> but my understanding of copyright is that if a person when into a clean room and wrote an new article from scratch, without having read any NYT, that just so happened to be exactly the same as an existing NYT article, it would still be a copyright violation.
It would not be. Independent creation is a complete defense against copyright infringement.
Life expectancy at birth is irrelevant, people who die in infancy don't consume many resources at all. What matters is the percentage of the population that's of working age, and here Germany is fairly typical for a developed country.
It matters in combination with the tax rate on the working population. There are huge transfers of wealth ongoing from the working class to the retired class of people, who are the richest generation in history. People need to wake up and protect their class interests an stop arguing for this German Ponzi scheme.
Here's a less galactic version. Suppose you're implementing a binary tree where every leaf has to have the same height - a toy model of a self-balancing search tree.
Here's an implementation using GADTs and type-level addition
data Node (level :: Nat) (a :: Type) where
Leaf :: a -> Node Zero a
Interior :: a -> (Node l a, Node l a) -> Node (l + 1) a
It's impossible to construct an unbalanced node, since `Interior` only takes two nodes of the same level, and every `Leaf` is of level 0.
> Yet Turing machines are about as far from abstract mathematics as one can get, because you can actually build these things in our physical universe and observe their behavior over time (except for the whole "infinite tape" part)
The infinite tape part isn't some minor detail, it's the source of all the difficulty. A "finite-tape Turing machine" is just a DFA.
Oh is that all? If resource bounded Kolmogorov complexity is that simple, we should have solved P vs NP by now!
I debated adding a bunch of disclaimers to that parenthetical about when the infinite tape starts to matter, but thought, nah, surely that won’t be the contention of the larger discussion point here haha
No, an LBA in general doesn't have a finite tape. It still has an infinite tape, to accommodate arbitrary length inputs, it's just that the tape cannot grow beyond the length of its input (or a constant multiple of it, which is equivalent by the linear speedup trick).
No they really are. Arbitrarily long means "pick a number any number." Infinite means, well infinite. And infinity is always bigger than "pick a number, any number."
I suggest that you read Michael Sipser's Introduction to the Theory of Computation. It's absolutely lovely and an easy read, considering the subject matter. It will help you to understand cardinality (and many other things) in a pragmatic computing science context.
It may be surprising to you that I've actually read a good portion of that text as well as other books on the subject. Or that I actually do understand what a cardinality is in a "pragmatic computing science context" (whatever that's supposed to mean).
> Infinite means, well infinite
There's a technical definition of "infinite cardinality" which is a bit more rigorous than "well infinite". If |S| > n for every natural number n, then S has infinite cardinality basically by definition.
Why don't you try implementing an LBA simulator in your favourite programming language with a finite-sized array and tell me how it goes? Anyway, the original point was that a Turing Machine with a finite tape is an FSM. This is true (well, or more technically: it's equivalent to an FSM) because you can encode all possible configurations of the tape with a finite set of states and thus don't need any extra memory.
"Geometric algebras" are Clifford algebras over the reals. Differential geometry is probably the field where you're most likely to see mathematicians discuss them, though they also come up in certain (closely related) areas of mathematical physics via spinors.
Imaginary numbers aren't a field, so there's no such thing. Clifford algebras over the complex numbers work fine, but it's usually not what the people talking about "geometric algebra" are doing.
How is complex / imaginary numbers not what they are doing? Numbers that square to -1, 0, and 1 are the bread and butter of the GA I know. Exploring different combinations of types of imaginary numbers and their products and space describing algebras. (including naturally quaternions, duel quaternions, i-rotation, nilpotent, ect)
Typically GA people are working with real algebras, meaning the coefficients are real, and things like a square root of -1 appear as some object in the algebra (like a 2-blade). But you could also have a Clifford algebra with coefficients in e.g. the complex numbers or fields of finite characteristic.
In fact using different coefficient rings is one way to write a compact recursive definition of real Clifford algebras:
As this shows, ei and ej are imaginary numbers. This confusion is the biggest issue in GA to me. Typically Linear Algebra users don't use imaginary numbers either. It's all connected when doing geometry though and its pointless to draw invisible lines between these concepts.
We have no way to talk about more general types of complex / imaginary numbers besides rigid math lingo that provides no geometric intuition or grace for geometric imagination.
You seem to be calling complex numbers imaginary numbers, but they're not the same thing. Imaginary numbers are a subset of the complex numbers consisting of the imaginary axis without 0, e.g. i, 2i, -3.1i. Complex numbers also include the real numbers and all combinations of real and imaginary.
I'm glad you seem to know what I mean. I love imaginary numbers like the Square Root of -1 or the Square Root of 0. They can only be used like:
nil²=0
i²=-1
j²=-1
And combining them into complex forms like:
(i + j)
(nil + i)
What can I call this general idea of using imagination to determine new number rules and combining them together?
If I call this GA or Clifford Algebra in a math community will it trigger rigor admins to ban me from talking because I'm not using their terms?
I wish we had artistic imaginary math communities for exploring Geometry without rigor turing everything into Semantics. Geometry literally doesn't need semantics if you agree on points, lines, planes ect. Algebra to me should just be simple maps from clifford numbers to examples of intuitive geometry / physics.
People should stop calling that set "imaginary numbers". It's net bad. That set isn't really worth it to be called anything. Maybe "pure imaginary numbers". Also "imaginary numbers" is used for the complex numbers already.
People also like colloquially saying "The Square Root of -1". Technically wrong but personally meaningful.
GA is mainstream once normal people start joking about the square root of zero. We need to imagine and teach even more imaginary numbers one day.
Not a physicist, but this seems to depend on the physical setting? Seems there are usually metric with physical meaning in continuum mechanics, e.g., elasticity or GR, but not so much if one is working in say geometric mechanics — one can define a Hamiltonian flow on a symplectic manifold without a metric.
To an extent. A truly completely formal proof, as in symbol manipulation according to the rules of some formal system, no. It's valid or it isn't.
But no one actually works like this. There are varying degrees of "semiformality" and what is and isn't acceptable is ultimately a convention, and varies between subfields - but even the laxest mathematicians are still about as careful as the most rigorous physicists.
