Skip to main content

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7805))

  • 1865 Accesses

Abstract

In §5.3, we noted that type distinctions among various elementary objects were needed in order to avoid two linguistic problems. The first of these was notational collision; we saw that if we took, say, natural numbers and sets to be of the same type, then our theory incorrectly predicted that expressions like ‘1 + 1’ had multiple readings. Constructing this type distinction was hard because the standard foundational account defines natural numbers as sets. The second problem was that of unwarranted distinctions; we saw, for example, that the actual use of mathematical language required real numbers and complex numbers to be of the same type in order to assign a reading to an expression such as ‘zz  ∗ . < 1’. Preventing a type distinction here was difficult because the standard foundational account defines complex numbers as particular structures containing real numbers.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Ganesalingam, M. (2013). Foundations. In: The Language of Mathematics. Lecture Notes in Computer Science, vol 7805. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37012-0_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-37012-0_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-37011-3

  • Online ISBN: 978-3-642-37012-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics