[thwayunion]

It's not a bijection either. If we take your level of pedantry seriously, it's nothing. There are no mathematical structures in play. I'm using natural language words to describe informal ideas outside of any mathematical system.

Let's use bloogidy-blop to avoid silly arguments over sequences of characters that have clear contextual meaning which you refuse to acknowledge for some reason :)

I'm not sure what your second point is supposed to be. Here's the relevant quotes:

>>> Humans can create notation and formalisms, but they do not invent the truths those mathematics represent.

>> The land represented by a map exists independently of humanity. Another intelligent species would have to come up with a roughly isomorphic representation if they wanted a similar tool.

> Which gets to the second point, if there is a true isomorphism between the map and the land, it doesn't matter that one isn't the other. That would mean that the land is constrained by the same axioms as the 'map', which gives some significance to them.

Right... but I didn't say that there was a bloogidy-blop between maps and land. I said that a useful alien map would have to be roughly bloogidy-blop to our maps. So I'm not really sure what you're trying to say here that is relevant to my original post.

[TheOtherHobbes]

This is clearly not true. If the aliens happened to be 10% or 1000% of our size, their concept of a relevant physical feature would be very different.

Maps are basically feature extraction -> data compression. The feature extraction part is subjective and depends on the experience of a species.

A map of cellphone towers is useless to a cat. A map of blobblytoids is useless to a human who doesn't know what a blobblytoid is or how to recognise one.

A human may have some vague awareness that something is there, but it's also possible that blobblytoids look like random noise, or like weird probabilistic anomalies that travel around inside a multidimensional space, or like something completely unimaginable.

So in the limit features can't be extracted because they are invisible to a different consciousness. They can still be physically present, but their meaning as a feature of interest depends on having a subjective referent for them.

This seems to be something many humans struggle with. We assume everyone else - including other humans - has the same set of referents, and therefore our personal feature maps are somehow universal.

Of course they aren't. They aren't even universal among humans, never mind a completely unknown alien species.

[mxkopy]

I think there is some sort of threshold analogous to Turing-completeness (it probably is just Turing-completeness) when it comes to intelligence, in that information is eventually accessible to any system that passes it.

If there is a species that thinks in terms of things that humans never could understand, then I would argue that species isn't part of our physical reality. If their expression of concepts is at all rooted in physical reality, then we, as physical beings, would have access to it. Maybe not the first person that sees it, nor the second, nor their great-great-great-grandchildren, but at some point their children could build things/begin to appreciate what was being communicated.

[JackFr]

> If there is a species that thinks in terms of things that humans never could understand, then I would argue that species isn't part of our physical reality.

What if they think can think in our terms AND they can think thoughts we are physically unable to, thoughts that are literally inconceivable?

[mxkopy]

If the thoughts have any interaction at all between each other, we would be able to leverage that to understand those higher-level thoughts to the degree that they affect the ones on our level. As an analogy, we can project an N dimensional object onto N! planes and get a complete, but not intuitive, description of what it is.

Maybe they know the exact value of Chaitin's constant, but at least we know its properties.

[thwayunion]

Yes, the exact sense in which maps are bloogidy-blop to one another may vary considerably, but if they cover the same geographic region there'll be some bloogidy-blop, ie common reference points, which would relate to the maps. Maybe blobblytoids always go in valleys. Or whatever.

We seem in agreement on the object question in any case.

[VirusNewbie]

Well, we can say 'an equivalence' (or adjoint) which would be accurate and still using natural language, no?

Onto my second point, you don't need an isomorphism, an equivalence works fine if you find that the axioms hold in both cases, no? So now you're not just working with a 'formalism' that has no basis in reality.

Otherwise, if reality wasn't constrained by axioms (or even meta axioms) we'd use it to do things we couldn't with our formalisms.

[thwayunion]

> Well, we can say 'an equivalence' (or adjoint) which would be accurate and still using natural language, no?

No, their maps could be VERY different but be used in roughly the same way and therefore useful in the same way. There is an operator involved -- reading/navigating/interpreting -- which is why I chose "isomorphism" instead of "equivalent".

Also, "isomorphism" IS natural language that is used in many fields outside of mathematics and also has a vernacular sense to it. The word happens to also be used to describe certain formal constructions by mathematicians from time to time, but it is natural language.

Again, this is all very pedantic and silly. I insist on bloogity-blop. If we're going to have silly arguments, let's use silly words :)

> Onto my second point, you don't need an isomorphism, an equivalence works fine if you find that the axioms hold in both cases, no? So now you're not just working with a 'formalism' that has no basis in reality.

Reality isn't constrained by maps.

Useful maps are constrained by reality.

> Otherwise, if reality wasn't constrained by axioms (or even meta axioms) we'd use it to do things we couldn't with our formalisms.

Huh? I can't use reality the way I use maps because I'm not always able to fly into the air and look around before navigating to the supermarket or a trail head.

Reality is not constrained by my inReach. I promise you it's the other way around. And I promise you I can't fly, which means I need maps, even if flying high into the air would make it way easier to find a trail head than following a not-great trail map.

Maps are useful. Very useful. But they DO NOT constrain reality.

A belief to the contrary in the case of mathematics and physics is quite spiritual. Which was kind of my original point :)