Online talks

(Provisional list)

• Curtis Franks (University of Notre Dame)

• Elaine Landry (UCDavis)

• Marco Panza (Chapman University)

• Luiz Carlos Pereira (PUC - Rio)

• Stewart Shapiro (Ohio State University)

• Wilfried Sieg (Carnegie Mellon University)

• Will Stafford (Kansas State University)

• Neil Tennant (Ohio State University)

• Gabriel Uzquiano (University of Southern California)

• Kai F. Wehmeier (UC Irvine)

• March 31st - Elaine Landry (UC Davis), Abstraction, Structures, Satisfiability and Applicability

• April 28th - Will Stafford (Kansas State University)

• May 26th - Gabriel Uzquiano (University of Southern California)

• June 30th - Curtis Franks (University of Notre Dame)

• July 28th - Neil Tennant (Ohio State University)

• August 25th - Marco Panza (Chapman University)

• September 29th - Wilfried Sieg (Carnegie Mellon University)

• October 27th - Luiz Carlos Pereira (PUC - Rio)

• November 10th - Stewart Shapiro  (Ohio State University)

• November 24th - Kai F. Wehmeier (UC Irvine)

Elaine Landry
Abstraction, Structures, Satisfiability and Applicability

In this paper I use a Plato-inspired methodological notion of mathematical axioms to first connect to the modern Hilbert-inspired structural notion of the axiomatic method and then use this to argue that this methodological notion of structure further allows us to use satisfaction to give an account of the applicability of mathematics.I have argued that Plato was not a mathematical metaphysical realist, rather he was a methodological realist. That is, for the purpose of solving a problem, we act as if our mathematical hypotheses are true principles and, in so doing, we commit to the stably defined objects that our hypotheses employ. For example, in solving the Meno problem, we act as ifthe hypothesis “the length of the line that doubles the area of the square of length two is the diagonal of the square of length two” is true, we then construct the square and reason down to the conclusion that such a square will have double the area. In virtue of our having solved the problem we then commit to the “objects of thought” that are needed to solve the problem, i.e., to stably defined “squares themselves” and “diagonals themselves”. Thus, the methodological realistmoves from truth to existence, whereas the metaphysical realist moves from existence to truth.
I next apply this to the structuralist notion the axiomatic method: we take consistent mathematical axioms as-if they are true first principles for the purpose of solving now a metamathematical problem, that is, for the purpose of stably defining the objects of mathematics in terms of the structures they satisfy. In line with both Dedekind and Hilbert, we again methodologically move from truth to existence. For example, taking the Peano axioms as if they were true first principles that stably define natural numbers themselves, we are thereby committed to numbers as positions in any system that satisfies the axioms. Thus, we arrive at our methodological account of mathematical structuralism: a mathematical object is anything that satisfies the structural relations as set out by the axioms.
Finally, I turn to consider how this methodological account of mathematical structuralism can be further used to give a methodological structuralist account mathematical applicability. Simply, when we use a mathematical theory to solve a physical problem we act as if our physical objects/relations are mathematical objects/relations. As with Plato, in their solving of physical problems physicists are to reason down from mathematical objects/relations as defined by a mathematical theory, and if they are able to solve the problem, they then commit to taking the physical objects/relations as if they were mathematical objects/relations, where such objects/relations are now taken as anything that satisfies the structural relations as set out by the now physically interpreted mathematical theory. Thus, in science, as methodological structural realists, if a theory is successful, we commit not to the objects of the physical theory but only to the successive structural relations they satisfy.