[FOM] The Gold Standard

[Harvey Friedman]

On Feb 22, 2006, at 3:04 AM, Nik Weaver wrote:

> That isn't my definition of platonism.  My definition is: belief
> that mathematical "set" talk literally refers to a special kind
> of abstract object.

So you take "Platonism" to refer only to "set talk"?

Does 5 refer to a special kind of abstract object?

> If you're not admitting this, please specify an impredicative
> system and explain how it can be justified on non-platonistic
> grounds.

Already it can be perfectly well argued that the series of natural 
numbers is already Platonistic. In fact, one can already argue, if one 
wants, that the number 0 is Platonistic.

So under this view, you cannot even begin any predicative development 
of mathematics without Platonism. Hence under this view, one has, at 
most, "levels of Platonism", and predicativity is simply one particular 
level of Platonism among many levels of Platonism. So once again, 
predicativity has no special place in the hierarchy of levels.

In fact, it can be equally argued that the number 0 is already a 
Platonistic construction, so you can't even start classical finitism.


>
> Why I'm not a platonist: I've answered this several times
> already; see
> http://www.cs.nyu.edu/pipermail/fom/2006-January/009520.html
> http://www.cs.nyu.edu/pipermail/fom/2006-February/009818.html
> or better yet read Hartley Slater's analysis in
> http://www.cs.nyu.edu/pipermail/fom/2006-February/009887.html

Some of this argues with equal force that the natural number series 
should be rejected. Some even argue against 0.
>
> Yes.  In order to believe that PA has a model we need to believe
> in structures of type omega.  For example, I have suggested a
> structure involving marks on paper.  There is no need to posit
> the existence of natural numbers as some special kind of abstract
> entities.  You can doubt that omega structures exist but this is
> not a doubt about platonism.

It's not actual marks on actual paper. If it were, then only you have 
natural numbers, because you can't relate your marks on your paper with 
my marks on my paper. Also, what happens when your marks on your paper 
disappear b because that paper was accidentally thrown into the trash 
and destroyed by the refuse company? What about those many numbers 
below 1 billion that were never represented by any marks on anyone's 
paper?

It sounds like you are a confirmed Platonist who has some abstract 
concept of marks on paper that is independent of actual paper, actual 
marks, actual possession, actual destruction of paper, etcetera.

I always suspected you of some sort of Platonism. If you adhere to some 
sort of Platonism, why do you stop somewhere above Gamma_0? You could 
instead stop at 0.

On Feb 22, 2006, at 3:16 AM, Robert Lindauer wrote:

> "There is a bunch of things with nothing in it."  And this seems to be
> the basics of the empty set.  To my mind, if there are no things,
> there is no bunch either.

Incorrect. Clearly we all agree that there is the "idea of nothing", 
for you have been talking about this idea. Even simpler yet, there is 
the "set with no elements in it".
>
> Now, admittedly common English usage is not a good place to begin for
> authority,

This is an essential point. common English usage can be argued not to 
embrace the empty set. However, common English usage also does not 
embrace 0, and we know how important that has been for the history of 
mathematics and science.

On the other hand, common English usage does embrace the idea of an 
ordered pair through the notion

*marriage*

and the idea of an unordered pair through the notion of

*gay marriage*

So we can think of the Cartesian plane as consisting of

*marriages of real numbers*.

There is also the common English  usage of "polygamous marriage". From 
that, we can try to develop finite set theory.

There are documents in the local Governmental offices listing all of 
the marriages that have been filed locally. Sounds a lot like a set or 
list of ordered pairs to me.

> but when introducing a new object, it's important to first
> fix its reference or description sufficiently so that no confusions
> arise later in the development of the idea.

As was done by Cantor and earlier by Boole and others.

> It would be truly confusing indeed to talk about, in seriousness, the
> bunch of bananas on my desk right now, since there are in fact, no
> bananas on my desk.

This is standard in the development of science. E.g., the velocity of 
an object when it is not moving. We learn in school that this should 
not be singled out as a special case. In fact, we learn that velocity 
splits into more than one notion, and that the most important of them 
for many (most) purposes is

*velocity vector*

which is an ordered triple of real numbers, and certainly can be the 
ordered triple (0,0,0) representing no motion. Whoops! I mean 
instantaneously no motion. My gosh! I can't find instantaneously no 
motion in ordinary English usage, I can only find "not moving at all"!

Does such linguistic points affect the thinking of, say, Isaac Newton, 
and the course of mathematical physics?

> The problem is that we're trying to refer to the bunch -itself- not
> the members of the bunch (which in this case remain missing), but
> somehow by reference to what it "contains" or "does not contain".

Yes, this is called scientific progress. It has tremendous benefits, 
intellectually, culturally, economically, etcetera.
>
> But this loses our "bunch" metaphor and forces us to understand the
> "bunch" or "set" metaphor in terms of "contains", but here we remain
> at a loss, in the case of the empty set.

The only loss is that of people who truly are confused by elementary 
set theory - and certainly you are not among them.
>
> But here, too, we seem to be at a loss with the empty set.  The empty
> set "contains" nothing.

Set containment and set membership are to be distinguished. This is not 
hard to do with small amounts of training, even for people with 
ordinary intelligence.

> And it's hard to fathom what a relationship
> with "nothing" could mean.  One can't drive nails into nothing, nor
> can nothing be a member of something (like a club, say) nor can
> nothing be a basis for a metaphorical introduction of an ill-founded
> concept.

As I explain to non mathematicians, sets are not physical objects. They 
are what is called mathematical objects. They don't have any problem 
with this, because they are used to the following from grammar school:

numbers are not physical objects. they are mathematical objects.

Finite set theory is certainly the clearest and cleanest and powerful 
of all theories in the whole of science. For many purposes, one may 
instead wish to use somewhat different but closely related concepts, 
which also, arguably, are just as clear and clean and powerful. I'm 
thinking many of

finite list theory

so fundamental in computer science.
>
> I daresay the rest of the confusions surrounding set theory remain
> unresolved (like Cantor's Absolute)

There are no confusions in set theory at all. There are some issues 
about the correct way to resolve paradoxes, etcetera.

But even the most delicate of these issues are INCONSEQUENTIALLY MINOR 
compared with the logical and philosophical problems that are present 
in every single subject in science and engineering other than in pure 
mathematics.

Even the situation with regard to large cardinals is CRYSTAL CLEAR 
compared to the basics when one steps outside of pure mathematics.

> because the notion of "set" and
> "contains" remain metaphorically defined, and there badly.

I have never seen anything defined outside of pure mathematics that 
would in any way shape or form compare to the clarity of the notion of 
finite set. If this is bad, then everything else is far worse than 
horribly atrocious by comparison.

> A search
> for new axioms to further define them may yeild a "better" set
> concept, but the likelihood that it will serve the epistemological and
> ontological needs of foundational mathematics is unlikely

What needs? The foundations of mathematics looks like the ultimate of 
perfection compared to the foundations of anything else of depth.


On Feb 22, 2006, at 6:39 AM, slaterbh at cyllene.uwa.edu.au wrote:

> There is another problem, if one brings in ordered pairs - they do not
> differ in their members from the corresponding unordered pairs.

Members of order pairs? Do you mean coordinates of ordered pairs? Note 
that
"differ from" is ambiguous or meaningless in your paragraph above.

> So
> where, with two apples, for instance, are the *three* further objects -

First of all, I was simply talking about this. Let x,y be objects. Then 
there is x and y and <x,y>.

Of course, I see what you are referring to. One can also consider

x,y,<x,y>,<y,x>

and yet more:

<x,x>, <y,y>

and also more:

<x,<x,y>>

etcetera.

Mathematics is powerful.

> the pair of apples, and the two ordered pairs of them?  The bowl must
> be quite overflowing! (Note that if one eats both apples, the pair of
> them disappear, so one is talking about physical things all the time.)

No. An ordered pair of objects is not a physical object.
Sets are not physical objects either.


On Feb 22, 2006, at 3:29 AM, Nik Weaver wrote:

>
> I pointed out a fact that I find quite remarkable: although
> in general one cannot predicatively form the dual of a Banach
> space, all of the standard Banach spaces that functional
> analysts care about enough to have given them special names
>

I doubt the relevance of this, but it is in any case it seems to be 
false. See the posting 
http://www.cs.nyu.edu/pipermail/fom/2006-February/009998.html

Also, if you take seriously names, what about the following names 
invented by core mathematicians:

1. Topoi (Lawvere/Tierney style).
2. Universes (Grothendieck style).

In this regard, please comment on the postings
http://www.cs.nyu.edu/pipermail/fom/2006-February/009992.html
http://www.cs.nyu.edu/pipermail/fom/2006-February/009970.html
http://www.cs.nyu.edu/pipermail/fom/2006-February/009981.html
http://www.cs.nyu.edu/pipermail/fom/2006-February/009982.html

Harvey Friedman