![[CS Dept., U Chicago]](http://www.cs.uchicago.edu/i/header_background.jpg)
There are many modes in which we assert propositions:
opinion, belief, hunch, wild
guess, .... Knowledge is distinguished from these other
modes of assertion by some sense of certainty. Different sorts of
knowledge probably derive their senses of certainty from very
different sources. Try to describe the source of certainty in each of
the following types of knowledge.
- ``I know that my redeemer liveth.''
- ``But I do know that I love you, and if you would only love me
too, what a wonderful world this would be.''
- ``I know that the sun will rise tomorrow morning.''
- ``I know you will expedite an answer.''
- ``I know I can pull the train over that mountain.''
- ``You know that you know that nothing can be known! How do you
know that you know?''
- ``I knowed dad-blamed well they wa'n't no fox in that
sourwood.''
- ``I know that two plus two is four.''
In this lecture, I try to explain the source of certainty in our
knowledge that two plus two is four, as well as more impressive
mathematical assertions, and perhaps some assertions that are not so
obviously mathematical.
Mathematical certainty derives from formal systems. There
are a number of reasons why it is particularly satisfying to learn
about the contribution of formal systems to OoK.
- There is a fun surprise in the way that ``formal'' is used in a
sense that is probably not the one you thought of first.
- Most mathematicians misunderstand very badly the way that their
work is founded on formal systems. So, you can get even with them for
the weird things they understand that mystify you.
- Formal systems support a highly objective and reliable form of
certainty, perhaps the most objective and reliable form of certainty
achievable by human minds.
- We can trace the way that formal systems are designed to provide
objective certainty. They do it by filtering out sources of
uncertainty.
- The knowledge derived from formal systems appears to be eternal
and static. But the means by which we achieve certainty with formal
systems is interactive and dynamic.
- Formal systems provide a highly explicit organization of the
knowledge that they support.
- There is a delicious irony in the way that derivations in formal
systems provide a very strong sense of external a priori certainty,
yet the derivations themselves are best understood as a sort of mental
choreography that each thinker must perform for herself.
Just read Section 5 of my article for the discussion of Descartes
and Hilbert.
Last modified: Mon Dec 5 19:14:49 CST 2005
