# FOM: certainty; reply to Noria, Vorobey, Trybulec

Mon Jan 4 18:49:33 EST 1999

```F. Xavier Noria wrote:

>    Professor Davis:
>
>     | 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.
>
>    I think there are the same difficulties to understand both things. From
>    my point of view, the same sort of objections can be made to the
>    sentences:
>
>       * Let n be a natural number.
>       * Let x be a variable.

By my opinion there is a strong difference between these two
sentences.  As I wrote recently and as, I am sure, everybody
here realize, syntactic concepts like variables, formulas,
proofs should be distinguished from semantic ones. (Say, the
length of a formula or proof should be feasible, while natural
numbers considered in PA can be as big and unrealistic as it is
postulated there.) Of course, after Goedel we know that
syntactic concepts can be *imitated* (encoded, numbered) by
semantic ones. But this is a different theme.

(By the way, by this reason Consis(PA) does not represent
adequately consistency of PA. It is much stronger! Nevertheless,
I feel that it is reasonable to consider PA + Consis(PA), etc.
without asserting or believing that Consis(PA) is necessarily
true in the "intended" model of PA. Cf. also "our N's" below.)

>    The typographical character of syntax might bring near the concepts and
>    I feel that we suppose "less" properties to the formulas than the ones
>    we suppose to the naturals, but this is likely to be a psychological
>    matter.

I do not agree. Cf. above.

>    In my humble opinion, both natural numbers and sets of variables

But when you say about (arbitrary?) *sets* of variables, I agree
that they are rather like to sets of numbers.

>    are not
>    real objects and the N which is in the mind of two different mathematicians
>    is actually not the same N, because actually there is no such N anywhere.

I like very much this conclusion!!

>    We agree what is right about naturals and what not. All our N's have an
>    associative addition, and prime and composite numbers and questions to
>    know the answer, but I think that _truth_ have nothing to do with it.
>
>    Our N's satisfy that 2 + 2 is equal to 4, but, to my mind, this is our
>    _agreement_ about our abstraction. Saying "2 + 2 = 4 is true" sounds
>    quite different to me.

Again, I completely agree! "Our N's", etc.!!

Anatoly Vorobey wrote:
>
> You, Vladimir Sazonov, were spotted writing this on Wed, Dec 23, 1998 at 01:07:04AM +0300:
> > > Nevertheless, we would agree if your claim was that "+(ss0, ss0) = ssss0"
> > > is PA-demonstrable or the like.
> >
> > Yes, this is a very good syntactic analysis of this big problem. What
> > about semantics? Take, e.g. two drops of water (or vodka or what you
> > like) + again two drops. The result will be 2 + 2 = 1 (one big drop).
>
> It's raining now. I lift my eyes off the computer screen, look up at my
> window and see raindrops falling on the window, and racing down towards
> their doom. I mark two of them on the left side of the window, and in a few
> seconds, two more fall beside them and join the race. Together, they make up
> 4 lovely raindrops, each of them unique in its special way. And in a few more hours,
> there'll be no raindrops at all on my window.
>
> The point, to put it in plain words, is that there's nothing strange about
> 4 raindrops merging into one, and it isn't at all relevant to natural numbers.
> This particular "paradox" is mentioned, alas, all too often, probably due to its
> cute and laconic way of arriving to a "contradiction". Our intuition, of course,
> tells us there's no contradiction, and it's perfectly right for a change. The fallacy
> lies in being used to interpret semantically "adding" as "bringing together
> spatially", while the correct interpretation would be something like "perceiving
> conceptually as distinct parts of one whole". Bringing together spatially
> simply helps, it's a useful mental (or physical) operation to perform - it helps
> visualize the objects as belonging to one collection.

Etc. ...

It seems you took too seriously my joke. However, I hope it may
have some value for the discussion on certainty and on absolute
truth.  Of course, pebbles are more appropriate than drops of a
natural numbers.  That example was used to show that we should
first fix some meaning and only then say "2 + 2 = 4 is true with
respect to this meaning".  What does it mean that "2 + 2 = 4 is
absolutely true" is absolutely unclear to me.

Andrzej Trybulec wrote:
>
> On Mon, 21 Dec 1998, Michael Thayer wrote:
>
> > > I don't either. But "2+2=4" seems to be absolutely true.
> > >
> > >Andrzej Trybulec
> >
> >
> > Yes, but WHY does it seem absolutely true??
> > Is it any better grounded in our knowledge than either:
> >
> > 1."The Sun will rise tomorrow"
> > 2."Winston Churchill was a Prime minister of England"
>
> 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.

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?

> 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.

> Of course, what I wrote may be meaningless for some members of f.o.m.,
> sorry. I am a platonist (maybe not so extreme as Randall Holmes, but close)
> so it is easier for me.

Andrzej Trybulec wrote:
>
> 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.

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.