[FOM] A reply to Podnieks on Platonism and Formalism

Dmytro Taranovsky dmytro at mit.edu
Tue Oct 7 20:44:25 EDT 2003

On Tue, 2003-10-07 at 01:58, Karlis Podnieks wrote:
> I will try to represent the most extreme formalist point of view, filling in
> the f.o.m.-ism questionaire set up by Harvey Friedman
> (http://www.cs.nyu.edu/pipermail/fom/2003-September/007352.html).
> As the main principle, the formalist philosophy allows investigation of any
> formalizable theories, independently of what some other people might think
> of their "truth". 

A consistent formal theory in first order logic has a guaranteed truth
value only once we specify its semantics.  For example, the theory PA +
not Con(PA) is false in the standard model of PA, but true in some other
models.  Investigation of PA + not Con(PA) amounts to investigation of
the models in which it is true, and is a valid mathematical activity.

> Of course, not all theories of this kind are worth of
> investigation. How do we select? The most popular criteria:  a theory should
> be useful, fruitful, beautiful, successful, etc. 

The decisions to investigate various physical objects are also based on
whether the investigations are useful, fruitful, beautiful, etc. 
Investigation of mathematical theories amounts to investigation of the
(metaphysically) existing objects that satisfy them.

> But, saddly enough: such theoretical consistency
> proofs can be only relative, not absolute (my interpretation of Godel's
> Second Theorem).

There are two ways to demonstrate consistency.  A statement may be shown
true (projective determinacy is true for reasons that I will explain in
a later posting), and hence consistent.  Alternatively, a detailed
theory of the consequences of a statement, such as the existence of a
supercompact cardinal, combined with a philosophical argument that if
the statement is inconsistent then it should have a simple proof of
inconsistency, strongly suggests that the statement is consistent.

>  For me, accepting the extreme formalist philosophy makes the world of set
> theory more colorful. Of course, a formalist may follow the mainstream (ZFC,
> ZFC + there is an inaccessible cardinal, ..., ZFC + there is a measurable
> cardinal, ..., ZFC + there is a contradiction). But, in parallel, a
> formalist may also investigate seriously (at least) ZF+V=L and ZF+AD as
> well. 

ZF+V=L is true in the minimal inner model of ZF.  ZF+AD is true in
L(R).  The two theories are seriously investigated by mathematicians as
they reveal facts about constructible sets and about sets constructible
from the reals.

> From my sideview point, these new "worlds of sets" look even more
> interesting than the older mainstream "world".

These "new worlds of sets" are simply various interesting sets (or
classes). Platonism in no way constrains the study of interesting sets.

>  Formalists are regarding as very valuable the relative consistency proofs
> (like, Con(ZF)->Con(ZF+V=L)), and proofs that some consistency proofs are
> impossible (like Con(ZF) does not imply Con(ZFC + there is an inaccessible
> cardinal)). Why? As a necessary guidance in the world of possible axioms.

These proofs are very valuable because they provide much more than
consistency.  The proof of consistency of V=L also shows that L is the
minimal inner model of ZFC.  The proof that Con(ZFC) does not prove
Con(ZFC+there is an inaccessible cardinal) demonstrates that V(the least
inaccessible cardinal) is a natural and well behaved model of ZFC.  The
proofs through forcing of the consistency of the Continuum Hypothesis
(CH) and of its negation, also imply that the CH is independent of the
consistent large cardinal axioms.

> Yes! An extreme formalist would be pleased by a contradiction found (best of
> all) in PA (but ZFC + a measurable cardinal also will do). 

An inconsistency in PA arithmetic would be a disaster--most fields of
knowledge and practical constructions depend on arithmetic (although
most practical applications can actually be carried out in weaker
systems).  Fortunately, we know a priori that every axiom of PA is true,
and hence PA is consistent.

> And, of course, formalists are regarding as very valuable the projects of
> computer-aided formalized mathematics (see the Mizar project at
> http://mizar.uwb.edu.pl/).

I agree.  Some fields of knowledge, such as philosophy, are plagued by
disagreements among the scholars on even the most basic questions.
Mathematics is spared such disagreements because every proof can be
algorithmically verified, and while some philosophers claim that ZFC is
meaningless and has no relation to sets, they agree that ZFC proves the
formal version of "every set can be well ordered".  

> Formalism seems to be bolstered by: non-Euclidean geometries, continuous
> nowhere differentiable functions, paradoxes of set theory, Skolem's paradox,
> Banach-Tarski paradox, Godel's incompleteness theorems, non-standard models
> of PA, the formal concept of algorithm, the independence of CH and other
> famous hypotheses from ZFC, the unsolvability of the 10th Hilbert problem,
> the natural independence phenomenon. But, perhaps, any other -ism could
> declare almost the same list of bolsters?

Natural independence results raise the prospect that some important
questions about sets will be forever unresolved.  It is consistent with
formalism that there is one and only one **natural** solution to the
Generalized Continuum Hypothesis, Suslin Hypothesis, Kurepa Hypothesis,
etc.  It is also consistent with Platonism that we will never know which
of the hypotheses are true.  However, Platonism strongly suggests that
we should search for truth in the higher levels of the set theoretical
hierarchy rather than being content with studying various countable

> Best wishes,
> Karlis.Podnieks at mii.lu.lv
> www.ltn.lv/~podnieks
> Institute of Mathematics and Computer Science
> University of Latvia

Best Wishes,
Dmytro Taranovsky

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