The bulletin of mathematical biophysics

, Volume 16, Issue 4, pp 317–348 | Cite as

Topology and life: In search of general mathematical principles in biology and sociology

  • N. Rashevsky


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 fewprimordial spaces or complexes.

The appropriate representation of the relations between the different biological functions of an organism appears to be a one-dimensional 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 four-dimensional universe, biological phenomena may perhaps reflect some local topological properties of that universe.


Biological Function Transformation Rule High Animal Mathematical Biophysics Organic World 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Blair, H. A. 1932a. “On the Intensity-Time Relations for Stimulation by Electric Currents: I.”Jour. Gen. Physiol.,15, 709–29.CrossRefGoogle Scholar
  2. — 1932b. “On the Intensity-Time Relations for Stimulation by Electric Currents: II.”,15, 731–55.CrossRefGoogle Scholar
  3. Buchsbaum, R. 1938.Animals without Backbones. Chicago: University of Chicago Press.Google Scholar
  4. — and R. W. Williamson. 1942. “The Rate of Elongation and Constriction of Dividing Sea-Urchin Eggs as a Test of a Mathematical Theory of Cell Division.”Physiol. Zool.,16, 162–71.Google Scholar
  5. Cohn, David. 1953. “Optimal Systems: I. The Vascular System.”Bull Math. Biophysics,16, 59–74.CrossRefGoogle Scholar
  6. Culbertson, J. T. 1950.Consciousness and Behavior: A Neural Analysis of Behavior and Consciousness. Dubuque, Ia.: Wm. C. Brown Co.Google Scholar
  7. Einstein, A. 1952.The Meaning of Relativity. Princeton: Princeton University Press.Google Scholar
  8. Fuller, H. J. and O. Tippo. 1953.College Botany. New York: Henry Holt.Google Scholar
  9. Haldane, J. B. S. 1924. “A Mathematical Theory of Natural and Artificial Selection I.”Trans. Comb. Phil. Soc.,23.Google Scholar
  10. Harvey, E. N. and H. Shapiro. 1941. “The Recovery Period (Relaxation) of Marine Eggs after Deformation.”Jour. Cell. and Comp. Physiol.,17, 135–44.CrossRefGoogle Scholar
  11. Hearon, J. Z. 1949a. “The Steady State Kinetics of Some Biological Systems: I.”Bull. Math. Biophysics,11, 29–50.CrossRefGoogle Scholar
  12. — 1949b. “The Steady State Kinetics of Some Biological Systems: II.”,11, 83–95.CrossRefGoogle Scholar
  13. — 1950a. “The Steady State Kinetics of Some Biological Systems: III. Thermodynamic Aspects.”,12, 57–83.CrossRefGoogle Scholar
  14. — 1950b. “The Steady State Kinetics of Some Biological Systems: IV. Thermodynamic Aspects.”,12, 85–106.CrossRefGoogle Scholar
  15. — 1950c. “Some Cellular Diffusion Problems Based on Onsager's Generalization of Fick's Law.”,12, 135–59.CrossRefGoogle Scholar
  16. Hill, A. V. 1936. “Excitation and Accommodation in Nerve.”Proc. Roy. Soc. Lond. B,119, 305–55.CrossRefGoogle Scholar
  17. Hogben, L. 1946.An Introduction to Mathematical Genetics. New York: W. W. Norton & Company, Inc.Google Scholar
  18. Householder, A. S. 1939. “A Neural Mechanism for Discrimination.”Psychometrika,4, 45–58.zbMATHCrossRefGoogle Scholar
  19. — 1940. “A Neural Mechanism for Discrimination: II. Discrimination of Weights.”Bull. Math. Biophysics,2, 1–13.CrossRefGoogle Scholar
  20. Jacobson, E. 1930a. “Electrical Measurements of Neuromuscular States during Mental Activities. III. Visual Imagination and Recollection.”Am. Jour. Physiol.,95, 694–702.Google Scholar
  21. — 1930b. “Electrical Measurements of Neuromuscular States during Mental Activities. IV. Evidence of Contraction of Specific Muscles during Imagination.”,95, 703–12.Google Scholar
  22. — 1931. “Electrical Measurements of Neuromuscular States during Mental Activities. VII. Imagination, Recollection and Abstract Thinking Involving the Speech Musculature.”,97, 200–09.Google Scholar
  23. Karreman, George. 1951. “Contributions to the Mathematical Biology of Excitation with Particular Emphasis on Changes in Membrane Permeability and on Threshold Phenomena.”Bull. Math. Biophysics,13, 189–243.CrossRefGoogle Scholar
  24. —. 1952. “Some Contributions to the Mathematical Biology of Blood Circulation. Reflections of Pressure Waves in the Arterial System.”,14, 327–50.MathSciNetCrossRefGoogle Scholar
  25. König, D. 1936.Theorie der Endlichen und Unendlichen Graphen. Leipzig: Akademische Verlagsgesellschaft.Google Scholar
  26. Landahl, H. D. 1938. “A Contribution to the Mathematical Biophysics of Psychophysical Discrimination.”Psychometrika,3, 107–25.zbMATHCrossRefGoogle Scholar
  27. — 1939. “A Contribution to the Mathematical Biophysics of Psychophysical Discrimination: II.”Bull. Math. Biophysics,1, 159–76.zbMATHCrossRefGoogle Scholar
  28. — 1941a. “Studies in the Mathematical Biophysics of Discrimination and Conditioning: I.”,3, 13–26.CrossRefGoogle Scholar
  29. — 1941b. “Studies in the Mathematical Biophysics of Discrimination and Conditioning: II. Special Case: Errors, Trials, and Number of Possible Responses.”,3, 71–77.CrossRefGoogle Scholar
  30. — 1941c. “Theory of the Distribution of Response Times in Nerve Fibers.”,3, 141–47.CrossRefGoogle Scholar
  31. — 1942a. “A Kinetic Theory of Diffusion Forces in Metabolizing Systems.”,4, 15–26.MathSciNetCrossRefGoogle Scholar
  32. — 1942b. “A Mathematical Analysis of Elongation and Constriction in Cell Division.”,4, 45–62.CrossRefGoogle Scholar
  33. — 1942c. “An Expression for the Rate of Return of an Egg after Artificial Deformation.”,4, 139–47.CrossRefGoogle Scholar
  34. — 1942d. “An Analysis of the Shapes of a Cell during Division with Particular Reference to the Role of Surface Tension.”,4, 151–54.CrossRefGoogle Scholar
  35. — 1942e. “Equilibrium Shapes in Non-Uniform Fields of Concentration.”,4, 155–58.MathSciNetCrossRefGoogle Scholar
  36. Lefschetz, S. 1930.Topology. New York: Am. Math. Society.zbMATHGoogle Scholar
  37. — 1949.Introduction to Topology. Princeton: Princeton University Press.zbMATHGoogle Scholar
  38. Lewin, Kurt. 1936.Principles of Topological Psychology. New York: McGraw-Hill.Google Scholar
  39. —. 1951,Field Theory in Social Science: Selected Theoretical Papers. (Ed. Dorwin Cartwright.) New York: Harper & Bros.Google Scholar
  40. Lotka, A. J. 1922a. “Contribution to the Energetics of Evolution.”Proc. Nat. Acad. Sci.,8, 147–51.CrossRefGoogle Scholar
  41. — 1922b. “Natural Selection as a Physical Principle.”,8, 151–54.CrossRefGoogle Scholar
  42. Luria, S. E. 1953.General Virology. New York: John Wiley & Sons, Inc.Google Scholar
  43. Morales, M. and R. Smith. 1944. “On the Theory of Blood-Tissue Exchanges: III. Circulation and Inert-Gas Exchanges at the Lung with Special Reference to Saturation.”Bull. Math. Biophysics,6, 141–52.CrossRefGoogle Scholar
  44. — 1945a. “A Note on the Physiological Arrangement of Tissues.”,7, 47–51.CrossRefGoogle Scholar
  45. — 1945b. “The Physiological Factors which Govern Inert Gas Exchange.”,7, 99–106.CrossRefGoogle Scholar
  46. Neumann, J. von 1951. “The General and Logical Theory of Automata.”Cerebral Mechanisms in Behavior (Ed. Lloyd A. Jeffress.) New York: John Wiley & Sons, Inc.Google Scholar
  47. Pitts, Walter and Warren S. McCulloch. 1947. “How We Know Universals: The Perception of Auditory and Visual Forms”Bull. Math. Biophysics,9, 127–47.CrossRefGoogle Scholar
  48. Pollard, E. 1953.The Physics of Viruses. New York: Academic Press.Google Scholar
  49. Rapoport, A. 1948. “Cycle Distributions in Random Nets.”Bull. Math. Biophysics,10, 145–57.CrossRefGoogle Scholar
  50. Rashevsky, N. 1933. “The Theoretical Physics of the Cell as a Basis for a General Physicochemical Theory of Organic Form.”Protoplasma,20, 180.Google Scholar
  51. — 1934. “Physico-Mathematical Aspects of the Gestalt Problem.”Phil. of Sci.,1, 409.CrossRefGoogle Scholar
  52. — 1938a.Mathematical Biophysics. Chicago: University of Chicago Press.zbMATHGoogle Scholar
  53. — 1938b. “The Relation of Mathematical Biophysics to Experimental Biology.Acta Biotheoretica,4, 133–53.zbMATHCrossRefGoogle Scholar
  54. — 1939. “The Mechanism of Cell Division.”Bull. Math. Biophysics,1, 23–30.zbMATHCrossRefGoogle Scholar
  55. Rashevsky, N. 1943a. “Outline of a New Mathematical Approach to General Biology: I.”,5, 33–47.CrossRefGoogle Scholar
  56. — 1943b. “Outline of a New Mathematical Approach to General Biology: II.”,5, 49–64.CrossRefGoogle Scholar
  57. — 1943c. “On the Form of Plants and Animals.”,5, 69–73.zbMATHMathSciNetCrossRefGoogle Scholar
  58. — 1944. “Studies in the Physicomathematical Theory of Organic Form.”,6, 1–59.MathSciNetCrossRefGoogle Scholar
  59. — 1948.Mathematical Biophysics. Rev. Ed. Chicago: University of Chicago Press.zbMATHGoogle Scholar
  60. — 1952. “Some Suggestions for a New Theory of Cell Division.”Bull. Math. Biophysics,14, 293–305.CrossRefGoogle Scholar
  61. — and V. Brown. 1944. “Contributions to the Mathematical Biophysics of Visual Aesthetics.”,6, 163–68.CrossRefGoogle Scholar
  62. Reidemeister, K. 1933.Einführung in die Kombinatorische Topologie. New York: F. Ungar.Google Scholar
  63. Schmidt, George. 1953. “The Time Course of Capillary Exchange.”Bull. Math. Biophysics,15, 477–88.CrossRefGoogle Scholar
  64. Seifert, H. and W. Threlfall. 1934.Lehrbuch der Topologie. Leipzig and Berlin: B. G. Teubner.zbMATHGoogle Scholar
  65. Sheer, B. T. 1948.Comparative Physiology, New York, John Wiley & Sons, Inc.Google Scholar
  66. Thompson, D.'Arcy W. 1917.On Growth and Form. Cambridge: At the University Press.zbMATHGoogle Scholar
  67. Wright, Sewall. 1931. “Evolution in Mendelian Populations.”Genetics,16, 97–159.Google Scholar
  68. —. 1938a. “Size of Population and Breeding Structure in Relation to Evolution.”Science,87, 430–31.Google Scholar
  69. —. 1938b. “The Distribution of Gene Frequencies under Irreversible Mutation.”Nat. Acad. Sci.,24, 253–59.zbMATHCrossRefGoogle Scholar
  70. —. 1945. “The Differential Equation of the Gene Frequencies.”Proc. Nat. Acad. Sci.,31, 382–89.zbMATHMathSciNetCrossRefGoogle Scholar
  71. —. 1950. “Genetical Structure of Populations.”Nature,166, 247.CrossRefGoogle Scholar
  72. —. 1951. “The Genetical Structure of Populations,”Ann. Eugenics,15, 323–54.zbMATHMathSciNetGoogle Scholar

Copyright information

© University of Chicago 1954

Authors and Affiliations

  • N. Rashevsky
    • 1
  1. 1.Committee on Mathematical BiologyThe University of ChicagoChicagoUSA

Personalised recommendations