[FOM] On Platonism and Formalism

[Harvey Friedman]

Reply to Taranovsky.

On 9/30/03 8:38 PM, "Dmytro Taranovsky" <dmytro at mit.edu> wrote:

> Of the many mathematical papers that I have read, every one of them
> treats the mathematical objects as though they exist.

I know a Fields Medalist, through correspondence, who is very interested in
f.o.m., and of course writes mathematical papers treating the mathematical
objects as though they exist, but is a formalist, and says he is playing
nice games.

>In all papers
> deriving the consequences of ZFC that I have read, ZFC is a theory about
> sets.  It is difficult to even think about how to derive the
> consequences of ZFC without invoking the notion of a set.

That doesn't stop my correspondent from saying he is playing nice games.

>The thoughts
> and discussions on sets appear to refer to something, but one cannot
> refer to something unless it exists (or existed or will exist).

That is something my correspondent would surely deny. He discusses some
infinite sets of integers, also Fourier series, etcetera, but all the while
insisting that there is no sense of existence, and that it is a nice game.
> 
> Thus, sets appear to exist.  On the other hand, sets are not directly
> observable, ...
> 
> However, physical reality is not directly observable either:  All we
> observe are our feelings.  No experiment directly disproves the belief
> that the only objects that exist are human souls ...

This situation is not comparable to that of inaccessible cardinals.
> 
> We believe that electrons exist because the explanation of the patterns
> of our experience becomes much more natural if we assume their
> existence.  In other words, any natural explanation of say, chemistry,
> invokes the notion of an electron.

Quantum mechanics seems to suggest, perhaps, that the notion of existence is
naive.

>However, any such natural
> explanation also invokes the notion of a real number:  Physical theories
> are supposed to be quantitive--and hence mathematical.

The invoking of real numbers in physical reality may turn out to be a very
naive idea. We don't know yet. It certainly is suspicious. Perhaps a
discrete approach is better.

>Mathematics is
> extremely effective at making physical predictions.

Physics is extremely effective at making physical predictions. I do not
believe that any infinitary mathematics has gotten confirmed in this way -
***YET***. 

Perhaps it can argued that some infinitary mathematics gets "confirmed" in
that various finite consequences of it are seen to be correct by putting
them on a computer.

An important question is whether or not this kind of "confirmation" really
amounts to confirmation.

In physics, one has at least the feeling of genuine confirmation, because
there is the feeling that if the physics was even slightly wrong, one would
get a REFUTATION instead of a confirmation.

This doesn't happen in mathematics, certainly not yet. There may be good
reasons to believe that it can't happen in mathematics. I'm not sure.

>Such effectiveness
> would certainly be unreasonable if mathematics is the theory about
> nothing, that is if sets do not exist.

One additional point is that for the kind of mathematics you are talking
about, there is no problem giving a treatment that is not set theoretic.
Surely a completely computational finitistic treatment can be given,
avoiding any use of infinities, etc.

More generally, I believe that directly finitistic treatments of huge
portions of normally infinitary mathematics can be naturally and
systematically given, where interesting estimates emerge of genuine interest
that do not emerge in the normal infinitary treatment.

>Mathematical objects are as
> central to our understanding of physics as are physical objects.

I think that most physicists are formalists regarding mathematics, and do
not think that mathematical objects exist. I think they regard mathematics
as an aid for setting up appropriate computations, and has no independent
meaning.

>To
> claim that mathematical sets do not exist is as reasonable as claiming
> that the physical world does not exist.

This claim cannot be adhered to if physicists actually take the attitude
that I think that they take (see above).
> 
> It appears as though most mathematicians believe that mathematical
> objects semi-exist.

I'm not sure about this. For example, my correspondent says that they don't
even semi-exist, and one is playing a nice game. The Fields Medal committee
says he is a very notable player.

>They would deny that the empty set exists in the
> same sense that pens, pencils, and buildings exist, yet they would
> accept that ZFC is not merely a string of symbols but a theory about
> sets.  

Physicists and ZFC??

>However, what are sets if they do not exist?

My correspondent thinks that they are useful figures of speech in a very
nice game.

>The mere reference
> to sets implies that they are something, and hence exist.  An object
> either exists or it does not; there is no such thing as semi-existing
> object, and for the reasons stated above, mathematical sets do exist.
> 

Look, I don't subscribe to what my correspondent says. I just want to point
out just how unconvincing what you wrote it to some people of note.

I now turn to the next posting by Taranovsky - also not convincing.

On 10/3/03 3:57 PM, "Dmytro Taranovsky" <dmytro at mit.edu> wrote:

> In my last posting, I demonstrated that it is generally agreed that we
> can refer to mathematical objects, and I noted that this implies that
> mathematical objects metaphysically exist.

In the above, I argued that this was not a demonstration. I didn't say that
I agree with you, or disagree with you. Just that it was unconvincing.

>...  and although, technically speaking, the fact that
> we can meaningfully refer to a character in a work of fiction implies
> that the character exists, a fictitious character cannot cause anything,
> so, as a practical matter, we can assume its nonexistence.

An interesting line. Cause and effect. There is a chance you can get some
traction with this idea, but there are many problems.

One problem you have to deal with is that for a large class of set theoretic
statements, there are some well known results that indicate that they have,
demonstrably, no effect of the kind you need for your argument.

E.g., we know that the continuum hypothesis, over not only ZFC, ZFC with any
of the usual large cardinal axioms, has absolutely no consequences of a more
concrete nature - even absolutely no consequences as high up as the
projective hierarchy.

So it seems very difficult for you to justify a Platonistic attifude towards
the continuum hypothesis via cause and effect.

The same thing is true for gobs and gobs of other set theoretic statements.

On cause and effect for ZFC, and also large cardinal hypotheses, there may
be some sort of traction, given that we know from Godel that there is at
least a theoretical effect at very low levels, and from recent f.o.m.
developments, ever more mathematically significant such effects.

However, there is still a problem of identifying just what the cause is of
the cause and effect. E.g., the low down effects of a measurable cardinal
are known to be exactly the same as the low down effects of "the continuum
is real valued measurable" - according to a celebrated result of Solovay.

So are the low down consequences of a measurable cardinal "caused" by a real
valued measure on the continuum, or instead by a measurable cardinal?

Is it an object that is the cause of the effect, or is it the statement of
its existence?

In physics, one normally thinks of the object as the cause. Maybe the most
modern physics has already shown that this is terribly naive.

> An objection to Platonism is the claim that there are multiple universes
> of sets and that,  for example, the Continuum Hypothesis is meaningless
> because it is true in some universes and false in others, and in talking
> about the Continuum Hypothesis, we fail to specify which universe we are
> talking about. ... in talking
> about sets, we refer to the unique maximal universe that consists of all
> sets that exist.  Unlike a work of fiction, where parts of the plot may
> be unspecified, the invocation of maximality and totality causes the
> universe of sets to be unique.

Unfortunately, the very formulation of maximality and the proof of
uniqueness you talk about require a commitment to the objectivity of the
objects whose objectivity is in question.

>For example, suppose that the notion of
> the set of all real numbers is vague.  In that case, there are two
> different sets, R1 and R2, each of which is the set of all real
> numbers.  

You used the assumption that there is the set of all real numbers. Someone
questioning the notion of the set of all real numbers probably is
questioning the idea that all real numbers that ever exist exist now. More
real numbers come into existence later - perhaps, especially, when
mathematicians need new real numbers for their work, and "create" them. I
don't see how your argument refutes such a position.

>By the axiom of extensionality, there is a real number r that
> belongs to only one of the two sets.  However, the set that does not
> contain r cannot be the set of all real numbers, contradicting the
> assumption.

See above.

>  Only recently did it become clear that projective
> determinacy is the correct existence axiom for second order arithmetic.
> The vast realms of the set theoretical universe are yet to be explored.

Of course I know the technical results concerning projective determinacy to
which ou refer. The notion of axiom you use here is completely nonstandard.
Axiom normally refers to something that is self evident, and whose meaning
is at least intuitively clear.

Anyone who is not already a Platonist will simply say that PD creates a nice
game, but there are alternative nice games. Some like chess, others like go.

And now we move on to another posting by Taranovsky - also unconvincing.

On 10/10/03 8:18 PM, "Dmytro Taranovsky" <dmytro at mit.edu> wrote:

> Projective determinacy is perhaps the best example of a statement that
> is
> (1) generally accepted as true,
> (2) of fundamental importance in mathematics, and
> (3) unprovable in ZFC.

When I first read this, I felt it was absurd. But then I read further, and
saw that you also know it was absurd, and seriously modified it.
> 
> Note:  General acceptance only means general acceptance among those who
> work in the relevant fields; most people are aware of neither ZFC nor of
> projective determinacy.

With this modification, (1) is arguably true, but I am not really sure that
it is true. It depends on how broadly you construe "relevant fields". E.g.,
is the whole of mathematical logic a "relevant field"? Is f.o.m. a "relevant
field"? Is philosophy of mathematics a "relevant field"? Is "descriptive set
theory" a relevant field?

With (2), I question that even professional set theorists think that PD is
of fundamental importance in mathematics, if you mean mathematics taken as a
whole. I'm not sure they think this. I certainly think that they think it is
important intellectually.

Also what does "best example" mean? What about

1) any two non Borel analytic sets have a Borel bijection;
2) every projective set is Lebesgue measurable?

>Projective determinacy is of fundamental
> importance in second order arithmetic, that is in the study of real
> numbers.  

Identifying second order arithmetic with the study of real numbers is rather
unusual mathematically. Only a set theorist would make an identification
like that.

>It is probably not of fundamental importance to those who work
> on finite structures.  However, Boolean Relation Theory requires some
> axioms beyond ZFC, and projective determinacy would do.

Of course BRT and various new offshoots (not yet posted) are directed
towards finite structures, with increasing clarity. The latest versions (not
yet posted, still being developed and improved) are far far stronger than
projective determinacy, and go into the largest of the large cardinal
axioms. 

> In this posting, I will concentrate on explaining why projective
> determinacy is true.
> 
> First, for finite games, determinacy is obvious:  ...  by induction
> Projective determinacy is simply the transfer of our proven intuitions
> into the infinite.  Such transfer is not new:  ZFC itself is the
> transfer of our basic intuitions about sets into the infinite.

Now this is a really promising direction of research in which I have worked
and have obtained some partial results - transfer principles. But it is far
from clear just how ZFC is an extrapolation from the finite. It is very
promising...

>Of
> course, one has to be careful in making such transfers:  Since in all
> finite games, the payoff set and each position are definable, projective
> determinacy only asserts determinacy where positions can be coded by
> integers and payoff set definable in a simple way from real numbers.

How about an even more fundamental problem. In the finite, every ordinal has
an immediate predecessor. Try transferring that to the infinite...
> 
> Second, projective determinacy is an existence axiom:
> Theorem:  Projective Determinacy is true if and only if for every real
> number z and natural number n, there is an iterable transitive model
> that contains z and n Woodin cardinals.

That is quite a definition of an existence axiom!! We have come a long way
from such set existence axioms as the axiom of pairing, etc.!!!
>  
> Fact: Every statement in set theory that has a sufficiently strong set
> existence component implies projective determinacy.

It would be interesting to make some sort of precise statement along these
lines.
> 
> Third, projective determinacy provides a very nice canonical theory of
> second order arithmetic.   Every projective set is measurable, has Baire
> property, and has perfect subset property.  The pointclasses Pi - 1 -
> 2n+1 and Sigma - 1 - 2n have the scale property, which is a very strong
> structural property and implies the properties of prewellordering and of
> uniformization.  

Beauty is in the eyes of the beholder. People who like this kind of
descriptive set theory think that it is so beautiful that is must be true.
However, what about others?

A skeptic says: you are playing a game, which you choose because you think
that it is beautiful. That's also why people love to play chess. And the
better they are at chess, the more they tend to love chess. The better
people are at handling measurability, scales, prewellordering,
uniformization, the more beautiful they tend to think they are. The more
measurable sets, the more scales, the more prewellordings, the more
uniformization, the better.

>The proofs from determinacy are much more natural than
> proofs from V=L. 

Of course you recognize that V = L not only answers all of these questions,
many in the negative, but also answers continuum hypothesis, and all of
those other related pesky set theoretic questions that have proved so
difficult for the Platonists.

So you say "much more natural". However you should also say that they are
"much more difficult". In fact, the fact that they are much more difficult
is considered a plus by practitioners. So again the skeptic says: you are
playing a game that you like to play partly because it is so difficult. Good
chess players like to play chess partly because of how difficult and
challenging chess is...

>Finally, every nonrestrictive canonical theory of
> countable sets implies projective determinacy.

What does this mean?

I close with a comment about V = L. V = L is not best construed as an axiom.
Rather it is best construed as an announcement of intent. I.e.,

1. "Constructible sets are more than enough for almost all normal
mathematical purposes."

2. "I wish to work within the constructible universe because there is no
serious loss, and I do not have to be concerned with those famous set
theoretic problems such as the continuum hypothesis and its cousins. They
are all settled. Also projective descriptive set theory is settled.
Specialists don't like the way it is settled, but the fact that all of this
is settled, and settled easily, more than makes up for the fact that they
don't like it."

3. "I don't care if there are or are not nonconsructible sets. If there are
some, they are slippery, and I don't really need them - they cause more
trouble than they are worth. So I only consider constructible sets. In fact,
I have put a sign on my office door that reads: I use only constructible
sets."

3. "I can still accommodate what some of those set theorists do with large
cardinals, and what this crazy nuisance guy Friedman is doing by showing
that the discrete and finite provide absolutely no escape for us from those
damn large cardinals. This I can do simply by postulating, if necessary,
that there are systems of objects which have certain nice combinatorial
properties - and the systems are countable".

I am not advocating anything. I am merely putting V = L into its best light.

I will leave your ads in tact:
 
> In my paper,
> http://web.mit.edu/dmytro/www/DeterminacyMaximum.htm
> the section "Philosophy of Determinacy Hypotheses", explains in detail
> why (and which) determinacy hypotheses should be true.
> 
> A good introduction to projective determinacy can be found in Woodin's
> paper "The Continuum Hypothesis, Part I".

Harvey Friedman