FOM: The Foundational Exposition Project
Harvey Friedman
friedman at math.ohio-state.edu
Fri Mar 26 08:42:31 EST 1999
Here is a first draft of a manifesto. Feedback greatly welcomed and needed.
THE FOUNDATIONAL EXPOSITION PROJECT
by
Harvey M. Friedman
March 22, 1999
The Foundational Exposition Project (FEP) seeks to exposit crucial
mathematical and logical topics across the intellectual landscape. Initial
efforts will be confined to computer science, logic, mathematics, physics,
and probability/statistics. Later efforts will include economics, finance
and management science, electrical and mechanical engineering, linguistics,
mathematical psychology and political science, and law. Ultimately, it is
hoped that crucial mathematical and logical topics in the biological and
neuro sciences will be addressed.
The goal of the FEP is to produce a series of penetrating papers written
from a common approach to intellectual life. Each paper treats a crucial
topic from first principles, relying only on at most a few earlier papers
from the ongoing FEP. The papers are to be fully accessible to any
professional academic whose work involves mathematical and/or logical
considerations. Initial portions of each paper are to be fully accessible
to students with mathematical and/or logical sophistication. No substantial
knowledge of mathematics or logic is required for full accessibility.
In order to achieve this level of fully accessible uniform exposition,
substantial research is required at virtually every stage of the project.
The initial FEP papers are expected to treat the most classical of topics -
often considered to be completely understood by experts. The present state
of exposition of crucial topics in computer science, logic, mathematics,
physics, and probability/statistics all have serious drawbacks and
limitations that prevent them from having this level of fully accessible
uniformity:
1. In physics. When standard expositions are subjected to close
examination, serious ambiguities and/or hidden assumptions typically
appear. When subjected to intense philosophical examination, the standard
development loses meaning. E.g., consider Newton's laws based on force.
mass, space, time. In standard expositions, none of force, mass, space, or
time, is defined either theoretically or observationally. Their meaning is
recovered informally from the way that the laws are applied. However, after
the development is applied, experts fail to go back and redo the
development in a more meaningful, or philosophically honest, way. And
mathematical physicists tend to quickly assign sophisticated mathematical
structures to physical reality (manifolds, geometries, infinite dimensional
operators, etcetera). We believe that a suitably observational approach
should have greater philosophical clarity.
Tentative Initial Agenda:
a. Free particles. An observational treatment.
b. Free relativistic particles. An observational treatment.
c. Particles under gravitation. An observational treatment.
d. Relativistic particles under gravitation. An observational treatment.
2. In probability/statistics. Standard expositions ignore a number of
fundamental issues. One issue is randomness. This concept may best be
treated as a manifestation of symmetric ignorance. Another issue is that we
cannot measure to more than a finite amount of accuracy or conduct more
than finitely many trials, etcetera. Consequently models should be
finitary. For example, we are looking for a detailed philosophical analysis
of the meaning of such statements as: if I choose a sample of size k from a
population of size n, and the distribution in that sample is such and such,
then I know with confidence x that the distribution in the population has
such and such property. NOTE: The Bayesians emphasize the use of "priors,"
but there are important situations where priors are not needed, and one can
use "absolute ignorance."
Tentative Initial Agenda:
a. Probabilistic statements. A subjective approach.
b. Sampling theory. A subjective approach.
3. In computer science. This comparatively new field is already getting
fragmented. We need a clear unified treatment of the computer from the
circuit level through architecture through operating systems through
machine, assembly, and programming languages, and implementation.
Simplified languages should be constructed at these levels, and
verification given for some simple code. Reasoning involving protocols
should be analyzed formally. Complexity issues should be addressed in terms
of finite models since actual computer systems are finite.
Tentative Initial Agenda:
a. Circuit specification. Verification.
b. Architecture specification. Verification.
c. Programming languages and their implementation. Verification.
d. Protocols and reasoning.
e. Asymptotic and finite complexity.
4. In mathematics. Standard expositions typically cover a large number of
topics that are considered important by experts, and are loosely connected.
There is a concentration on efficient and elegant presentations, rather
than the realization of overarching intellectual goals of general
intellectual interest. Crucial definitions are typically introduced without
compelling justifications. Often compelling justifications can be given
such as "this is the only concept satisfying crucial conditions."
Tentative Initial Agenda:
a. Counting.
b. Measurement.
c. Shapes.
d. Linearity.
e. Symmetry.
5. In logic. Same general comments as for mathematics. At the most
elementary levels, logic and foundations of mathematics are closely
connected. Yet at more advanced levels, they have grown apart. So standard
expositions beyond the elementary levels do not concentrate on issues in
the foundations of mathematics.
Tentative Initial Agenda:
a. Propositional calculus.
b. Predicate calculus.
c. Set theory and the formalization of mathematics.
d. Axioms of set theory.
e. Fragments and extensions of the axioms.
f. Underivability results.
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