An apple is not an apple. An apple is a subjective construct that summarises the distinguishing features of a certain kind of object as it appears to our sense.
To a non-human consciousness those features may be uninteresting, irrelevant, or incomprehensible, so they might not see apples at all. But they could see " "s, which we don't even have a concept for, never mind a word. And which we either ignore or possibly don't see at all. (Imagine perceiving complex networked relationships directly instead of having to access them through symbolic models.)
There's no reason why math wouldn't be the same. From experiments we know that cats can't count, but they can distinguish sizes. So cat math likely wouldn't have integers as we know them, but would have some kind of size-based analogue.
I have a theory this is why Hilbert's Project failed and you always end up with an incompleteness theorem.
You cannot create an absolute internally consistent mathematics, because foundational axioms depend on subjective experience, not on objective logic.
So you can define integers in various more and more obscure ways. But fundamentally you have to start with the subjective experience of "integer" as a concept that matters to you. And you can't prove a subjective experience objectively.
> An apple is not an apple. An apple is a subjective construct that summarises the distinguishing features of a certain kind of object as it appears to our sense.
This is utter nonsense. An apple IS an apple. If I put one on a table, then obliterate every human being that’s capable of sensing it, the apple is utterly unaffected.
If aliens come down and experience the apple differently than we would have, that doesn’t change the apple one bit.
The world is a sea of particles and energy which behave according to certain patterns (both fundamental laws and emergent behavior). Some of these patterns are pertinent to us, so we name them, giving rise to a category. "Apple" is such a category.
The clump of molecules on the table we denote with the term "apple" doesn't care that our brains have deemed it similar enough to certain other clumps of molecules to be placed in the same category. If all humans cease to exist, the clump of molecules may still be on the table, but there's no one left to consider it part of any category.
If aliens then visit who can't eat the apple and aren't interested in botany, they may simply choose not to distinguish between apples and pears, or apples and any other fruit, or even apples and any other form of organic material. The same clump of molecules is there, but the categories it belongs to have changed.
> The clump of molecules on the table we denote with the term "apple" doesn't care that our brains have deemed it similar enough to certain other clumps of molecules to be placed in the same category. If all humans cease to exist, the clump of molecules may still be on the table, but there's no one left to consider it part of any category.
Correct. If all humans cease to exist, the human-made abstraction of "the apple" passes away. But an apple--the one on the table--is completely unaffected.
The difference here is that between an abstraction and a concrete instantiation of said abstraction. The abstraction of "the apple" is indeed a subjective construct which would pass away with humanity. But OP said "an apple". "An apple" is a part of the physical universe. It is, like you said, a clump of molecules. Molecules that can be measured objectively.
An apple is no more subjective than the orbital period of the earth or the length of the standard meter.
> So cat math likely wouldn't have integers as we know them, but would have some kind of size-based analogue.
Sorry, but no. Any species capable of actually creating some kind of math will have some mathematical structure isomorphic to the integers. If cats can't count, then that just says that cats are not capable of creating some kind of math.
Firstly, the reals contain the integers, so there is an isomorphism as I said.
Secondly, discretization absolutely is universal. It's literally in the laws of physics for one (particles are discrete, energy levels are discrete, etc.). For another, are you suggesting a physical alien species will have a continuous number of appendages, or organs, or that their population will somehow be continuous? I frankly don't see how you can possibly escape aliens capable of math developing a notion of basic counting.
Eh, you're not explaining universal truths here, you're just anthropomorphizing. Why must it have appendages, organs, or populations? Why presuppose that its conceptual model includes particles at all? What if a vast, hyper-continuous intelligence simply cannot comprehend the concept of being discrete?
Firstly, those were just examples of commonly countable structures, even if they're not universal (which is debatable). Discretely countable structures are literally everywhere and fundamentally inescapable, which is why I mentioned physics. I didn't presuppose physics, the discrete structure of physical reality is directly observable, it's not some fiction we made up.
Secondly, what we know must be bound by what we've observed. You can imagine any sort of being you like, but that doesn't make your imagined creature logically coherent or physically realizable.
Any physically realizable intelligence must:
a) Be differentiable from its environment: that means it must have some enclosing boundary separating an inside that's different than an outside.
b) Have internal structure: intelligence by necessity is structured thought. Structured thought entails differentiable physical structure to hold structured thoughts. Such structure by itself is necessarily countable, being made of matter.
> An apple is not an apple. An apple is a subjective construct that summarises the distinguishing features of a certain kind of object as it appears to our sense.
It was clearly not about the apple, but about distinct entities with similar features. Now for another being these might not be similar, but something else probably is, disregard of different dimensions, different senses or the likes. As the observed reality for an alien species or actually any other species on this earth is different of course.
> You cannot create an absolute internally consistent mathematics, because foundational axioms depend on subjective experience, not on objective logic.
> And you can't prove a subjective experience objectively.
I think the misconception comes from the fact that you need basic assumptions to build an abstraction. One of the most basic assumptions is that something like a shared reality exists and we're not for example in a virtual world or a dream.
You can happily deny this shared reality, however I would not necessarily encourage you to touch fire (literally and figuratively speaking).
To a non-human consciousness those features may be uninteresting, irrelevant, or incomprehensible, so they might not see apples at all. But they could see " "s, which we don't even have a concept for, never mind a word. And which we either ignore or possibly don't see at all. (Imagine perceiving complex networked relationships directly instead of having to access them through symbolic models.)
There's no reason why math wouldn't be the same. From experiments we know that cats can't count, but they can distinguish sizes. So cat math likely wouldn't have integers as we know them, but would have some kind of size-based analogue.
I have a theory this is why Hilbert's Project failed and you always end up with an incompleteness theorem.
You cannot create an absolute internally consistent mathematics, because foundational axioms depend on subjective experience, not on objective logic.
So you can define integers in various more and more obscure ways. But fundamentally you have to start with the subjective experience of "integer" as a concept that matters to you. And you can't prove a subjective experience objectively.