Aristotle also as a criticism of 
Parmenides’ theory of the universe which assumes only 
one being and nothing else. That what changes, moves 
was thought to be contingent (not necessary) in respect to 
the not changing (or even not changeable) necessary 
principle or law. In general this idea belongs to the Greek 
Ideal of Science which was more or less manifest in 
several Greek thinkers from Thales on but was elaborated 
in detail by Plato and Aristotle: 
To describe and explain the visible (observable), concrete, 
particular, changing, material contingent world by non-
visible (non-observable) abstract, universal, non changing 
immaterial and necessary principles. 
Second Level: Space Time Invariance 
Whereas the first level of understanding a law is con-
cerned with invariance in general the second level of 
understanding is concerned with finding out a specific kind 
of invariance. Among these the oldest and most famous 
one is the invariance w.r.t. space and time. Or in other 
words: the invariance under changes of place and time. 
Third Level: Dynamical Laws 
D1 the state of the physical system S at any given time ti is 
a definite function of its state at an earlier time ti-1. A 
unique earlier state (corresponding to a unique solution of 
the differential equation) leads under the time evolution to 
a unique final state (again corresponding to a unique 
solution of the equation). 
D2 condition D1 is also satisfied for every part of the 
physical system, especially for every individual body 
(object) as part of the system even if the individual objects 
may differ in the classical or in the quantum mechanical 
sense. 
D3 the physical system S is periodic, that is the state of S 
repeats itself after a finite period of time and continues to 
do so in the absence of external disturbing forces. 
D4 the physical system S has a certain type of stability 
which obeys the following condition: Very small changes in 
the initial states, say within a neighbourhood distance of ε 
lead to proportionally small (no more than in accordance of 
linearly increasing function of time) changes h(ε) in the 
final state. This kind of stability which survives small 
perturbations and leads to relaxation afterwards is called 
perturbative stability and holds in many linear systems. 
Fourth Level: Statistical Laws 
S1 the state of the physical system at ti is not a definite 
function of an earlier state at ti-1 . The same initial state 
may lead to different successor states (branching). 
S2 Statistical laws describe and predict the states of the 
whole physical system but they do not describe or predict 
the individual parts (objects) of this system. 
S3 Statistical laws describe only physical systems which 
are non-periodic, i.e. systems with extremely un-probable 
recurrence of the whole state of the system. 
S4 the loss of information (and consequently the difficulty 
of prediction) about the state of an individual object (or a 
small part) of the whole system increases exponentially 
with the complexity of the system. On the other hand: the 
(accuracy of the) information about the average values of 
magnitudes (parameters) of the state of an individual 
object (or small part) increases also with the complexity of 
the system. 
Fifth Level: Laws and Causality 
Causal condition for dynamical laws:  
CD  The same initial state leads – under the same 
conditions – to the same series of successor 
states. 
On the Cognition of Laws of Nature - Paul Weingartner 
 
 
 368 
Causal condition for statistical laws: 
CS  The same initial state may lead to different 
 successor states. But those successor states 
 which belong to the same initial state obey the 
same statistics. 
Sixth Level: Chaotic Motion 
The discovery of chaotic motion showed something new 
which was not understood in connection with laws of 
nature so far. It showed that initial conditions, boundary 
conditions or mathematical proportions are not necessarily 
accidental but can play an important role w.r.t. several 
properties of the laws. The new discovery was that even 
within the area of relatively simple physical systems which 
perfectly obey dynamical laws of Classical Mechanics, like 
the spherical pendulum, such systems can change 
radically its behaviour. Thus a dynamical system obeying 
Newton’s laws with strict predictability can become chaotic 
in its behaviour and practically unpredictable just by 
changing slightly some initial conditions. Experiments 
which prove such a behaviour of dynamical systems have 
been made since the seventies. 
Seventh Level: Initial Conditions and 
Constants of Nature 
The understanding of what a law is depends on the 
distinction between laws and initial conditions. This goes 
back to the Greeks (see first level of understanding). The 
deeper problems of such a distinction are very well 
described by the following quotations of Wigner and 
Wheeler (Wigner (1967), 3; Milne (1948), 4): 
“The world is very complicated and it is clearly impossi-
ble for the human mind to understand it completely. Man 
has therefore devised an artifice which permits the 
complicated nature of the world to be blamed on some-
thing which is called accidental and thus permits him to 
abstract a domain in which simple laws can be found. 
The complications are called initial conditions; the 
domains of regularities, laws of nature. … The artificial 
nature of the division of information into “initial condi-
tions” and “laws of nature” is perhaps most evident in the 
realm of cosmology. Equations of motion which purport 
to be able to predict the future of a universe form an 
arbitrary present state clearly cannot have an empirical 
basis. It is, in fact, impossible to adduce reasons against 
the assumption that the laws of nature would be different 
even in small domains if the universe had a radically 
different structure.”  


On the Cognition of Laws of Nature

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