Brillouin, L.

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INFOrmATION A~D CONTnOL I, 1-5(1957) 
Mathematics, Physics, and Information 
(An Editorial) 
L. BRI~LOUIN 
Columbia University, New York 
I. INFORMATION IN SC IENCE 
The  theory of information raises a number  of fundamenta l  questions 
about the first postulates of science and the limits of their validity. It 
is hoped that among its many other functions, this new journal will offer 
a possibility for free discussion of these problems and for a fruitful 
comparison of points of view. As an exampIe of such problems, let us 
examine the opinions of a pure mathematician and of a physicist about 
the foundations of geometry. The mathematician starts with dimension- 
less points, infinitely thin curves and surfaces, and continuous pace 
time. Atomic  science denies any  real mean ing  to these definitions. 
Consider a very thin tinfoil, and look at it with X-rays: you  dis- 
cover an atomic lattice, with isolated atoms separated by  large empty  
intervals. The  foil has a finite thickness and is not continuous. Even  a 
monomolecu lar  layer exhibits similar properties. 
2. EXPERIMENTAL ERRORS 
The  mathemat ic ian  very carefully defines irrational numbers .  The  
physicist never meets  any  such numbers .  Whatever  he measures is 
represented by  a finite number ,  with so many figures, and a certain 
amount  of uncertainty. The  mathemat ic ian  shudders at uncertainty 
and tries to ignore experimental errors. 
Open a book of pure mathematics and consider a theorem. It  is always 
built on a typical scheme: given certain conditions A, B, C, which are 
assumed to be exactly fulfilled, it can be proven rigorously that conclu- 
sion Q must be true. Here the physicist starts wondering: how can we 
know that A, B, C, are exactly fulfilled? No observation can tell us that 
much. The only thing we may know is that A, B, C, are approximately 
satisfied within certain limits of error. And then what does the theorem 
L. BRILLOI/IN 
prove? Very small errors on A, B, C, may result in a very small error on 
the final statement Q,or they may destroy it completely. The discussion 
is not complete until the problem of the stability of the theorem has been 
investigated, and this is another story! 
3. INFORMATION IS F IN ITE 
In a reeent book, 1 I discussed at great length the problem of experi- 
mental errors. The theory of information gives us a possibility for 
stating eorreetly these questions and getting consistent answers. We are 
now able to define the amount of information obtained from a eertain 
experiment, and to measure it in a precise way. We only need to know 
the field of uncertainty before and after the observation. The logarithm 
of the ratio of these two uncertainties yields the amount of information. 
If the final uncertainty is very small (very accurate measurement) the 
information obtained is very large. 
The mathematician dreams of measurements of infinite accuracy, 
defining for instance the position of a point without any possible rror. 
This would mean an experiment yielding an infinite amount of informa- 
tion and this is physically impossible. One of the most important results 
of the theory is known as the "negentropy principle of information." 
It states that any information obtained from an experiment must be 
paid for in negentropy. As D. Gabor states it :"you cannot get something 
for nothing, not even an observation." If an experiment yields an in- 
formation AI, there must have been increase of entropy 
AS > A[ (1) 
in the apparatus or in the laboratory where the experiment has been per- 
formed. An increase AS in entropy means a decrease 
AN = - AS (2) 
in the total "negentropy." The information AI is paid for by a larger 
amount AN of negentropy. 
AN + AI < 0 
A very large amount of information shall cost a very high price, and 
an infinite amount of information is unattainable. No wealth could pay 
for it. Another important point refers more specifically to the definition 
1 L. Brillouin, "Science and Information Theory." Academic Press, New York, 
1956. 
MATHEMATICS~ PHYSICS, AND INFORMATION 3 
of a continuum in space and time (loc. cit., Ch. XVI). A very direct dis- 
cussion leads to the result that the measurement of a very small length 
Ax (and by "very small," a length smaller than 10 -I3 em is meant) re- 
quires the use of a total energy AE such that 
zXEAx > ½hc (3) 
This represents a new limitation, different from the uncertainty prin- 
ciple and completely independent from the conditions pecified in (1). 
Extremely small distances cannot be measured, unless a source of very 
high energy is used. This energy may not be completely dissipated, but 
it is needed for the experiment. Only a fraction AQ of this total energy 
is dissipated, according to Eq. (1) 
AS = AQ/T  ~ A I  (4) 
if the experiment is performed under most economical conditions. 
Both results lead to similar conclusions: an infinite amount of infor- 
mation can never be obtained. An infinitely small distance cannot be 
measured. Geometrical and mathematical definitions are only dreams, 
but the physicist cannot rust them and we should especially emphasize 
the impossibility of physically defining a continuum in space and time. 
4. TItE VIEWPOINT OF M. BORN 
It  is not necessary to repeat here many comments or explanations 
which were given in my book. Let us only recall that the negentropy 
principle of information gives a precise meaning to an old remark of 
J. yon Neumann saying that an observation is an irreversible process. 
Condition (1) specifies the amount of irreversibility. 
I t  may be interesting to quote another author, Max Born, and to dis- 
cuss a paper he recently published in the jubilee-book 2 presented to 
Niels Bohr for his seventieth birthday. Under the title "Continuity, 
determinism and reality," Max Born presents ome general remarks 
very close to those of the present paper, although his line of reasoning 
is different: 
" I  maintain that the mathematical concept of a point in a continuum 
has no direct physical significance. It  has no meaning to say the value 
of a coordinate x -- ,  has a value x = %/2 inch or x = ~r era." (This dis- 
poses of irrational numbers !)
"Modern physics has achieved its greatest successes by applying the 
2 Kgl. Dansl~e Videnslcab. Selsl~ab, Mat. fys. Medd. (1955) 30(2). 
L, BI~ILLOUIN 
methodologicM principle that concepts which refer to distinctions be- 
yond possible experience have no physical meaning and ought to be 
eliminated . . . .  The most glaringly successful cases are Einstein's foun- 
dation of relativity based on the rejection of the concept of aether.. .  
and tteisenberg's foundation of quantum mechanics . . . .  I think that 
this principle should be applied also to the idea of physical continuity." 
And M. Born explains that experimental errors should be taken into 
account right from the beginning, in the field of classical physics where 
they have been too often ignored. 
He does not want to reject he mathematical concept of a real number, 
but specifies that"  the situation demands a description of haziness." The 
probability for the value of a physical quantity to be in a given interval 
should be specified, instead of pretending that the value of the quantity 
can be known exactly. 
Many examples discussed by Born ought o be given here, but we hope 
the reader will refer to the original paper. 
Let us remember, that the general principle invoked by Born is often 
quoted as Bridgman's operational point of view. 
5. THE PROBLEM OF DETERMINISM 
This is again a problem which, in our opinion, belongs to the domain 
of metaphysics but not of physics. The famous Laplace's tatement is 
well known: an infinitely clever fellow measures with infinite accuracy 
all the positions of the atoms in the whole world, and computes the be- 
havior of this world for any distant future. This is a dream and corre- 
sponds to no physical problem. Bridgman would dismiss it as contrary 
to the operational method. Born discusses a very simple example which 
proves the fallacy of the statement: he considers one single atom, mov- 
ing back and forth along a straight line, between two reflecting walls 
located at x = 0 and x = 1. Here the prediction looks easy if the initial 
position x and velocity v are given. But v is measured only within an 
error Av and this means an error tar in x after an interval of time t. 
Very soon tar will become larger than the length l of the strip and no- 
body can predict where to find the particle. A similar example is dis- 
cussed in E. Borel's lectures on statistical mechanics (written down by 
F. PerrinS). You may believe in determinism, or you may deny it, or 
simply doubt it. Classical science can give you no answer: it is an act of 
E. Borel, "Mecanique statistique classique," p. 22. Gauthier-Villars, Paris, 
1925. 
MATHEMATICS, PHYSICS, AND INFORMATION 5 
faith. And when we come to quantum mechanics and atomic structures 
then we know that determinism does not apply at all. 
6. PHYSICAL THEORIES 
The difficulty comes when you consider that mathematics is used as a 
tool for building physical theories or discussing technical problems. 
Here we are back on the earth. We cannot measure any quantity with 
infinite accuracy and we correct he stiffness of mathematical statements 
by using statistical methods and the calculus of probability. This is 
practically what Born recommends ( ee section 4). 
The methods of quantum mechanics have been extremely useful in 
many problems, but they definitely failed on others. The most typical 
example is the impossibility of computing a variety of quantities, which 
appear as divergent integrals. Some of these troubles have been par- 
tially connected by the use of "renormalization" methods, but this again 
is only a remedy. Every physicist knows where the trouble comes from. 
The integrals extend from zero to infinity, where zero means the imme- 
diate neighborhood of a particle. And it is impossible to measure any- 
thing too close to the particle. The "exact" position of the center of the 
particle cannot be measured, and the geometry of a continuum is to 
blame, but we have no other method for the time being, and we are 
waiting for a bright idea, telling us how to rebuild mathematics on a 
practical basis. 
The information theory may help in discovering the actual weak points 
of present methods. Information theory and quantum theory are in full 
agreement and emphasize the need for a strict obedience to operational 
definitions. 
A practical theory seems to be shaping up progressively with the use 
of new discontinuous operators, rather similar to the spin operator. All 
these methods ignore the usual space-time continuum, and they may 
represent the beginning of the new theory we need. 


Mathematics, physics, and information (an editorial) 

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