Mathematics and Science

The first of two semesters of field science, this course is an introduction to natural science through field study that puts students in direct contact with the local natural environment. Through the direct experience and methodical observation of the heavens, geological formations, flora, and fauna, observational skills are sharpened and a sense of wonder at nature and natural history is cultivated. Students spend much time outdoors, drawing and recording data in sketchbooks. Shorter scientific readings supplement the books listed below.

Euclid’s Elements is the foundational text of mathematics in Western civilization. Without an understanding of this text, one cannot fully understand the most significant scientific and mathematical works of later eras, such as those by Galileo and Newton. This course studies Books I–VI, beginning with basic definitions, axioms, and common notions as established in Book I, and proceeding through introductory plane geometry, with a treatment of rectilinear figures (Books I–II) and circles (Books III–IV), as well as proportion theory, considered first in itself (Book V), then as applied to plane geometry (Book VI). Given the crucial role played by student-led demonstrations at the board, this semester and the following are skills classes.

This course continues with Books XI–XIII of Euclid’s Elements, completing the geometrical works with a treatment of solid geometry and the method of exhaustion, and culminating in the investigation of the five Platonic solids. Building on this foundation, the course then engages selections from the works of Apollonius and Archimedes on conic sections.

This course gives students a formal foundation in the nature of mathematics, including its relationship to natural science and metaphysics. Following on previous courses’ focus on the mathematics of continuous quantity, the course begins by investigating discrete quantity through several important proofs in number theory. The course then turns to the modern conception of number, including rationals, irrationals, complex, algebraic, and transcendental numbers, and more generally, the modern effort to unify discrete and continuous mathematics, which gave rise to analytic geometry and the calculus. This unification makes possible the proof that several famous problems (e.g., doubling the cube), which were unsuccessfully attempted by the ancients, are in fact impossible to solve. These considerations allow for reflection on Euclid’s accomplishment, as well as the very nature of mathematics and axiomatic reasoning, in a much more profound way. This conversation is further deepened by St. Thomas’s reflections on the nature of mathematics. Students sharpen their mathematical reasoning skills and develop problem solving techniques through frequent exercises, and engage in discussions on the nature of numbers, the mathematical conception of logic, infinity, limits, and mathematical beauty.

This course explores the mathematical methodology of empirical science through a consideration of the philosophical underpinnings of this methodology and through the study of probability and statistics. Students consider the processes of evaluating experimental evidence used to confirm or disconfirm testable hypotheses and learn to identify various fallacies in evidential reasoning.

The second of the two semesters of field science, this course provides students with the opportunity to take up field studies with the analytical, quantitative approach of modern empirical science. The tools of statistical reasoning acquired in the preceding semester allow for a more rigorous, firsthand experience of the application of the scientific method to natural observation.

Having seen the method of modern science in action (SCI 301 and 302), students in the senior year pay closer attention to the relations between science, philosophy, and theology. Introductory readings in SCI 401 consider the relation between science and philosophy explicitly. The course traces the development of atomic theory from the ancients through Aquinas, Dalton and other 19th century experimentalists, Mendeleev, and progenitors of the 20th-century atomic and quantum-theoretical viewpoints. Apparent tensions between classical and modern views are resolved, and the contributions of both science and philosophy validated.

In this course, students continue to ponder developments in modern science and their philosophical ramifications, with special attention to biology. Topics addressed include the relation of biology to the physical sciences, basic cell theory and genetics, and evolutionary theory, considered in light of scientific, philosophical, and theological reasoning. Focus is maintained on the ultimate goal of achieving a coherent synthesis of faith and reason.