Mathematical biology/Signed Articles/Biology's next microscope: Mathematics
 This article is an adaptation of
 Mathematics is biology’s next microscope, only better; biology is mathematics’ next physics, only better,
 originally published by Joel E. Cohen in PLoS Biol 2(12): e439 (2004).^{[1]}
Although mathematics has long been intertwined with the biological sciences, an explosive synergy between biology and mathematics seems poised to enrich and extend both fields greatly in the coming decades ^{[2]} ^{[3]} ^{[4]} ^{[5]} ^{[6]} ^{[7]}. Biology will increasingly stimulate the creation of qualitatively new realms of mathematics. Why?
In biology, ensemble properties emerge at each level of organization from the interactions of heterogeneous biological units at that level and at lower and higher levels of organization (larger and smaller physical scales, faster and slower temporal scales). New mathematics will be required to cope with these ensemble properties and with the heterogeneity of the biological units that compose ensembles at each level.
How the discovery of the microscope challenged biology
The discovery of the microscope in the late 17th century [— by Anton van Leeuwenhoek (16321723) —] caused a revolution in biology by revealing otherwise invisible and previously unsuspected worlds.^{[8]} Western cosmology from classical times through the end of the Renaissance envisioned a system with three types of spheres: the sphere of man, exemplified by his imperfectly round head; the sphere of the world, exemplified by the imperfectly spherical earth; and the eight perfect spheres of the universe, in which the seven (then known) planets moved and the outer stars were fixed.^{[9]} The discovery of a microbial world too small to be seen by the naked eye challenged the completeness of this cosmology and unequivocally demonstrated the existence of living creatures unknown to the Scriptures of Old World religions.^{[8]}
Mathematics as microscope
Mathematics broadly interpreted is a more general microscope. It can reveal otherwise invisible worlds in all kinds of data, not only optical. For example, computed tomography can reveal a crosssection of a human head from the density of Xray beams without ever opening the head, by using the Radon transform to infer the densities of materials at each location within the head (Hsieh 2003). Charles Darwin was right when he wrote that people with an understanding “of the great leading principles of mathematics... seem to have an extra sense” (F. Darwin 1905). Today’s biologists increasingly recognize that appropriate mathematics can help interpret any kind of data. In this sense, mathematics is biology’s next microscope, only better.
Role of biology in the development of mathematics
Conversely, mathematics will benefit increasingly from its involvement with biology, just as mathematics has already benefited and will continue to benefit from its historic involvement with physical problems. In classical times, physics, as first an applied then a basic science, stimulated enormous advances in mathematics. For example, geometry reveals by its very etymology (geometry) its origin in the needs to survey the lands and waters of Earth. Geometry was used to lay out fields in Egypt after the flooding of the Nile, to aid navigation, to aid city planning. The inventions of the calculus by Isaac Newton and Gottfried Leibniz in the later 17th century were stimulated by physical problems such as planetary orbits and optical calculations.
In the coming century, biology will stimulate the creation of entirely new realms of mathematics. In this sense, biology is mathematics’ next physics, only better. Biology will stimulate fundamentally new mathematics because living nature is qualitatively more heterogeneous than nonliving nature. For example, it is estimated that there are 2,000–5,000 species of rocks and minerals in the earth’s crust, generated from the hundred or so naturally occurring elements (Shipman et al. 2003; chapter 21 estimates 2,000 minerals in Earth’s crust). By contrast, there are probably between 3 million and 100 million biological species on Earth, generated from a small fraction of the naturally occurring elements. If species of rocks and minerals may validly be compared with species of living organisms, the living world has at least a thousand times the diversity of the nonliving. This comparison omits the enormous evolutionary importance of individual variability within species. Coping with the hyperdiversity of life at every scale of spatial and temporal organization will require fundamental conceptual advances in mathematics.
The past interactions between mathematics and biology
William Harvey introduces ‘offtheshelf’ mathematics into biology
The interactions between mathematics and biology at present follow from their interactions over the last half millennium. The discovery of the New World by Europeans approximately 500 years ago—and of its many biological species not described in religious Scriptures—gave impetus to major conceptual progress in biology.
The outstanding milestone in the early history of biological quantitation was the work of William Harvey, Exercitatio Anatomica De Motu Cordis et Sanguinis In Animalibus (An Anatomical Disquisition on the Motion of the Heart and Blood in Animals) (Harvey 1847) [generally referred to as De Motu Cordis], first published in 1628. Harvey’s demonstration that the blood circulates was the pivotal founding event of the modern interaction between mathematics and biology. His elegant reasoning is worth understanding.
From the time of the ancient Greek physician Galen (131–201 C.E.) until William Harvey studied medicine in Padua (1600–1602, while Galileo was active there), it was believed that there were two kinds of blood, arterial blood and venous blood. Both kinds of blood were believed to ebb and flow under the motive power of the liver, just as the tides of the earth ebbed and flowed under the motive power of the moon. Harvey became physician to the king of England. He used his position of privilege to dissect deer from the king’s deer park as well as executed criminals. Harvey observed that the veins in the human arm have oneway valves that permit blood to flow from the periphery toward the heart but not in the reverse direction. Hence the theory that the blood ebbs and flows in both veins and arteries could not be correct.
Harvey also observed that the heart was a contractile muscle with one way valves between the chambers on each side. He measured the volume of the left ventricle of dead human hearts and found that it held about two ounces (about 60 ml), varying from 1.5 to three ounces in different individuals. He estimated that at least oneeighth and perhaps as much as onequarter of the blood in the left ventricle was expelled with each stroke of the heart. He measured that the heart beat 60–1 00 times per minute. Therefore, the volume of blood expelled from the left ventricle per hour was about 60 ml × 1/8 × 60 beats/minute × 60 minutes/hour, or 27 liters/hour. However, the average human has only 5.5 liters of blood (a quantity that could be estimated by draining a cadaver). Therefore, the blood must be like a stage army that marches off one side of the stage, returns behind the scenes, and reenters from the other side of the stage, again and again. The large volume of blood pumped per hour could not possibly be accounted for by the thenprevalent theory that the blood originated from the consumption of food.
Harvey inferred that there must be some small vessels that conveyed the blood from the outgoing arteries to the returning veins, but he was not able to see those small vessels. His theoretical prediction, based on his meticulous anatomical observations and his mathematical calculations, was spectacularly confirmed more than half a century later when Marcello Malpighi (1628–1694) saw the capillaries under a microscope.
Harvey’s discovery illustrates the enormous power of simple, offtheshelf mathematics combined with careful observation and clear reasoning. It set a high standard for all later uses of mathematics in biology.
Mendel’s mathematics and the discovery of genes
Mathematics was crucial in the discovery of genes by Mendel (Orel 1984) and in the theory of evolution.
Mathematics was and continues to be the principal means of integrating evolution and genetics since the classic work of R. A. Fisher, J. B. S. Haldane, and S. Wright in the first half of the 20th century (Provine 2001).
The enrichment of biology by mathematical progress in geometry, topology, algebra, and analysis
Over the last 500 years, mathematics has made amazing progress in each of its three major fields: geometry and topology, algebra, and analysis. This progress has enriched all the biological sciences.
In 1637, René Descartes linked the featureless plane of Greek geometry to the symbols and formulas of Arabic algebra by imposing a coordinate system (conventionally, a horizontal xaxis and a vertical yaxis) on the geometric plane and using numbers to measure distances between points. If every biologist who plotted data on x–y coordinates acknowledged the contribution of Descartes to biological understanding, the key role of mathematics in biology would be uncontested.
Another highlight of the last five centuries of geometry was the invention of nonEuclidean geometries (1823–1830). Shocking at first, these geometries unshackled the possibilities of mathematical reasoning from the intuitive perception of space. These nonEuclidean geometries have made significant contributions to biology in facilitating, for example, mapping the brain onto a fl at surface (Hurdal et al. 1999; Bowers and Hurdal 2003).
In algebra, efforts to find the roots of equations led to the discovery of the symmetries of roots of equations and thence to the invention of group theory, which finds routine application in the study of crystallographic groups by structural biologists today. Generalizations of single linear equations to families of simultaneous multivariable linear equations stimulated the development of linear algebra and the European reinvention and naming of matrices in the mid 19th century. The use of a matrix of numbers to solve simultaneous systems of linear equations can be traced back in Chinese mathematics to the period from 300 B.C.E. to 200 C.E. (in a work by Chiu Chang Suan Shu called Nine Chapters of the Mathematical Art; Smoller 2001). In the 19th century, matrices were considered the epitome of useless mathematical abstraction. Then, in the 20th century, it was discovered, for example, that the numerical processes required for the cohortcomponent method of population projection can be conveniently summarized and executed using matrices (Keyfi tz 1968). Today the use of matrices is routine in agencies responsible for making official population projections as well as in populationbiological research on human and nonhuman populations (Caswell 2001).
Finally, analysis, including the calculus of Newton and Leibniz and probability theory, is the line between ancient thought and modern thought. Without an understanding of the concepts of analysis, especially the concept of a limit, it is not possible to grasp much of modern science, technology, or economic theory. Those who understand the calculus, ordinary and partial differential equations, and probability theory have a way of seeing and understanding the world, including the biological world, that is unavailable to those who do not.
Biology enriches mathematics
Conceptual and scientific challenges from biology have enriched mathematics by leading to innovative thought about new kinds of mathematics. Table 1 lists examples of new and useful mathematics arising from problems in the life sciences broadly construed, including biology and some social sciences. Many of these developments blend smoothly into their antecedents and later elaborations. For example, game theory has a history before the work of John von Neumann (von Neumann 1959; von Neumann and Morgenstern 1953), and Karl Pearson’s development of the correlation coefficient (Pearson and Lee 1903) rested on earlier work by Francis Galton (1889).

Present and future interactions of mathematics and biology
To see how the interactions of biology and mathematics may proceed in the future, it is helpful to map the present landscapes of biology and applied mathematics.
The biological landscape may be mapped as a rectangular table with different rows for different questions and different columns for different biological domains. Biology asks six kinds of questions.
 How is it built?
 How does it work?
 What goes wrong?
 How is it fixed?
 How did it begin?
 What is it for?
Those are questions, respectively, about structures, mechanisms, pathologies, repairs, origins, and functions or purposes. The former teleological interpretation of purpose has been replaced by an evolutionary perspective. Biological domains, or levels of organization, include molecules, cells, tissues, organs, individuals, populations, communities, ecosystems or landscapes, and the biosphere. Many biological research problems can be classified as the combination of one or more questions directed to one or more domains.
In addition, biological research questions have important dimensions of time and space. Timescales of importance to biology range from the extremely fast processes of photosynthesis to the billions of years of living evolution on Earth. Relevant spatial scales range from the molecular to the cosmic (cosmic rays may have played a role in evolution on Earth). The questions and the domains of biology behave differently on different temporal and spatial scales.
The opportunities and the challenges that biology offers mathematics arise because the units at any given level of biological organization are heterogeneous, and the outcomes of their interactions (sometimes called “emergent phenomena” or “ensemble properties”) on any selected temporal and spatial scale may be substantially affected by the heterogeneity and interactions of biological components at lower and higher levels of biological organization and at smaller and larger temporal and spatial scales^{[10]} ^{[11]}
The landscape of applied mathematics is better visualized as a tetrahedron (a pyramid with a triangular base) than as a matrix with temporal and spatial dimensions. (Mathematical imagery, such as a tetrahedron for applied mathematics and a matrix for biology, is useful even in trying to visualize the landscapes of biology and mathematics.) The four main points of the applied mathematical landscape are data structures, algorithms, theories and models (including all pure mathematics), and computers and software.
 Data structures are ways to organize data, such as the matrix used above to describe the biological landscape.
 Algorithms are procedures for manipulating symbols. Some algorithms are used to analyze data, others to analyze models.
 Theories and models, including the theories of pure mathematics, are used to analyze both data and ideas. Mathematics and mathematical theories provide a testing ground for ideas in which the strength of competing theories can be measured.
 Computers and software are an important, and frequently the most visible, vertex of the applied mathematical landscape. However, cheap, easy computing increases the importance of theoretical understanding of the results of computation. Theoretical understanding is required as a check on the great risk of error in software, and to bridge the enormous gap between computational results and insight or understanding.
The landscape of research in mathematics and biology contains all combinations of one or more biological questions, domains, time scales, and spatial scales with one or more data structures, algorithms, theories or models, and means of computation (typically software and hardware).
The following example from cancer biology illustrates such a combination: the question, “how does it work?” is approached in the domain of cells (specifically, human cancer cells) with algorithms for correlation and hierarchical clustering.
An example: Gene expression and drug activity in human cancer
Suppose a person has a cancer. Could information about the activities of the genes in the cells of the person’s cancer guide the use of cancertreatment drugs so that more effective drugs are used and less effective drugs are avoided? To suggest answers to this question, Scherf et al. (2000) ingeniously applied offtheshelf mathematics, specifically, correlation— invented nearly a century earlier by Karl Pearson (Pearson and Lee 1903) in a study of human inheritance—and clustering algorithms, which apparently had multiple sources of invention, including psychometrics (Johnson 1967). They applied these simple tools to extract useful information from, and to combine for the first time, enormous databases on molecular pharmacology and gene expression (http:⁄⁄discover.nci.nih.gov/arraytools/). They used two kinds of information from the drug discovery program of the National Cancer Institute. The first kind of information described gene expression in 1,375 genes of each of 60 human cancer cell lines. A target matrix T had, as the numerical entry in row g and column c, the relative abundance of the mRNA transcript of gene g in cell line c. The drug activity matrix A summarized the pharmacology of 1,400 drugs acting on each of the same 60 human cancer cell lines, including 118 drugs with “known mechanism of action.” The number in row d and column c of the drug activity matrix A was the activity of drug d in suppressing the growth of cell line c, or, equivalently, the sensitivity of cell line c to drug d. The target matrix T for gene expression contained 82,500 numbers, while the drug activity matrix A had 84,000 numbers.
These two matrices have the same set of column headings but have different row labels. Given the two matrices, precisely five sets of possible correlations could be calculated, and Scherf et al. calculated all five:
 The correlation between two different columns of the activity matrix A led to a clustering of cell lines according to their similarity of response to different drugs.
 The correlation between two different columns of the target matrix T led to a clustering of the cell lines according to their similarity of gene expression. This clustering differed very substantially from the clustering of cell lines by drug sensitivity.
 The correlation between different rows of the activity matrix A led to a clustering of drugs according to their activity patterns across all cell lines.
 The correlation between different rows of the target matrix T led to a clustering of genes according to the pattern of mRNA expressed across the 60 cell lines.
 Finally, the correlation between a row of the activity matrix A and a row of the target matrix T described the positive or negative covariation of drug activity with gene expression.
A positive correlation meant that the higher the level of gene expression across the 60 cancer cell lines, the higher the effectiveness of the drug in suppressing the growth of those cell lines. The result of analyzing several hundred thousand experiments is summarized in a single picture called a clustered image map (Figure 1). This clustered image map plots gene expression–drug activity correlations as a function of clustered genes (horizontal axis) and clustered drugs (showing only the 118 drugs with “known function”) on the vertical axis (Weinstein et al. 1997).
What use is this? If a person’s cancer cells have high expression for a particular gene, and the correlation of that gene with drug activity is highly positive, then that gene may serve as a marker for tumor cells likely to be inhibited effectively by that drug. If the correlation with drug activity is negative, then the marker gene may indicate when use of that drug is contraindicated.
While important scientific questions about this approach remain open, its usefulness in generating hypotheses to be tested by further experiments is obvious. It is a very insightful way of organizing and extracting meaning from many individual observations. Without the microscope of mathematical methods and computational power, the insight given by the clustered image map could not be achieved.
The future
Box 1. Challenges Here are five biological challenges that could stimulate, and benefit from, major innovations in mathematics
Here are five mathematical challenges that would contribute to the progress of biology
From: Cohen JE (2004) Mathematics is biology’s next microscope, only better; biology is mathematics’ next physics, only better. PLoS Biol 2(12): e439. 
To realize the possibilities of effective synergy between biology and mathematics will require both avoiding potential problems and seizing potential opportunities.
Potential problems
The productive interaction of biology and mathematics will face problems that concern education, intellectual property, and national security.
Educating the next generation of scientists will require early emphasis on quantitative skills in primary and secondary schools and more opportunities for training in both biology and mathematics at undergraduate, graduate, and postdoctoral levels (CUBE 2003).
Intellectual property rights may both stimulate and obstruct the potential synergy of biology and mathematics. Science is a potlatch culture. The bigger one’s gift to the common pool of knowledge and techniques, the higher one’s status, just as in the potlatch culture of the Native Americans of the northwest coast of North America. In the case of research in mathematics and biology, intellectual property rights to algorithms and databases need to balance the concerns of inventors, developers, and future researchers (Rai and Eisenberg 2003).
A third area of potential problems as well as opportunities is national security. Scientists and national defenders can collaborate by supporting and doing open research on the optimal design of monitoring networks and mitigation strategies for all kinds of biological attacks (Wein et al. 2003). But openness of scientificc methods or biological reagents in microbiology may pose security risks in the hands of terrorists. Problems of conserving privacy may arise when disparate databases are connected, such as physician payment databases with disease diagnosis databases, or health databases with law enforcement databases.
Opportunities
Mathematical models can circumvent ethical dilemmas. For example, in a study of the household transmission of Chagas disease in northwest Argentina, Cohen and Gürtler (2001) wanted to know—since dogs are a reservoir of infection—what would happen if dogs were removed from bedroom areas, without spraying households with insecticides against the insect that transmits infection. Because neither the householders nor the state public health apparatus can afford to spray the households in some areas, the realistic experiment would be to ask householders to remove the dogs without spraying. But a researcher who goes to a household and observes an insect infestation is morally obliged to spray and eliminate the infestation. In a detailed mathematical model, it was easy to set a variable representing the number of dogs in the bedroom areas to zero. All components of the model were based on measurements made in real villages. The calculation showed that banishing dogs from bedroom areas would substantially reduce the intensity of infection in the absence of spraying, though spraying would contribute to additional reductions in the intensity of infection. The model was used to do an experiment conceptually that could not be done ethically in a real village. The conceptual experiment suggested the value of educating villagers about the important health benefits of removing dogs from the bedroom areas.
The future of a scientific field is probably less predictable than the future in general. Doubtless, though, there will be exciting opportunities for the collaboration of mathematics and biology. Mathematics can help biologists grasp problems that are otherwise too big (the biosphere) or too small (molecular structure); too slow (macroevolution) or too fast (photosynthesis); too remote in time (early extinctions) or too remote in space (life at extremes on the earth and in space); too complex (the human brain) or too dangerous or unethical (epidemiology of infectious agents).
Box 1 summarizes five biological and five mathematical challenges where interactions between biology and mathematics may prove particularly fruitful.
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Notes cited as superscripts
Citations with author names bolded represent those added by Citizendium.
 ↑ This article represents a permissible (Creative Commons Attribution License) adaption and modification of an article by Joel E. Cohen of Rockefeller University published in the openaccess journal, PLoS Biology 1(12):e439 in 2004, under the full title: Mathematics is biology’s next microscope, only better; biology is mathematics’ next physics, only better.
 Copyright: © 2004 Joel E. Cohen. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 Joel E. Cohen is at the Laboratory of Populations, Rockefeller and Columbia Universities, New York, New York, United States of America. Email: cohen (at) rockefeller.edu.
 In the original PLoS Biology article, Dr. Cohen offered the following acknowledgements:
 This paper is based on a talk [by Dr. Joel Cohen] given on February 12, 2003, as the keynote address at the National Science Foundation (NSF)–National Institutes of Health (NIH) Joint Symposium on Accelerating Mathematical–Biological Linkages, Bethesda, Maryland; on June 12, 2003, as the first presentation in the 21st Century Biology Lecture Series, National Science Foundation, Arlington, Virginia; and on July 10, 2003, at a Congressional Lunch Briefing, cosponsored by the American Mathematical Society and Congressman Vernon J. Ehlers, Washington, D.C. Dr. Cohen thanks Margaret Palmer, Sam Scheiner, Michael Steuerwalt, James Cassatt, Mike Marron, John Whitmarsh, and directors of NSF and NIH for organizing the NSF–NIH meeting, Mary Clutter and Joann P. Roskoski for organizing his presentation at the NSF, Samuel M. Rankin III for organizing the American Mathematical Society Congressional Lunch Briefing, and Congressman Bob Filner for attending and participating. Dr. Cohen expresses gratitude for constructive editing by Philip Bernstein, helpful suggestions on earlier versions from Mary Clutter, Charles Delwiche, Bruce A. Fuchs, Yonatan Grad, Alan Hastings, Kevin Lauderdale, Zaida LutheySchulten, Daniel C. Reuman, Noah Rosenberg, Michael Pearson, and Samuel Scheiner, support from U.S. NSF grant DEB 9981552, the help of Kathe Rogerson, and the hospitality of Mr. and Mrs. William T. Golden during this work. Any opinions, findings, and conclusions or recommendations expressed in this material do not necessarily reflect the views of the NSF.
 Article reformatted to correspond to Citizendium style, initially by Citizendium Biology Editor Anthony.Sebastian("A.S"), with subheaders added, with Editor explanatory interpolations in square brackets, annotations of references, and a few editorial notes in References section.
 Citizendium makes no claim that the originator of the openaccess article, Joel E. Cohen, endorses Citizendium's modification of the original article, cited above.
 ↑ Levin S. (1992) Mathematics and biology: The interface.
 "The application of mathematics to biology has in turn had considerable effect on the development of new areas of mathematics. This may seem surprising, because of the different natures of biology and mathematics. Mathematics strongly prizes rigor and precision. Mathematical fact is immutable, and successful mathematical theories have lifetimes of hundreds or thousands of years. By contrast, most of our knowledge of biological systems is recent, and most biological theories evolve rapidly. Nonetheless, the interface between mathematics and biology has initiated and fostered new mathematical areas. This report highlights areas of mathematics that have been influenced greatly by biological thinking in the past, and presages future developments by identifying some areas of biology that will require the development of new mathematical tools."
 ↑ Murray, J.D. (1993). Mathematical Biology, Second, Corrected Edition. SpringerVerlag, Berlin.
 ↑ Jungck JR. (1997) Ten Equations That Changed Biology: Mathematics in ProblemSolving Biology Curricula. Bioscene 23:1136.
 "[For biology students] there is a need to identify a succinct list of achievements that represent the power of mathematics in biology. Hence, ten equations and a brief description of their historical importance are presented here in order to draw students' and faculty's attention to a variety of mathematical models that have been intrinsic to many of the significant discoveries in biology in the twentieth century." (See Jungck’s article for the actual equations and a discussion of their historical importance.)
 Fisher's "Fundamental Theorem of Natural Selection"
 Cormack Computer Assisted Tomography
 Genetic Mapping (The Haldane Function)
 FitchMargoliash Little Maximum Parsimony Algorithm for Phylogenetic Tree Construction
 LotkaVolterra Interspecific Competition Logistic Equations
 HodgkinHuxley (Goldman) Equations for Neural Axon Membrane Potential
 Michaelis  Menten Equation for Enzyme Kinetics:
 Allometry: E.g., MacArthurWilson Theory of Island Biogeography SpeciesArea Law
 Hypothesis Testing: LuriaDelbrück Fluctuation Test for Mutations in Viruses
 Crick  Griffith  Orgel Commafree Coding Theory for the Adaptor Hypothesis of Genetic Coding
 ↑ Hastings, A.; Arzberger, P.; Bolker, B.; Ives, T.; Johnson, N.; Palmer, M. (2004). Quantitative biology for the 21st century. Retrieved on 20080606.
 ↑ Palmer, M.A.; Arzberger, P.; Cohen, J.E.; Hastings, A.; Holt, R.D.; Others, (2003). "Accelerating mathematicalbiological linkages". Report of a joint National Science FoundationNational Institutes of Health Workshop: 1213.
 ↑ Hastings, A.; Palmer, M.A. (2003). "Mathematics and Biology: A Bright Future for Biologists and Mathematicians?". Science 299 (5615). Retrieved on 20080606.
 ↑ ^{8.0} ^{8.1} Waller J. (2002) The Discovery of the Germ: Twenty Years that Transformed the Way We Think About Disease. Series: Revolutions in Science. Series Ed., John Turney. Icon Books Ltd., Cambridge: UK. ISBN 1840463732. Available as eBook (Adobe Digital Editions)
 ” The first recorded dental floss took place in September of 1683. It was performed by Anthony van Leeuwenhoek, a minor city official in Holland, who became one of the finest microscope builders of his day. Having scraped some of the 'white matter' from between his 'usually very clean' teeth, Leeuwenhoek examined it beneath his microscope. Only a few months earlier this eminent Dutchman had been the first human to see microbes. But the sheer scale and diversity of the 'small living Animals' he witnessed after this historic floss simply astonished him.”
 ↑ Nicolson MH. (1960) The breaking of the circle: Studies in the effect of the “new science” upon seventeenthcentury poetry. Revised ed. New York: Columbia University Press. 216 p.
 ↑ Anderson PW. (1972) More is Different. Science 177:393396.
 "The behavior of large and complex aggregates of elementary particles, it turns out, is not to be understood in terms of a simple extrapolation of the properties of a few particles. Instead, at each level of complexity entirely new properties appear, and the understanding of the new behaviors requires research which I think is as fundamental in its nature as any other."
 ↑ Anderson PW. (1995) Physics: The opening to complexity. Proc Natl Acad Sci USA 92: 6653–6654. PMID 11607557.
 "Emergent complex phenomena are by no means in violation of the microscopic laws, but they do not appear as logically consequent on these laws."