[FOM] polemic

[Arnon Avron]

There were 3 official responses on FOM to my reaction to
Friedman's posting "Role of Polemics" from January 21
(There have been  also a couple of others, of a 
different type, personal responses). One of them was from Friedman
himself. The others were from Neil Tennant and Joe Shipman
who both questioned my understanding of what Friedman
was saying in the message I was reacting to. In the present message
I'll reply to the claim made by Neil and Joe that I misinterpret
Harvey's polemic. I'll reply to  Harvey's philosophical claims 
(and to Neil and Joe's comments supporting a part of them)
in another message. 

So Neil Tennant writes:

> Might you be misinterpreting the dialectical background 
> to Harvey's "polemic"?

He then explains that Harvey's "polemic" (Neil's quotation)
is directed against core
mathematicians who refuse to see the importance of Godel
theorems to their work, and adds that

> Whether you, as a predicativist, choose to be impressed by the Pi-0-1
> statements and/or salivating to get it, is not really relevant to the
> assessment of such a profound breakthrough.

Well Neil, I was not reacting to the background of Harvey's polemic
(not "polemic"), but to the polemic itself, as it was presented in that
message. That message was very explicitly directed against 
"fom polemics" of the following form:

> There are restrictions placed on allowable methods of proof. 
> This can either be a restriction on the rules of inference, or 
> a restriction on the kinds of mathematical objects allowed, 
> or more subtle restrictions. 

Neil, do you really think that the people Harvey was talking about here
and who disallow some mathematical methods are the "core 
mathematicians" he seems to adore so much? Do you really think 
that Harvey would have classified *them* as "polemicists"? Anyway, 
even if you do, Harvey was very explicit in his examples who 
are the people he is attacking. He gave two examples: intuitionists and 
predicativists, and it is clear that his main attack is on 
the predicativists. So my message was concentrating mainly on
the relevance of Harvey's results to predicativism. Whether or not
they are important for other goals or subjects (as you obviously think)
is irrelevant to the subject of his polemical message, or to 
my reply to it (by the way: for a long time I have my own opinion 
on this issue. I kept it to myself, since it is not productive 
to attack mathematical work even if one doubts its stated goals.
But since I was anyway carried into debate by Harvey's attack
on predicativists, I'll address this subject too in the very 
near future).

  Let me add that I have found Harvey's attack on predicativists 
quite brutal. First classifying our claims as "polemic" and us as 
"polemicists" is already pure polemic (not "polemic"). Saying
that predicativists have given no "convincing" explanations what 
is wrong with the methods disallowed by pure predicativism 
is another strongly polemic assertion. I am sure that Friedman
has read many texts in which such explanations are given in detail.
The fact that he does not find them convincing is his right. To
categorically claim as a matter of fact that predicativists
have given no such explanation and this shows the polemical
nature of their claims is *not* his right. Finally, consider the
following paragraph from Harvey's polemic posting:

>It has always been considered VERY SAFE to complain about big set
>theory such as ZFC, and more so about the biggest set theory -
>ZFC with large cardinals.

The only image I could have got from this sentence was of a group 
of people (to which I belong)  who are seeking to
complain about some mathematical method, and here they find
an area in which they believe they can safely do so.
I was truly offended by this description, as well as by the talk about
"SAFE POLEMICS" ("considered to be") in the next sentence in
Friedman posting::

>In particular, it has been considered to be a SAFE POLEMIC that the
>underlying set concept behind ZFC and especially ZFC with large
>cardinals, is useless for something "good".

One clarification: in his reply Harvey noted that he 
has never mentioned my name in his posting. This is absolutely 
true, of course. Moreover: I never thought his message 
was directed at me personally, or even
that Harvey was thinking at all about me when he was writing it. 
I reacted as a member of the group he was attacking. That's all.

Arnon Avron