The very notion of discreteness depends on subjective definitions of "objects". We take concepts of objects for granted because they make interacting with the world tractable, but it's really hard to define them outside of minds.

As far as we know, the universe is made up of discrete units and any other type of math is an abstraction over discrete math.

I thought it was a smooth continuous manifold

To what extent are the Planck length and Planck second confirmed smallest discrete units?

I was referring to spacetime in GR is modeled as smooth continuous manifold. In case you're serious though, planck length are not some fine-grained pixels/voxels in the cartesian 3d world, at least not confirmed; in-fact planck units are derived scales.

Where could you recommend I read more about how they are derived scales?

Wikipedia is a good start, but if it's too terse/advanced there's always simple wikipedia which seems to have an explanation as well https://simple.wikipedia.org/wiki/Planck_units

I suppose it was pointless of me to ask if what I received was a wiki link, no offense. Thanks.

you received a link to simplified wikipedia, which is not wikipedia. https://simple.wikipedia.org/wiki/Simple_English_Wikipedia

I can't tell now if you're being intentionally condescending.

I'm not a physicist, but I think those are the smallest units in the sense that they are the smallest units we could theoretically interact with/measure, not some hard limit. It's just that it's moot to consider anything smaller because there's no way for us to ever know.

Is that because we see no way to bootstrap equipment down so many magnitudes of scale, not even close - or is it something else?

Any given model has less fidelity than reality. An atlas map of the US has less detail than the actual terrain. The Planck constants represent the maximal fidelity possible with the standard model of physics. We can’t model shorter timeframes or smaller sizes, so we can’t predict what happens at scales that small. Building equipment the can measure something so small is difficult too… how do you measure something when you don’t know what to look for?

It may be that one day we come up with a more refined model. But as of today, it’s not clear how that would happen or if it’s even possible.

Imagine going from 4K to 8k to 16k resolution and then beyond. At some point a “pixel” to represent part of an image doesn’t make sense anymore, but what do you use instead? Nobody currently knows.

One addendum / clarification:

It may also be that "space" and "time" are emergent properties, much like an "apple" is "just" a description of a particular conglomeration of molecules. If we get past Planck scales it may turn that out that there are no such things as "space" and "time" and the Planck constants are irrelevant. We currently don't know but there _are_ a few theoretical frameworks that have yet to be empirically verified, like string theory.

As far as we know, the universe is a single unity, and any discrete units and any other type of math are human distinctions overlaid upon that unity.

Can you explain what you mean here? I mean yes there’s a universe so it can be see as a unit. There’s also quantum mechanics, telling us we can only distinguish discrete objects at the bottom of the scale. Can you give an example of a non-human distinction, or explain what you mean by that concept?

I assume the post is missing its "/s".

Yes. It was a parody of yunwal's post (which I possibly should have cc'ed). . Reread yunwai's post and mine together and you will understand.

Endless "unity...discrete" discussions have arisen in both science and philosophy since the beginning.

FWIW my use of the phrase "human distinction" was merely for emphasis: I make no claims about non-humans.

In any case, any distinction I have interest in or speak of would be human, whether made by a human or otherwise.

No, discrete math is exactly the same regardless of your definition of "object". It is completely independent of that. Discrete math is important to any theoretical beings that have any concept of "objects" whatsoever. It would be mostly irrelevant to entities that have no such conception, but those entities are not writing math papers.

Which is exactly why I initially suggested that the structure of primes has more to do with how theoretical beings count than with how the universe propagates state.

No, the discreetness comes from physical experiments. I do see a problem defining something outside of one’s mind or outside the universe though :)

Discreetness only comes from experiments carried out in secret laboritories.

Pronounce "lah-BORE-i-tories", obviously.

I for one never saw a "number 2" in the wild. But I'm a homebody.

If I handed you 1 apple, and then handed you another apple, you wouldn't be surprised to find that you had 2 apples. The same trick works with oranges and pears.

> If I handed you 1 apple,

At this point I hold one object that we agree to label "apple". Note that even seeing it as a single object is a layer of abstraction. In reality it's a clump of fundamental particles temporarily banding together

> and then handed you another apple,

What's "another apple"? What does it have in common with the thing I'm already holding? We label this thing to be also an apple, but it's a totally different set of atoms, from a different tree, perhaps from the other side of the planet. Perhaps the atoms formed in stellar processes light years away from that of the other apple.

Calling both of these things "apple" is a required first step to having two of them, but that is an of abstraction, a mental trick we use to simplify the world so we can represent it in our minds.

I'm not a particle physicist but I hear electrons *can* be counted without any unwitting help from our lower-level neural circuitry.

I wouldn't even go with particles. I'd call it a stream of sensations.

I’m not sure I am intended to understand what your problem is.

Personally, I suffer from whatever it is that drives a person to think about what numbers mean.

The numbers themselves aren't a problem, I'm just pointing out that our cognition involves many overlapping layers of abstraction, and we're doing mathematics and every other mental activity in one or more of those layers.

That this seems to correlate strongly to real-world phenomena speaks well of the types of abstraction that nature has equipped us with.

Very nicely said.

However, what about virtual things, e.g. apples in a computer game.

There are no virtual things! Every computer / imaginary apple is represented by real electrons on a drive / chemicals in a brain. Even if we are a simulation, we also have to ultimately be represented by something real.

There are no real things either. As far as we know, every real thing is represented by another real thing. Even electrons are made of quarks, which are probably made of other things we don't know yet.

Addition is abstract phenomena based on math which itself is abstract, so it can only function in abstract setup.

But not necessarily with rabbits. Can easily end up with dozens of 'em when you ony started out with two.

There are a dozen leaps of abstraction occuring before you arrive at "2 apples".

You are differentiating, classifying, etc.

Zero, one, infinity.

Infinity, aka 2 or more. I agree that those are truly three distinct classes of quantity/identity

Whoa

I've never seen gravity, but here I am, stuck to the ground

bears shit in the woods so they're out there if you look in the right place

I guess you saw two things in the wild though.