ࡱ> 5@ bjbj22 &XXݝ3""""8LlDE"@@@@@@+E-E-E-E-E-E-E$bFRHdQE8@@88QE"jL@@fE+C+C+C8@@+E+C8+E+C+CDD@ $.;?DD,|E0EDJAJD""JD(@"&X+Cz,1 @@@QEQEpBpMATHEMATICS AND THE MORAL SCIENCES: The Value of Value-Free Mathematics Bonnie Shulman Bates College, USA  HYPERLINK "mailto:bshulman@abacus.bates.edu" bshulman(at)abacus.bates.edu Introduction Most modern practitioners of the art of mathematics, as well as those who only admire it from afar (some with not a little trepidation), share an image of mathematics as a realm of abstract thought, where one breathes a pure air, unsullied by emotions, value-laden judgments, or ethical considerations. Indeed, some mathematicians, like myself, were attracted to the field, in part, because unlike the applied sciences, where an ethical stance might lead one to be troubled by a concern for the uses of ones research, pure mathematicians, as a rule, do not have to worry about potential (mis)uses of the arcane structures and spaces their imaginations inhabit. I say as a rule, because one cannot always predict what will be useful. The pure mathematician G.H. Hardy (1877-1947) wrote There is one comforting conclusion which is easy for a real mathematician. Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years. (Hardy, 1967: 140) Ironically his specialty, number theory, proved to have significant applications in cryptography. But potential good or bad applications aside, we still believe that the mathematics itself is value-free. In fact, it is precisely this assumed value-free nature of mathematics that leads us to hold it up as the standardthe preferred methodology to be followed in any honest inquiry into the truth of the matter, whatever the matter may be. However, over the years, my own experiences, informed by readings in the history and philosophy of mathematics, led me to wonder if it is really possible to separate out a purely cognitive knowledge that is in some absolute sense completely free of the values embedded in the socio-political-cultural context from which it is distilled. And, even were this possible, looking at some of the applications and consequences of knowledge for its own sake purged of values, I began to question the assumption that the quest for such value-free knowledge was desirable. In asking what is the value of value-free mathematics? I am really asking several questions, each arising out of a slightly different meaning of the word value. First, there is the notion of value as worth. In what ways is the search for a value-free methodology a worthwhile project? Then there is the idea of value as utility or usefulness. To what ends can we put a value-free methodology? In particular, can such a methodology have anything useful to say about ethics? Finally, there is the association of value with that which we hold in high regard or esteem. The high esteem accorded to mathematics because of its purported objectivity and value-free nature, is itself a cultural value held by particular people, in particular times and places, serving particular purposes. What are the cultural contexts and intellectual milieus in which this value flourishes? The Mathematical Ethic The writings of the philosopher C.S. Peirce (1839-1914) both reflected and were influential in forming 20th century attitudes towards mathematics, in particular the proliferation of mathematical models in the moral as well as the natural sciences. Peirce emphasized the importance of applying the exactitude of mathematics to philosophy. His commitment to the search for truth and logical thinking was rooted in values he held dear. He maintained that [to] call an argument illogical, or a proposition false, is a special kind of moral judgement. (Peirce, 1958: 191) This deference to mathematical expertise is familiar to us. We take as self-evident the efficacy and desirability of using mathematical models to describe, to explain and to solve economic and social problems. Intrinsically there is nothing wrong with mathematizing other fieldsindeed mathematical models and formalism have a lot to offer. But some object to what they perceive as the imperialistic drive of mathematics to colonize other disciplines and the uncritical acceptance of mathematical formalism. They resent the imposition of what might be called the mathematical ethic, the assumption that any honest inquiry after the truth must employ mathematical arguments. Peirces views reflect a kind of mathematical fundamentalism, a belief in mathematics as the most fundamental discipline, the only discipline which ha[s], and need[s], no foundations. (Hookway, 1995: 649) How does it happen that a collection of descriptive rules of argument used to help us obtain reliable knowledge become moral prescriptions for how we ought to proceed in creating knowledge in any domain? The authority of mathematics is so deeply embedded in our worldview and difficult to challenge that it verges on a religious article of faith. The quest for methods of maximum reliability for obtaining pure unadulterated knowledge is an ethical search. According to Peirce, [t]he method of science is intrinsically ethical insofar as it entails the rejection of personal prejudice in favor of the good of the community. (Corrington, 1993: 35) The imposition of the mathematical ethic through the mathematization of the moral sciences is intimately connected with metaphysical and ethical commitments to absolute, certain, pure, impersonal knowledge. Indeed the authority of mathematics is inextricably linked to metaphysical commitments of individuals and their communities. This relationship helps explain the deep-seated power of our attraction to mathematics and deference to mathematicians when seeking results of the greatest reliability in any endeavor. The road to value-free knowledge is evidently paved with values. The implicit assumption behind the building of mathematical models to gain knowledge about reality (whether physical or social) is that there is a hidden subtext in nature to be dug out and displayed and that this subtext is written in the language of mathematics. The principles underlying our trust in the efficacy of mathematical modeling are accepted and remain for the most part unquestioned by natural and social scientists: there is a lawlike mathematical order in nature, and we can discern this order through the clever procedures we use to design our models; simple laws explain complicated things; universal principles exist and are mathematical in form; reality is accessible to the human mind through the power of reason. These assumptions are inherited in a direct line of descent from the beliefs of the Presocratic philosophers and the three central concepts of their cosmology. Their universe (kosmos) was ordered; it was simple and reducible to a small number of first principles (archai); and the first principles were accessible to rational analysis by the use of reason (logos). The commitment to the existence of fundamental principles (archai) has the character of a religious belief. It is more like a faith commitment than an empirically based system of thought warranted by its success in dealing with the material world. (Van Brummelen, 2001: 52) A couple millennia later, the Nobel laureate physicist Eugene Wigner (1902-1995) called the empirical law of epistemologythe appropriateness and accuracy of the mathematical formulation of the laws of naturean article of faith of the theoretical physicist. (Wigner, 1960: 17) He concludes that the unreasonable effectiveness of mathematics in the natural sciences is akin to a miracle and a wonderful gift which we neither understand nor deserve. (Wigner, 1960: 13) A Logic of Ethics Peirces work was an integral part of the new logic that was part of the climate and in the air breathed by an entire generation of intellectuals, and which continues to influence their heirs. A similar commitment to applying the exact thinking of mathematics to the human and social sciences, in particular to ethics, appears in the work of the Viennese mathematician Karl Menger (1902-1985). In 1934 he finished his book Morality, Decision and Social Organization: Toward a Logic of Ethics in which he claims logico-mathematical studies are an indispensable training . . . and, strange as it may seem to you, especially for the treatment of questions of morality. Indeed, the applicability of exact thinking to ethics appears to me to be an aspect of science that is of some importance for human life. (Menger, 1974: 1) The writings of Menger, reflect a distinguishing characteristic of mathematical and philosophical thinking in the early 20th-century: a supreme faith in the power of mathematical formalism to explain and solve problems in all fields. What originally sparked Mengers interest in ethical questions? His motivation is probably best understood against the backdrop of social unrest and civil war that characterized Austria (and indeed much of Europe) in the years when he was writing his Logic of Ethics. As he wrote in a postscript to the 1974 edition, the winter of 1933 and the first half of 1934 during which this book was written belong to the darkest periods in Austrian history. (Menger, 1974: 114) Mengers interest in ethics was aroused amidst the social upheaval he witnessed all around him. His response was to retreat from the darkness into the clear light of exact thinking as an antidote. He wielded mathematics as a sword to cut through the morass of political intrigue and conflict and constructed an abstract system that treated the logic of relationships. It would seem to be a paradox that a treatise on ethics should purport to be value-free, however in his remarkably dispassionate book (Cornides, 1983: 9) Menger set out to completely eliminate value judgmentsimplicit as well as explicit, hidden as well as open evaluations. (Menger, 1974: 94) He sought to cleanse scientific thinking of all extra-scientific speculations about human abilities and motivations. The author has regarded statements [of a certain form] as value judgments and as such relegated them to the biographies of the proponents while suggesting that the expression mathematical theory be used for the transformationfree from praise and censure, license and proscriptionof certain propositions into other propositions with the help of precisely stated rules. (Menger, 1974: 21) It is in this context that Menger wanted very much to offer a positive contribution to the conduct of human affairs. To this end, he enlisted his pure mathematics to create models for the compatibility of individuals and social groups. His ethics consisted of an abstract analysis of the consequences of the voluntary choices of groups of individuals and he bracketed discussions of the value of these decisions as extra-ethical concerns. Any group of individuals can be divided into classes based on their attitudes towards rules of social behavior. For instance, suppose the rule is No Smoking in Public Places. An individual might follow this rule all of the time, some of the time, or never. Similarly, an individual might agree with this rule (in principle, whether they follow it or not) always, sometimes, or never. Menger organized groups into equivalence classes based on their positions towards given norms, and using combinatorial arguments drew logical conclusions about the formation of socially compatible groups. He summarized the good features of his scheme. First, the ideas governing each group are determined by its voluntary decisions, and all the decisions are explicit, not hidden. Second, decisions can be carried out with the full energy of the group and a minimum of internal friction. Third, in a system composed of several groups following different self-imposed regulations, one can see clearly the effects of the diverse ideas in the contest of plans. (Menger, 1974: 90) At this point, one might be inclined to agree with Ludwig Wittgenstein (1889-1951) who said it was strange that you could find books on ethics in which there was no mention of a genuine ethical or moral problem. (Rhees, 1965: 21) Menger himself fully realized that ethics thus externalized would be regarded by most philosophers as quite superficial. On the other hand, it would lend itself to sound applications of the logic and mathematics of classes and relations. . . (Menger, 1994: 183). Can logical analysis devoid of moral content tell us anything useful about ethics? What is the value of building mathematical models of ethics? Mathematical Models and the Moral Sciences It was not only Menger whose attitudes towards the application of mathematics to the moral (or, as they were now being called the social) sciences were affected by the social and political climate of the interwar years. Karl Menger (who was the son of the noted economist Carl Menger) had an enormous impact on the economist Oskar Morgenstern (1902-1977), who came under his influence in the 1930s (Leonard, 1994: 5). The publication in 1934 of Mengers Logic of Ethics had a profound effect on him. Mengers insistence on precision and rigor appealed to Morgenstern, who argued tirelessly for the necessity of applying the exact thinking of mathematics to economics. Today the situation in the social sciences is such that authors do not want to be taken at their word. Only when one takes men really at their wordas the mathematicians at their symbols!can real progress ever be made The origin of most difficulties in the social sciencesas for instance illustrated in the theory of capitalresides in the lack of rigor in the language. (Morgenstern, 1976: 400) Morgenstern worked to extend Mengers theory of ethics to economic decisions, and like Menger, his work was strongly motivated by a desire to offer a framework for clear thinking about social issues that was based solely on logical principles and not dependent on a particular system of values. This commitment to a mathematics that was value-free was not ephemeral, cured by a change in political climate, but remained in force throughout their lives. Even four decades later, Morgenstern wrote in a foreword to a book on game theory: [I]t was also difficult to avoid the injection of personal value judgments in the discussion of emotionally laden social and economic problems. The present book is entirely free from such faults; it explains, it analyzes and it offers precepts to those who want to take them; but the theory it describes and develops is neutral on every account. (Morgenstern, 1970: ix) Given the direct influence Mengers book on ethics had on the authors of the seminal book on game theory, it is ironic, though perhaps not completely unexpected, to learn that game theory is now being used to model ethical decision making and even make moral discoveries. Mathematics as a way of thinking can help clarify complex tangled issues. Ken Binmore, (1940- ), a self-described mathematician-turned-economist, writes in Game Theory and the Social Contract (1994), a modern treatise of epic proportions detailing the applications of (evolutionary) game theory to ethics, my belief [is] that much insight can be gained into ethical matters by approaching them systematically from a game-theoretic perspective. (Binmore, 1994: vii) To build our models, we must simplify, strip down to the bare essentials, and abstract the key features of a phenomenon or situation. This process is worthwhile if one believes, as does Binmore, that what remains to be studied after the gross simplifications that are made remains worth studying. (Binmore, 1994: viii) But we must always remember what was left out. I think Karl Menger would agree wholeheartedly with contemporary game theorist Robert Axelrod that the goal of modeling is not to arrive at prescriptions about how to act, but to achieve some clarity in complicated situations. The value of any formal model, including game theory, is that you can see some of the principles that are operating more clearly than you could without the model. But its only some of the principles. You have to leave off a lot of things, some of which are bound to be important. (quoted in Poundstone, 1992: 253) And here is a juncture where one might conceivably uncover values stowing away, hiding in the tool chest of apparently neutral and value-free methodologies. The decisions made (as to what the essential features are, and what can be omitted) embed the values of the modeler in the model. Game theory is a kaleidoscope that can only reflect the value systems of those who apply it. (Poundstone, 1992: 170) The Ethic of Mathematics The work in foundations of mathematics and the new logic, exemplified in the writings of Peirce, became a cornerstone in the philosophy of science known as logical empiricism or logical positivism, as developed by the Vienna Circle. Several members of the Circle believed that science, as exemplified by mathematics, embodied profound moral lessons on how to live well with others. They subscribed to what might be called an ethic of mathematics/science based on applying values inherent in their scientific disciplines to societal relations. As early as 1907, a group that included the mathematician Hans Hahn (1879-1934) and the political economist and sociologist Otto Neurath (1882-1945) met to discuss philosophy of science. Ernst Mach (1838-1916) and Pierre Duhem (1861-1916) were two early 20th-century scientists whose writings had a significant influence on this proto-group. Both Mach and Duhem expressed mixed feelings about the scientific revolution, and broke ranks with most of their scientific colleagues in raising objections to science becoming the secular religion of the modern nation-state. (Fuller, 2000:189) They were concerned that importing scientific and mathematical methods of inquiry divorced from societal values into the human and social sciences would make people feel alienated from and disenchanted by science. The first and most essential aim of Enlightenment science was to debunk superstition and myth in favor of a literal, true description of the world. Hahn and Neurath took this idea further. In a 1930 pamphlet Occams Razor, (1980a: 1-19) Hahn lambasted theology and metaphysics as world-denying philosophies that functioned like political opium, fragmenting the workers and consoling the masses. Religion encouraged belief in a reality not attainable by the senses, and a refusal to come to terms with the world as it presents itself. Hahn and Neurath championed the rational expert knowledge of true science as a means to liberate the people from the stranglehold of religion. However, not all members of the Circle subscribed to this ethic of mathematics/science. Karl Menger (who was mentored by Hahn in mathematics) was profoundly disturbed by what he saw as a conflation of science and politics. Indeed, it was partly in reaction to Hahn and Neurath that Menger retreated into his externalized or denaturalized ethics, developing his abstract combinatoric analysis of groups of individuals and the consequences of their normative choices. His insistence on making a sharp distinction between value judgments and mathematics grew out of his frustration with discussions then taking place in the Vienna Circle. Debates over the ethic of mathematics/science reappear in succeeding generations of thinkers, including our own, whether they, like Menger, explicitly reject this coupling of moral and scientific concerns, or actively embrace it. Ethical Practice of Mathematics It seems paradoxical that Mach, Duhem and their intellectual heirs, philosophers, mathematicians and scientists who challenged certitude, were also often leaders in movements for social reform and popular scientific education, filled with the fervor of firm unshakeable convictions. (Hacohen, 1998:718) How is one to reconcile the seeming disparity between the strict rejection of objective values and the objective justification of values, shared by all thinkers of this group [the Vienna Circle], and their forceful declamatory moral and political will, their urge to promote the liberation and increased happiness of humanity? (Rutte, 1991:143) And what is one to make of the prevalence of eugenical ideas in vogue throughout the international community of biologists at this time, and especially predominant amongst British intellectuals? (Wersky, 1978: 103) The specter of Nazi scientists application of the theories of eugenics will no doubt continue to haunt us well into the 21st-century. Particularly acrimonious debates within and about the field of sociobiology reflect this ambivalence over the moral implications of scientific research. Ullica Segerstrale, a Harvard-trained sociologist who witnessed first-hand the genesis of the conflict over the emerging field of sociobiology, presents an in-depth analysis of the controversy as it developed over twenty-five years in her book Defenders of the Truth. (Segerstrale, 2000b) As a sociologist, she focused on understanding the various types of scientific, moral and political convictions that motivated the two camps. (2000b:35) What makes the story especially dramatic is that two of Edmund O. Wilsons (1929- ) main opponents in the controversy, Richard C. Lewontin (1929- ) and Stephen J. Gould (1941-2002), were colleagues of his in the same department at Harvard. Segerstrale sets the scene for her compelling narrative in the second chapter: In the early summer of 1975 the distinguished Harvard entomologist Edward O. Wilson published a very large tome, Sociobiology, the New Synthesis. In his book, Wilson defined sociobiology as a new discipline devoted to the systematic study of the biological basis of all social behavior In November 1975, a group called the Sociobiology Group, composed of professors, students, researchers and others from the Boston area launched an attack on Wilsons Sociobiology. (2000b:13) As Segerstrale makes eminently clear in her nearly 500-page study, although the underlying agendas of the key protagonists conflicted, they shared in common a striving to combine the pursuit of moral and scientific truth. Describing Wilsons personal moral agenda, she writes: Wilsons zeal in making sociobiology a truly predictive science, encompassing all of social behavior, was intimately tied to an old desire of his: to prove the (Christian) theologians wrong. He wanted to make sure that there could not exist a separate realm of meaning and ethics which would allow the theologians to impose arbitrary moral codes that would lead to unnecessary human suffering. He believed that there must exist a natural ethics for humans and was on the lookout for it. (2000b:38) Like Hahn and Neurath, but for very different reasons, Wilson is committed to scientific materialism as a means of reining in the authority of religion. His search for a quantitative explanation of moral conduct, was driven by a desire to wrest from religion its privileged status as dispenser of correct ethical norms. Hahn and Neurath saw the clarity of the new logic as a means of undermining the political role of world-denying religious philosophies in rationalizing the status quo. Institutions in power have a vested interest in maintaining the social, economic and political structures that support them. In the time of the Vienna Circle, science was attempting to wrest power from the Church and science was used to advance a liberatory progressive social agenda that challenged the status quo (things as they are). Later, when science was on the ascendant, Wilsons critics accused him of using science (sociobiology) to create just-so stories that could be used to justify the status quoin particular the existing power relations that supported social inequities based on gender, race and class. The metaphysical commitments of each of these characters are not merely individual personal and idiosyncratic positions, but rather they epitomize (and are embedded in) historical trends, notably the power struggle between science and religion. Wilson was strongly influenced by the tradition of Harvard entomologist W.M. Wheeler (1865-1937), an expert on the behavior of social insects (especially ants, also Wilsons specialty). Wheeler, who had his own agenda, studied the evolution of social systems. As a socialist, he was particularly interested in explaining altruism, and saw cooperation amongst insects as a good model for human society. In this context, Wilson came to view altruism as the central problem in sociobiology. But his work is also historically linked to that of the British eugenicists of the 1930s. Two of the founders of mathematical population genetics, J.B.S. Haldane (1892-1964) and R.A. Fisher (1890-1962), both strong supporters of eugenics, were among the first to attempt genetic explanations of altruism. (Wersky, 1978 and MacKenzie, 1981:183) Population genetics is regarded as the hardest (i.e., most mathematical) branch of evolutionary biology, and Wilson began his quantitative studies with Fishers equations. (Segerstrale, 2000b:40) The criticisms of the new discipline by the Sociobiology Group (SSG) were based on exactly this linkage with earlier biological determinist theories, which had been so misused by the German Reich. An excerpt from the first public statement by the SSG, the infamous letter published in the New York Review of Books that was the opening salvo in the controversy, states unequivocally that [t]hese theories provided an important basis for the enactment of sterilization laws and restrictive immigration laws by the United States between 1910 and 1930 and also for the eugenics policies which led to the establishment of gas chambers in Nazi Germany. (Allen, 1975: 182) It is clear that both camps in the sociobiology dispute pursued science, in part, to advance a moral agenda. But the roots of their often vehement, even personal disagreements were not over which particular moral values they espoused. Indeed, Wilson considered himself a liberal and anti-racist and claimed he was genuinely shocked and hurt by the accusations of his critics. (Segerstrale, 2000b: 25-6, 47) Rather, we can best understand their opposition by distinguishing between two distinct interpretations of what it might mean to practice a moral sciencethe difference between adhering to what I have termed an ethic of mathematics/science and the ethical practice of ones discipline. Wilsons practice of a moral science was based on a positivist view of science harking back to the writings of the Vienna Circle. His was the Enlightenment project in modern dress: the search for a universal ethical standard (based on reason) that is applicable to the social as well as the natural sciences. For Wilson, the norms that separated good science from bad science also embodied moral principles for ethical behavior. However Lewontin and many of the members of SSG did not view scientific practice as inherently moralindeed good science could lead to morally repugnant outcomes! For this group, practicing moral science meant taking responsibility for the uses to which science was put. Reflexivity and the Feminist Critique of Objectivity Most mathematicians and scientists recoil in horror at the notion of values intruding into their work. This is understandable considering the historical conflict between science and religion for authority, with the fate of Galileo emblazoned in collective memory as a cautionary legend. The desire for a pure description untainted by the emotional investments or idiosyncracies of individual researchers (or entire research groups), is a laudable goal, when understood as an attempt to reduce bias, to prevent vested interests from controlling knowledge for their own ends. The essence of the ideal of objectivity is that if ideas are to be used as efficient tools of explanation and prediction, they must not be allowed to become tools of anything else. (Horton, 1970:161) But, desirable though it may be, the attempt to produce knowledge that is true independent of human values or social, economic, political, psychological or other factors is probably an impossible dream. Efforts to ensure the neutrality of the knower are a moral striving approaching an ideal that may be unrealizable, but is no less worthy for being unachievable. Perhaps the real danger lies not in the contamination of objective science by subjective values, but rather in refusing to face up to and take responsibility for the value decisions that are made, and in making them unconsciously. This is the position held by a group of feminist scholars trained in diverse disciplinary perspectives (including history, philosophy, sociology, anthropology, psychology, women studies, physics, mathematics and biology) who situate their work in the new field of science studies. Their project, to understand, in conceptually nuanced and empirically specific terms, the difference contextual values make to science, both positive and negative, (Nelson and Wylie, 1998) calls for a radical reworking of the ideal of scientific objectivity, a stronger brand of objectivity. What is being proposed here is that objectivity for science lies at least in becoming precise about what value judgments are being and might have been made in a given inquiryand even, to put it in its most challenging form, what value decisions ought to be made; in short that a science of ethics is a necessary requirement if sciences progress toward objectivity is to be continuous. (Rudner, 1988:332) The great irony is that Enlightenment science, the debunker of superstition and myth in favor of a literal, true description of the world, created a new myththe myth of science as a true literal description of the world. In the same way that the positivists sought to de-mythologize science by purging it of metaphysics, science studies scholars seek to de-mythologize science by critiquing the illusion of objectivity. Neurath, primary author of the manifesto of the Vienna Circle, wrote that [t]he scientific world conception is characterized not so much by theses of its own, but rather by its basic attitude, its points of view and direction of research. (Neurath, 1973:305-6) Throughout the manifesto, the vital role of mathematics (in particular the new logic) in defining this point of view is evident. The method that distinguished their version of empiricism was logical analysis. Quoting Bertrand Russell (1872-1970), Neurath says that this method has gradually crept into philosophy through the critical scrutiny of mathematics It represents, I believe, the same kind of advance as was introduced into physics by Galileo. . . . (Russell, 1914:4, quoted in Neurath, 1973:306) These remarks and others scattered throughout the works of Neurath and Hahn, convey a set of values and an attitude towards life that are basic to the empiricist worldview. These attitudes, derived from elevating the methodologies of science to a program that prescribes moral conduct, are freely expressed, but never really analyzed and may be all the more potent for remaining unexamined. The feminist critique of objectivity insists that if we really want to reduce bias, then we must ferret out and acknowledge all possible sources of bias, especially hidden assumptions. Unexamined myths. . .have a subterranean potency; they affect our thinking in ways we are not aware of , and when we lack awareness, our capacity to resist is undermined. (Keller, 1995:28) The point is that it is not irrational or wrong to admit non-cognitive values into mathematics or science, but it is irrational to do so unself-consciously without careful consideration. Strong objectivity is about valuing reflexivity over neutrality and building self-reflection into the methodologies of mathematics and science, especially in the notion of objectivity itself. The criteria for choosing among theories, for deciding what to put in and what to leave out in a model, can be liberatory or oppressive, feminist or androcentric or neutralit all depends on the socio-political context (situatedness) of the research and the researcher. (Longino, 1995:396) If the role of values in scientific inquiry is inescapable, then we should be asking lots of tough questions. What are the reasons offered for considering a particular set of assumptions, criteria, or results as valid (for what purposes) or useful (to whom)? Feminist theorists admonish that a genuinely ethical practice requires that in any knowledge production we must always ask: who benefits from this knowledge and who suffers; what are the material consequences and material ambitions supported by this model; what (and whose) purposes are served by pursuing this investigation? Tough Questions We started by asking, what is the value of value-free mathematics? What have we learned by asking this question in the first place? Clearly, mathematical models are powerful tools that can be used to predict probable outcomes, given a certain set of assumptions. However, if our aim goes beyond mere academic knowledge production and we wish to self-consciously use our insights in order to intervene, then it behooves us to pay attention to historical examples of bad interventions (the German and Soviet cases in the 20th century) and learn the lessons well. Rather than denying the values intrinsic to and embedded in the models we create, it is imperative that we make explicit the assumptions on which the models are based, and engage in thoughtful critique of different sets of assumptions. Investigating multiple sets of assumptions embodying different values, and comparing outcomes based on these diverse storylines will encourage us to be conscious of our decision-making and help ensure the future accurately reflects the things we value. (Frodeman, 2001:4) Perhaps we can reframe questions based on alternative value systems and come up with new ways of conceptualizing. For instance, an alternative to viewing abstraction in mathematics as the stripping away of meaning is to see it as the expression of commonality (within the appearance of difference). (Confrey, 2000:20) What are the underlying (often hidden) presuppositions, prejudices, illusions, assumptions, values, and metaphysical commitments implicit in our (modern, Western) ways of creating mathematical knowledge? Integrating research methods and results from anthropology and history to compare cross-cultural and historical case studies across time of the different mathematics humans have created to solve similar problems, can lead us to reflect on the purported universality and inevitability of Western mathematics. This, in turn, may lead us to challenge the assumption that Western rules of mathematical reasoning or ethical behavior are preordained eternal truths. The German historian and philosopher Oswald Spengler (18801936) wrote [t]here is no mathematic but only mathematics. (Spengler, 1927:2319, italics in original) Spengler believed that the style of a soul was revealed in the world of numbers, and that there were as many number-worlds as there were cultures. Mathematics has been called the science of patterns. The patterns of mathematics are threads discernable against a background of metamathematicsan insiders account of their beliefs about the nature, purpose and value of mathematicswithout which the practice is worthless. Most mathematicians would agree with Menger that these background concerns belong outside the purview of mathematics proper. However, an alternative to accepting this statement would be to expand the definition of mathematics so that it encompasses the whole cloth. This would mean including the Spenglerian math worlds, the sociopolitical and cultural contexts, as well as the mathematical practices and symbols in the definition of mathematics. If we reconfigure the roles and relationships between the practice of mathematics and the cultural setting in which it occurs, might we enable an entirely different set of questions to be asked and different modes of answering to be accepted? This is not as radical or far-fetched a proposition as it may first appear. Different cultures and time periods have included material in mathematics that we would not include today, for instance, music. (Wilder, 1981:41) In the 17th- through the 19th- centuries, the term mixed mathematics was used to denote much that today would be called applied mathematics. But whereas the connotation of applied is that of imposing one area on another, the term mixed connotes a blurring of distinctions and a fluidity of boundaries. (Richards, 1995:127) Should mathematicians worry about the potential applications of their work? Should they actively pursue particular applications to benefit (by whose standards?) society? Is it arrogance to feel free (as Wilson did) to ignore the power of hidden and unexamined assumptions, and deny their influence on public perceptions of the implications of your research? Understanding and acknowledging the motivations that led to particular kinds of knowledge doesnt invalidate the resultseven bad motivations can lead to good mathematics/science. On the other hand, good results may originate in mindsets we have since abandoned or rejected as bad. The methods we develop are not necessarily marked by the motivations and desires that led to their invention. It is possible to separate our insights from the standpoint in whose service they are used. They are not the exclusive property of that standpoint, even though they may have inspired it. One of the senses of the notion of objectivity is that knowledge claims can travel, that tools developed to solve problems in one context can be applied in another. A sword may be used to cut grass and a scythe may be seized upon as a weapon. Our belief in the validity of the mathematics/science does not necessarily entail acceptance of the mindset that inspired our investigations. However, even if the social consequences of a philosophy or set of assumptions can be disentangled from questions of its validity, that doesnt make the social consequences unimportant, or relieve us of the obligation to take responsibility for the implications of our work. Mathematics and the sciences depend on the material support of national governments for their continued survival. The discipline and its practitioners adapt themselves to the changing political climate in order to preserve their existence. Although the repressive policies of the Third Reich and the loss of some of its best (many of them Jewish) intellectuals posed a threat to the continued health of German mathematics, the technological and imperialistic demands of the regime posed interesting challenges for the mathematicians who remained. They were able to offer their services to advance technological needs (by developing theories of aerodynamics and producing calculations for the V2 rocket) and offer means of social control (through eugenics and the statistics of inheritance). Their work was good mathematics and they were able to offer these services without becoming outcasts in the disciplineau contraire, these same mathematicians were actively recruited by the allied victorious powers after the war (to continue their work on these same projects)! I suggest that it is irresponsible for mathematicians to hide behind a faade of value-free mathematics, and leave consideration of the consequences of their research to others. Sometimes the questions we ask are more important than the answers we get, because the answers are rooted in the assumptions inherent in the questions. We should be asking why are we asking these questions and not others? Or, as a friend of mine put it: Mathematics is among the most beautiful and elegant hammers weve got, and we should be careful to use it on the right nails. (Kempster, 2002) PAGE  PAGE 1  The term moral science, used especially in the 17th and 18th centuries, encompassed a range of questions associated with the study of human nature, including a systematic analysis of human ethical behavior.  Galileos oft-quoted dictum that the Book of Nature is written in mathematics is based on a deep metaphysical commitment to a universe that conforms to simple, mathematical laws. (Lloyd, 1995:358)  Galileo Galilei (1564-1643) wrote: Philosophy is written in this grand book the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and to read the alphabet in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth. (1623:237-8)  As G.H. Hardy wrote, [T]here is one purpose at any rate which the real mathematics may serve in war. When the world is mad, a mathematician may find in mathematics an incomparable anodyne. For mathematics is, of all the arts and sciences, the most austere and the most remote, and a mathematician should be of all men the one who can most easily take refuge where, as Bertrand Russell says, one at least of our nobler impulses can best escape from the dreary exile of the actual world. (Hardy, 1967: 143)  A personal anecdote from a colleague confirms the allure of the retreat into mathematics in perilous times. When I was living with Raya, a Russian immigrant, she claimed that Russian pure mathematics was so well developed in the post-World War II years because all the most creative minds of Russia sought expression there where they would be safe from political persecution. (Martin, 2002) Also see (Maritz 2003:66) The Soviet government repressed and controlled various forms of intellectual inquiry during those years [post World War II]. However, in the midst of this repressive climate, the divisions of Mathematics and Mechanics, the Faculty of Mathematics at Moscow State University (MSU) commonly known as Mekh-Mat was a haven where the objective value of ones research was ones best asset. According to (Saul, 1992:6) Russian mathematicians seem to develop a passion for their subject quite early, retain it for their entire professional career, and are anxious to impart it to a new generation. These cultural elements, particularly the last, are much less developed in American mathematics education. What characteristics of the surrounding culture produced this remarkable blend of intensity and evangelism? . . . . Many Soviet mathematicians trace its origins to a strategy for intellectual survival in a totalitarian environment. In such surroundings, mathematics is a relatively safe intellectual activity.  A fellow mathematician wrote: Reading this I, at least, take a moment and chuckle at Mengers naivete. It is so clearly wrong to bracket what would seem to most to be the central concerns of ethics as extra-ethical. Yet I dont have the same moment when physicists or chemists, say, bracket off huge chunks of what were formerly their disciplines as extra-scientific. Why dont I see the second as equally clearly wrong? Loss of the baby with the bath water? (Martin, 2002) One might argue that Mengers naivete, if such it be, lay in applying his rules of what gets to count as mathematics to a non-scientific field like ethics. However, perhaps the difference is that most of us think we know what ethics is (and that it should rightly include consideration of values) and we are content to let mathematicians define what properly belongs in mathematics (proscribing values and human motivations). Seeing the absurdity of the extension of this proscription to ethics might lead us instead to question its validity when applied to mathematics as well. Who gets to decide what mathematics is and what it is not?  Robert Leonard, noted historian of economics writes: . . . [John V]on Neumanns personal history too, left him with a strong sensibility to the contingencies of social structure. (Leonard, 1995: 757)  Menger himself wrote, I also showed the manuscript [of this book] to the economist O. Morgenstern, who even then began studying mathematics and expressed great interest in the book. (Menger, 1974: 114)  In 1944 Morgenstern collaborated with John von Neumann (1903-1957) as coauthor of the seminal book Theory of Games and Economic Behavior. It is not an exaggeration to claim that the theory of games was born with the publication of von Neumann and Morgensterns book.  Indeed, the understanding of logic and mathematics has always been the main crux of empiricism; for any general proposition that has its origin in experience will always carry with it an element of uncertainty, whereas in the propositions of logic and mathematics we find no such uncertainty. (Hahn, 1980b: 21)  On the so-called first Vienna Circle, whose members later formed part of the circle that became known as The Vienna Circle, see (Haller, 1991:95-108).  And so the world-denying philosophy proves to be a means, employed over and over again, of consoling the masses of those who had reason to be somewhat dissatisfied with this world with the prospect of another world. . . . (Hahn, 1980a: 1)  Wilson was a grand-student of Wheelers. He was a student of Wheelers student, C.M. Carpenter.  One can also read this distinction as that between an essentialist and existentialist viewpoint, or rationalism vs. empiricism. The first horn of each dilemma locates the source of knowledge or truth within, and the second position locates it in the outer world.  Some feminist philosophers of science have argued for a new notion of objectivity, called Strong Objectivity, which requires knowledge of the situatedness of the knower, in order to fully evaluate the knowledge produced. The term originated with Sandra Harding (1996). Sociologists of scientific knowledge (SSK) at the Edinburgh School, in particular proponents of the Strong Programme of David Bloor and Barry Barnes, argue for related notions. They take as one of their main tenets that the interaction between the knower and the knowledge they seek is an essential part of the knowledge itself (Bloor, Barnes and Henry, 1996).  This pamphlet, The Scientific Conception of the World, first sold at a Prague conference on theory of knowledge in the exact sciences in September, 1929, contained the essential common tenets, publications and antecedents of the school and may be regarded as the christening of the Vienna Circle. (McGuiness, 1979:18)  Much of this scholarship takes place in the newly emerging discipline of ethnomathematics. See (Frankenstein and Powell, 1997). References Allen, E. et al. (1975) Letter. The New York Review of Books 13:182, 184-6. Binmore, K. (1994) Game Theory and the Social Contract, Vol. 1: Playing Fair. Boston: MIT Press. Bloor, D., B. Barnes and J. Henry (eds) (1996) Scientific Knowledge : A Sociological Analysis. Chicago: University of Chicago Press. Cohen, R.S. and Neurath, M. (eds) (1973) Empiricism and Sociology. Dordrecht: Reidel. Confrey, J. Leveraging Constructivism to Apply to Systemic Reform, Nordish Mate-Matik Didaktik (Nordic Studies in Mathematics Education) 8(30). Cornides, T. (1983) Karl Mengers Contributions to Social Thought, Mathematical Social Sciences 6(1):1-11. Corrington, R.S. (1993) An Introduction to C.S. Peirce. Lanham, Md: Rowman & Littlefield. Frankenstein, M. and A. Powell (eds) (1997) Ethnomathematics: Challenging Eurocentrism in Mathematics Education. Albany: SUNY Press. Frodeman, R., M. Bullock, et al. (2001) Global Climate Change: The State of the Debate, Science, Technology and Society Newsletter 130(Winter):1-4. Fuller, S. (2000) Science Studies Through the Looking Glass, Chapter 9 in (Segerstrale, 2000a). Galileo Galilei (1623) Il Saggiatore, trans. S. Drake (1957), Discoveries and Opinions of Galileo. New York: Doubleday Anchor. Hacohen, M.H. (1998) Karl Popper, the Vienna Circle, and Red Vienna, Journal of the History of Ideas 59(4):711-734. Hahn, H. (1980a) Superfluous Entities, or Occams Razor, (McGuinness, 1980:1-19). Hahn, H. (1980b) The Significance of the Scientific World View, Especially for Mathematics and Physics, (McGuinness, 1980:20-30). Haller, R. (1991) The First Vienna Circle, (Uebel , 1991:95-108). Harding, S. (1996) Rethinking Standpoint Epistemology: What is Strong Objectivity? in (Keller and Longino, 1996). Hardy, G.H. (1967) A Mathematicians Apology. Cambridge: Cambridge University Press. Hookway, C.J. (1995) Entry on C.S. Peirce in Honderich, T. (ed.) The Oxford Companion to Philosophy. Oxford: Oxford University Press. Horton, Robin (1970) African Traditional Thought and Western Science Chapter 7 in (Wilson, 1970). Howell, R. and J. Bradley (eds) (2001) Mathematics in a Postmodern Age: A Christian Perspective. Cambridge, UK: Eerdmans. Keller, E.F. (1995) Gender and Science: Origin, History and Politics, Osiris 10:27-38. Keller, E.F. and H.E. Longino (eds) (1996) Feminism & Science. Oxford: Oxford University Press. Klemke, E.D., R. Hollinger and A.D. Kline (eds) (1988) Introductory Readings in the Philosophy of Science. Buffalo: Prometheus Books. Kempster, M. (2002) Personal communication, 3/14/02. Leonard, R.J. (1994) Reason, Ethics and Rigour: Morgenstern, Menger, and Mathematical Economics, 1928-1944, Cahiers de recherche du departement des sciences economiques de lUQAM, Cahier no. 9403. Montreal, Quebec. Leonard, R.J. (1995) From Parlor Games to Social Science: Von Neumann, Morgenstern, and the Creation of Game Theory 1928-1944, Journal of Economic Literature 33(2):730-761. Lloyd, E. (1995) Objectivity and the Double Standard for Feminist Epistemologies, Synthese 104:351-381. Longino, H. (1995) Gender, Politics, and Theoretical Virtues, Synthese 104:383-397. MacKenzie, D. A. (1981) Statistics in Britain: 1865-1930. Edinburgh : Edinburgh University Press. Martin, J. (2002) Personal communication, 5/30/02. McGuiness, B. (ed.) (1979) Wittgenstein and the Vienna Circle, Conversations recorded by Friedrich Waismann. Oxford: Basil Blackwell. McGuinness, B. (ed.) (1980) Empiricism, Logic, and Mathematics. Dordrecht: D. Reidel Publishing Co. Menger, K. (1974) Morality, Decision and Social Organization: Toward a Logic of Ethics. Boston: D. Reidel Publishing Co. Menger, K. (1994) Reminiscences of the Vienna Circle and the Mathematical Colloquium, L. Golland, B. McGuinness and A. Sklar (eds), Boston: Kluwer Academic Publishers. Morgenstern, O. (1970) Foreword in M. Davis, Game Theory, A Nontechnical Approach. New York: Basic Books. Morgenstern, O. (1976) Logistics and the Social Sciences, in Selected Economic Writings of Oskar Morgenstern, Andrew Schotter, (ed.) New York: New York University Press. Neurath, O. (1973) The Scientific Conception of the World (Cohen and Neurath, 1973). Nelson, L.H. and A. Wylie (1998) Coming To Terms with the Value(s) of Science: Insights from Feminist Science Scholarship, paper presented at Workshop on Science and Values, Center for Philosophy of Science, University of Pittsburgh, Oct. 9-11, 1998. Peirce, C.S. (1958) Collected Papers of Charles Saunders Peirce. Vol. 8, A.W. Burks (ed.). Cambridge, MA: Belknap Press. Poundstone, W. (1992) Prisoners Dilemma. New York: Doubleday. Rhees, R. (1965) Some Developments in Wittgensteins View of Ethics, The Philosophical Review 74(1):17-26. Richards, J.L. (1995) The History of Mathematics and Lesprit humain: A Critical Reappraisal, Osiris, 10:122-135. Russell, B. (1914) Our Knowledge of the External World. Chicago: Open Court. Saul, M. (1992) Love Among the Ruins: The Education of High-Ability Mathematics Students in the USSR, Focus 12(1):1, 6-7. Segerstrale, U. (ed.) (2000a) Beyond the Science Wars. Albany: SUNY Press. Segerstrale, U. (2000b) Defenders of the Truth, The Sociology Debate. Oxford: Oxford University Press. Spengler, O. (1927) Meaning of Numbers, in The Decline of the West, reprinted in J.R. Newman (ed.) The World of Mathematics, Vol. 4. New York: Simon and Schuster. Rudner, R. (1988) The Scientist Qua Scientist Makes Value Judgments, (Klemke, Hollinger and Kline, 1988:327-33). Rutte, H. (1991) Ethics and the Problem of Value in the Vienna Circle, in (Uebel, 1991:143-157). Uebel, T. (ed.) (1991) Rediscovering the Forgotten Vienna Circle. Dordrecht: Kluwer Academic Publishers. Van Brummelen, G. Mathematical Truth: A Cultural ̒. Chapter 2 in (Howell and Bradley, 2001). Von Neumann, J. and O. Morgenstern (1944) Theory of Games and Economic Behavior. Princeton: Princeton University Press. Wersky, G. (1978) The Visible College. New York: Holt Rinehart and Winston. Wigner, E. The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Communications in Pure and Applied Mathematics 13:1-18. Wilder, R. L. (1981) Mathematics as a Cultural System. New York: Pergamon Press. Wilson, Bryan, (ed.) (1970) Rationality. New York: Harper and Row. HJYZglmn , - TXLܾϪϕ hmW6] hmWNH hmW5\hV&OJQJ^JhV& hV&hV&'jhV&hV&OJQJU^J!jhV&hV&OJQJU^JhV&hV&OJQJ^JhV&hmWOJQJ^JhmW#hV&hmW5CJ OJQJ^JaJ 2$HIJYZm01z&{&& $da$gdV& $da$gdV&$a$gdV&$a$gdV&$a$gdV&ݥLM_`YZ  JKvwwx^_jk?B}~:;st9 : !!!! 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