Skip to main content
SpringerLink
Log in

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

  • Published:
The bulletin of mathematical biophysics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature

  • Blair, H. A. 1932a. “On the Intensity-Time Relations for Stimulation by Electric Currents: I.”Jour. Gen. Physiol.,15, 709–29.

    Article  Google Scholar 

  • — 1932b. “On the Intensity-Time Relations for Stimulation by Electric Currents: II.”,15, 731–55.

    Article  Google Scholar 

  • Buchsbaum, R. 1938.Animals without Backbones. Chicago: University of Chicago Press.

    Google Scholar 

  • — 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 

  • Cohn, David. 1953. “Optimal Systems: I. The Vascular System.”Bull Math. Biophysics,16, 59–74.

    Article  Google Scholar 

  • Culbertson, J. T. 1950.Consciousness and Behavior: A Neural Analysis of Behavior and Consciousness. Dubuque, Ia.: Wm. C. Brown Co.

    Google Scholar 

  • Einstein, A. 1952.The Meaning of Relativity. Princeton: Princeton University Press.

    Google Scholar 

  • Fuller, H. J. and O. Tippo. 1953.College Botany. New York: Henry Holt.

    Google Scholar 

  • Haldane, J. B. S. 1924. “A Mathematical Theory of Natural and Artificial Selection I.”Trans. Comb. Phil. Soc.,23.

  • Harvey, E. N. and H. Shapiro. 1941. “The Recovery Period (Relaxation) of Marine Eggs after Deformation.”Jour. Cell. and Comp. Physiol.,17, 135–44.

    Article  Google Scholar 

  • Hearon, J. Z. 1949a. “The Steady State Kinetics of Some Biological Systems: I.”Bull. Math. Biophysics,11, 29–50.

    Article  Google Scholar 

  • — 1949b. “The Steady State Kinetics of Some Biological Systems: II.”,11, 83–95.

    Article  Google Scholar 

  • — 1950a. “The Steady State Kinetics of Some Biological Systems: III. Thermodynamic Aspects.”,12, 57–83.

    Article  Google Scholar 

  • — 1950b. “The Steady State Kinetics of Some Biological Systems: IV. Thermodynamic Aspects.”,12, 85–106.

    Article  Google Scholar 

  • — 1950c. “Some Cellular Diffusion Problems Based on Onsager's Generalization of Fick's Law.”,12, 135–59.

    Article  Google Scholar 

  • Hill, A. V. 1936. “Excitation and Accommodation in Nerve.”Proc. Roy. Soc. Lond. B,119, 305–55.

    Article  Google Scholar 

  • Hogben, L. 1946.An Introduction to Mathematical Genetics. New York: W. W. Norton & Company, Inc.

    Google Scholar 

  • Householder, A. S. 1939. “A Neural Mechanism for Discrimination.”Psychometrika,4, 45–58.

    Article  MATH  Google Scholar 

  • — 1940. “A Neural Mechanism for Discrimination: II. Discrimination of Weights.”Bull. Math. Biophysics,2, 1–13.

    Article  Google Scholar 

  • Jacobson, E. 1930a. “Electrical Measurements of Neuromuscular States during Mental Activities. III. Visual Imagination and Recollection.”Am. Jour. Physiol.,95, 694–702.

    Google Scholar 

  • — 1930b. “Electrical Measurements of Neuromuscular States during Mental Activities. IV. Evidence of Contraction of Specific Muscles during Imagination.”,95, 703–12.

    Google Scholar 

  • — 1931. “Electrical Measurements of Neuromuscular States during Mental Activities. VII. Imagination, Recollection and Abstract Thinking Involving the Speech Musculature.”,97, 200–09.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • —. 1952. “Some Contributions to the Mathematical Biology of Blood Circulation. Reflections of Pressure Waves in the Arterial System.”,14, 327–50.

    Article  MathSciNet  Google Scholar 

  • König, D. 1936.Theorie der Endlichen und Unendlichen Graphen. Leipzig: Akademische Verlagsgesellschaft.

    Google Scholar 

  • Landahl, H. D. 1938. “A Contribution to the Mathematical Biophysics of Psychophysical Discrimination.”Psychometrika,3, 107–25.

    Article  MATH  Google Scholar 

  • — 1939. “A Contribution to the Mathematical Biophysics of Psychophysical Discrimination: II.”Bull. Math. Biophysics,1, 159–76.

    Article  MATH  Google Scholar 

  • — 1941a. “Studies in the Mathematical Biophysics of Discrimination and Conditioning: I.”,3, 13–26.

    Article  Google Scholar 

  • — 1941b. “Studies in the Mathematical Biophysics of Discrimination and Conditioning: II. Special Case: Errors, Trials, and Number of Possible Responses.”,3, 71–77.

    Article  Google Scholar 

  • — 1941c. “Theory of the Distribution of Response Times in Nerve Fibers.”,3, 141–47.

    Article  Google Scholar 

  • — 1942a. “A Kinetic Theory of Diffusion Forces in Metabolizing Systems.”,4, 15–26.

    Article  MathSciNet  Google Scholar 

  • — 1942b. “A Mathematical Analysis of Elongation and Constriction in Cell Division.”,4, 45–62.

    Article  Google Scholar 

  • — 1942c. “An Expression for the Rate of Return of an Egg after Artificial Deformation.”,4, 139–47.

    Article  Google Scholar 

  • — 1942d. “An Analysis of the Shapes of a Cell during Division with Particular Reference to the Role of Surface Tension.”,4, 151–54.

    Article  Google Scholar 

  • — 1942e. “Equilibrium Shapes in Non-Uniform Fields of Concentration.”,4, 155–58.

    Article  MathSciNet  Google Scholar 

  • Lefschetz, S. 1930.Topology. New York: Am. Math. Society.

    MATH  Google Scholar 

  • — 1949.Introduction to Topology. Princeton: Princeton University Press.

    MATH  Google Scholar 

  • Lewin, Kurt. 1936.Principles of Topological Psychology. New York: McGraw-Hill.

    Google Scholar 

  • —. 1951,Field Theory in Social Science: Selected Theoretical Papers. (Ed. Dorwin Cartwright.) New York: Harper & Bros.

    Google Scholar 

  • Lotka, A. J. 1922a. “Contribution to the Energetics of Evolution.”Proc. Nat. Acad. Sci.,8, 147–51.

    Article  Google Scholar 

  • — 1922b. “Natural Selection as a Physical Principle.”,8, 151–54.

    Article  Google Scholar 

  • Luria, S. E. 1953.General Virology. New York: John Wiley & Sons, Inc.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • — 1945a. “A Note on the Physiological Arrangement of Tissues.”,7, 47–51.

    Article  Google Scholar 

  • — 1945b. “The Physiological Factors which Govern Inert Gas Exchange.”,7, 99–106.

    Article  Google Scholar 

  • 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 

  • Pitts, Walter and Warren S. McCulloch. 1947. “How We Know Universals: The Perception of Auditory and Visual Forms”Bull. Math. Biophysics,9, 127–47.

    Article  Google Scholar 

  • Pollard, E. 1953.The Physics of Viruses. New York: Academic Press.

    Google Scholar 

  • Rapoport, A. 1948. “Cycle Distributions in Random Nets.”Bull. Math. Biophysics,10, 145–57.

    Article  Google Scholar 

  • 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 

  • — 1934. “Physico-Mathematical Aspects of the Gestalt Problem.”Phil. of Sci.,1, 409.

    Article  Google Scholar 

  • — 1938a.Mathematical Biophysics. Chicago: University of Chicago Press.

    MATH  Google Scholar 

  • — 1938b. “The Relation of Mathematical Biophysics to Experimental Biology.Acta Biotheoretica,4, 133–53.

    Article  MATH  Google Scholar 

  • — 1939. “The Mechanism of Cell Division.”Bull. Math. Biophysics,1, 23–30.

    Article  MATH  Google Scholar 

  • Rashevsky, N. 1943a. “Outline of a New Mathematical Approach to General Biology: I.”,5, 33–47.

    Article  Google Scholar 

  • — 1943b. “Outline of a New Mathematical Approach to General Biology: II.”,5, 49–64.

    Article  Google Scholar 

  • — 1943c. “On the Form of Plants and Animals.”,5, 69–73.

    Article  MATH  MathSciNet  Google Scholar 

  • — 1944. “Studies in the Physicomathematical Theory of Organic Form.”,6, 1–59.

    Article  MathSciNet  Google Scholar 

  • — 1948.Mathematical Biophysics. Rev. Ed. Chicago: University of Chicago Press.

    MATH  Google Scholar 

  • — 1952. “Some Suggestions for a New Theory of Cell Division.”Bull. Math. Biophysics,14, 293–305.

    Article  Google Scholar 

  • — and V. Brown. 1944. “Contributions to the Mathematical Biophysics of Visual Aesthetics.”,6, 163–68.

    Article  Google Scholar 

  • Reidemeister, K. 1933.Einführung in die Kombinatorische Topologie. New York: F. Ungar.

    Google Scholar 

  • Schmidt, George. 1953. “The Time Course of Capillary Exchange.”Bull. Math. Biophysics,15, 477–88.

    Article  Google Scholar 

  • Seifert, H. and W. Threlfall. 1934.Lehrbuch der Topologie. Leipzig and Berlin: B. G. Teubner.

    MATH  Google Scholar 

  • Sheer, B. T. 1948.Comparative Physiology, New York, John Wiley & Sons, Inc.

    Google Scholar 

  • Thompson, D.'Arcy W. 1917.On Growth and Form. Cambridge: At the University Press.

    MATH  Google Scholar 

  • Wright, Sewall. 1931. “Evolution in Mendelian Populations.”Genetics,16, 97–159.

    Google Scholar 

  • —. 1938a. “Size of Population and Breeding Structure in Relation to Evolution.”Science,87, 430–31.

    Google Scholar 

  • —. 1938b. “The Distribution of Gene Frequencies under Irreversible Mutation.”Nat. Acad. Sci.,24, 253–59.

    Article  MATH  Google Scholar 

  • —. 1945. “The Differential Equation of the Gene Frequencies.”Proc. Nat. Acad. Sci.,31, 382–89.

    Article  MATH  MathSciNet  Google Scholar 

  • —. 1950. “Genetical Structure of Populations.”Nature,166, 247.

    Article  Google Scholar 

  • —. 1951. “The Genetical Structure of Populations,”Ann. Eugenics,15, 323–54.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Committee on Mathematical Biology, The University of Chicago, Chicago, USA

    N. Rashevsky

Authors
  1. N. Rashevsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rashevsky, N. Topology and life: In search of general mathematical principles in biology and sociology. Bulletin of Mathematical Biophysics 16, 317–348 (1954). https://doi.org/10.1007/BF02484495

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02484495

Keywords

Search

Navigation