Warren Weaver

The Mathematical Theory of Communication

HE recent development of various methods of modulation such as PCM and PPM which exchange bandwidth for signal-to-noise ratio has intensified the interest in a general theory of communication. A basis for such a theory is contained in the important papers of Nyquist1 and Hartley2 on this subject. In the present paper we will extend the theory to include a number of new factors, in particular the effect of noise in the channel, and the savings possible due to the statistical structure of the original message and due to the nature of the final destination of the information. The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages. The system must be designed to operate for each possible selection, not just the one which will actually be chosen since this is unknown at the time of design. If the number of messages in the set is finite then this number or any monotonic function of this number can be regarded as a measure of the information produced when one message is chosen from the set, all choices being equally likely. As was pointed out by Hartley the most natural choice is the logarithmic function. Although this definition must be generalized considerably when we consider the influence of the statistics of the message and when we have a continuous range of messages, we will in all cases use an essentially logarithmic measure. The logarithmic measure is more convenient for various reasons:

Science and complexity.

Science has led to a multitude of results that affect men’s lives. Some of these results are embodied in mere conveniences of a relatively trivial sort. Many of them, based on science and developed through technology, are essential to the machinery of modern life. Many other results, especially those associated with the biological and medical sciences, are of unquestioned benefit and comfort. Certain aspects of science have profoundly influenced men’s ideas and even their ideals. Still other aspects of science are thoroughly awesome.

The Mathematical Theory of Communication.

T HE recent development of various methods of modulation such as PCM and PPM which exchange bandwidth for signal-to-noise ratio has intensified the interest in a general theory of communication. A basis for such a theory is contained in the important papers of Nyquist and Hartley on this subject. In the present paper we will extend the theory to include a number of new factors, in particular the effect of noise in the channel, and the savings possible due to the statistical structure of the original message and due to the nature of the final destination of the information. The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages. The system must be designed to operate for each possible selection, not just the one which will actually be chosen since this is unknown at the time of design. If the number of messages in the set is finite then this number or any monotonic function of this number can be regarded as a measure of the information produced when one message is chosen from the set, all choices being equally likely. As was pointed out by Hartley the most natural choice is the logarithmic function. Although this definition must be generalized considerably when we consider the influence of the statistics of the message and when we have a continuous range of messages, we will in all cases use an essentially logarithmic measure. The logarithmic measure is more convenient for various reasons:

A Debate on the Theory of Relativity.

Even though this volume of Proceedings omits such papers read at the Congress as have elsewhere been published, the material is so rich that one cannot but be impressed with the quality and the range of philosophic thought in present day India. In addition to Rabindranath Tagores presidential address, the Proceedings contain eighteen papers in the field of Indian philosophy, ten papers on logic and metaphysics, eight papers relating to the philosophy of religion, four papers read in the Section of History of Philosophy, and three in the Section of Ethics and Social Philosophy. The volume should be made accessible to all serious students of philosophy.

Rededication to Liberal Decency.

THERE is at present a sickniess in our countrya sickness of rumor and anxiety, of suspicion and distrust, and, at its worst, of fear and tragedy. In part, this sickness is the result of overemnphasis on caution for the past rather thani on constructive courage with respect to the future. In part, it is an antiintellectualism, a strange and dang,erous lack of faith in scholarly competence. In part, it is, we all realize, the misguided groping of sincere persons who really want to be good citizens but who have been misled as to what good citizenship is. In its worst part it is the horrid result of political pressure, of personal selfishness, and of the pathological arrogance of demagogs with small and nasty minds. One of the most dangerous and wicked results of this disease is the destruction of confidence-confidence that honest, capable, and devoted service will be rewarded as such; confidence that governmental promises can be trusted to be stable, confidence that the precious Anglo-Saxon tradition of due process will be observed. This sickness attacks our society at all levels and in all fields of activity. Science is in a position to be particularly aware of the dangers in this sickness, but science asks for no special privilege or protectioni. Science voices its concern but priimiarily because the problem is a universal one. Many of the dread disorders of mens bodies involve the production of antibodies which furnish future immunity from the disease. But a most irrational and discouraging fact about our societys present disease is that apparently one can never stop worrying, can never be sure that the disease will not strike again and again and again. Some of us-partly oinly because of the acceide,tal character of our activities, partly because we attest to our liberalismii in issore discreet (or should I say more timid) ways-have so far avoided this disease. But no one of us is imI1muine. The timie has b)een reached, many of us are convinced, when it is no longer defenlsible to fail to take a stand. We must use all our wits and patience, all our reasonableness and courage, to see to it that we take a really sensible, constructive stand, and in particular that we do iiot fight fire with fire. Although there oftein is miiuch incentive to protest without restraint, we must not do this. Freedom is too precious to deserve rash or stupid support. There is some enieouraginig evidence that this past year may have seen the worst of this disease. There are promising signls that at appropriately high levels in our government a concern now exists to improve the whole loyalty-security-secrecy setup. If this is in fact done with promuptness and candor, then we can hope that the disease will also begin to wane in other places and in other fields of interest. So there is a chance that, at this particular session of the AAAS, we are actually celebrating the upturn, the overdue rededication to liberal decency. If so, then there is a poetic appropriateness to this occasion. It is miy very great pleasure, and my special honor, to present to you Edward Uhler Condon, the retiring president. WARREN WEAVER, President American Associationi for the Advancement of Science

Die Sedimentationszeit kleiner Teilchen in einer Flüssigkeit

ZusammenfassungIm Anschluß an eine Arbeit von Fürth wird die Frage der oberen Grenze für die Sedimentationszeit kleiner Teilchen nochmals diskutiert.

Die Diffusion kleiner Teilchen in einer Flüssigkeit

ZusammenfassungWenn kleine Teilchen in einer Flüssigkeit schwimmen, führen sie eine Brownsche Bewegung aus und erleiden einen ständigen Fall, dessen Geschwindigkeit aus dem Stokesschen Gesetz folgt. Wie auch die Anfangsdichtigkeit der Teilchen über die Flüssigkeit verteilt sein mag, der Endzustand wird immer ein stationärer sein, in dem die Dichtigkeit eine Exponentialfunktion der Tiefe ist. In mannigfachen Problemen ist es eine wichtige Frage, wie lange es dauern wird, bevor dieser stationäre Zustand erreicht wird. Diese Frage ist vor kurzem von Fürth behandelt worden. Seine Schlüsse sind weit verschieden von denen des Verfassers, der früher gefunden hatte, daß der stationäre Zustand in allen Fällen hauptsächlich in derselben Zeit erreicht wird, in der ein Teilchen mit der Stokesschen Geschwindigkeit durch die doppelte Tiefe der Flüssigkeit fallen würde. Dies Resultat wird bestätigt, indem der Fehler in Fürths Behandlungsweise klargelegt wird.