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

- 488 Downloads
- 139 Citations

## 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 few*primordial* 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.

## Keywords

Biological Function Transformation Rule High Animal Mathematical Biophysics Organic World## Preview

Unable to display preview. Download preview PDF.

## Literature

- Blair, H. A. 1932a. “On the Intensity-Time Relations for Stimulation by Electric Currents: I.”
*Jour. Gen. Physiol.*,**15**, 709–29.CrossRefGoogle Scholar - — 1932b. “On the Intensity-Time Relations for Stimulation by Electric Currents: II.”,
**15**, 731–55.CrossRefGoogle 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.CrossRefGoogle 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**.Google Scholar - 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 - Hearon, J. Z. 1949a. “The Steady State Kinetics of Some Biological Systems: I.”
*Bull. Math. Biophysics*,**11**, 29–50.CrossRefGoogle Scholar - — 1949b. “The Steady State Kinetics of Some Biological Systems: II.”,
**11**, 83–95.CrossRefGoogle Scholar - — 1950a. “The Steady State Kinetics of Some Biological Systems: III. Thermodynamic Aspects.”,
**12**, 57–83.CrossRefGoogle Scholar - — 1950b. “The Steady State Kinetics of Some Biological Systems: IV. Thermodynamic Aspects.”,
**12**, 85–106.CrossRefGoogle Scholar - — 1950c. “Some Cellular Diffusion Problems Based on Onsager's Generalization of Fick's Law.”,
**12**, 135–59.CrossRefGoogle Scholar - Hill, A. V. 1936. “Excitation and Accommodation in Nerve.”
*Proc. Roy. Soc. Lond. B*,**119**, 305–55.CrossRefGoogle 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.zbMATHCrossRefGoogle Scholar - — 1940. “A Neural Mechanism for Discrimination: II. Discrimination of Weights.”
*Bull. Math. Biophysics*,**2**, 1–13.CrossRefGoogle 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.CrossRefGoogle Scholar - —. 1952. “Some Contributions to the Mathematical Biology of Blood Circulation. Reflections of Pressure Waves in the Arterial System.”,
**14**, 327–50.MathSciNetCrossRefGoogle 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.zbMATHCrossRefGoogle Scholar - — 1939. “A Contribution to the Mathematical Biophysics of Psychophysical Discrimination: II.”
*Bull. Math. Biophysics*,**1**, 159–76.zbMATHCrossRefGoogle Scholar - — 1941a. “Studies in the Mathematical Biophysics of Discrimination and Conditioning: I.”,
**3**, 13–26.CrossRefGoogle Scholar - — 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 - — 1941c. “Theory of the Distribution of Response Times in Nerve Fibers.”,
**3**, 141–47.CrossRefGoogle Scholar - — 1942a. “A Kinetic Theory of Diffusion Forces in Metabolizing Systems.”,
**4**, 15–26.MathSciNetCrossRefGoogle Scholar - — 1942b. “A Mathematical Analysis of Elongation and Constriction in Cell Division.”,
**4**, 45–62.CrossRefGoogle Scholar - — 1942c. “An Expression for the Rate of Return of an Egg after Artificial Deformation.”,
**4**, 139–47.CrossRefGoogle Scholar - — 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 - — 1942e. “Equilibrium Shapes in Non-Uniform Fields of Concentration.”,
**4**, 155–58.MathSciNetCrossRefGoogle Scholar - Lefschetz, S. 1930.
*Topology*. New York: Am. Math. Society.zbMATHGoogle Scholar - — 1949.
*Introduction to Topology*. Princeton: Princeton University Press.zbMATHGoogle 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.CrossRefGoogle Scholar - — 1922b. “Natural Selection as a Physical Principle.”,
**8**, 151–54.CrossRefGoogle 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.CrossRefGoogle Scholar - — 1945a. “A Note on the Physiological Arrangement of Tissues.”,
**7**, 47–51.CrossRefGoogle Scholar - — 1945b. “The Physiological Factors which Govern Inert Gas Exchange.”,
**7**, 99–106.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar - — 1938a.
*Mathematical Biophysics*. Chicago: University of Chicago Press.zbMATHGoogle Scholar - — 1938b. “The Relation of Mathematical Biophysics to Experimental Biology.
*Acta Biotheoretica*,**4**, 133–53.zbMATHCrossRefGoogle Scholar - — 1939. “The Mechanism of Cell Division.”
*Bull. Math. Biophysics*,**1**, 23–30.zbMATHCrossRefGoogle Scholar - Rashevsky, N. 1943a. “Outline of a New Mathematical Approach to General Biology: I.”,
**5**, 33–47.CrossRefGoogle Scholar - — 1943b. “Outline of a New Mathematical Approach to General Biology: II.”,
**5**, 49–64.CrossRefGoogle Scholar - — 1943c. “On the Form of Plants and Animals.”,
**5**, 69–73.zbMATHMathSciNetCrossRefGoogle Scholar - — 1944. “Studies in the Physicomathematical Theory of Organic Form.”,
**6**, 1–59.MathSciNetCrossRefGoogle Scholar - — 1948.
*Mathematical Biophysics*. Rev. Ed. Chicago: University of Chicago Press.zbMATHGoogle Scholar - — 1952. “Some Suggestions for a New Theory of Cell Division.”
*Bull. Math. Biophysics*,**14**, 293–305.CrossRefGoogle Scholar - — and V. Brown. 1944. “Contributions to the Mathematical Biophysics of Visual Aesthetics.”,
**6**, 163–68.CrossRefGoogle 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.CrossRefGoogle Scholar - Seifert, H. and W. Threlfall. 1934.
*Lehrbuch der Topologie*. Leipzig and Berlin: B. G. Teubner.zbMATHGoogle 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.zbMATHGoogle 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.zbMATHCrossRefGoogle Scholar - —. 1945. “The Differential Equation of the Gene Frequencies.”
*Proc. Nat. Acad. Sci.*,**31**, 382–89.zbMATHMathSciNetCrossRefGoogle Scholar - —. 1951. “The Genetical Structure of Populations,”
*Ann. Eugenics*,**15**, 323–54.zbMATHMathSciNetGoogle Scholar