Is Category Theory a Theory?

[Patrik Eklund <peklund@cs.umu.se>]

Thank you, Hidekazu-san, for you reply.

You use several important keywords that we could debate much more, and I
sincerely hope Catlist will.

One word you use is "role". I find that very appealling and intriguing.
What is the role of zero in the natural numbers signature? What is the
role of zero as a term and number in the axiomatized number system. Does
that role change and does it appear as a 'subrole' in another role? I
don't know, and I am certainly overly intuitive saying so, but if we do
not allow ourselves to do that, we will forever remain applying
"ignorabimus", something Hilbert fiercely rejected, ending his famous
radio speech with "We must know! We will know!".

When I said "Is Category Theory a Theory? I think not. At least not in a
logical sense." I obviously refer to "theory" as logically defined by
the set of all 'clauses' we can 'infer' starting from all acceptable
clauses ('axioms'), iteratively throwing them into the acceptables, and
exhaustively doing that over and over again. Is Category Theory such a
Theory. Of course it isn't. It's much more. If it isn't, why are we
here? A monoidal category is a good example. It's not a category. It
contains one, but as a structure it's not a category. Yet, monoidal
categories are part of category theory. I allow myself to say that the
category in a monoidal category plays a certain Role in that structure.
So does the tensor, and soon we almost feel more algebraic than
categorical. If we do, well, is a monoidal category then a foundation
for a signature?! Signature in a broader sense, of course. Why not? We
obtain expression, words, and so on. Maybe somewhere along that line
somebody starts to think "What is a Turing Category?". What I try to say
is that if we disallow ourselves to think in these ways, we apply that
"ignorabimus". Your intuition about "role" seems to be quite rich, and I
surely do not provide it with the appreciation it deserves, but I hope
to hear more about it in years to come.

You also use "properties", which in logic often means fixing how certain
expression relate to each other. Algebra may be more fundamental than we
think, and myself I am inclined to explore this more than I have before.
Indeed, when in 'lativity' I underline that "signatures come first, and
then we create sentences, and so on, latively", am I not actually
speaking warmly about algebra being among the first ones in line?
"Algebra studying itself" may be interesting to further explore, and to
compare it with "logic studying itself" and "set theory studying
itself". Clearly we also have "logic studying set theory", "algebra
studying logic", "logic studing algebra", and in fact all combinations.
Do we not even have "algebra studying how logic is studying itself", and
so on? I think we do, and this really to me is one of the reasons why we
sometimes so little understand each other, because we do not always
clearly acknowledge what we really are doing and why, when we e.g.
describe category theory as a logic, logic in a topos, categories as
algebras, logic over a monoidal closed category, and so. Why do we do
these things. I develop logic over monoidal closed categories because I
see how underlying rich structures of a certain information scope (like
health ontology) suitable can be represented because of such a category
so that reasoning involving that information enables us to say more than
we have used to be hearing (in health care). I am seriously starting to
believe that we are able to save lives by enriching health language with
category theory.

So thank you once again again, Hidekazu-san, for sharing your thoughts.
It is not unskillful at all, and certainly has no mistakes. It's
Excellent Harmony.

Looking forward to more debate on these aspects.

Best regards,

Patrik

PS My origical catlist posting was forwarded also to FOM, in the hope
that the FOM community would be even more interested to bridge
foundations with aspects learned within Category Theory. That forward
was rejected:
   "Your message was deemed inappropriate by the moderator after
   consulting our editors. One of them wrote: "This message seems to be
   mainly an expression of Eklund's personal opinions, with no supporting
   arguments and little clarity. I recommend rejecting it.""
I replied to the fom-owner as follows:
Thank you for your response.
   The reason why I sent it to Catlist was to raise some debate more than
presenting my personal opinion. Catlist accepted it, and I thought I
would bridge it over to FOM, since recently FOM has taken up an interest
to undestand the role of category theory for foundations.
   Obviously the content was less kind and more provocative to the FOM
readers, and my guess was it will be rejected.
   Anyway, I appreciate very much to be part of dialogue within FOM and I
take this opportunity to thank you for all acceptances so far of many of
my postings.
   My gut feeling is that foundations discussions are finding and
exploring new dimensions. There are Pandora boxes around some corners,
and there is resistence to open some of these.
   Anyway, it's all up to us, and we will all jointly continue to do the
best we can. I feel very confident about what will come in 50 years from
now, yet I feel sad not to live to experience it.


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