Rarely. Practical example: ground truth. Ground truth is the structure of truth about something. It’s an actual measurement taken at the ground level of a structure and drawn in a binary image to precise scale. This is used as a way to compare photographs that are taken from a distance and assists artificial intelligence in matching images to what objects are. But these ground truths are not the structure of the thing. It is the structure of truth about something. Depending on purpose it is useful and pragmatic. But it is not without flaw. It is a transfer – an abstraction lifted from – and put into alternate service. The service of the structure of truth is distinct from the structure of the thing. I am taking a practical approach because that’s just how I think. I hope it provides some insight into the question. If not insight then a nice distraction. Note: my first thought was kernel/cokernel but that is over my head.

Rarely.
Practical example: ground truth.
Ground truth is the structure of truth about something. It’s an actual measurement taken at the ground level of a structure and drawn in a binary image to precise scale.

This is used as a way to compare photographs that are taken from a distance and assists artificial intelligence in matching images to what objects are.

But these ground truths are not the structure of the thing.

It is the structure of truth about something.

Depending on purpose it is useful and pragmatic. But it is not without flaw.

It is a transfer – an abstraction lifted from – and put into alternate service.

The service of the structure of truth is distinct from the structure of the thing.

I am taking a practical approach because that’s just how I think.

I hope it provides some insight into the question. If not insight then a nice distraction.

Note: my first thought was kernel/cokernel but that is over my head.

—————

As long as one stays in the world of definition, you’re may be right. My response, which got a sad emoji from you and I understand why, is more of a reflection of the granularity of nature and imperfection of our capabilities.

Alas, though, I also think that continua _may_ be but an averaging – a smoothing function that occurs due to how our minds compress information and not a reflection of a smoothness to be found in actuality.

As long as you don’t leave the world of definition, you may be right. But that is because in the world of definition, the structure of truth and the structure of about-truth are both structured of definition-like-things.

—-

infinite granularity in aleph null is pushing the issue back into what is untouchable on a theoretical level.

But in a practical level, what it does is amounts to an averaging function. A smoothing. A ~. A blur. Gaussian in nature. Then There’s a kernel and this is the top hat and we will strip away the fuzzy parts to leave the sharp edge.

A, Fuzz, sharpen, B.
Now A ~ B.

—-

If you work with infinitesimals, you’ll get there. I understand that. But on a pragmatic level, working with infinitesimals and a smoothing function (fuzz (to fill in the little infinitesimal bits) , sharpen) are the same.

This is the kind of thing done in image analysis all of the time. It’s how you get to exact from incomplete.

====

In computational complexity, a certificate:

” A certificate is often thought of as a solution path within a verification process, which is used to check whether a problem gives the answer “Yes” or “No”.”

So, it ensures a binary Yes/No answer.

===

I’m glad I’m not alone in this thing. It wasn’t until I looked into mathematical morphology (ImageJ / Fiji), then learned it was a complete lattice, my world opened up. I could accept discrete space as “ok” as I could not escape some form of granularity and some kind of necessary cutoff choices.

If I switch to waves for everything, for example, I still need to chose bandwidth and once I chose bandwidth, I can easily switch to a discrete space, adjusting grain as needed.

I had to give up on the notion of ultimate answers for some things for now but I’ve gained flexibility and agility.

It works better with my brain too. I was raised with computing rather than mathematics. I can’t read most math or logic but I can read pseudo-code and think things through in terms of circuits / loops operating over time.

Yet I also know that the universe is not a chessboard and when I treat it like a chessboard, I have to occasionally look at the cracks between the wooden squares, peel the squares up and look underneath, study the surface of the square I’m on, etc.

Still, I would love it to unify.

====

 

This made choosing discrete ok for me. From this, I looked up what I’d be missing in constructive mathematics and in predicative mathematics. Was I ok with that? I am ok with that as I had trouble understanding either as it was.
 
https://ncatlab.org/nlab/show/complete+lattice
 
Complete lattices are harder to come by in constructive mathematics and nearly impossible in predicative mathematics (at least if they are to be small). In particular, one must use the Mac Neille reals (and be a bit careful about infinity) for the analytic examples to work constructively.
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