[FOM] Multiple types of foundation for mathematics (cognitive, biological, mathematical...)

Patrik Eklund peklund at cs.umu.se
Fri Mar 17 06:56:18 EDT 2017

Hi Aaron,

All these views and efforts are basically fine. What I find tricky in 
all this is how to arrange a forum which embraces but doesn't control 
the flow of the discussion.

FOM is controlled by Davis, and Davis is a student of Church. Martin 
Davis therefore kind of acts as being obliged to control the FOM forum 
the way he does.

Church did the same thing with the J Symb Logic. He had some 
"co-editors", like Kleene and others. Some were his PhD students, but 
others not, like Bernays. Church took the journal and its content into 
certain directions and foundations evolved the way it did for a couple 
of decades after WWII. Grundlagen I and II are known but not so much 
used, and Bernays was not a promoter. Clearly, he also did his post-WWII 
outside the US.

I've tried to post a few things on the Liar, in order to test if there 
is an interest to open up a debate. My postings have been rejected. One 
reason for rejection is probably also that I suggest to allow category 
theory to stay within the foundations picture. FOM rejects this.


If I understand your mail correctly, you suggest a forum for discussing 
foundations more broadly. It's fine, but I guess sooner or later comes 
the question about boundaries and priorities. What is and what isn't.

Under the Catlist I have been able to post a few things, and I have been 
thinking about similarly suggesting to have a category theory based 
forum for foundations, but it's not clear how to start it. And it's not 
clear if there is enough interest to provide a critical mass.


In your content I also like "biological" and I like it because I believe 
we need some attachment to reality and real world problems. Physics 
without the physical world makes no sense, and the physical world is 
well-known. Logics without the logical world makes no sense but the 
logical world makes no sense.

On applications I prefer health, in which biology plays its role. 
However, a human is more than biology. Simple things like "disorder", 
"health condition" or "intervention" have never been formally explained 
or defined. In medicine, there is also mix of things involving different 
levels like the molecular, cellular, tissue and organ levels of 

I think a foundation discussion connected to something that embraces a 
"logical or metaphysical world" would add value, but it requires that 
participants understand e.g. biological or medical problems in more 
depth as compared to anyone on the street.


Basically what I say is that the FOM forum is "small". It will never 
have any societal impact of any kind. But it could have such an impact, 
and it would if we just could arrange a forum where the objectives are 
shared, and we really try to bring things forward.




On 2017-03-17 01:58, Aaron Sloman wrote:
> Some notes comparing and contrasting investigations into several 
> different
> kinds of foundation for mathematics, including cognitive,
> biological/evolutionary, mathematical and metaphysical foundations:
> http://www.cs.bham.ac.uk/research/projects/cogaff/misc/maths-multiple-foundations.html
> (Also pdf)
> Comments, criticisms, suggestions, and pointers welcome.
> I would particularly welcome pointers to work on how biological 
> evolution
> was able to produce brains able to make ancient mathematical 
> discoveries in
> geometry and topology, e.g. leading to Euclid's Elements. Can those
> processes be replicated in AI theorem provers?
> The answers should refer to explanatory mechanisms not the competitive
> advantages of having mathematical abilities.
> Thanks.
> Aaron Sloman
> a.sloman at cs.bham.ac.uk
> http://www.cs.bham.ac.uk/~axs
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom

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