The Bulletin of Mathematical Biophysics. University of Chicago. 9” x 6”. Original wrappers. Long series of individual issues of the Bulletin, in original wrappers. The BMB was the only journal of its kind in the world at that time, and ‘‘the most important and active center in the US for mathematical biology” according to Warren Weaver (Tara Abraham, “Nicolas Rashevsky’s Mathematical Biophysics”, Journal of the History of Biology 37: 333–385, 2004).
Provenance: all of these issues have one thing in common in terms of condition—they are all from the Committee on Mathematical Biology Library at the University of Chicago—unfortunately someone used black magic marker to strike a line through each of the threelined library stamp on the cover of each issue. The rubber stamp is 90% covered, though on close inspection you can clearly see the owner of origin. The Committee was headed by Nicholas Rashevsky, and was responsible for publishing the BMB.
Rashevsky was “...a Russian [Ukrainian] émigré theoretical physicist who developed a program in “mathematical biophysics” at the University of Chicago during the 1930s. Stressing the complexity of many biological phenomena, Rashevsky argued that the methods of theoretical physics – namely mathematics – were needed to “simplify” complex biological processes such as cell division and nerve conduction. A maverick of sorts, Rashevsky was a conspicuous figure in the biological community during the 1930s and early 1940s...”
Rapoport, A. “Nets with Distance Bias”, pp 8593. WITH: “Connectivity of Random Nets” (with Ray Solomonoff) [cited 473 times]. Pp 107119. $125 This issue has no stamps from U Chicago June, 1951; 13/2.

A distance bias is imposed on the probability of direct connection between every pair of points in a random net. The probability that there exists a path from a given point in the net to another point is now a function of both the axone density and the distance between the points. A recursion formula is derived in terms of which this probability can be computed.

The rate of spread of an epidemic where probability of contact depends on the distance between the individuals can also be computed from the recursion formula.
Rapoport, Anatole. “Spread of Information through a Population SocioStructural Bias I & II”, pp 323557. December 1953, 15/4. $125

“A previously derived iteration formula for a random net was applied to some data on the spread of information through a population. It was found that if the axon density (the only free parameter in the formula) is determined by the first pair of experimental values, the predicted spread is much more rapid than the observed one. If the successive values of the “apparent axon density” are calculated from the successive experimental values, it is noticed that this quantity at first suffers a sharp drop from an initial high value to its lowest value and then gradually “recovers”.

An attempt is made to account for this behavior of the apparent axon density in terms of the “assumption of transitivity”, based on a certain sociostructural bias, namely, that the likely contacts of two individuals who themselves have been in contact are expected to be strongly overlapping. The assumption of transitivity leads to a drop in the apparent axon density from an arbitrary initial value to the vicinity of unity (if the actual axon density is not too small). However, the “recovery” is not accounted for, and thus the predicted spread turns out to be slower than the observed.”Abstract [Cited 443 times.]
Rapoport, A. “Contribution to the Theory of Random and Biased Nets”, pp 257279. December 1957, 19/4. $125

The probabilistic theory of random and biased nets is further developed by the “tracing” method treated previously. A number of biases expected to be operating in nets, particularly in sociograms, is described. Distribution of closed chain lengths is derived for random nets and for nets with a simple “reflexive” bias. The “island model” bias is treated for the case of two islands and a single axon tracing, resulting in a pair of linear difference equations with two indices. The reflexive bias is extended to multipleaxon tracing by an approximate method resulting in a modification of the random net recursion formula. Results previously obtained are compared with empirical findings and attempts are made to account for observed discrepancies.”Abstract
Rapoport, A. “Nets with Reciprocity Basis” pp 191203 and with Rashevsky, “A Comparison of SetTheoretical and GraphTheoretical in Topological Biology”, pp 267275. September 198, 20/3. $45
Rashevsky, “On Imitative Behavior” pp 7385 and with Rapoport, A. “A Derivation of a Rate Learning Curve from the Total Uncertainty of a Task”, pp 8595. With: and Rashevsky “Contributions to a Relational Biology”, pp 7385. $45 March 1960, 22/1
Rashevsky. “Contributions to the Theory of Imitative Behavior: the Number of Political Parties as Determined by Biological and Social Factors”. Pp 15. March 1962. (With Robert Rosen “A Note on Abstract Relational Biologies”, pp 3139. 24/1 $45

“The theory of imitative behavior as applied tow mutually exclusive behavior patterns (N. Rashevsky,Mathematical Biology of Social Behavior, Rev. Ed., 1959; The University of Chicago Press) leads to the possibility of any number of different behavior patterns existing in a social group. Mutually inhibitory effects suppress the effectiveness of behavior of groups that are very small numerically. The manner in which the different biological and social parameters that enter into the theory of imitative behavior determine the number of different effective behaviors is discussed. The results are applied to the problem of what determines the number of political parties in different countries. This number is expected to increase with increasing spread of the distribution curves for the tendencies towards different behaviors, with decreasing imitation factors, and with increasing instability of psychophysical judgments of the average individuals.”Abstract
Rashevsky, “Some Remarks in the Mathematical Aspects of Automobile Driving”, pp 299309; Rosen, Robert. “Some Further Comments on the DNAProtein Coding Problem”, pp 289299; Rashevsky “Suggestion for a Possible Approach to Molecular Biology”, pp 309328. September 1959. 21/3
Rashevsky, “Mathematical Biology of Division of Labor Between Individuals or Two Social Groups”, pp 213229, September 1952. 14/3. $45
Rashevsky, “Imitative Behavior in Nonuniformally Spatially Distributed Populations”, pp 6373. March 1953. 15/1. $45
Rashevsky, “Outline of a Mathematical Approach to History”, pp 197235. June 1953, 15/2. $75

A mathematical model for the development of human society, beginning with the earliest stages of urban cultures, is outlined. In the early stages of history, behavior was characterized largely by adherence to a number of beliefs and prejudices of different kinds, which were accepted on faith and not subject to critical rational analysis. Due to psychobiological variability a very small number of individuals spontaneously appear at all times who challenge the accepted beliefs and prejudices and do not follow the accepted patterns of social behavior. The effect of these individuals upon the rest of the society, especially upon the younger generation, depends on the facilities with which information spreads in society. In earliest societies, when modern methods of mass communication were unknown, the channels of communication were practically identical with the channels of economic transport. The latter in its turn depended on the nature of the roads, and especially on the presence of waterway, which facilitated transportation. The sizes of the earliest cities and the distances between them were largely determined by relative ease of transportation
Rashevsky, “Some Quantitative Aspects of History”, pp 339361, September 1953, 15/3. $45
Rashevsky, “A Contribution to the Search of General Mathematical Principles in Biology”, pp 7195. March 1958, 20/1. $95

A somewhat different approach to the principle of biotopological mapping, discussed in previous publications, is given. The organism is considered as a set of properties, each of which is in its turn a set of numerous subproperties which are logically included in the corresponding properties. Topology is introduced by an appropriate definition of neighborhoods, and four postulates are stated which concern the mapping of the spaces corresponding to higher organisms on those of lower ones. A number of conclusions are drawn from the postulates. Some of them correspond to wellknown facts. For example, in man and some higher organisms appropriate emotional stimuli should produce gastrointestinal or cardiovascular disturbances; or some microorganisms should produce substances harmful to other microorganisms (antibiotics). Some other conclusions are still awaiting verification. One of them is, for example, that there must exist unicellular organisms which produce antibodies to appropriate antigens.
Rashevsky, “A Note on the Origin of Life”, pp 185195. Also: Robert Rosen, “A Relational Theory of Biological Systems II”, pp 109129. June 1959, 21/2. $75

The general Theory of Categories is applied to the study of the (M, R)systems previously defined. A set of axioms is provided which characterize “abstract (M, R)systems”, defined in terms of the Theory of Categories. It is shown that the replication of the repair components of these systems may be accounted for in a natural way within this framework, thereby obviating the need for an ad ho postulation of a replication mechanism.

A timelag structure is introduced into these abstract (M, R)systems. In order to apply this structure to a discussion of the “morphology” of these systems, it is necessary to make certain assumptions which relate the morphology to the time lags. By so doing, a system of abstract biology is in effect constructed. In particular, a formulation of a general Principle of Optimal Design is proposed for these systems. It is shown under what conditions the repair mechanism of the system will be localized into a spherical region, suggestive of the nuclear arrangements in cells. The possibility of placing an abstract (M, R)system into optimal form in more than one way is then investigated, and a necessary and sufficient condition for this occurrence is obtained. Some further implications of the above assumptions are then discussed.
Rashevsky, N. “Some Suggestions for a New Theory of Cell Division”, pp 293307. H.D. Landahl, “Mathematical Biophysics of Color Vision”, pp 317327. December 1952, 14/4. $45
Rashevsky, N. “Topology and Life: in Search of General Mathematical Principles in Biology and Sociology”, pp 317349. December 1954, 16/4.

Mathematical biology has hitherto emphasized the quantitative, metric aspects of the physical manifestations of life, but has neglected the relational or positional aspects, which are of paramount importance in biology. Although, for example, the processes of locomotion, ingestion, and digestion in a human are much more complex than in a protozoan, the general relations between these processes are the same in all organisms. To a set of very complicated digestive functions of a higher animal there correspond a few simple functions in a protozoan. In other words, the more complicated processes in higher organisms can be mapped on the simpler corresponding processes in the lower ones. If any scientific study of this aspect of biology is to be possible at all, there must exist some regularity in such mappings. We are, therefore, led to the following principle: If the relations between various biological functions of an organism are represented geometrically in an appropriate topological space or by an appropriate topological complex, then the spaces or complexes representing different organisms must be obtainable by a proper transformation from one or very fewprimordialspaces or complexes.

The appropriate representation of the relations between the different biological functions of an organism appears to be a onedimensional complex, or graph, which represents the “organization chart” of the organism. The problem then is to find a proper transformation which derives from this graph the graphs of all possible higher organisms. Both a primordial graph and a transformation are suggested and discussed. Theorems are derived which show that the basic principle of mapping and the transformation have a predictive value and are verifiable experimentally.

These considerations are extended to relations within animal and human societies and thus indicate the reason for the similarities between some aspects of societies and organisms.

It is finally suggested that the relation between physics and biology may lie on a different plane from the one hitherto considered. While physical phenomena are the manifestations of the metric properties of the fourdimensional universe, biological phenomena may perhaps reflect some local topological properties of that universe.”Abstract [Cited by 255.]
Rosen, Robert. “The Representation of Biological Systems from the Standpoint of the theory of Categories”, pp 317343. December, 1958, 20/4. Good condition, only. $125

“A mathematical framework for a rigorous theory of general systems is constructed, using the notions of the theory of Categories and Funetors introduced by Eilenberg and MacLane (1945, Trans. Am. Math. Sac., 58, 23194). A short discussion of the basic ideas is given, and their possible application to the theory of biological systems is discussed. On the basis of these considerations~ a number of results are proved~ including the possibility of selecting a unique representative (a "eanonieal form") from a family of mathematical objects, all of which represent the same system. As an example, the representation of the neural net and the finite automaton is constructed in terms of our general theory.”Abstract
Rosen, Robert. “On a Logical Paradox Implicit in the Notion of a SelfReproducing Automaton”. $150 pp 387395. With: Rashevsky, “Mathematical Biophysics of Automobile Driving”, pp 375387.

For Rosen: “The notion of automaton as used by J. von Neumann is formalized according to methods previously described (Rosen, 1958,Bull. Math. Biophysics 20, 245–60; 317–41). It is observed that a logical paradox arises when one attempts to describe the notion of selfreproducing automaton in this formalism. This paradox is discussed, together with some of the recent attempts to construct automata which exhibit selfreproduction. The relation of these results to biological problems is then investigated.”Abstract
Rosen, Robert. “The DNAProtein Coding Program”, pp 7197. With Rashevsky, “A Note on Topological Biology” and “A SetTheoretical Approach to Biology”. March, 1959. 21/1. $75

“The DNAprotein coding problem is given a general algebraic formulation, the consequences of which are then explored by standard mathematical methods. To keep the treatment selfcontained, the mathematical techniques to be used are explained in detail. It is demonstrated that there exist a priori a countably infinite number of different abstract DNAprotein codes, thereby showing that inductive attempts to construct such a code will most likely be fruitless. A notion of ergodicity is then introduced, which imposes a number of restrictions on the admissible codes, and, in fact, these considerations enable us to derive a small portion of a code which, if our hypothesis of ergodicity is correct, must occur in nature. Finally, we discuss briefly the problem as to whether there can exist more than one DNAprotein code in nature.”Abstract
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