[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
diseases.
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.
---
Best,
Patrik
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
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