FOM: Comment on Hersh

Soren Riis sriis at
Sun Mar 1 11:42:28 EST 1998

Comment on Hersh 
Hersh wrote:

>     I propose a way of thinking about the reality and
> existence of mathematics which lets us keep our mathematical objects
> really existing, really meaningful, without resort to mysticism.
>        The key observation is that in our world there are not two
> but three main kinds of reality. 

I suggest Hersh begin to give proper credit for his ideas. This 
observation is certainly found in Carl Popper's writings. Popper 
talk about World 1, World 2 and World 3. Examples of World 3 objects
are numbers, theories, designs, works of art, institutions etc.
Wisely enough Popper realizes the danger of putting loaded labels 
(i.e. labels which might carry preconceptions) on these worlds.

> Mind and matter are familiar.  But they
> do not help with our puzzle, because mathematical objects are not
> material, and they are not mental, in the sense of being
> part of anyone's private subjectivity.
>        But they are not the only things that are neither
> mind nor matter.
>        For instance, your job.  The money in your bank account.  Your
> degree. Your career....  
>        Catholicism, Lutheranixm, Judaism, Islam, Bolshevism, Fascism,
> racism, nationalism, rationalism,. liberalism, feminism....
> Beethoven's fifth symphony, Shakespeare's Hamlet, The
> Bible, Darwin's theory of evolution, Einstein's theory of relativity....
>        This journal. Its editorial policy, its standards of
> publication, its mailing list, its backlog.  This mathematical
> society.  Its history. Its traditions....   
>        All these things are real. They are the substance of our lives.
> None of them is material, none of them is mental.  What then?
> I call them social-cultural-historical, or just social for short.
>        Once these examples are  pointed out, it is hardly questionable
> that there is another level of existence besides the mental and 
>        Now, our problem was--what sort of existence or reality has
> mathematics?
>        We have not two but three choices.  Material and mental are
> wrong.  What about social?

You can call it social or social-cultural-historical. You could also 
call this choice "World 3" (Popper), "A", "X" or you name it. 
These are just a labels. You want to place "French grammar" and much
more under the same heading as Mathematics. Fine - as long as you 
are not smuggling in preconceptions and connotations attached to 
the label you choose. 
Why do you call the world social-cultural-historical?? Why do you 
not call it social-cultural-historical-mathematical??? 

To keep things simple I will only briefly address the other flaws 
in Hersh posting. 

> I propose a way of thinking about the reality and
> existence of mathematics which lets us keep our mathematical objects
> really existing,

Notice the "existence" here is used on two different levels. The
existence of comic strips / mathematics is not to be confused with
the question of the existence of Mickey Mouse / non-abelean groups.

Hersh are trying to comfort us that Riemanns theta-function have the 
same kind of existence as Donald Duck.

Hersh ideas are not new, neither are the flaws in his thinking.


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