HELEN S. LANG

Trinity College

In the spring of 1994, the Philosophy Department at Trinity College (Hartford, Connecticut) applied for and was awarded funding for two years (1995-96; 1996-97) from the NEH under a program entitled "Leadership Opportunity in Science and Humanities Education." The purpose of that grant was to add mathematics and science laboratories to philosophy courses. Historically, philosophy, mathematics and science have always been closely related, e.g., physics was often called "natural philosophy" and work in theories of consciousness is closely related both to developments in neuroscience and in Artificial Intelligence. Furthermore, a major part of philosophy's task as a discipline has been to account for the successes and failures of mathematics and science. Thus, we proposed, our laboratories would develop specific problem-solving skills and techniques in mathematics and science as they are associated with philosophy; they would include problems in physics, astronomy, biology and engineering as well as geometry, algebra, and calculus. In short, as we argued in our proposal, these laboratories would increase literacy in mathematics and science in a setting appropriate to these disciplines and resting on their historical and/or systematic relations to philosophy. By the end of the 1996-97 academic year, laboratories had been attached to ten courses in the Philosophy Department at Trinity College, including historical courses such as Ancient Philosophy, Medieval Philosophy, and Early Modern Philosophy, as well as systematic courses such as Medical Ethics, Philosophy of Science, Philosophy of Art, Philosophy of Sport, and two courses, Introduction to Cognitive Science and Minds and Brains, which also serve the Neuroscience major.

These laboratories proved enormously instructive both for faculty and students. Discussions arose concerning the possibility of developing laboratories in other courses in the humanities and in the arts, including archeology, art history, history, linguistics, and modern languages. In each case there was, once we saw it, an obvious relation between the humanities subject and specific related problems in science and mathematics. For example, the physiology of the human voice mechanism and the range of possibilities for producing sound as defined by that mechanism bear a direct relation to issues of language and speech formation that constitute a central topic for linguistics archeology utilizes techniques of surveying (and hence requires geometry) as well as basic principles of geology. So the question became one of extending and supporting science and mathematics laboratories into these areas of the humanities.

Unfortunately, the NEH initiative under which we had been funded was terminated by reductions at the NEH. So with a specific plan, but frankly little hope, for maintaining laboratories in the Philosophy Department and also extending them into other areas of the humanities, the Philosophy Department applied to the National Science Foundation. In the spring, of 1997 we received notification that this second proposal had been funded for a two-year period, i.e., September 1997 through August 1999. We are now moving ahead by developing further science/mathematics laboratories attached to philosophy courses and initiating laboratories in other departments in the humanities.

I would like to discuss here the idea, history, and work done by the laboratories as they have been (and are being) taught in conjunction with philosophy courses in particular (and humanities or arts courses more generally): what they are intended to achieve, what they look like, who forms their audience, and why they should be taught. I shall then turn to the experiences of those who have taught laboratories thus far, and draw some conclusions concerning these laboratories as constituting a natural bridge between disciplines that have come to live in quite separate academic homes. Science and mathematics laboratories attached to philosophy courses form this bridge not by pretending to be, or claiming to replace, science or mathematics

courses, courses with agendas of their own, but by recognizing, the coherence of modern culture itself, even as expressed within-or perhaps across-these disciplines. Here, for me at any rate, lies the most important lesson taught by these laboratories: by introducing topics in science and mathematics into philosophy courses, these courses have been improved as philosophy courses.

To begin at the beginning, then, our concern is that while many students who major in one of the natural sciences or mathematics regularly take courses in the humanities-in effect, they feel free and confident to take courses across the curriculum-the converse is not true. Most students who major in one of the humanities or arts, including philosophy, typically take the minimal requirements in science and mathematics (and that only because they are required), often because they lack confidence in their ability to perform in such courses. As a consequence, a major portion of the curriculum is closed to them at the starting gate. Mathematics and science departments have responded by designing imaginative and engaging courses in which these students fulfill the college requirements in symbolic reasoning and natural science and, perhaps more importantly, do achieve some contact with these disciplines. But these courses virtually never count toward the major and are perceived by the students themselves as "courses for incompetents." Furthermore, and most telling, the students for whom they are designed virtually never proceed beyond them. Herein lies the problem: it is hard to see how, within a four-year program, a single course in an area as broad as natural science, or mathematics, imparts skill or even minimal literacy, especially given the starting point and perception of these students; but it is impossible to increase the requirement or to ask of these departments that they do anything, more or differently than what they are now doing. These are just the students and these are just the problems targeted by the "laboratories in philosophy and the humanities" initiative.

At Trinity College, which is in this respect typical of many liberal arts colleges, the humanities occupy a central position. Thus humanities students represent a significant portion of the population. Indeed, before Trinity instituted its current requirements, students often told me that they chose Trinity because they "did not have to take science or mathematics." By integrating topics from these areas into the humanities, laboratories attached to philosophy courses locate science and mathematics centrally within the College and so grant these students the confidence and training to turn directly to the sciences and mathematics. Indeed, the laboratories make philosophy students both better philosophers and candidates for further work in mathematics and the natural science.

Courses in the Philosophy Department at Trinity College, as at most American colleges and universities, divide into historical courses, e.g., ancient or early modern philosophy, and systematic courses, e.g., philosophy of mathematics or cognitive science. The impact of laboratories is significant for both kinds of courses. First the laboratories introduce students to topics in science/mathematics and require them to develop problem-solving skills traditionally associated with these disciplines. But they do so within a humanities context. Consequently, methods and problems of science and mathematics are presented within a comfortable environment to students who are often reluctant to pursue such work. Because of their close historical and systematic relations, laboratories linked to specific philosophy courses provide a completely natural home to topics from mathematics and science.

Each laboratory takes up work appropriate to the topics of the course to which it is attached. For example, the famous sign over Plato's academy reads "who knows no geometry do not enter here" because mathematics provides the model for all knowledge. But students can enter a course on Ancient Philosophy with no knowledge of geometry-indeed with a sense of fear, if not loathing, of mathematics. The laboratory, which I shall discuss in detail in a moment, addresses an obvious need here. Again, problems in cognitive science have developed from and work with neuroscience; thus, a laboratory that is shared by Introduction to Cognitive Science and its more advanced cousin, Minds and Brains, takes up neural network theory and the use of computers for modeling and simulating brain function. And again, Philosophy of Sport considers a variety of ethical issues associated with sports as a professional activity and has developed a "wet-lab" that includes, for example, experiments with rats on the use of steroids. Here the question is clear: if a philosopher claims to take up ethical issues concerning the use of "performance enhancing" drugs, is it not important to have some basic understanding of the drug, itself and its physiological effects? These examples can easily be multiplied.

Here we arrive at what is, I believe, the central issue concerning these laboratories, an issue to which I shall return in my conclusion. Philosophy has become isolated from its origins in mathematics and science because it has developed a full agenda of its own theoretical material. Thus courses almost universally fail to consider specific scientific or mathematical problems, even though these problems lie directly behind philosophical claims. Consequently, the theoretical claims of philosophy as a discipline are considered in abstraction from the substantive issues on which those claims bear. Laboratories dedicated to relevant mathematics and science extend, indeed amend, the curriculum by adding appropriate science and/or mathematics directly into the humanities curriculum; and in doing so they enable philosophy to conduct it own work in a fully informed way. In short, laboratories provide a forum in which relevant mathematics and science can be explored because basic literacy about substantive issues in physics and biology renders questions such as "can one base a theory of knowledge on a method derived from physics or mathematics" meaningful-and answerable-in a sense that is impossible without such literacy.

Mathematics is of special importance in this regard. For example, interests in geometry are central to ancient philosophy; fourteenth-century philosophy is marked by the recovery of Arabic numbers, the development of algebra and the ability to write (and hence work with) exponents; Descartes, "the father of modern philosophy," was first (some say foremost) a mathematician; Leibniz is a co-inventor of the calculus. A recent (and highly acclaimed) study of Kant argues that one cannot understand his theories of space or time as ideal apart from his understanding, of the calculus.' More recent examples would include Russell, Wittcenstein and Quine. Yet claims regularly made by philosophers about mathematics and the mathematical basis of reality often have virtually no specific content for humanities students. Some students are "afraid" of mathematics and see the humanities as a "haven of safety"; even students who know some mathematics have learned it in a setting so sharply cut off from that of philosophy, and the humanities generally, that it disappears within a humanities setting. By focusing on the problem-solving strategies

of mathematics within a humanities setting, science/mathematics laboratories attached to philosophy courses allow students access to the unique strengths of mathematics and the reason why it plays so pivotal a role in science and philosophy. In my own laboratory (which I shall discuss in a moment), a number of students not only found that they could solve problems using, mathematical tools, but some have reported that now "at last" they understand the point of doing, proofs and why mathematics is so important.

Although the work performed in the various laboratories has differed considerably, I would like to describe in some detail a laboratory, that has recently been taught so as to give a clear idea of what it entails. Two courses, one in ancient philosophy and the other in medieval philosophy, are generally offered simultaneously and draw students with varying interests and backgrounds-usually not in mathematics or science. I have developed a laboratory required for all students from both courses, following, the model of those offered in the sciences. That is, it meets once every week for three hours, carries .25 course credit² and focuses on solving problems deriving from mathematics, physics, and astronomy, as they relate to philosophical concepts within these courses. In effect, the courses explain philosophical concepts connected to science and mathematics, the laboratory actually solves problems, as they were done at the time.

(1) In ancient and medieval philosophy, mathematics is in a strong sense the model of knowledge for Plato, Augustine, Anselm and Bonaventure. What we might think of as empirical science for these philosophers resembles applied mathematics. In the laboratory, we first take up problems in mathematics. We consider the problem-solving strategy of Euclidean geometry, especially constructions-in Plato's Timaeus god "constructs" the world according to mathematical intervals and Augustine's account of Genesis follows Plato's Timaeus, we learn some definitions and consider the mathematical character of definitions-Socrates seeks definitions for justice or courage and Anselm analyses a "perfect definition" of God in order to prove that He must exist; we take up problems including, for example, treating numbers as lines, the discovery of incommensurables and the Pythagorean theorem. Students prove that the internal angles of a triangle equal 180⁰. Most importantly, we discuss the concept of a proof, an issue central to both mathematics and physics. When the works of Aristotle, which had been lost to the West, were recovered around 1200, they were accompanied by Arabic commentaries that introduced Arabic numbers and a new kind of mathematics, i.e., algebra (an Arabic word) into Europe. The goal of this section of the laboratory is to understand what is at stake between constructions of geometry and manipulations of algebra and how a shift in mathematical strategy produces a corresponding, shift in the conception of problems to be solved within mathematics. This point prepares the way for work in astronomy and physics.

(2) Plato and Aristotle developed the first known proofs of god and these proofs go on to a long, and important history in both medieval and modern intellectual history. Although both proofs utilize notions developed in geometry, e.g., that a circle is "the perfect figure," they clearly rest on certain assumptions derived from astronomy, e.g., all the stars are equidistant from the earth and appear to move in perfect circles. Greek and medieval astronomy may be thou-ht of as basic observations, e.g., the motions of the stars and planets look circular, the sun appears to move northward and southward, which are constructed on a geometrical model. With this model, a number of problems can be accurately and successfully solved. So for example, one can prove that the earth must be spherical, calculate its circumference at the equator, explain eclipses and the motions of the sun, and show that the morning star and the evening star (Venus) are the same body. (Students actually solve this problem in the laboratory and find it particularly satisfying.) We also consider problems of planetary (retrograde) motion and why they cannot be solved on ancient/ medieval models. This raises for us the issue of when (if ever), or why, a scientist changes from one model to another.

(3) Mathematics and astronomy raise issues concerning how we interpret the world and our experience of it. Astronomy considers the heavens and physics takes all bodies in motion. So for Aristotle (and Thomas Aquinas puts Aristotle's argument to work for his own purposes), physics founds an important proof of god. The laboratory concludes with a series of problems concerning bodies in motion, including problems in ancient atomism, the geometry of Plato's

elements, and Aristotle's so-called law of motion, "everything moved is moved by something." In the fourteenth century, ingenious thou-ht experiments completely redefine Aristotle's physics and in many respects prove it wrong.³ These thought experiments develop the notion of variables and are generally thought to have been precursors of "physical" laboratories. They demonstrate quantitative problem-solving strategies and show how late medieval physics prepared the way for the scientific revolution of the Renaissance. Students design their own "anti-Aristotelian" thought experiments, identify their variables, and define the conclusions that follow from them. Thought experiments return us to the problem of models. In the final laboratory, students are shown a set of 16th century celestial maps which mix Ptolemaic and Copernican astronomy and physics; they identify the specific characteristics of each model, name the technical features of each map and on one map find the mistake that mixes the two models in an impossible way.

At a more practical level, the first hour of each laboratory establishes the problems at hand and the tools necessary for their solutions. The remainder of the laboratory involves collaborative problem solving and discussion of the relation of these problems and their solutions to more theoretical implications of the concept at work. Each week, every student must write a laboratory report consisting, of solutions to the problems posed for that laboratory and a one-page thought paper on the concepts and methods used in producing the solutions. The laboratory appears on the transcript as a course carrying .25 credit and receives a separate grade.⁴

At its conclusion, a laboratory is evaluated as is any other course. The laboratory described here has been taught twice; the student evaluations were extremely positive and may be briefly summarized. (1) The students found the laboratory intrinsically interesting, and enjoyed developing and exercising problem-solving skills. The collaborative character of the work within the laboratory was especially important. (2) Upwards of half the students in the laboratory first reported themselves "humanities types." The laboratory reports of these students show not only a significant growth of confidence with quantitative skills, but also a willingness to abandon the rigid line between humanities, philosophy in this particular case, mathematics, physics, and astronomy. (3) The laboratories consisted of almost half women, many of whom initially felt unable to do this work. As the laboratory continued, they expressed a strong sense of achievement not only at succeeding in the laboratory, but at finding it interesting. Several ultimately assumed leadership roles within the working groups of the laboratory. (4) The criticism voiced most concerns the credit awarded for the laboratory-that it should be .5 rather than .25 credits. Frankly, I think that the standard set by the sciences is right and I am untroubled by this criticism.

All the laboratories in the Philosophy Department are developed by, members of the Philosophy Department working together with partner(s) from mathematics and science departments to design the specific experiments to be conducted in them. The grant pays each partner for their time commitment. Two tasks are being addressed simultaneously: (a) the science related to the philosophy course must be identified and developed in a series of experiments. (I may note here that two laboratories were ultimately taught by, or team-taught with, scientists.) (b) The second task concerns coordinating the laboratories with one another and also with other programs. So for example, some philosophy courses serve other programs, most importantly Neuroscience. Laboratories developed in the Philosophy Department complement and extend the work of these programs.

We believe that our laboratories, first supported by our NEH grant and now by the NSF, have been quite successful and promise a means for accomplishing several goals, which I shall briefly mention here, but return to more fully in my conclusion. (1) They significantly increase problem-solving skills associated with mathematics and natural science because they target students in the humanities (and they reach a large number of women), who benefit most from such integration. (2) They provide a model for treating the humanities, sciences and mathematics as intrinsically related disciplines. (3) In some cases they have led to a constructive rethinking of the teaching, of philosophy itself. Members of the Philosophy Department initially expected the theoretical issues of philosophy to be applied within the problem-solving context of the science and mathematics in the laboratory. But our experience has been exactly the opposite: as problem-solving, skills increased, students become both more demanding and more skeptical of theoretical claims taken apart from problems. In short, the science/ mathematics laboratories drive the classroom, not the other way around. This relation is in part intellectual and in part rests on the collaborative character of work in the laboratories.

The issues raised by laboratories attached to philosophy courses (and courses in other departments as well) are in part addressed by a recent report entitled "Quantitative Reasoning for College Graduates." Prepared by the Subcommittee of the Committee on the Undergraduate Program in Mathematics of the Mathematical Association of America, it concludes that:

Basic quantitative literacy depends on students being, introduced to the foundations of quantitative reasoning and then given reinforcement experiences which develop and deepen in the student the habits of thinking which the student has been encouraged to develop. Taking one course is not enough to endow a student with a habit of mind, but completing a carefully devised program can provide sufficient practice to make a pattern of thou-ht part of the student's intellectual tools.

Our project of laboratories for courses in Philosophy (and other departments in the humanities) presents, we believe, a distinctive and effective means for achieving this goal-making "a pattern of part of the student's intellectual tools"-through the integration of knowledge and the "decompartmentalization" of knowledge that has resulted from its division into departments.

With our current grant from the National Science Foundation, our experiment in laboratories is entering its second phase, which, building on the first phase, will extend the laboratories into other departments, including Classics, Modern Languages, and History within the humanities, where there is a "natural fit" between humanities, science and mathematics. They represent more fully than can Philosophy alone the sense in which much can be done to "decompartmentalize" the relation between science/mathematics and the humanities. In these cases, each individual instructor must develop his or her own laboratory as attached to a course in the appropriate discipline-philosophy has nothing to say on this matter; but our pedagogical experience in crossing the standard boundaries of science, mathematics, and humanities can be helpful as, too, can our experience with various administrative issues, e.g., those involving, the awarding of credit. The entire community benefits from the broadening of the experiment of the Philosophy Department, the broadening of the conversation among members of the humanities and the sciences.

Adding laboratories to courses in the Philosophy Department has suggested a number of conclusions to the philosophers who have participated in them. Perhaps the most direct route to these conclusions lies in a consideration of the single most difficult question that vexed us at the outset of this "experiment." Is the Philosophy Department (and now other departments in the humanities as well) claiming to teach (and to "do") science? Would we, for example, one day ask that completion of a certain number of our laboratories count toward completion of the science requirement? If we answer "yes we are teaching, and doing, science," then what are our qualifications and why should we take over this work, when it is already being, done and done better by our colleagues in the sciences? But if we answer "no," then what are we doing? I believe that the answer is mixed: yes and no. Scientists have a full agenda of their own and they proceed to conduct their work in a systematic way, according to this agenda. And the science requirement recognizes the character and importance of this work as systematic and as disciplinary. Likewise philosophers (and members of other humanities disciplines, such as classicists, historians, and linguistics) have their own agendas, their own issues that they pursue according to their own disciplines. The laboratories take up problems from the natural sciences and mathematics, but not according to the systematic agenda of sciences or mathematics; the laboratories are by definition subordinated to topics in Philosophy (or Classics or History) which is a discipline in the humanities. What are we doing? We are doing for the natural sciences and mathematics the same job that they do for English and composition: impart basic skills and habits within the intellectual agenda set by one's own discipline. Students who study chemistry or biology or physics must learn to write "lab reports," which are in their way a sophisticated form of writing; but we do not feel that they should receive credit for a composition course. Basic literacy in language is necessary for, achieved through, and put to work in science laboratories for a larger purpose which is itself defined by the science or mathematics course. Likewise laboratories attached to philosophy courses help students achieve basic "habits of thought" in solving problems defined by the natural sciences and mathematics; but they do so for the larger purposes defined immediately by the philosophy course and ultimately by the issues raised by philosophy as a discipline.

This brings us to the most important conclusion that follows from our laboratories. Language is a universal skill-and is recognized as such-but quantitative reasoning skill and the problems that it best addresses have, for complicated historical reasons, come to be seen as something, dispensable, something which "humanities types" can largely live without-and live well. And to some extent this view is simply false. To talk, for example, about mathematics as a model for knowledge in the complete absence of skills for solving problems in mathematics-and a sense of what is involved in those problems-is at once to perform a 'Very dangerous abstraction and to impoverish the domain of one's own work, whether one is the teacher or the student.

Earlier this term I asked a philosophy major in my laboratory whether he had had any other laboratories in the Philosophy Department. "Yes" he said, "in fact I have not had a course without a laboratory. And they have all been so different and so interesting. I really had no idea how connected everything is." My sample, while admittedly limited, is nonetheless important. And philosophy is the best place to be-in seeing these connections because they lie at the heart of the discipline and so should lie at the heart of our teaching.

1, Michael Friedman, Kant and The Exact Sciences (Cambridge: Harvard

University Press, .1992). Cf. the review of Friedman by Gary Hatfield, "Review

Fssay: The Importance of the History of Science for Philosophy in General" in I Syntheses 1995, pp. 10-18.

2. Different institutions use different counting systems. At Trinity College, philosophy courses, like most courses in the humanities, meet 150 minutes a week and carry one credit toward the 36 credits required for graduation.

3. For an example of how this physics was done, cf. Helen S. Lang, Aristotle's Physics And Its Medieval Varieties (Albany: S.U.N.Y. Press, 1992), pp. 173-187.

4. This matter has in fact been somewhat complicated and the practice of a separate course number and grade for the laboratory applies only to laboratories enrolling students from more than one course. When the laboratory is attached to a single course, the course credit is shown as 1.25 credits and the laboratory is shown as 0.00 credit. Thus in this case, there is a single grade for the course and. the laboratory. At, Trinity College, this practice follows that of the natural sciences.

Helen S. Lang, Department of Philosophy, Trinity College, Hartford, CT 06106-3100