[FOM] New Umbrella?/big picture

[Charlie]

	  See interspersed comments below:

On Nov 5, 2014, at 4:07 PM, Mitchell Spector <spector at alum.mit.edu> wrote:

> Harvey Friedman wrote:
>> ...
>> There is one quite friendly modification of the usual set theoretic
>> orthodoxy of ZFC and its principal fragments, and that is Flat
>> Foundations:
>> 
>> http://www.cs.nyu.edu/pipermail/fom/2014-October/018337.html
>> http://www.cs.nyu.edu/pipermail/fom/2014-October/018340.html
>> http://www.cs.nyu.edu/pipermail/fom/2014-October/018344.html
>> 
>> With regard to the last of these links, I think Leibniz is credited
>> for the general principle
>> 
>> x = y if and only if for all unary predicates P, P(x) iff P(y)

	   Leibniz is truly complex and writes his so-called Identity of Indiscernibles
principle in different places frequently using different terminology.  I am unaware, 
however, of his ever referring to “unary predicates”.  To me, the closest he comes 
to expressing this is by referring to “properties,” but how are they to be explicated?  

	Admittedly, he does seem to think all individual concepts (i.e., monads) are both 
distinguished by a single property which also (somehow) “reflects” all properties 
of all other objects (over all time) in the same world (thereby creating an 
equivalence class of worlds in which every element belongs to a single partition.
God seems to select out a given representative of each world, then decides which
one reflects the “best possible world,” which is often explicated in terms of
“plentitude”. <— Maybe this is suggestive mathematically (??)




>> 
>> whereby giving a way of defining equality in practically any context
>> (supporting general predication).
>> ...
> 
> 
> 
> Leibniz' principle, in this context, appears to be essentially a minimizing principle, making the mathematical universe as small as possible.  ("The universe is so small that there can be no two distinct objects that look alike.”)

	   This point seems possibly correct (I’m hedging because Leibniz is complicated), though Mates says for Leibniz "[t]o say that x is the same as y does not mean that they fall under the same concepts, but the principle guarantees that (fortunately for us) if x is different from y, then it is at least in principle possible for some mind to tell them apart.” 


> This is in contrast to the maximizing principle which is a prime motivation of many large cardinal axioms -- saying that the mathematical universe is so very large that many objects in it are indistinguishable from one another (indistinguishable according to some specific criterion, of course, depending on the large cardinal axiom).
> 
> 
> 
> However, I'd like to throw out a different way of viewing Leibniz's principle that just might make it a maximizing principle after all, in an approach that may be more in line with Harvey's Flat Foundations.
> 
> 
> 
> The traditional view involves taking a specific collection of predicates (formulas in some language) and then saying that the universe of objects is so large that there are objects that are indistinguishable relative to that collection of predicates.  So we're starting with a fixed collection of predicates and making sure the universe of objects is large relative to that collection of predicates.
> 
> On the other hand, what happens if one starts with the collection of objects and then looks at predicates or relations as first-class mathematical objects, not as extensions of formulas in some language?  One might then say that to maximize the mathematical universe, one would want the collection of predicates to be so very large that, given any two distinct objects, there's a predicate that distinguishes between those two objects.
> 
> 
> 
> Let's continue in this vein, carrying the line of reasoning to its natural conclusion. If understanding the type-0 objects requires type-1 relations as a separate sort, it would seem that understanding the type-1 relations would require a third sort, that of type-2 relations (predicates on the collection of type-1 relations, rather than predicates on objects).  Leibniz' principle would then suggest that there are so many type-2 relations that, given any two distinct type-1 relations, there's a type-2 relation that distinguishes between those two type-1 relations.
> 
> If we do this, we wouldn't stop at type-2 relations, of course.  As we added in type-3 relations, type-4 relations, etc., we would seem to find ourselves reconstructing at least Russell's theory of types.  Eventually we'd build up to Z and then ZF, on the principle that one would want to iterate the type hierarchy into the transfinite, so that we have as many type levels as possible (maximizing the universe again).
> 
> 
> 
> For what it's worth, the name "Flat" Foundations may no longer be appropriate, since the structure is now tiered rather than flat.
> 
> I'd be interested in hearing any thoughts on this.  It seems to develop the Flat Foundations idea in a natural way -- but at the same time this extension to more than two sorts appears to obviate the purpose of Flat Foundations, since we're reconstructing what has become the traditional mathematical universe, tiered by rank.
> 
> 
> Mitchell Spector
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