Skip to main content Accessibility help
×
Home
Hostname: page-component-99c86f546-7mfl8 Total loading time: 0.334 Render date: 2021-11-29T10:00:59.691Z 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 III - GEOMETRY

Published online by Cambridge University Press:  19 January 2018

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

Summary

Themoremodern interpretation:- Geometry treats of entities which are denoted by the words straight line, point, etc. These entities do not take for granted any knowledge or intuition whatever, but they presuppose only the validity of the axioms, such as the one stated above, which are to be taken in a purely formal sense, i.e. as void of all content of intuition or experience. These axioms are free creations of the human mind. All other propositions of geometry are logical inferences from the axioms (which are to be taken in the nominalistic sense only). The matter of which geometry treats is first defined by the axioms. Schlick in his book on epistemology has therefore characterized axioms very aptly as ‘implicit definitions.’ [Einstein 2002]

We have identified Einstein's ‘modern interpretation’ with Hilbert. But we take Einstein's ‘free creations’ in a limited sense. The axioms represent and sharpen prior intuitions. In this part we examine the historical relationship between certain intuitions, often formed by earlier axiomatizations, and new sets of axioms.We aim to evaluate axiomatizations of the geometric continuum.

In Chapter 9.1, we consider several accounts of the purpose of axiomatization and adjust Detlefsen's notion of descriptive completeness by fixing a criterion for evaluating axiom systems: modest descriptively complete axiomatization.

We lay out in Chapter 9.3 various sets of axioms, crucially formulated in different logics, for geometry and correlate them with the specific sets of propositions from Euclid that they justify.We emphasize those propositions of Euclidean, Cartesian, and Hilbertian geometry which might be thought to require the Archimedean or Dedekind axiom but do not; Hilbert's proof that the first order axioms suffice to define a field yields these geometric propositions. In particular, the notions of similarity and area of polygons are so grounded. This leads to the conclusion argued in Chapter 11 that Hilbert's full axiomatization is immodest. Such a formula as A = πr2 is not justified on the basis of Hilbert's first order axioms (even with Archimedes); but in Chapter 10, we expand the first order theory of Euclidean geometry EG, by adding a constant π which allows us to compute the area and circumference of a circle. Invoking o-minimality we do the same for the Descartes/Tarski geometry.

Type
Chapter
Information
Model Theory and the Philosophy of Mathematical Practice
Formalization without Foundationalism
, pp. 201 - 202
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.

  • GEOMETRY
  • John T. Baldwin, University of Illinois, Chicago
  • Book: Model Theory and the Philosophy of Mathematical Practice
  • Online publication: 19 January 2018
  • Chapter DOI: https://doi.org/10.1017/9781316987216.013
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.

  • GEOMETRY
  • John T. Baldwin, University of Illinois, Chicago
  • Book: Model Theory and the Philosophy of Mathematical Practice
  • Online publication: 19 January 2018
  • Chapter DOI: https://doi.org/10.1017/9781316987216.013
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.

  • GEOMETRY
  • John T. Baldwin, University of Illinois, Chicago
  • Book: Model Theory and the Philosophy of Mathematical Practice
  • Online publication: 19 January 2018
  • Chapter DOI: https://doi.org/10.1017/9781316987216.013
Available formats
×