FOM: listing foundational issues

Vaughan Pratt pratt at cs.Stanford.EDU
Sat Mar 7 17:33:03 EST 1998

From: Stephen G Simpson <simpson at>
>I have therefore proposed the following tentative list of basic
>mathematical concepts:
>    number
>    shape
>    set
>    function
>    algorithm
>    mathematical definition
>    mathematical axiom
>    mathematical proof

Presumably "shape" is intended here as a suitably generic term that
includes "line", "surface", "sphere", etc.

But in that case it would seem equally reasonable to include "object" as
an equally generic term that includes "relational structure", "topological
space", "manifold", etc.  After all, what would the first-order definition
of "set" mean without a relational structure to furnish it with a meaning?

Just how basic objects are depends on where one views mathematics as
beginning.  While it is reasonable to consider objects as being not so
basic when they are defined set theoretically, it seems unreasonable
to insist that this perspective is absolute.  From the categorical
perspective on mathematics, objects are the most basic of all entities,
being the first thing a typical categorical foundation talks about.

If objects are not basic I do not understand what criterion excludes
them while admitting algorithms, definitions, axioms, and proofs as basic.

Vaughan Pratt

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