**January 19, 2017, **

**Erick H. Reck, University of California, Riverside**

*Note this event was cancelled.

Title: Dedekind and the Structuralist Transfomation of Mathematics

Abstract: In recent history an philosophy of mathematics, "structuralism" has become an important theme. There are two different ways in which this theme is typically approached: (a) by focusing on a structuralist methodology for mathematics (concerning the tools used, the ways in which various parts of mathematics are organized, etc.); (b) in terms of a structuralist semantics for mathematics ( a conception, or several related conceptions, of what we talk about when we study "the natural numbers", "the real numbers", "the cyclic group with five elements", etc.). In this talk, I will show that Richard Dedekind's writings from the nineteenth century should be seen as one of the main historical sources for both strands. Moreover , the two strands are intimately connected for him, as i will also try to establish. This will involve comparing this foundational work (on the natural and real numbers) with his contributions to algebra and number theory (his investigation of the notions of algebraic number, group, field, ring, lattice, etc., including his famous theory of ideals), Dedekind built on the work of his teachers and mentors, to be sure, especially Gauss, Dirichlet, and Riemann. But he went significantly beyond them precisely by initiating a structuralist transformation of modern mathematics, one that was later continued by Hilbert, Noether, Bourbaki, and in category theory.

**October 20, 2016,**

**Toke Knudson of SUNY, Oneonta**

During the first decades of the 20th century in Denmark the didactic method of geometry was discussed intensely. Early on it became clear that there were only two possible paths when it came to the teaching of geometry. One path was to uphold the Euclidean ideal and teach geometry according to the axiomatic method. The other path was to accept geometry as a natural science in which connections are seen through experiments. The “experimental method” was outlined in textbooks already from 1904. According to this method, the pupils were to get as far as they could through experiments, then switch to deduce new results from the set of “axioms” brought forth by the experiments. Johannes Hjelmslev, who was professor of mathematics at the University of Copenhagen, went further and constructed what he called “the geometry of reality,” which had as its only axioms the existence of graphing paper and rectangular blocks. Hjelmslev’s geometry directly contradicted classical geometry, which Hjelmslev considered a crude and poor approximation to the actual conditions of reality—the opposite view of a classical mathematician, who tells us that what we draw in geometry is only a crude approximation to pure geometry. Some of Hjelmslev’s claims, including that a tangent of a circle has a line segment in common with the circle, were rejected by some, but others took to his ideas. In particular, his followers wrote school textbooks according to his geometry. My talk will trace the discussion of geometry’s didactics in Denmark with an emphasis on the contributions by Hjelmslev.

**September 15, 2016,**

**Dr. Fred Rickey, Professor Emeritus, West Point Military Academy**

Title: E228

Abstract: How's that for the shortest title ever? How can you decide if a number is the sum of two squares? Euler begins with the dumbest possible algorithm you can think of : Take the number, subtract a square, and check if the remainder is a square. If not, repeat, repeat, repeat. But Euler, being Euler, finds a way of converting all those subtractions into additions. He applies this to 1,000,009, and ---in less than a page--finds that there are two ways to express this as a sum of squares. Hence, by earlier work in E228, it is not a prime. Amusingly, when he later described how to prepare a table of primes"ad millionem et ultra" (E467), he includes this number as prime. So he then feels obliged to write another paper, E699, using another refinement of his method, to show that 1,000,009 is not prim**e. **

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