Re: Paper and slides on indefiniteness of CH

Dear Sol,

A good place to generate fruitful discussion would be your ten points in your reference.

  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.

  1. 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.

  1. 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.

  1. 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.

  1. 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.

  1. 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.

  1. Pure (theoretical) mathematics is a body of thought developed systematically by successive refinement and reflective expansion of basic structural conceptions.
  2. 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.
  3. 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.
  4. 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.

Re: Paper and slides on indefiniteness of CH

John –

I am not sure that you are looking at this the right way – with regard to “one semester graduate course”.

Everybody, including mathematicians, have a wide range of affinities. Generally, they are very comfortable with and find certain subjects and topics very easy to absorb properly, whereas with other subjects and topics, the opposite is true. Here are some examples.

  1. One Fields Medalist said to me when I was starting to explain some work on graphs: I don’t know what a graph is. When I said “an irreflexive symmetric relation”, his eyes glazed over, clearly not absorbing that definition, and he said “I never want to know what a graph is”. Obviously, he had come into contact with some sort of things related to graphs in some passive setting, and he found it disgusting, intellectually, and that was that.
  2. Another very famous Fields Medal equivalent has tremendous geometric intuition. Yet he professes to be very uncomfortable with anything in discrete math. He took me aside and said he just couldn’t understand what a Turing machine is. I thought this was going to be easy. So I started with a picture of Turing machine squares on the blackboard, and then I put an arrow toward – and a little inside – one of the squares, and said that was the reading head, and that the reading head is at all times on exactly one square of tape. He stopped me and said that is where he just can’t understand what is going on. He asks me: what really is a reading head?
  3. You and I had some nice interactions on the piano, and I know at one time you wanted to play. You were OK, but you are now about as old as I am – I just set up for my Social Security checks. I know you would like to play a perfectly listenable rendition of say the entire Moonlight Sonata by Beethoven. Thousands of people worldwide can do that, including a few 6 year olds. Now what if I said that you could do that just attending piano lectures/demonstrations/lessons and paying serious attention for one semester?

With regard to 1,2, these are obviously somewhat unusual cases at the Fields Medal level, but I doubt if either one could after one semester get any useful understanding of even Goedel’s work on CH. They would not be able to even sit still for more than a minute in a serious course in set theory without becoming completely disgusted and wandering off mentally into the math they find so fundamental and congenial and for which they are duly very famous for devouring with little effort at the highest levels.

Harvey

PS: I apologize if I insulted you about the piano. Are you still working on the piano?

Re: Paper and slides on indefiniteness of CH

Obviously Pen and Sy are sort of talking past each other, which is understandable given the abstract nature of a discussion on general strategy of research in set theory with almost no mathematical specifics presented. In order to help the discussion, I have a few suggestions.

  1. Woodin and other close associates like John Steel have definite specific proposals for “settling” CH through explicit or reasonably definite conjectures. The statements are very complex, even for most set theorists, let alone logicians generally, or philosophers.
  2. Sy has raised various kinds of objections to their proposals for “settling” CH. Sy has been offering an alternative plan for “settling” CH. Sy is claiming some substantial advantages of his plan over Woodin/Steel plans. In particular, either explicitly or by inference, Sy is claiming that his plans are comparatively straightforward.
  3. I was intrigued by the offering up of a comparatively straightforward plan to “settle” CH. So I asked for Sy to provide some account of these comparatively straightforward plans here in this forum, so people can comment on them. I was hoping that this would not only help the present discussion, but also bring in some other people to comment on the advantages and disadvantages of Woodin/Steel versus Sy et al. I don’t know why I have been ignored. Sy?

Now here are some other matters related to the discussion.

  1. People seem to be blackboxing the idea of “good mathematics” or “good set theory.” These notions are desperately in need of some sort of elucidation particularly by people with foundational sensibilities. My working definition during my entire career has been “mathematics with a clear foundational purpose”. I do recognize that there can be “good mathematics” that does not – at least not obviously – fit that criterion. This is a crucial issue – what is good math or good set theory – as on at least one kind of reading, almost none of it has any clear foundational purpose.
  2. Sol’s use of the phrase “mental picture” in his earlier email reminded me of some ideas that I had left up in the air for some time. I buckled down and wrote the following posting to the FOM email list. I close with a copy of the substantive part.

Harvey Friedman

I have an extended abstract on some mental pictures, which can be used to arguably justify the consistency of certain formal systems.

#84  August 11, 2014

SOME MENTAL PICTURES I
by Harvey M. Friedman*
August 11, 2014
EXTENDED ABSTRACT

*This research was partially supported by the John Templeton Foundation grant ID #36297. The opinions expressed here are those of the author and do not necessarily reflect the views of the John Templeton Foundation.

Abstract. All mathematicians rely on mental pictures of structures. Some can be used to offer justifications for certain axiom systems. Here we use them to make arguable justifications ranging from Zermelo set theory to ZFC to various large cardinal hypotheses.

Re: Paper and slides on indefiniteness of CH

Can you “reasonably briefly, without getting into the weeds” state for the group, in email:

  • one or two of your favorite important set theoretic problems and one or two specific readily understandable set theoretic (maybe countable models of set theory) conjectures, answers to which you claim would “settle” these questions?

If you do this, we can see if any of us have any ideas about these set theoretic conjectures (or countable models of set theory conjectures). Also we can see what we think of whether they “settle” the  important set theoretic problems you choose.

There is an old saying KISS = keep it simple, stupid. (smile)

Harvey