[FOM] polemic

Harvey Friedman friedman at math.ohio-state.edu
Tue Jan 31 19:17:18 EST 2006

This is a reply to 

The part of what I wrote that Avron most objects to is my claim that

*The usual f.o.m. polemics asserting that such and such mathematical methods
are legitimate, whereas stronger methods are illegitimate, give no
convincing explanations of the illegitimacy. Instead, they only contrast
differences between the "legitimate" methods and the "illegitimate" methods.
Sensing the weakness of such illegitimacy claims, often there is an implicit
or explicit claim that no "good" mathematics, in some sense, can come from
the use of "illegitimate" methods that can't come already from "legitimate"
methods. There are now a growing list of stronger and stronger examples that
are making it impossible to maintain that position.*

(The above is a paraphrase of what I wrote).

The main f.o.m. content of what I wrote deals with these examples, and how
they have grown in strength as these usual forms of f.o.m. polemics modify
over time. I include some predictions of future results along these lines
that are much stronger than anything to date.
> 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?

You are seriously misinterpreting what I wrote, if you think that I "adore"
core mathematicians. In fact, I often complain about core mathematicians as
thinkers. I don't complain about core mathematicians as phenomena.

In part, f.o.m. treats mathematical activity as a given phenomenon.
Physicists do not complain about atomic particles as phenomena.

I regard the relevant discussions I have with core mathematicians as
"running f.o.m. experiments".

> Anyway, 
> even if you do, Harvey was very explicit in his examples who
> are the people he is attacking.

I am not attacking people. Just pointing out the obvious - as in * above.

>He gave two examples: intuitionists and
> predicativists, and it is clear that his main attack is on
> the predicativists.

I gave these examples mainly because they are very famous and involve
several famous great core mathematicians, not only logicians.

The point that I make - that banning mathematical methods is not supported -
applies to a wide variety of f.o.m. polemics. Certainly not just
constructivity and predicativity.

> 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").

I have been very recently advised by colleagues on the f.o.m. that "being a
polemic" is not a negative assertion.

The valuable part of these f.o.m. polemics is the discussion of the
differences between methods - not claims of illegitimacy, which do not seem

>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,

I have no idea whether or not you are a member of this "group" you refer to.
If you say you are, then I believe you.

>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".

Perhaps if you want to continue this exchange, instead of complaining about
being offended because you are a member of some "group", you should give a
brief example of an argument in favor of ***banning*** certain mathematical
methods - right here on the FOM?

Harvey Friedman

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