from PART I - REFINING THE NOTION OF CATEGORICITY
Published online by Cambridge University Press: 19 January 2018
Bourbaki wrote ‘We emphasize that it [logical formalism] is but one aspect of this [the axiomatic] method, indeed the least interesting one.’ In this chapter, we begin the argument that Bourbaki have it backwards. Formal methods are not a pedantic puddle on the path to find the methods and principles that underlie a mathematical result but rather a stepping stone to understanding.
We explore the epistemological purposes of formalization along the lines of Manders (Chapter 2.1) and contrast Bourbaki's views with the contemporary mathematician, Kazhdan (Chapter 2.2). In Chapter 2.3 we take up Detlefsen's questions (page 15), and provide a criterion for the virtue of a property of a theory. In Chapter 2.4 we begin the explanation of how virtuous formal properties can provide a more fruitful way of organizing mathematics than that proposed by Bourbaki.
The Process of Formalization
In Chapter 1.1 we gave a static description of a formalization because such full formalizations of various topics are taken as the data for the organization of mathematics via classification theory. Here, we take a somewhat broader view. Lakatos writes:
For more than two thousand years there has been an argument between dogmatists and skeptics … The core of this case-study will challenge mathematical formalism but will not challenge directly the ultimate positions of mathematical dogmatism. Itsmodest aim is to elaborate the point that informal, quasi-empirical,mathematics does not grow through a monotonous increase of the number of indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations. ([Lakatos 1976], 4–5)
In contrast, I take formalization as part of that ‘incessant improvement’ and one that aims toward ‘indubitably established theorems’. Our goal here is to move beyond case studies, to show how formalization can be a tool for systematizing the changing definitions of mathematical notions. In this section, I expand more on how this tool will be developed in the book and where it is applied.
This kind of systematic analysis is developed byMarkWilson for physical science. My approach is less ambitious than Wilson's; I deal only with mathematics, not science writ large.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.