[FOM] Axioms and self-evidence

Arnon Avron aa at tau.ac.il
Sun Mar 24 05:28:21 EDT 2013

First, I want to explain why I have changed the subject line: the
initial discussion on the axioms of infinity and their proofs
has spread into several direction, and what we are discussing
here has only weak connection with it. So I thought it might
be useful to make this fact explicit.

Second, and more important: I agree that the way I put my point
in my previous short message was too simplistic. Actually,
a central idea in my personal views on the nature
of mathematics and its foundations is that there are several
degrees of certainty in mathematics, and not all
mathematical concepts and theorems have been created equal.
Thus you could have pointed out that the fact that
among all of Euclid's postulates and axioms only
the fifth has caused dissatisfaction (and desperate
attempts to prove it from the other axioms) shows
that it looks to mathematicians as less self-evident
than the other axioms (even though they still
saw it as obviously true).

Having said that, I still insist that mathematical
propositions that start their career as conjectures
should never be accepted as true from the pure
mathematical  point of view until they are proved.
Note that I am not talking here
about their acceptance in applications in physics
or technology. There is no reason I can see to
demand the mathematical component of theories used
in such applications to have higher degrees of certainty
than the other components, and I can imagine
a state in which the scientific evidence for p!=Np
will be no less compelling than the evidence
we now have for the theory of relativity. However
inside mathematics we still have (and hopefully will
continue to have) stricter criteria.

Now the two examples you brought do not contradict
the view I have just expressed (and if I understand
you correctly, you do not claim that they do). Let
me add a few words of them nevertheless, since
they are interesting.

I admit that I feel certain that ZF is consistent-
but unfortunately, I am even more certain that each 
of us will die sooner or later. In both cases
at least my certainty is not pure mathematical certainty. 

As for AC - this is not the place and time to start
deep discussions of it. So I just state that not
only it is not (and never has been, and never will be)
evidently true, I am not even sure that it is meaningful
(neither do I understand in what sense the real numbers
"can be well-ordered" even though we shall never be able
to well-order them...). What is more: by Godel's consistency proof,
there are cases in which using ZFC is as safe as using ZF.
However, there are also cases in which AC leads to
very doubtful results (Like Banach-Tarski, or (to a lesser
degree) the existence of non-measurable sets). So maybe
you are right about the sociological fact that most
mathematicians accept it now freely. But this does
not mean that they are entitled to do so.



Quoting "Timothy Y. Chow" <tchow at alum.mit.edu>:

> Arnon Avron wrote:
>> On Wed, Mar 20, 2013 at 10:23:27PM -0400, Timothy Y. Chow wrote:
>>> The difference between wanting proof and having doubt can be seen
>>> even in the context of famous conjectures, e.g., P != NP or the
>>> Riemann hypothesis.  Although there is not quite enough consensus
>>> about these statements for them to achieve axiomatic status, in
>>> practice they are treated much like axioms, in that people feel free
>>> to assume them whenever they need to.  There's still an intense
>>> desire to find proofs for them, even among people who are totally
>>> convinced that the statements are true.
>> I prefectly agree with the content of the two first paragraphs above.
>> However, I think that the two examples given in the third are bad.
>> The difference should be clear: the truth of those given in the
>> first two examples had never been in doubt before they were adopted
>> as axioms. The consensus reached in their case was about the need
>> to take them as axioms, not about their truth (and even that need
>> has actually been *proved*!). But concerning the two last examples
>> the "not quite enough consensus" is about their *truth* -
>> and once a proposition has not been self-evident,
>> its truth can be established in mathematics only by a *proof*.
>> No "consensus" can be a substitute for this!
> There's certainly a qualitative difference between famous  
> conjectures and axioms.  I wasn't giving P != NP or the Riemann  
> hypothesis as examples of potential axioms, but as examples of the  
> distinction between wanting proof and having doubt.
> Having said that, I'm not sure the distinction you're drawing is  
> quite as sharp as you make it out to be.  Consider something like  
> the axiom of choice or the consistency of ZF.  For a while, these  
> propositions were not self-evident, so by your argument, it would  
> seem that we would never reach the point of accepting them as  
> axioms, and would forever insist on finding proofs.  Well, as we all  
> know, the trouble is that in these particular cases, we have since  
> learned that finding proofs is a hopeless enterprise. Given that,  
> mainstream mathematics has in effect granted them axiomatic status  
> because of a consensus about their truth.
> In the case of P != NP or the Riemann hypothesis, the difference is  
> that we have no reason at all to believe that the search for a proof  
> is hopeless.  But if in the unlikely event we were to come to  
> believe that a proof was in principle unattainable, I could imagine  
> them coming to be accepted as axioms.
> Tim
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