As late as the development of any resource gathering society
How inevitable is the concept of numbers?
121–130 of 211 posts
Re: How inevitable is the concept of numbers?
#122Repetition or iteration is fundamental to mathematics. Symbolic mathematics is iteratively applying axioms and schemas to elements of a language. Computation is iteratively applying rules or functions to sets. The correlation between numbers and iterations is strong; zero iterations is identity, one iteration is the unit of computing/application/doing-something. More iterations yield ordinal numbers, and there we are at numbers being inevitable.
In theory a universe could be entirely continuous with no need to "compute" in any discrete steps but quantization in our universe suggests that numbers also have a place in correspondence to quantized aspects of reality.
Re: How inevitable is the concept of numbers?
#123Earlier quoted context omitted.
Because of the limitation of language there's only a countable number of mathematical truths that can be proven or written down. So for all practical purposes there's countably many.
Could you prove for every real value between 0 and 1 that it's greater than or equal to 0? That's an uncountable amount of proofs
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Re: How inevitable is the concept of numbers?
#124I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting, what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero. Maybe related, while I can also partially understand multiplication (syntactic sugar for adding) I could never understand multiplication by zero, meaning how come when you mu…
So I'm no philosopher of math, but the intuitive way I think of it is that negative numbers are like borrowing a place holder. Think about how electrical charge and currents work at the physical level. There's electrons, which have negative charge, or holes, which have positive charge. Holes are just an empty place an electron can go, rather than an extant particle. Similarly, when we think of negative numbers in rel…
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Re: How inevitable is the concept of numbers?
#125Earlier quoted context omitted.
Because of the limitation of language there's only a countable number of mathematical truths that can be proven or written down. So for all practical purposes there's countably many.
But we can make new words.
Only an (unbounded) finite number of them.
Re: How inevitable is the concept of numbers?
#126Earlier quoted context omitted.
Because of the limitation of language there's only a countable number of mathematical truths that can be proven or written down. So for all practical purposes there's countably many.
Could you prove for every real value between 0 and 1 that it's greater than or equal to 0? That's an uncountable amount of proofs
Only a countable subset of such potential proofs exist, which more the enough to answer every question of that type that can ever be asked.
Countability is not just a limit on our ability, it's also a limit on our needs.
Re: How inevitable is the concept of numbers?
#127Earlier quoted context omitted.
> 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 thi…
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.
Re: How inevitable is the concept of numbers?
#128Earlier quoted context omitted.
It's not. It's a single proof about a set, a set that's assumed to be uncountable in standard ZF set theory. The "axiom system" that (supposedly) contain a countable number of axioms. But these too are constructs of set theory. We still create proofs one by one of theories about axiom systems with infinite axiom - so we have a countable/enumerable set of such theories. The proof systems to we can see or touch involve…
I think you have a slightly stricter definition of "a proof" than me. I would consider a proof that all the numbers in (0,1) are positive to also be a proof that the number 0.5 is positive, as well as the number 1/e, and Champernowne's constant. Since the original question was about uncountably many mathematical truths I would say we have one proof that proves uncountably many mathematical truths.
It is an abstract proof of a generator for concrete proofs of specific assignments to variables. The potential is uncountable, but only a countable subset will ever be invoked.
Re: How inevitable is the concept of numbers?
#129I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting, what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero. Maybe related, while I can also partially understand multiplication (syntactic sugar for adding) I could never understand multiplication by zero, meaning how come when you mu…
We can informally derive other types of number by extending operations on our existing set of natural numbers, seeking a "closure" for that operation, and seeing if we get consistent results. So extend the concept of differences between natural numbers, by subtracting a large number from a smaller number and defining the result as belonging to a new class of number. Similarly, get fractions by defining non-integer ra…
0^0 is near universally regarded as 1, except in specific domains. If you define your terms carefully, you usually get a reasonable answer.
0^-1 = 0^0/0 = 1/0 is indeterminate, as are all the negative powers of 0.
Re: How inevitable is the concept of numbers?
#130I think numbers are wonderful and cool abstract thingies. We grow up being taught them (in my case Arabic numerals in base 10) and they become such part of our being that we start to think that we know what “5” is. Then I was introduced to Roman Numerals. Didn’t like those much. That one taught me that 4 is more related to 5 than it is 3. Then some of us learn a base 2 number system or base 16 number system and for a…
> Did we really know what “5” was then? What is five-ness really?
Ah, but we do, and it’s very simple: it is what five cows an five fingers have in common in the most obvious sense. There’s no mystery there.
Or, stated in another way, “fiveness is something you’ll be very sad not to see when you look at your hand.”