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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]