From the article, "But if we ignore these technicalities we can still appreciate the general principle, that each small part contains the information about the whole." Hmm. And maybe if we're not excessively worried about what "information" is, exactly.

We might rephrase using the time-honored locution, "get theah from heah." The idea being - the question being - whether given any here and there, you can or cannot, "get theah from heah."

And this is a question in elementary education. It is generally held that below a certain age the student simply cannot "grok" a lot of math. Against this is the also general belief that subject matter can be taught and learned incrementally, that the boundaries on what can be learned can be extended and expanded. I favor the incrementalists because 1) it's already demonstrated many times over, 2) teaching itself is very far from perfect, and 3) I think that brains are not limited according to what educators believe.

That leaves the question if there is any body of knowledge (and let's rule out here any involved discussion or debate as to what knowledge is) so isolated that we "can't get theah from heah." A key to this answer is the notion of specialized knowledge. I have fingers and can hum: in principal, then, I can learn to play the piano, as can everyone else with fingers who can hum. But with some qualification on what it means to be able to play the piano, most will acknowledge that most folks cannot learn to play the piano. Or that bouncing a ball makes a basketball player, and so forth.

But this would seem to concern learned skills and most folks can learn to do most things at least badly.

Perhaps the best approach to an answer is to consider the learners as a whole: is there any knowledge they - we - cannot aspire to? This groups all knowledge as that which can be known, with the common element being the knower writ large. And since of this whole everything is a part, it would seem that from any part we can get to any other part, eventually.

Let's note, though, that while the holograph contains immediately information about the whole in every part, the progress of knowledge is instead mediate and sequential. Or, "You can get theah from heah, you just ain't theah yet."

It is generally held that below a certain age the student simply cannot "grok" a lot of math. — tim wood

Not in conflict with anything you've written, but it brought to mind some studies done by Karen Wynn and others that show that even infants have a sense of quantity and some very basic "arithmetic" skills.

Welcome to the forum. This general subject is something I've thought a lot about. Some thoughts:

Calling knowledge "holographic" seems a bit highfalutin to me. It gives the concept a veneer of exotic science that I don't think is needed.

This gives me a chance to bring out my favorite quote from Franz Kafka:

You do not need to leave your room. Remain sitting at your table and listen. Do not even listen, simply wait, be quiet, still and solitary. The world will freely offer itself to you to be unmasked, it has no choice, it will roll in ecstasy at your feet. — Franz Kafka

When I've posted this quote before, people respond that obviously you can't know about Newton's laws sitting alone in your room, but that's not the point. What you can learn is how the world works, how we know what we know, how we fit in the world. I think this is the kind of thing we learn "holographically." Not something specific, but an awareness of our own thinking process and how to use it to know things.

The other thing we get from small bits of knowledge is bricks for our wall. In my experience, I have a mass of connected knowledge that builds a model of the world I live in. Each small piece of knowledge is connected to others to make a structure of interconnected pieces. I have a vivid visual image of this model as a cloud, lit from within, which contains everything - dogs, cats, protons, love, poundcake, values, Donald Trump, and oxygen. Areas where I have a lot of knowledge are more in focus than those where my knowledge is lacking. As I understand it and experience it, this model is the source of my intuition. I can make judgments about ideas based on how they fit into my model, even if I have relatively little knowledge. That doesn't mean I don't have to go back and verify things with more formal methods of justification, but it helps tell me where to get started and what is worth worrying about.