What is an Object?
Negation vs. Good for
It’s what it is, says it all. Unfortunately, the sayer doesn’t happen to be the beatified war-skulled Biden, a zombie eating dead-bodies, all set to be conferred sainthood by the NYTimes interns running/ruining the show.
Oh well, there isn’t much I can do about dem things hellbent on qualifying for admission into purgatory ;) So, here we move on ….
There seems to be two mathematical methods:
vs.
II. Universal mapping properties (UMP) / Good for
for constructing objects/operations abstracted from conscious participating in mathematical practices.
The Cantorian method of contrasting/negation, championed, plausibly along with many others, by Professor F. William Lawvere and Grothendieck, did not get to the long-sought ‘totally devoid of structure’ needed to model cohesively varying everyday objects, but got us very close: Kardinalen (of lauter Einsen / all ones), which with their almost structurelessness served as a background to represent various mathematical objects / operations as structures.
I am wondering if the Bourbaki method in terms of UMP can get us to the pure background ‘good for’ modeling any / every quality of cohesively varying objects populating our perceptual experience. Admittedly, my question has more to do with my lack of understanding of how the two methods: I. negation and II. UMP of defining mathematical constructs relate to each other. In the spirit of making a fool of myself, do we have to be confined to contrasting — cohesion vs. variation — as means of understanding them; can’t we approach them — cohesion, discreteness, variation, constancy — from a perspective(s) of what they are good for?
Please pardon me if it’s all nonsense /\
I’d be truly grateful to you, your time permitting, for helping me develop a proper understanding of the mathematical objectification of concepts abstracted in the course of conscious participation in mathematical practices /\ /\ /\
Happy Weekend :)
Thanking you, yours truly, poison
P.S. In my home turf of neuroscience, contrast is the basic building block of visual objectification, while Gibson (whom, for reasons unbeknownst to me, I have been keeping at arms distance ;) equated perception to its functions / uses; the above ‘good for’ method can be thought of as a refinement of the method of defining things in terms of their uses. The non-triviality of the refinement that is ‘good for’ can be readily seen in a toy exercise of defining ‘pen’. Without invoking any cosmic conspiracies, we invariably find a basketful of undesirables such as ‘scratching’ figuring in the definition of PEN, for there is no fatwa against using a pen to scratch my itchy back. Thanks to Bourbaki/Samuel, defining X in terms of what X is ‘good for’, or in terms of that which wouldn’t be but for X, we arrive at a sensible-and-reasonable definition of PEN in terms of writing, while shooing away scratching et al., to distant realms of oblivion. In a lighter vein / tangential note, one thing I wasn’t able to do is: dethrone Gibson from the demigod that he is to design students (no amount of ‘good for’ was good enough to deGibsonize the minds of students at the most prestigious design school in India ;)
