[FOM] to Sazonov about Certainty and Inquisition

[Vladimir Sazonov]

Dear Arnon,

First of all, believe me that there was nothing
personal towards you in my posting. Also no
feeling to be offended from my part. Please
consider this as an emotional, but purely scientific
discussion with some exaggerations (on the inquisition
- sorry about that) Emotions are related with the interest
to the subject matter only. I am glad that we agreed about that in a 
personal e-mail exchange.

Now, let me reply on the subject matter where we can struggle (still 
remaining good friends, or what you prefer).


On 20 Oct 2007 at 22:32, Arnon Avron wrote:

>
> On Sat, Oct 20, 2007 at 04:14:44PM +0100, Vladimir Sazonov wrote:

You cited me:

> > Aha, now I understand that mathematicians or those people working on
> > f.o.m. should be unified into a big sect (the bigger the better - more
> > "objectively") who "know" or believe that they know what is absolutely
> > certain (mathematical truth or a special class of mathematical proofs
> > of absolutely certain truths) and what is not.

But you missed the following continuation of this paragraph:

> > Yes, they realize that > their personal knowledge is not absolute, 
> but they believe in the > highest "divine" ideal. The typical 
> questions of people from such a > sect are "do you believe?" [snip]

where I intended to anticipate your:

> You surely understand the big logical difference between "recognizing
> that there are *some* mathematical propositions that are
> absolutely certain and *some* that are not" and
> "know what is absolutely certain and what is not" (especially
> that I explicitly said that I don't pretend to know the exact
> line).

The question is not on exact line. You missed the key question from my 
posting "in which objective way" would you fix   "some absolutely 
certain mathematical propositions and some that are not"?

> So why did you change what I had said in
> such a drastic way?

Sorry again about "inquisition". But the point is that you (and, I 
feel, many) take *beliefs* on the absolute mathematical truth or the 
like as the base of the philosophy of mathematics. Then non-believers 
should be probably considered as heretics and even the inquisition 
could be imagined as a natural consequence. (Reductio ad absurdum - 
nothing personal.)

What is interesting, whatever are our beliefs/non-beliefs and views, we 
are equally able to understand mathematical proofs and have very 
similar intuitions related with these proofs and applications of 
mathematical results to the real world. This is indeed rather objective 
fact realizing of which requires virtually no philosophy. This is just 
everyday mathematical practice. And this could be a point for mutual 
understanding even when we discuss the philosophy of mathematics. 
Moreover, analyzing this objective entities (mathematical proofs, 
mathematical intuition, applicability to the real world) can be taken 
as starting point of a philosophy of math. No myths on absolute truth 
or the like are necessary. Even if you prefer such myths (let you call 
them differently as you like), I see no reason for you not to be able 
to understand this simple view.

>  Still, since it is clear that you were offended, I apologize
> for talking about "strange people", and take these words back.

No, no, please! No need to apologize! Our discussion is about 
objectivity and subjectivity. Nothing personal.

> However, I am not apologizing about claiming that the only
> thing that somebody which share my goals in f.o.m can do about
> people who doubt even that there are infinitely many primes
> (even though they know and understand Euclid's proof) - is to ignore
> these people and their doubts. I do not understand these people,
> and it is obvious that their mind and language are completely
> different than mine.

Nothing completely different! In my posting I acknowledged (the trivial 
fact) that Euclid's proof (speaking in the contemporary language, from 
such and such axioms by using such and such formal derivation rules) is 
and will remain syntactically correct forever. Moreover, it is also 
intuitively understandable. (Otherwise this would not be mathematics - 
just a play with symbols.) All of us have different intuitions because 
we are different people and, also because our philosophical views on 
mathematics may have some influence on our intuitions. Anyway, when we 
discuss a mathematical proof we distract from these differences and 
find something joint. Otherwise it would be impossible to communicate 
on mathematical proofs and ideas. The great achievement of mathematics 
is that it excluded any subjectivity from mathematical proofs. In 
principle, there is no problem of mutual understanding about 
correctness of mathematical results and even the most essential 
intuitions on which they rely.

Your (and many others') point of view is that mathematics is about 
(absolute) truth or certainty. I do not understand and do not accept 
this, but let it be. I can interpret this as truth in my personal 
imaginary universe of ZFC or, if necessary, a stronger theory. In this 
way I can understand you sufficiently adequately, making always 
appropriate corrections in case of your references to the "absolute". 
All your mathematical results on predicativity or whatever else you 
will rely on the idea of some "absolute" will be appropriately 
reinterpreted. So, I am able to understand your results quite 
adequately by these reinterpretations (consisting in appropriate style 
of "ignoring" any your references to the "absolute"). Formally 
speaking, you will not notice any difference.

I, on the contrary, consider that mathematics is about formal systems 
and proofs related with (and improving, strengthening, correcting, 
organizing, governing, etc) our vague, poor, dreamy, but really 
existing intuition (the replacement of your absolute "truth").

You rely on some beliefs (something subjective, despite your claims 
that you intended to find objective grounds), whereas I rely on really 
objective things (formal proofs, our intuitions related with proofs, 
and applicability to the real world).

I know, you will always attach to these objective things your beliefs. 
But this is unnecessary, non-visible (something from your internal 
mental world), mathematically indistinguishable.

Infinity of prime numbers - for me this is some formally derivable 
sentence having some intuitive meaning in my imaginary, and vague (as 
in the dream) world of natural numbers governed by PA + classical 
logic; it is vague also because I somehow include in this world the 
idea of feasible numbers what makes the raw of natural numbers even 
more vague than feasible numbers themselves; somebody could include 
something else...; all of us are different).  But I can imagine also 
that in some other (imaginary) world of numbers this theorem might not 
hold. (Recall Euclidean and Imaginary - the excellent term of 
Lobachevsky - geometry.) At least in the real world it is unclear how 
to interpret the infinity of primes. Therefore I see no need to call 
this absolute truth. I even do not understand what this means at all. 
(Derivable here, underivable there..., it depends...) Could you explain 
your "absolute" without referring to beliefs or opinions (let even by 
majority) of mathematicians? I am not interested in opinion of the 
majority. I want to understand the objective roots of mathematics.

What can be here so unacceptable for you that we are unable even to 
understand one another? Your beliefs prevent that? Why beliefs if you 
pretend to be objective? If you do not pretend, just say: "I am 
creating a sect of believers". (Sorry, again reductio ad absurdum". How 
else to explain you what I mean?)


With my best regards and full respect!

Vladimir

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