The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

10
Finitism and Intuitive Knowledge
CHARLES PARSONS
1. INTRODUCTION
In the introduction to his paper of 1958 on an extension of the finitary method in proof theory, Kurt Gödel remarks that 'finitary mathematics is defined as that of intuitive evidence',1 by which he evidently means:
(1.1) If a proposition has been proved by the finitary method, then it is intuitively evident.
(1.2) If a proposition is intuitively evident, it can be given a finitary proof.
(1.1) and (1.2) formulate in a succinct way a thesis about the significance of the finitary method that did, in my view, belong to the outlook of the Hilbert school although it is not stated quite so directly in their writing. I have stated it in the way I have because I think it useful to consider (1.1) and (1.2) separately. It is also useful to consider them in connection with two theses that together constitute a mathematical characterization of finitism:
 (1.3) Proofs in primitive recursive arithmetic (PRA) are finitist; hence any theorem of PRA is finitistically provable. (1.4) If a proposition in the language of primitive recursive arithmetic is finitistically provable, then it is a theorem of PRA.
(1.3) is clearly expressed in writings of Hilbert and Bernays; an analysis of finitism that did not yield it would be hard put to it to show that it was

I am grateful to the participants in the Munich conference for their comments, especially to Geoffrey Hellman for subsequent correspondence. Since then the paper has been presented to other audiences, which I also wish to thank; in some cases I am conscious of not having done justice to the comments. I am grateful to Jaakko Hintikka for the invitation to speak at a symposium on Hilbert's Philosophy of Mathematics at Boston University, which led me to focus the paper on finitism.

____________________
1
"'Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes'", Dialectica, 12 ( 1958), 280-7, at 280. The paper is reprinted with an English trans, in Collected Works, ii: Publications 1938-1974, Solomon Fefermanet al. (eds), Oxford University Press, 1990. However, translations of quotations are my own. In the 1972 English version of the paper, what corresponds to the quotation in the text is the remark that 'finitary mathematics is defined as the mathematics of concrete intuition' (ibid. 272, emphasis Gödel's).

-249-

If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

Items saved from this book

This book has been saved
Highlights (0)
Some of your highlights are legacy items.
Citations (0)
Some of your citations are legacy items.
Notes (0)
Bookmarks (0)

You have no saved items from this book

Project items include:
• Saved book/article
• Highlights
• Quotes/citations
• Notes
• Bookmarks
Notes

Cited page

Style
Citations are available only to our active members.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited page

The Philosophy of Mathematics Today

Settings

Settings

Typeface
Text size Reset View mode
Search within

Look up

Look up a word

• Dictionary
• Thesaurus
Please submit a word or phrase above.

Why can't I print more than one page at a time?

Help
Full screen
/ 646

How to highlight and cite specific passages

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

Cited passage

Style
Citations are available only to our active members.

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.