Skip to main content Accessibility help
×
Home
Hostname: page-component-99c86f546-n7x5d Total loading time: 0.33 Render date: 2021-12-05T15:38:14.771Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

PART I - REFINING THE NOTION OF CATEGORICITY

Published online by Cambridge University Press:  19 January 2018

John T. Baldwin
Affiliation:
University of Illinois, Chicago
Get access

Summary

We reported in the introduction (page 15) several questions raised by Mic Detlefsen. In less precise fashion, they were: Is it better for a theory to be categorical or not? How do we justify whether categoricity is a virtue for a theory? Is completeness a good approximation to categoricity? This part has several themes that together respond to these questions.

  • The exact modern meaning of such terms as vocabulary, theory, and logic significantly influences the answer to these questions. These meanings were developed to address epistemological concerns.

  • The answers to such questions as these are highly dependent on the logic in which the theory is formalized. While a strong logic makes it easier to find categorical theories, this may in fact be a disadvantage. Too many theories may be categorical. The axiomatizationmay obscure the fundamental ideas of the area.

  • We precisely define our notion of a ‘virtuous property’.

  • We argue that ‘categoricity in power’ and ‘completeness’ are virtuous properties that spawned a family of others resulting in the role of modern model theory as both a mathematical tool and a schema for organizing mathematics.

  • Chapters 1 and 2 lay out the basic mathematical and philosophical (respectively) terminology of this book. Chapters 1.1 and 1.2 primarily address theme (1). Chapter 1.3 clarifies the role of various logics while properties of theories and axioms are examined in Chapter 1.4. Chapter 2 addresses philosophical issues about our notion of formalization. First we stress that it is a process and then we distinguish two possible goals: foundational and instrumental. Chapter 2.3 expounds the criterion of theme (3). Chapter 2.4 outlines how these virtuous properties can serve as organizing principles for mathematics and introduces the stability hierarchy, a set of virtuous properties that provide a specific method for such an organization that has powerful consequences for finding invariants for models.

    These distinctions underlie the argument in Chapters 3.1 and 3.2, which deal with theme (2). Chapters 3.3 and 3.4 develop the notion of categoricity in power, whose powerful consequences in finding invariants for models signal the importance of studying classes of theories. Thus we initiate the study of the paradigm shift which occupies Part II.

    Type
    Chapter
    Information
    Model Theory and the Philosophy of Mathematical Practice
    Formalization without Foundationalism
    , pp. 29 - 30
    Publisher: Cambridge University Press
    Print publication year: 2018

    Access options

    Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

    Send book to Kindle

    To send 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 sending to your Kindle.

    Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent 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.

    Available formats
    ×

    Send book to Dropbox

    To send 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 sending content to Dropbox.

    Available formats
    ×

    Send book to Google Drive

    To send 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 sending content to Google Drive.

    Available formats
    ×