FOM: certainty; reply to Noria, Vorobey, Trybulec

[Andrzej Trybulec]

On Tue, 5 Jan 1999, Vladimir Sazonov wrote:

...
> Andrzej Trybulec wrote:
> >
> > On Mon, 21 Dec 1998, Michael Thayer wrote:
> >
> > > Yes, but WHY does it seem absolutely true??
...
> > It is not the point. At least, my point was that even if we cannot define
> > what it means "true" in mathematics, there are sentences that are true
> > and we have no doubts about it. And in this sense they are absolutely true.

I should rather write "It seems absolutely true, because we have no doubts
about it". I looks like a joke. To be serious, I doubt if there is any
difference between "true" and "absolutely true" in mathematics, at least
when we talk about natural numbers.
... 
> Which kind doubts do you mean if you *postulated* (or agreed) by 
> your free will (in one or other way, say, by referring to 
> pebbles rather to drops of a liquid) that "2+2=4" should hold 
> for *your* natural numbers? 

I have no doubt, so it is difficult to chose one.
I had been taught what natural numbers are, it has nothing to do
 with free will.
They are not mine. I have no patent. I did not copyright them, either.

No offence intended. You do not understand what I am writing,
it is not more difficult to misunderstand you.
... 
> > There are some other sentences (Continuum Hypothesis?) that sometimes are
> > true, sometimes not (and sometimes, or rather for some people, are
> > meaningless). The truth depends on some additional factors (the
> > interpretation, the model chosen as the standard model, philosophical views)
> > and in this sense they are relatively true (if or when they are true).
> 
> After this it is unclear for my the next paragraph on your platonism. 

You are right. It is very unprecise what I wrote. Maybe we just do not
know if Continuum Hypothesis is true or not, (we know only that it is 
independent) or maybe there are different kinds of sets, for some of them 
it is true, for some not. I hope in the future we will discover what
is true.

> > On Tue, 22 Dec 1998, Martin Davis wrote:
> >
> > > I fail to understand why the formulas of PA, the set of axioms, and the
> > > notion of a proof in PA are considered to be easier to understand than the
> > > set of natural numbers and its members.
> >
> > OK. But try to convince these guys that they have to try to think.

It was rude. I apologize.

> 
> Yes, all of us are trying to think! When working in PA we need 
> not have arbitrary *sets* of axioms and *sets* of proofs, etc. 
> We need only a small number of axiom schemes and rules and a 
> (growing) number of proofs. These are very concrete objects in 
> comparison with the abstract natural numbers about which these 
> axioms and proofs "say" something. This seems to me the only 
> real way to deal with any abstract mathematical objects.

I am affraid, that my answer will be long, I want to state clearly my 
position.

1. The theorem "2+2=4" is rather old. I believe it was old in ancient Egypt.
I do not believe that it is accepted because of social agreement. I think
that Maya or Australian Aborigens and so on never consulted it. It was just
independently discovered.

Nobody in modern countries may claim sincerely that he does not know
that "2+2=4". Not if he wants to shop and has to pay taxes. It does not
matter, if he ever heard about natural numbers, not to mention PA.

It is true that we all try to think? It seems that people playing
the game called "Analytic Philosophy" try to do just the opposite. 
Nothing wrong. It is the gist of the game. But do not press other people 
to join it.

2. It was a trap. Not on the purpose. 
Because "2+2=4" is a concrete computation. It is easy 
to design a simple Turing machine that adds natural numbers represented
e.g. as sequences of "|". Or to develop equivalent formal system with
the rules
  A + |B           A +
 -------    and   -------
  A| + B           A

(A,B are strings of bars).

And it may be as feasible (tractable?) as you want. At least for "2+2=4". 
You may protest against Turing machine. Let us take an abacus, OK.

So, what it is about PA? Why so strong theory?

I do not want the argument to be misconstrued. I do not write
that I agree to reduce natural numbers to abacus. I only claim
that there is no argument that the fact that "2+2+4 may be proved
in PA" is a simpler argument that the mere computation.

This is the reason that I wrote that "2+2=4" is a trap. By accident.

3. Of course the argument 2. is not valid if we talk about a universal
sentence.  We all agree, there is a difference between a general formula 
proved in PA (one concrete computation) and intuitively accepted universal
sentence that is basically based on the "so on" rule. Here we all agree
that we need proofs and PA works pretty well.   

But, could we try to think?

What is PA? Peano just observed that many theorems proved about natural
numbers are consequences of a small number of axioms and the induction 
scheme.

And axioms seems to be obvious ("absolutely true ?").

One of the simplest: the associativity of the addition. Actually I was 
tempted to use it as an example of an "absolute truth", but then I thought
there are people that prefer pebbles, let us stick to "2+2=4".

On the other hand, maybe 
   (x+y)+1 = x+(y+1)
(so sorry, I should write "succ(x+y) = x +  succ y")
is a better example. Nobody would order me to prove it in PA. 

At least I hope that there a better reason for its avlidity than the fact, 
that the string (of length 1)

   (x+y)+1 = x+(y+1)

is a proof of it in PA.

Now if we use above mentioned Turing machine (well, a bit more complicated,
we allow for more than one "+" so we need parhetheses) 
or corresponding formal system (again it gets more complicated, but you 
know how to do it). To see that the addition is associative we have to check 
if we can derive

   (A+B)+C  from A+(B+C)

or rather to reduce both to ABC. We can do it easily with concrete
strings of bars (A,B, and C). However, we can easily observe that
all we have to do is to remove "+" signs and parentheses as well. 
It is not exactly an "incomplete induction" (the "so on" rule). I do not
know what it is. A meta reasoning, an experiment. Anyway, it is good enough
to believe in the asociativity. At least for me.

To wit, it is easy to claim that we have only PA (or any other formal
system). What is difficult, it is to explain why PA describes what we
do with the pebbles. 

4. Eventually, we have to get rid of pebbles (to use pebbles in bank accounts
or in a computer would be clumsy). We may still try to reduce 
the whole thing to physical objects. No reason to do that, so we agree
that we have a lot of natural numbers that enjoy properties
described by PA. 

In this moment, it is true that we have concrete proofs in PA and
abstract natural numbers. For a while.

Quite soon we want to prove something about PA and we need a precise
definition of PA. I not necessary to model it in its semantics.
But we need to describe it formally e.g. to prove incompleteness.
We may need also the formal description to prove any THEOREM ABOUT
NATURAL NUMBERS that cannot be proved in PA.

(What you think about such THEOREMs?)

And now the Martin Davis' argument is valid. Both natural numbers and
PA-proofs are abstract objects. Only, natural numbers are much simpler.

5. I believe that this abstract PA is a real PA. So I cannot accept
argument that instead checking by a computation
      "2+2=4"
I have to prove in PA "SS0+SS0=SSSS0". It sounds as the argument
that I have rather to prove that "SS0+S00=SSSS0 is provable in PA"
in PRA.

Sorry, the message gets too long.

Andrzej Trybulec