The seventeen thought-provoking and engaging essays in this collection present readers with a wide range of diverse perspectives on the ontology of mathematics. The essays address such questions as: What kind of things are mathematical objects? What kinds of assertions do mathematical statements make? How do people think and speak about mathematics? How does society use mathematics? How have our answers to these questions changed over the last two millennia, and how might they change again in the future? The authors include mathematicians, philosophers, computer scientists, cognitive psychologists, sociologists, educators and mathematical historians; each brings their own expertise and insights to the discussion.
The present authors contributed the article “Experimental computation as an ontological game changer: The impact of modern mathematical computation tools on the ontology of mathematics.” We point out that computational tools, and the hardware they run on, are becoming so powerful that in many cases they constitute a mode of research equally compelling to the traditional mode of axiomatic proof. At the least, these tools raise fundamental questions about what is the nature of secure mathematical knowledge and how we discover and confirm that knowledge.
Other essays in the collection include the following:
- Ursula Martin and Alison Pease, “Hardy, Littlewood and polymath”
- Philip J. Davis, “Mathematical products”
- Ernest Davis, “How should robots think about space?”
- David Berlinski, Mathematics and its applications”
- Jody Azzouni, “Nominalism, the nonexistence of mathematical objects”
- Donald Gillies, “An Aristotelian approach to mathematical ontology”
- Jesper Lutzen, “Let G be a group”
- John Stillwell, “From the continuum to large cardinals”
- Jeremy Gray, “Mathematics at infinity”
- Jeremy Avigad, “Mathematics and language”
- Micah T. Ross, “The linguistic status of mathematics”
- Kay L. O’Halloran, “Mathematics as multimodal semiosis”
- Steven T. Piantadosi, “Problems in philosophy of mathematics: A view from cognitive science”
- Lance J. Rips, “Beliefs about the nature of numbers”
- Nathalie Sinclair, “What kind of thing might number become?”
- Helen Verran, “Enumerated entities in public policy and governance”
Additional details, previews and other material are available at the Springer site.