[FOM] Continuum/Feferman
Harvey Friedman
hmflogic at gmail.com
Tue Aug 12 01:48:03 EDT 2014
Dear Sol,
A good place to generate fruitful discussion would be your ten points in
your reference http://math.stanford.edu/~feferman/papers.html, #85.
1. The basic objects of mathematical thought exist only as mental
conceptions, though the
source of these conceptions lies in everyday experience in manifold ways,
in the
processes of counting, ordering, matching, combining, separating, and
locating in space
and time.
I WONDER IF THESE PROCESSES ARE THE PRODUCTS OF EVOLUTION, PECULIAR TO OUR
BRAINS, OR WHETHER THEY ARE MORE FUNDAMENTAL. IF BRAIN RELATED, CAN
SOMETHING BE SAID ABOUT THE BRAIN MECHANISMS INVOLVED, AND THE RELEVANT
BRAIN ORGANIZATION? IF MORE FUNDAMENTAL, THEN CAN WE REPLACE OUR EXISTING
FOUNDATIONAL SCHEMES, WHICH ARE VERY POWERFUL AND ROBUST AND ARGUABLY
CONVINCING, WITH NEW MORE FUNDAMENTAL FOUNDATIONAL SCHEMES? OBVIOUSLY,
THERE MAY BE A COMBINATION OF THE EVOLUTIONARY AND THE FUNDAMENTAL, A
MESSIER SITUATION TO DEAL WITH.
2. Theoretical mathematics has its source in the recognition that these
processes are
independent of the materials or objects to which they are applied and that
they are
potentially endlessly repeatable.
OUR FOUNDATIONAL SCHEMES HAVE BEEN CAREFUL TO REMOVE ANY REFERENCE TO SUCH
MATERIALS OR OBJECTS. WHAT IS TO BE GAINED BY CAREFULLY PUTTING THEM BACK
IN? THE CRUDEST FORM OF PUTTING THEM BACK IN IS TO SIMPLY ADD INERT
URELEMENTS TO SET THEORY. SOME WELL KNOWN INTERESTING THINGS HAPPEN WHEN
YOU DO THIS.
3. The basic conceptions of mathematics are of certain kinds of relatively
simple ideal world
pictures which are not of objects in isolation but of structures, i.e.
coherently
conceived groups of objects interconnected by a few simple relations and
operations.
They are communicated and understood prior to any axiomatics, indeed prior
to any
systematic logical development.
AGAIN, WHAT IS TO BE GAINED BY DEALING MORE DIRECTLY WITH THESE IDEAL WORLD
PICTURES? IN MY
http://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#64, I CONSIDER THIS BUT ONLY WITH THE DELIBERATE GOAL OF GETTING VERY
STRONG INTERPRETATION POWER.
4. Some significant features of these structures are elicited directly from
the world pictures
which describe them, while other features may be less certain. Mathematics
needs little to get started and, once started, a little bit goes a long way.
THE LESSON OF
http://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#64 IS SIMPLY THAT EXTREMELY LITTLE IS NEEDED TO GO A VERY VERY VERY LONG
WAY IN TERMS OF INTERPRETATION POWER. PI01 STATEMENTS COME FROM THE
INTERPRETATION POWER. THUS IF A STRONG SET THEORY T IS INTERPRETED IN WHAT
IS ARGUABLY INHERENT IN A SIMPLE MENTAL PICTURE, (PICTURE ACCOMPANIED WITH
A LITTLE TEXT), THEN THIS CONSTITUTES A PROOF OF ANY PI01 CONSEQUENCE OF
THE STRONG SET THEORY.
NOTE THAT YOU CANNOT USE JUST THE SOUNDNESS OF A PICTURE (WITH A LITTLE
TEXT) TO PROVE EVEN PI02 STATEMENTS THIS WAY - AT LEAST NOT WITHOUT MAKING
THE PICTURE MUCH MORE INVOLVED AND PROBLEMATIC.
5. Basic conceptions differ in their degree of clarity. One may speak of
what is true in a
given conception, but that notion of truth may be partial. Truth in full is
applicable only
to completely clear conceptions.
"COMPLETELY CLEAR CONCEPTIONS". THIS IS AN ENORMOUSLY IMPORTANT NOTION, AND
IT IS TO ME HIGHLY DESERVING OF GREAT ATTENTION.
YOU THINK THAT THE RING OF INTEGERS IS A COMPLETELY CLEAR CONCEPTION. SO
YOU THINK THAT TRUTH IN FULL IS APPLICABLE TO THE RING OF INTEGERS. YOU
THINK THAT (N,POW(N),EPSILON) IS NOT A COMPLETELY CLEAR CONCEPTION, AND IN
FACT FAR FROM A CLEAR CONCEPTION. AND THAT IT IS NOT THE CASE THAT TRUTH IN
FULL IS APPLICABLE. I BELIEVE THAT AS A CONSEQUENCE, YOUR BELIEF APPLIES
EQUALLY WELL TO THE RING OF REAL NUMBERS WITH A DISTINGUISHED PREDICATE FOR
THE INTEGERS, (R,Z,+,DOT). I CERTAINLY SEE A GREATER LEVEL OF CLARITY ABOUT
THE RING OF INTEGERS THAN ABOUT THE RING OF REALS WITH THE INTEGERS. BUT I
ALSO SEE A GREATER LEVEL OF CLARITY ABOUT THE INTEGERS OF MAGNITUDE <= 2^
2^100 UNDER PARTIAL ADDITION AND MULTIPLICATION, THAN I SEE ABOUT THE RING
OF INTEGERS. IN FACT, I ALSO SEE A GREATER LEVEL OF CLARITY ABOUT THE
INTEGERS OF MAGNITUDE <= 2^100 UNDER PARTIAL ADDITION AND MULTIPLICATION,
THAN I SEE ABOUT THE INTEGERS OF MAGNITUDE <= 2^2^100 UNDER PARTIAL
ADDITION AND MULTIPLICATION. A GREATER LEVEL OF CLARITY ABOUT THE INTEGERS
OF MAGNITUDE <= 100 UNDER PARTIAL ADDITION AND MULTIPLICATION THAN I SEE
ABOUT THE INTEGERS OF MAGNITUDE <= 2^100 UNDER PARTIAL ADDITION AND
MULTIPLICATION.
6. What is clear in a given conception is time dependent, both for the
individual and
historically.
IT IS MY IMPRESSION THAT YOU ARE HOPEFUL THAT AN ARBITRARY FIRST ORDER
STATEMENT ABOUT A "COMPLETELY CLEAR CONCEPTION" - AS A RELATIONAL STRUCTURE
- CAN, WITH HARD WORK AND HARD REFLECTION AND TIME BE PROVED OR REFUTED.
OF COURSE THERE IS THE PROBLEM THAT THE PROPERTY IN QUESTION MAY BE
INCOMPREHENSIBLE FOR VARIOUS REASONS, INCLUDING BEING TOO LONG TO STATE, OR
TOO CONVOLUTED.
SO LET'S SAY FOR THE SAKE OF ARGUMENT THAT THE PROPERTY IN QUESTION IS
STATED IN A FORM THAT IS COMPLETELY "NORMAL" FOR TYPICAL MATHEMATICS BEING
WORKED ON BY PROFESSIONALS.
THEN WHAT OPTIMISM DO YOU HAVE? THE STATEMENT IN QUESTION, EVEN IF TYPICAL
MATHEMATICALLY, MAY BE KNOWN TO BE EQUIVALENT TO THE CONSISTENCY OF SOME
EXTREMELY STRONG SET THEORY.
THUS THERE IS THE REAL PROSPECT OF US NEVER BEING ABLE TO PROVE OR REFUTE
INTERESTING STATEMENTS IN THE RING OF INTEGERS? DOES THAT PROSPECT, AND THE
WAY IT ARISES, CAUSE YOU TO RETHINK YOUR FEELINGS ABOUT "COMPLETELY CLEAR
CONCEPTIONS"?
I BELIEVE THAT THE SAME KIND OF PROFOUND NATURAL INCOMPLETENESS IS ALREADY
PRESENT IN THE INTEGERS OF MAGNITUDE AT MOST 2^100. THAT EVEN THIS CONTEXT
IS INEXTRICABLY LINKED UP WITH THE PRESENT LARGE LARGE CARDINALS.
7. Pure (theoretical) mathematics is a body of thought developed
systematically by
successive refinement and reflective expansion of basic structural
conceptions.
8. The general ideas of order, succession, collection, relation, rule and
operation are pre mathematical;
some implicit understanding of them is necessary to the understanding of
mathematics.
9. The general idea of property is pre-logical; some implicit understanding
of that and of
the logical particles is also a prerequisite to the understanding of
mathematics. The
reasoning of mathematics is in principle logical, but in practice relies to
a considerable
extent on various forms of intuition in order to arrive at understanding
and conviction.
10. The objectivity of mathematics lies in its stability and coherence
under repeated
communication, critical scrutiny and expansion by many individuals often
working
independently of each other. Incoherent concepts, or ones which fail to
withstand critical
examination or lead to conflicting conclusions are eventually filtered out
from
mathematics. The objectivity of mathematics is a special case of
intersubjective
objectivity that is ubiquitous in social reality.
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