[Alta-Logic] March 3: Potential Infinity: A Modal Account (Calgary Mathematics & Philosophy Lecture)

[Richard Zach]

Please mark your calendars: http://ucalgary.ca/mathphil/


  <http://ucalgary.ca/mathphil/files/mathphil/cmp-shapiro-art.pdf>


  Potential Infinity: A Modal Account


    Stewart Shapiro
    The Ohio State University


    Thursday, March 3, 2016, 3:30 pm
    ST 141

Beginning with Aristotle, almost every major philosopher and 
mathematician before the nineteenth century rejected the notion of the 
actual infinite.  They all argued that the only sensible notion is that 
of potential infinity.  The list includes some of the greatest 
mathematical minds ever.  Due to Georg Cantor’s influence, the situation 
is almost the opposite nowadays (with some intuitionists as notable 
exceptions).  The received view is that the notion of a merely potential 
infinity is dubious:  it can only be understood if there is an actual 
infinity that underlies it.

After a sketch of some of the history, Prof. Shapiro will analyze the 
notion of potential infinity, in modal terms, and assess its scientific 
merits.  This leads to a number of more specific questions.  Perhaps the 
most pressing of these is whether the conception of potential infinity 
can be explicated in a way that is both interesting and substantially 
different from the now-dominant conception of actual infinity.  One 
might suspect that, when metaphors and loose talk give way to precise 
definitions, the apparent differences will evaporate.

A number of differences still remain. Some of the most interesting and 
surprising differences concern consequences that one’s conception of 
infinity has for higher-order logic.  Another important question 
concerns the relation between potential infinity and mathematical 
intuitionism.  In fact, as will be shown, potential infinity is /not/ 
inextricably tied to intuitionistic logic. There are interesting 
explications of potential infinity that underwrite classical logic, 
while still differing in important ways from actual infinity.  However, 
on some more stringent explications, potential infinity does indeed lead 
to intuitionistic logic.

(The lecture is based on joint work with Øystein Linnebo.)

*Stewart Shapiro <https://philosophy.osu.edu/people/shapiro.4>* is 
O'Donnell Professor of Philosophy at The Ohio State University. He is an 
eminent logician and philosopher of mathematics, well-known for his work 
on second-order logic and on mathematical structuralism. He is the 
author of numerous articles and of five books, including most recently 
/Vagueness in Context/ (2006) on vagueness and logic, and /Varieties of 
Logic/ (2014) on logical pluralism.

/This talk is the second annual Calgary Mathematics & Philosophy 
Lecture, co-sponsored by PIMS <http://www.pims.math.ca/>, the Pacific 
Institute for the Mathematical Sciences, and the Department of 
Philosophy <http://phil.ucalgary.ca/>. The Mathematics & Philosophy 
Lectures aim to introduce topics at the intersection of mathematics and 
philosophy to a general academic audience. The event is free & open to 
the public; a reception follows./


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