Biological Evolution and Mathematics - Research Paper Example

Biological Evolution and Mathematics The evolution of mathematics as a body of knowledge is a thoroughly documented process over the of many generations. Voluminous treatises have been written as forms and concepts have been introduced and established to generate mathematics as we know it today. Theorists generally believe that Mathematics is immutable, such as Hermann Hankel and Burt Nanus (in De Pillis, 2002) and Aaboe (1998). The concept of the universality of mathematics has, however, been challenged by reputable philosophers and mathematicians D’ Ambrosio (1994), Dehaene (1997) and Ernest (1998). The following anecdote in Dehaene (1997) was, thus, chosen to introduce the arguments which tend to favor that mathematical forms and concepts are also affected by man’s biological evolution: “A nobleman wanted to shoot down a crow that had built its nest atop a tower on his domain. However, whenever he approached the tower, the bird flew out of gun range, and waited until the man departed. As soon as he left, it returned to its nest. The man decided to ask a neighbor for help. The two hunters entered the tower together, and later only one of them came out. But the crow did not fall into this trap and carefully waited for the second man to come out before returning. Neither did three, then four, the five men fool the clever bird. Each time, the crow would wait until all the hunters had departed. Eventually, the hunters came as a party of six. When five of them had left the tower, the bird not so numerate, after all, confidently came back and was shot down by the sixth hunter” (Dehaene, 1997, p. 13). From the foregoing anecdote borrowed from Dehaene (1997), the moral is that - had the bird known how to count at least until six, it would have survived the trap prepared by the nobleman. Accordingly, smarter crows would have evolved which are capable of recognizing numbers larger than five. Dehaene affirmed that the evolution of mathematics and the whole body of knowledge attributed to it is thoroughly studied and evidenced throughout its long history. But mathematics, according to Dehaene is not a rigid body of knowledge where its forms and concepts “and even its modes of reasoning have evolved over the course of many generations” (p. 246). A similar belief was maintained by Steiner (1998) who claimed that evolution encompasses not only the biological sphere but the entire universe, and that the laws of evolution function everywhere in the universe governing even the evolution of the mind. Additionally, Restivo (1993) argued that evolution, even biological evolution, is a knowledge process and that the paradigm of Darwin’s natural selection for the evolution of knowledge can be generalized to such cases as learning, thought, science and mathematics. While Dehaene (1997) believes that even when the Darwinian concept of evolution was adopted as reference for biologists, both biological and cultural evolution impacted, and is still influencing the development of mathematics. It should, therefore, be but reasonable to believe that, in this regard, “mathematics is not static and God-given ideal, but an ever changing field of human research” (p. 4). In a way, Dehaene acknowledges evolution in the development of various number systems: “even our digital notation of numbers … is the fruit of a slow process of invention over thousands of years” (p. 4). The same evolution is also credited in such mathematical concepts as the multiplication algorithm, square root, the set of real and imaginary (or complex) numbers. As recounted in Dehaene (1997), the evolution of mathematical objects is ascribed to be: “… a product of a very biological organ, the brain, that itself represents the outcome of an even slower biological evolution governed by the principles of natural selection. The same selective pressures that have shaped the delicate mechanisms of the eye, the profile of the hummingbird’s wing, or the miniscule robotics of the ant have also shaped the human brain. From year to year, ever more specialized mental organs have blossomed within the brain to process the enormous flux of sensory information received, and to adapt the organism’s reactions to a competitive and even hostile environment” (Dehaene, 1997, p. 4). In the same sense as Dehaene (1997), Changeux and Connes (1998) described mathematical objects or forms as “uniquely the product of the human brain” (p. 180). Dehaene’s thoughts pertaining to biological evolution and mathematics were shared by Budnik (2009) who maintained that “biological evolution created the mathematically capable human mind” (p. 290). So diverse is such evolution which continuously takes place since the very first reproducing molecules until the biosphere where we live now has developed; and without these diversity, Budnik claimed the human mind may not have evolved with such profound depth and intricacy. As Budnik observed, through the human mind, evolution has become conscious of its own process and endeavors to secure tools to enable it to control its future destiny. One of the most important tools devised by man to secure its control of the future and in anticipation of the many needs which can be addressed by such tool is the development of the computer. Originally designed for America’s war efforts, the first electronic computer applied for calculation and solution of practical problems was called the ENIAC which stands for Electronic Numeric Integrator and Calculator. ENIAC was built during the period from 1943 to 1946 and was funded by the US Army to aid in the calculation of ballistic tables (Allan, 2001). In the middle of the 1950s, man dreamed to advance a step further when experts of that time decided to take on the challenge of simulating intelligent behavior by machines, otherwise called artificial intelligence (AI). Research into artificial intelligence led scientists in an attempt to create mechanistic devices which “took the drudgery out of human intellectual endeavor … and eliminated some of the errors to which it is prone” (Garnham, 1987). Ever, since, man has consistently directed the enhancement of the computer with such ingenuity that the contraption may now be used to make everyday processes easier and faster. Cellucci (2008) also supports the relationship between biological evolution and mathematics. Through natural selection, all organisms have innate capabilities including space sense, number sense, size sense, shape sense and order sense, which although mathematical in nature, possess biological function as a result of biological evolution which embodied the various senses in organisms. To wit, natural selection is believed to have hardwired organisms in general to make it possible for them to “build mathematics in several features of their biological structure” (Cellucci, 2008, p. 16). By virtue of such biological features, humans are able to process mathematical operations necessary to search for nourishment, stay away from danger or find a suitable mate. Cellucci, however, classified mathematics into natural mathematics, which results from biological evolution, and artificial mathematics, which results from cultural evolution. Budnik (2006) credits biological evolution for creating the human mind, which in turn builds and develops mathematical forms such as axioms and models. In other words, biological evolution spawned the creative nature of mathematics. Casey (2005) singled out Galileo’s genius to have espoused the theory that mathematics is a construction of the human mind. Similarly, Casey indicated that Kant’s realization that “we can and do know the world, not in spite of, but precisely because we are deeply implicated in the creation of its causal structure” (p. 43). This is facilitated by the language of mathematics. One of such forms or constructs in mathematics are numbers. Courant, Robbins, and Stewart (1996) maintained that numbers were created by the human mind to enable man to count objects in different collections. The fact that numbers do not possess any reference to the individual characteristics of the entities counted by virtue of being an abstraction evidences the safe ground on which the structure of mathematics is founded. Speaking on mathematical forms and abstracts, Nuñez (2006) suggests that most of the idealized abstract technical entities in mathematics are created through the human cognitive mechanisms which tend to broaden the structure of bodily experience, whether it be thermic, spatial or chromatic, as it preserves the inferential organization of such domains. Yet, notwithstanding the abstractions of mathematical forms and concepts, Piccolomini (in Mancosu, 1996) argued that the conceptual nature of such mathematical forms possess the greatest clarity and certainty being created by the human mind. In a way, Ornstein and Ehrlich (1989) support the position of this paper, but in a very pessimist sense in the context of the modern world. They argued that the human mental system is struggling, almost in vain, to understand the ever changing and dangerous world we are in. The duo predicts that occurrences will continue and may be out of control until mankind realizes “how selectively the environment impresses the human mind and how our comprehension is determined by our biological and cultural history” (in Marien, 1991, p. 9). Approximating poetical language, Kaput and Shaffer (2002) vividly described the foundations of mathematical knowledge as emanating from the cognitive faculties of the human mind: “… the existence of these books meant that ideas were being stored and transmitted in a more robust, permanent form … studied by generations of students and debated, refined and modified. A collective process of examination, creation and verification was founded. The process was taken out of biological memory and placed in the public arena, out there, in the media and structures of the external symbolic storage system … They founded the process of externally encoded cognitive exchange and discovery” (Kaput and Shaffer, 2002, pp. 285-286). From Kaput and Shaffer’s account, it was made explicitly evident that mathematical knowledge from books was transcribed knowledge from information provided by the mind, nurtured by scholars and preserved through available technology to serve as guide for the following generations. It was, however, Feist (2006) who offered a more scientific account of how the human mind created mathematics. Initially Feist established that mathematics, as well as science, are not evolved adaptions of the human mind. He clarified that “evolution did not produce brains so they could do systematic and explicit math …, but it did evolve a sophisticated central nervous system that organizes and interprets sensory information and is able to reflect upon experiences and put thought between impulse and behavior” (Feist, 2006, p. 217). The human brain is said to be able to recognize patterns during the process where sensory input is organized and interpreted. When patterns are recognized, the brain then makes causal connections and forms both expectations and predictions or hypotheses. These processes are actually the domain of the central nervous system. As the human brain evolves, particularly the cortex and the frontal lobes, the brain of the human species very gradually began an upward asymptotic trend into performing ancient and implicit organization of acquiring experience from the senses, or in other words, gaining knowledge. In its creation of the forms and concepts of mathematics, the human mind in the cognitive realm devised precise units of measurements and its own innate tools. Hence, mathematical knowledge, as well as of the sciences, are a direct result of very long evolutionary, historical and cultural processes (Feist, 2006). Still another proof that mathematics is a creation of the human mind is Ernst Mach’s (in Newton, 1997) generalization that “mathematics serves as an immensely efficient set of mental short-cuts” (p. 141). As regards to evolution, Cellucci (2008) argued that since mathematics is logic-based, it is therefore, bound by reason, and is a function of man’s biological structure. However, with all such complexities presented, one may still wonder what these suggest in as far as the actual process of cognitive evolution in humans are concerned. Van Huyssteen (2006) explains that the evolution of the human brain may be seen as a natural process of providing human beings with the internal machinery to manage and control information so that mankind can enhance its versatility or adaptability to the environment. In this regard, and in consonance with Rescher, Huyssteen maintained that biological accounts of the beginnings of human intelligence and rational thought effectively scaffold the origins and development of mental operations. While it is evident that many thinkers support the proposition that the human mind created mathematics, there are also differing views on how the mind is able to accomplish the feat. Two of the better known positions are those of the Kantians and the modernists. Based on the work of Kline (1982), Immanuel Kant and his followers placed the source of mathematics in the organizing faculty of the mind. The other view supported by modernists is that mathematics originated in the activity of the mind, and not in the morphology or physiology of the mind. Kline (1982) suggests that conflicting views on the nature of mathematics and that mathematics is not anymore a universally accepted and indisputable body of knowledge tend to favor the position that mathematics is man-made. Contrary to the early Greek belief that mathematics existed independently of humans like the mountains and planets and that humans just continuously attempted to discover more and more of the mathematical forms and structures over time, the more popular notion is that “mathematics is a human creation … [and that] mathematics is entirely a human product” (Kline, 1985, p. 27). The mathematical forms, structures and concepts such as the axioms and the theorems are created by humans to help them understand the environment, to give way to their artistic inclinations and to engage in intellectual activity (Kline, 1985). It should also be safe to state that mathematics was created by the human mind as man ventured to describe, expand and order the universe (Brumbaugh and Rock, 2001). The invention of various kinds of number systems enabled man to record numbers and facilitated the foundations of mathematics. Since then, numbers make up an essential part of everyday thought and language. The invention of mathematical concepts such as geometry by Euclid furthered the cause of surveying, engineering and exploration (Carlisle, 2004). The invention of weights and measures contributed to flourishing trade and commerce. Later on, man learned to use numbers to represent mathematical relationships, which gave rise to the now famous theorems such as that of Pythagoras. To further aid mankind in exploring other lands or solving flooding problems with the use of dams, plane and spherical trigonometry was also invented. Man also attempted to study the seasons and the equinoxes spearheaded by Greek and Egyptian astronomers. With the binary number system in place, man also invented a very important device in the computer. Fueled by imagination and the drive to conquer nature, man’s knowledge of mathematics, coupled with advances in the sciences brought the first man in space and later to the moon. There is, therefore, seemingly no end to what the human mind is capable of reaching powered by mathematics. Rescher (in Gale, 1998) attributes knowledge acquisition among humans to evolution and validated the position that acquiring knowledge is in the core of our being as Homo sapiens. Additionally, Rescher also argued that “human intelligence is the product of a prolonged process of biological evolution” (p. 50). There is an avalanche of theoretical evidence to support the position that mathematics is, indeed, a creation of an evolved human mind. While, it is ethically not possible to open up man’s brain to practically prove this paper’s position, a literary elucidation should be sufficient to convince the thinking rationale man that mathematics and the various mathematical forms and concepts are a product of the human mind. Otherwise, for the skeptics, I close this exposition not with a period but with a question mark, as I quote Chapman (1988), “…[referring to mathematical forms and concepts] if they are not the products of the mind, then how do they become known to begin with?” (p. 411). References Aaboe, A. (1998). Episodes from the early history of mathematics. Washington, DC: Mathematical Association of America. Allan, R. A. (2001). A history of the personal computer: the people and the technology. Ontario: Allan Publishing. Brumbaugh, D. K. & Rock, D. (2001). Teaching secondary mathematics. 2nd ed. Mahwah, NJ: Lawrence Erlbaum Associates, Inc. Budnik, P. (2006). 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The applicability of mathematics as a philosophical problem. Cambridge, MA: Harvard University Press. Van Huyssteen, J. W. (2006). Alone in the world: human uniqueness in science & theology. Cambridge, UK: Wm. N. Eerdsmans Publishing. Read More