A good place to generate fruitful discussion would be your ten points in your reference.
- 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.
- 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.
- 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.
- 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.
- 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.
- What is clear in a given conception is time dependent, both for the individual and
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.
- Pure (theoretical) mathematics is a body of thought developed systematically by successive refinement and reflective expansion of basic structural conceptions.
- 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.
- 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.
- 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.