[FOM] On Platonism and Formalism

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Thu Oct 2 15:43:24 EDT 2003

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

Why not? When we read, say, Tolstoy or Dostoevsky or some tales 
(about Baba Yaga - witch flighing on a broom, etc.) we can discuss 
on the personages knowing very well that they never existed and 
was pure creation by the authors or just a folklore. The difference 
with mathematics is only that in these stories there are no formal 
rules of behavior of these personages. In both these cases everything 
exists only in our imagination. 

> Thus, sets appear to exist.  On the other hand, sets are not directly
> observable, and there is no scientific experiment that will directly
> prove the existence of an inaccessible cardinal, so formalists claim
> that the notion of the mathematical universe is superfluous, 

If you mean the (imaginary) universe for ZFC, then of course not! 
It is NOT superfluous. Doing mathematics without intuition is 
impossible and meaningless. The role of imaginary worlds (of sets, 
of natural numbers) is extremely important from the point of view 
of formalism (as I understand it). Formalists, unlike Platonists, 
clearly realize the difference between imagination and reality 
and never consider mathematical concepts as absolute. 

Did you read my postings where I wrote about this many times? 
If you do not agree with me, what exactly is the disagreement? 

and that,
> by Ockham's razor, there is no such thing as a real number.

No, it exists as a vague idea, no more than that. That these 
ideas are subject to formal, rigid rules is not enough to 
assert that these ideas have some rigid, absolute character. 
If there is some rigidity, it has only relative character. 
Just relative to formal rules which these ideas should obey 
(because we want this, because this is a way as mathematics 
works, not because THEY are such). 

> 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 but their feelings are
> subject to the same patterns as they would if the physical universe
> existed.  The belief is non-empirical and hence not subject to
> experimental refutation.  Yet, we believe that the physical world
> exists.  

Why do we need these subtle considerations whether the physical 
reality exists or not? It definitely exists in such a strong sense 
which cannot be comparable with so called absolute existence of 
the standard model for PA or the like. It is much more reasonable 
to accept existence of the real world and "pljasat' ot etoj pechki" 
(in Russian; this literally means "to dance from this stove"). 
I see, you considerations on "doubts" on the physical reality have 
the evident goal to equate it with non-physical: 

> It exists because it feels like it exists and there is no
> evidence to the contrary.   Inaccessible cardinals metaphysically exist
> because once one has spent a lot of time studying them, it appears that
> they exist and there is no evidence to the contrary.

"one has spent a lot of time studying them" - this is your argument? 

> We believe that electrons exist because the explanation of the patterns 
> of our experience becomes much more natural if we assume their
> existence.  

Why to invoke electrons? They really have close relations to 
some mathematical theories, as you correctly write below. 
Electrons are a different story. If you do not want to go into 
vicious circle, consider something such evidently real as 
cobble-stones. Sorry, again, "dance from the stove". All 
normal sciences start from such kind of things. Electrons 
appear on a much later stage. 

In other words, any natural explanation of say, chemistry,
> invokes the notion of an electron.  However, any such natural
> explanation also invokes the notion of a real number:  Physical theories
> are supposed to be quantitive--and hence mathematical.  Mathematics is
> extremely effective at making physical predictions.  Such effectiveness
> would certainly be unreasonable if mathematics is the theory about
> nothing, that is if sets do not exist.  Mathematical objects are as
> central to our understanding of physics as are physical objects.  To
> claim that mathematical sets do not exist is as reasonable as claiming
> that the physical world does not exist.

You are making permanently too big jumps in your reasoning. 

If sets really exist, what is the solution to the Continuum Problem, 
or is ANY solution at all? 

For physical objects, even like electrons, there are quite concrete 
experiments to confirm corresponding physical theory. For mathematical 
theories with related vague ideas on real numbers, or the like, there 
are also such kind of concrete experiments. These are formal proofs 
having essentially the same reality as cobble-stones. THIS is 
the first and the main relation of mathematics with the reality. 
After realizing this fact we can try to approach the question 
how mathematics can be applied in physics and in other sciences. 
I think, it is reasonable to postpone this more difficult question. 

> It appears as though most mathematicians believe that mathematical
> objects semi-exist.  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.  

Of course it is theory about (imaginary) sets. No problem. 

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

Imaginary objects. What else? They exist ONLY in our imagination. 

The mere reference
> to sets implies that they are something, and hence exist.  

What strange logic? Natural language allows to discuss 
about how many devils can stay on the end of a needle. 
And so what? 

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.

What are you ever talking about? 
Something from the Middle Ages. Just about that needle. 

But seriously, let me imitate you, may be with some exaggeration. 
By the logical law of excluded middle, an object either exists 
(here) or it does not. Electron is an object. By modus ponens rule 
we conclude that electron either exists (here) or it does not. 
But we know from physics that it is not the case. A contradiction. 
Who is wrong, or what is wrong here? 

Of course the existence of electrons and of cobble-stones 
has not exactly the same status. The same about our ideas 
and the real world. Both exist, but differently. The world 
is not black-and-white as, it seems, you see it and should 
not obey the logical laws you like. In particular, the 
application of the low of excluded middle in the way as you 
do and in such a context is not acceptable. The discussion 
on mathematics is not itself a mathematical subject to use 
logical laws concerning nobody knows what. 

Also you seems want to say that there is no difference between 
physics and mathematics. Say, electrons and sets exist in the 
same sense. But the difference is that mathematical (imaginary) 
objects are exclusively OUR  creations. They behave according 
to those laws of logic or according those definitions which WE 
will prescribe. Of course, we want to have something intuitively 
interesting, coherent and probably applicable, say, in physics. 
Thus, our freedom to create these arbitrary imaginary worlds 
(obeying formal rules) is not complete. But, in principle we 
can create anything. Nobody knows in advance what could be 
applicable. In this sense we have a lot of freedom for our 
imagination and choosing formal rules in mathematics. This is 
one of the main features of mathematics. That is why mathematics 
has its own life rather independent from the life of physics an 
the real world. The only straightforward relation of mathematics 
to the real word is via formal (rigorous) proofs which are 
essentially physical objects. Applications of mathematics to 
the real world, however extremely important and also relating 
mathematics to the real world have a different character, not 
so straightforward. This could be discussed separately.  
Also, corresponding mathematical notions having a relation 
to applications, like real numbers, are only some approximations 
to the reality. There is NOTHING in the real world what would 
correspond to the real (and even natural) numbers PRECISELY. 

In contrast to mathematics, we start in physics with some 
quite real experiments. And electrons and other physical 
objects with "strange" behavior are so closely related to 
these experiments which (experiments) are as real as 
cobble-stones. Physicists have also a lot of fantasy. 
But they are strongly restricted by experiments. They cannot 
consider ARBITRARY theory nor related in any way to experiments 
only on the base that it is nice. In this sense physicists are 
investigating the real world, unlike mathematicians. 

> Dmytro Taranovsky

Vladimir Sazonov

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