I like that he keeps it open-minded, but I've thought a lot about this and to me the case for numbers being inevitable is decisively yes. Every adaptable system evolves to adapt to a changing environment by detecting "modes" (categories) and adapting to each mode. Then we start noticing categories come in instances. There's a tree, there are more trees. So now it's useful to count them... Then it's useful to have fractions. And measurements (like weight and length). Basic operations like + - emerge, and with that multiplication. And with that division... and so on.
I think the fundamental properties of math are one of those things that will keep emerging for every thinking system. Not because numbers are fundamental to the universe. Rather they're fundamental aspect of intelligently adapting to (i.e. thinking about and making predictions about) the universe.
> There's a tree, there are more trees. So now it's useful to count them...
It's not that useful to count them. My personal interactions with trees have likely never involved counting them; not even when I was involved in woodland management.
Trees control their numbers according to whether or not they can reasonably live in a particular density of tree coverage.
Only humans want to count trees, perhaps in general to claim (abstract) ownership of a particular large number.
Lets say, based on your nick, you are teaching two classes. Both consist of individuals.
If we were to count the classes, we would say that one class consist of 2 people, and the other of 20000.
But those are just numbers.
My point is that numbers are useful for us in understanding how to interact with the world. If you don't plan to interact with the trees, you probably don't need to count them. But if you are going to use them for firewood or building materials, the knowing the difference between 2 trees and 20000 is useful.
We agree that the difference between 2 and 20000 is significant, but I can easily observe this difference without counting all the way up to 20000.
Identifying the difference between 19,999 and 20,000 is certainly not significant in this way. 19,999 is already "too many" students; or, "plenty" of firewood. I don't need to count them to know this.
I'd say the difference between 5 and 6 is already relatively unimportant in these contexts.
My point is not that you need the number to be exactly accurate, my claim is that for the number 2 you don't even condider the student body as a quantity, but for 20000 it very much becomes a quantity.
Also, if 20000 students need training, as a teacher you can still provide training, you just need to do so in other ways. Either, you can recruit more teachers to help you (and then it matters if the number is 10000 students or 20000 students), or you can switch to another medium (a book, a video, on-line training).
Treating a student body as a number will make it more managable for you to provide/organize training to them than just hireing more teachers at random until each student gets enough attention.
Likewise, 2 trees may keep you warm for a week, 20000 may keep you and half your (small) village warm for a year, with enough trees left over for the forest to replenish.
> Every adaptable system evolves to adapt to a changing environment by detecting "modes" (categories) and adapting to each mode. Then we start noticing categories come in instances. There's a tree, there are more trees. So now it's useful to count them...
I really like this part of the explanation, although not all adaptive systems are intelligent. E.g if I add a grain of salt on a heap of salt and it suddenly collapses, that is an adaptive system but not as a result of intelligent modeling of how it should behave.
> Not because numbers are fundamental to the universe. Rather they're fundamental aspect of intelligently adapting to (i.e. thinking about and making predictions about) the universe.
Here the problem is assuming intelligent adaptivity and the agent that has it is something separate from the universe. Being able to have a model of the universe inherently requires internally resembling the structural functional organization of the universe, a mutual conformity if you will, and therefore if numbers, counting, modes, categories etc are fundamentally useful constructs, I think they also at least resemble a fundamental part of the universe.
Well, in a very primitive and non-accurate sense, it counts/measures the number of (photo synthesis generating) photons hitting each leaves, and then grows in directions that maximizes its photo synthesis.
Since a tree has a much lower number of branches and leaves than there are photons, there has to be some aggregation of information involved in this computational process.
Another perspective could be that we are creatures emergent out of the physical and chemical processes of the universe.
There is some similarity or equivalence to some of these processes with mathematics, which allows us to model a subset of physics/chemistry with mathematics.
Because physics/chemistry can create us using rules that follow math (or even some undiscovered rules), this same physics/chemistry gives us the faculties of reason that lets us think about and apply math in some universal way
The physical, chemical, and mathematical substrate to which we are born allow us to even discover that the rules of math can be applied to some subset of the laws (or chaos?) of the universe.
> Do we really know if adaptation is responding to discrete classifications of environmental pressure?
Yeah we do know.
Take any system of variables that interact, and change them smoothly, and inflection points will emerge. This is true literally any way you look at it. In space. In time.
Heck aggregate states of matter represent "categorical adaptation" of matter to smooth alteration of temperature.
Point being this process happens even before we even can call a system "thinking". It's absolutely inevitable.
We can keep our options open forever and say "I don't know", that's fine. But if we analyze what we do know... turns out we DO do know.
You are just saying that interacting systems have phase transitions.
I think this is a far cry from "emergent numbers".
Edit:
Maybe self organizing matter that lives on a phase transition boundary benefits from being aware of the boundary? Thats the only analogy I could think of that could logically connect.
> You are just saying that interacting systems have phase transitions.
Right.
> I think this is a far cry from "emergent numbers".
Because it was just a clarification remark on everything else I said before that, which did connect it to numbers.
> Maybe self organizing matter that lives on a phase transition boundary benefits from being aware of the boundary? Thats the only analogy I could think of that could logically connect.
That's also what I initially said, but think about it a little more broadly. Whether you "live on a boundary", or you live in a system that experiences such boundaries, or your own system internally has such boundaries, or the INTERACTION between you and your environment creates these boundaries, it doesn't matter. You benefit from being aware of them.
And here's the thing. You will experience such boundaries, because it takes infinitely more "resistance" to change, in order to survive a changing environment without changing yourself, than it takes energy to adapt to a changing environment so you resist less, and it resists you less (you become more compatible).
When leaves drop in winter it doesn't matter there's a specific day and hour and second we switch from summer to winter, but trees do benefit from recognizing the overall "shift" over time and adapting to it through its own shift. I'm deliberately using "non-thinking" adaptations to show that categories are precursors to how thinking works, rather than thinking inventing the idea of categories for no fundamental reason.
Our recognition of objects and entities are the same phenomenon. We recognize a boundary (inside and outside the entity/object) where there's a shift of overall behavior in that local timespace, compared to its surroundings. We benefit tremendously from recognizing that a field of grass looks and acts more like a hungry lion in a specific region of it. Technically objects/entities are not a perfectly defined thing. A lion is in consistant exchange and interaction with its environment, it's not a closed system. And anyway I don't feel like repeating the rest of this again.
TLDR; There be phases/categories in N dimensions/parameters. There be instances of them (repetition of patterns). There be adaptation by recognizing the phases/categories and their repetition and counting and measuring them, in order to optimize our predictions quality.
I think the fundamental properties of math are one of those things that will keep emerging for every thinking system. Not because numbers are fundamental to the universe. Rather they're fundamental aspect of intelligently adapting to (i.e. thinking about and making predictions about) the universe.