I'm not too keen on the title of this question but for somewhat ill-defined reasons I don't feel it should be closed just yet. It does not have the ring of crankery, and the text of the question refers to explicit remarks/ideas of Friedman.

Seeing as the question is on the verge of being closed, but with at most two people (at time of writing) giving reasons why, perhaps people who have seen more of the work/arguments on foundations could give their feedback on this thread?

I disagree with Andres: I think that even finding the name of a philosphy student who is working on these issues will be helpful to the original poster, and a gain to others who are learning of "meta"- metamathematics. If nothing else, one can comment on the success one has with using such heuristics to develop parts of set theory. Much of the work in the last century in foundations was experimenting with and refining certain philosophical notions and giving some of them a firm mathematical footing. If Harvey Friedman is looking at IP and PP and producing some results, that gives me an idea of what this century's work will be like.

Gerhard "Ask Me About System Design" Paseman, 2010.11.11

It seems to me that "PP: Anything that can happen will" cannot possibly be a mathematical statement. So the question:

> My main questions are: How confident are we that PP and IP are “true” ? More specifically, is it possible to “prove” or justify PP and IP rigorously? If yes, how? If not, why not?

seems like it cannot possibly be a research-level question in mathematics. That said,

> My secondary question is: are there any logician (beside Friedman) who are working on this kind of research?

seems perfectly reasonable, perhaps rephrased as "where can I read about the work being done based on Friedman's program?" without the philosophical agenda. MO seems to allow reference requests, no?

I think closing my previous question (link text) on the basis of its not being mathematics would be a mistake. At least there are two famous theorems of contemporary mathematics echo the principle of plenitude (PP), namely The Infinite Monkey Theorem and Kolmogorov's zero-one law (See e.g. link text).

In his most recent posting on FOM, Harvey Friedman used PP to formulate the system OEU and proved the mutual interpretability of ZF and OEU. That means anything that you can prove in ZF, you can prove in OEU as well. Therefore if mathematics can be interpreted in ZF, then mathematics can be interpreted in OEU as well. See his complete posting on FOM (link text).

Prof. Caicedo, it sounds like you see (and please, please correct me if I am wrong) Friedman as essentially using an extended metaphor to explain his motivation for considering various mathematical statements or axioms. If so, I would like to retract my unwarranted comparison with Badiou - at first read, I was extremely put off by many sections from Lianna's linked pdf, in particular

(p.3):

In particular, there are contexts in which there are orderings but where the very idea of quantitative measurement (as presently construed) is inappropriate or even absurd, and so there will not be or cannot be any associated real number orthodoxy. For example, "x is more beautiful than y". Or "idea x is more interesting than idea y". Or "act x is morally preferable to act y". Or "agent x is morally superior to agent y". Or "outcome x is more just than outcome y". Or "act x is more pleasurable than act y." Or "activity x is preferable to activity y" or "state of affairs x is preferable to sate of affairs y".

(p.10-11):

From some points of view, one can criticize Real Existence of section 1.2 on the grounds that one is asserting the existence of a real object using a formula that involves things that are not real. By the claim after Theorem 1.2.1, we have met this objection to a considerable extent.

There are few things more irritating to me than seeing these words or concepts mixed in with math, and I clearly have strong philosophical disagreements with Friedman, but if thinking about these things is how he develops his mathematical ideas, then I understand why they are included.

Lianna, I see from the link in your second question (which I will point out was not included in your first question) that the formal axiom of plentitude Friedman states is:

PLENTITUDE. Any subset of D, bounded (<=) above by some element of D, and definable by a formula of L (parameters allowed), is the set of values of F at the elements of some open interval (x,y), x,y in D.

Now, I'm sure I will never understand the mental process Friedman uses to get from "Anything that can happen will" to this, but without a doubt the above is a mathematical statement. If you had asked for reasons why one might choose to, or choose not to, take the above as an axiom in a logical system (or whatever the appropriate technical phrase is), I would have no complaints, no opinions, and certainly no knowledge, about the matter. But where I make the distinction between math and philosophy (and where I am imagining I can make a judgment without any background in this area) is asking whether some piece of an extended metaphor is correct or not. Is "Anything that can happen will" a true statement? Regardless of whether one's answer is yes, no, or "that's nonsense", the question isn't mathematical. It's not even a matter of formalist vs. Platonist views of mathematics - regardless of whether there is a fact of the matter as to whether the above mathematical statement Friedman calls Plentitude is true or not, the truth or falsity of the metaphysical claim "Anything that can happen will" is irrelevant to mathematics, and mathematics is irrelevant to it. That is my fundamental complaint here.

Zev, all I can say is that I think that is what Friedman is doing: Using these intuitive principles as a sort of guide from which he is extracting mathematical results. Maybe not. Maybe he has the results first, and then he sees that they can be "motivated" by this framework. Maybe not even that. In any case, I have never been able to make much sense of these motivations, and I doubt they are at all useful to anybody other than the person who formulates them.

I have seen erroneous comparisons between, say, Friedman's note and the "philosophy" behind the work of other researchers in set theory. (Woodin's name has been mentioned.) As far as I can see, these erroneous comparisons come with at least a misunderstanding of the nature of the mathematical work that is carried out. It seems to me that short of understanding the mathematics at play, some people think that it is (vague) philosophical ideas that are guiding the research. This is a serious, potentially damaging misconception.

Of course, there are some not-formalizable guiding principles behind research in set theory. But that is true in any field. It is blatantly evident in set theory, as one of our goals is to understand
the consistency strength hierarchy. However, for most of us, the philosophical ideas, however dimly understood they are, are set to the side, and if they interfere with the mathematics, they are ignored, discarded, or changed. Not the other way around.

I feel, however, that to avoid misrepresenting what (we or some of us) do would take much more time and energy from me than I currently have, so I will stop here.

But is reopening the question before the question has been rephrased to meet the objection putting the cart before the horse?

I will pick out Simon Thomas and Andres Caicedo as two knowledgeable people who seem to have problems with the question as it currently stands; perhaps the question should first be changed, and then reopened.

I have no technical ability to vote to open currently. I would like to see this question get more attention, so I am bumping this thread again to get a few more high reputation members to consider reopening, especially those in the foundations community. The reasons I posted earlier in this thread I still hold, and encourage potential reopeners to consider them and the other posts in this thread.

EDIT 2011.07.29 The question is open and has at least one declared closer. I hope the closing does not succeed. If it does succeed, I hope it is with more care and consideration on the part of those voting to close. END EDIT 2011.07.29

Gerhard Paseman, 2011.07.28