FOM: The Foundational Exposition Project/preamble
Harvey Friedman
friedman at math.ohio-state.edu
Mon Apr 12 16:21:12 EDT 1999
I have beefed up my original manifesto of March 22, 1999 with a preamble.
This posting contains both the preamble and the original manifesto.
Comments appreciated!!
PREAMBLE
by
Harvey M. Friedman
April 11, 1999
Below you will find a short "manifesto" about the Foundational Exposition
Project. I have a lot of enthusiastic optimism about it and some definite
ideas of how to get it off the ground.
The project is designed to directly address the continued and growing
intellectual alienation between different academic cultures. I think that
we are witnessing greater intellectual compartmentaliztion between at least
the areas I am most familiar with than I have ever seen since entering the
profession in 1967. The project seeks to address this at the most
elementary and fundamental levels possible.
I have a rough model of some important aspects of this phenomenom. The
usual presentations of the basics of all major subjects are logically and
philosophically unclear (strictly speaking, incoherent) - in various senses
and levels. Most people - even those who go on to become professional
academics - are uncomfortable with these presentations. At least not
sufficiently comfortable so as to consider a professional acadmic career in
that subject. They don't really "get it" - they just don't feel at ease
with the culture of that subject.
However, some people are somehow comfortable with one (or sometimes more
than one) of these presentations - despite their logical and philosophical
murkiness. These people are somehow "immune" to the difficulties that
others have.
Typically, for example, A may feel entirely comfortable with the basics of
subject X, but feel quite uncomfortable with the basics of subjects Y,Z,W.
And B may feel entirely comfortable with the basics of subject Y, but feel
quite uncomfortable with the basics of subjects X,Z,W.
The project seeks to create a series of fundamental expositions and
reworkings of the basics of a wide variety of subjects from a logically and
philosophically sophisticated point of view which aims to have the
following effect: That any professional academic (or suitably sophisticated
intellectual) prefers these expositions in every subject, except, perhaps,
the subject in which they have become an expert - i.e., except the subject
for which they have an "immunity" and somehow don't need logically and
philosophically coherent presentations.
I have two slogans that pinpoint the point of view of this project.
1. START EVERYTHING OVER FROM THE BEGINNING.
2. EXPLAIN EVERYTHING TO EVERYBODY (OUTISDE THEIR AREA OF EXPERTISE).
The intention is to first create an initial collection of logically and
philosophically sophisticated presentations that is sufficient to test
these ideas with a suitable audience. It is expected that the nature of the
logical and philosophical approach will be clear from these presentations,
so that the principles of exposition become clearly understood. Uniformity
of approach is a crucial issue, but this uniformity is rather subtle. I
have no intention of falling into oversimplistic traps that some feel have
plagued such efforts - of the philosophy of science community - earlier in
the century. Extremely close contact with top professionals who have a
logical/philosophical bent is absolutely essential.
The implications of the successful completion of this project are very
substantial:
A. Putting an end to this profoundly crippling lack of common understanding
between academic communities. They don't understand and/or appreciate each
other's assumptions, goals, value systems, or methodologies.
B. Facilitating the emergence of key interdisciplinary fields of the
future. As a profound understanding of the basics of most fields other than
one's own becomes reality, the discovery of key interdisciplinary insights
will be greatly facilitated.
C. Overhauling the educational process at the University level. General
education will attain an entirely new meaning. There will be a level of
integration in education that is currently unimaginable.
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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