You’re reducing things to a single issue and have the gall to say my political world is narrow. You’re unreal.

Something has already happened and they didn’t touch my rates. I’ve been saving hundreds of dollars a year. I’ve saved well into the thousands of dollars at this point. I’m not saying the insurance companies are my friends and while I am better off using the tracker than not using it, that wasn’t even my point. My point was that the trackers all function differently and some are better than others.

It’s crazy how most of those programs work. The way my insurance handles it is way better. For example, no matter how bad you are at driving, they never raise the premiums above the normal rate, so it almost always makes sense to get the tracker from a finance perspective. (The only exception is that they will raise your rates if you drive farther in 6 months than you estimated on your initial application. The flip side is that they lower your rates if you don’t drive very much. I only drive about 1000 miles every 6 months, so my premium is really low.) They also have a Bluetooth device that stays in your car that your phone must be connected to in order for it to record trip data, and if you happen to be riding as the passenger in the car, the app has an option that allows you to clarify for each trip that you weren’t the driver. I was surprised to learn they aren’t all like that.

It’s funny you say the philosophy is simple when strategic voting requires multiple layers of analysis and voting for bubblegum ice cream just amounts to what feels good. You can’t bring yourself to accept the reality of the situation, so you pretend like the problem is easy to solve if you just ignore it. That’s truly simple minded. Pathetic projection on your part.

Doesn’t matter where the track leads if the trolley can’t get to it. It could lead to rainbows and sunshine, but that isn’t where the trolley is headed because there is no possibility that someone other than Trump or Biden is elected president. A few cry babies voting third party won’t get some third person elected. A vote for the third track is a vote for a track that will not be ridden.

Language parsing is a routine process that doesn’t require AI and it’s something we have been doing for decades. That phrase in no way plays into the hype of AI. Also, the weights may be random initially (though not uniformly random), but the way they are connected and relate to each other is not random. And after training, the weights are no longer random at all, so I don’t see the point in bringing that up. Finally, machine learning models are not brute-force calculators. If they were, they would take billions of years to respond to even the simplest prompt because they would have to evaluate every possible response (even the nonsensical ones) before returning the best answer. They’re better described as a greedy algorithm than a brute force algorithm.

I’m not going to get into an argument about whether these AIs understand anything, largely because I don’t have a strong opinion on the matter, but also because that would require a definition of understanding which is an unsolved problem in philosophy. You can wax poetic about how humans are the only ones with true understanding and that LLMs are encoded in binary (which is somehow related to the point you’re making in some unspecified way); however, your comment reveals how little you know about LLMs, machine learning, computer science, and the relevant philosophy in general. Your understanding of these AIs is just as shallow as those who claim that LLMs are intelligent agents of free will complete with conscious experience - you just happen to land closer to the mark.

Imaginary numbers are no more imaginary than real numbers. The name trips a lot of people up. If you want to call imaginary numbers “dark unicorns” then you really should say the same thing of the numbers 1, 2, and all other numbers as well.

You’re thinking of topological closure. We’re talking about algebraic closure; however, complex numbers are often described as the algebraic closure of the reals, not the irrationals. Also, the imaginary numbers (complex numbers with a real part of zero) are in no meaningful way isomorphic to the real numbers. Perhaps you could say their addition groups are isomorphic or that they are isomorphic as topological spaces, but that’s about it. There isn’t an isomorphism that preserves the whole structure of the reals - the imaginary numbers aren’t even closed under multiplication, for example.

You’re mistaken unfortunately. The books don’t start that way. They start by describing Arthur Dent’s house.

Of course! I’m always excited for an opportunity to discuss these sorts of things, so I should be thanking you instead.

I’ll preface this with the fact that I am also not a physicist. I’m also simplifying a few concepts in modern physics, but the general themes should be mostly accurate.

String theory isn’t best described as a genre of physics - it really is a standalone concept. Dark matter and black holes are subjects of cosmology, while string theory is an attempt to unify quantum physics with general relativity. Could string theory be used to study black holes and dark matter? Sure, but it isn’t like physicists are studying black holes and dark matter using methods completely independent from one another and lumping both practices under the label string theory as a simple matter of categorization.

You are correct to say that string theory is an attempt at a theory of everything, but what is a theory of everything? It’s more than a collection of ideas that explain a large swath of physical phenomena wrapped into a single package tied with a nice bow. Indeed, when people propose a theory of everything, they are constructing a single mathematical model for our physical reality. It can be difficult to understand exactly what that means, so allow me to clarify.

Modern theoretical physics is not described in the same manner as classical Newtonian physics. Back then, physical phenomena were essentially described by a collection of distinct models whose effects would be combined to come to a complete prediction. For example, you’d have an equation for gravity, an equation for air resistance, an equation for electrostatic forces, and so on, each of which makes contributions at each point in time to the motion of an object. This is how it still occurs today in applied physics and engineering, but modern theoretical physics - e.g., quantum mechanics, general relativity, and string theory - is handled differently. These theories tend to have a single single equation that is meant to describe the motion of all things, which often gets labeled the principle of stationary action.

The problem that string theory attempts to solve is that the principle of stationary action that arises in the quantum mechanics and the principle of stationary action that arises in general relativity are incompatible. Both theories are meant to describe the motion of everything, but they contradict each other - quantum mechanics works to describe the motion of subatomic particles under the influence of strong, weak, and electromagnetic forces while general relativity works to describe the motion of celestial objects under the influence of gravity. String theory is a way of modeling physics that attempts to do away with this contradiction - that is, string theory is a proposal for a principle of stationary action (which is a single equation) that is meant to unify quantum mechanics and general relativity thus accurately describing the motion of objects of all sizes under the influence of all known forces. It’s in this sense that string theory is a standalone concept.

There is one caveat however. There are actually multiple versions of string theory that rely on different numbers of dimensions and slightly different formulations of the physics. You could say that this implies that string theory is a genre of physics after all, but it’s a much more narrow genre than you seemed to be suggesting in your comment. In fact, Edward Witten showed that all of these different string theories are actually separate ways of looking at a single underlying theory known as M-theory. It could possibly be said that M-theory unifies all string theories into one thus restoring my claim that string theory really is a standalone concept.

In discussions about intelligence we’re always talking about the ability to acquire knowledge, not knowledge itself.

I’m not talking about either of these things. I have already stated that I’m not referring to knowledge. Additionally, I do not agree that intelligence is merely the ability to acquire knowledge. Intelligence is famously difficult to define - but I’m working with a definition akin to a capacity for problem solving and pattern recognition. If we can’t see eye to eye there, then we’re clearly talking past each other.

Thanks for the interesting conversation. I wish you well.

You’re taking my analogy too far. Learning isn’t your ability to exercise intelligence. It’s simply the acquisition of knowledge or skills usually through study or training. You’re going to have to provide an argument or a source to back up the claim that intelligence is innate and that it can’t be changed by adjusting our behavior. You’re going to have to show that intelligence is nearly 100% determined by genetics. Those are the types of claims that eugenicists make regarding intelligence by the way, and I’m pretty sure that would make you uncomfortable given your other comment on IQ tests.

I don’t see how this could be true. It would be analogous to observing a species of bone-thin weaklings that becomes interested in body building over the course of a few hundred years, gaining more muscle mass on average with each passing year, and making the claim that the strength of this species has not changed. Maybe if one of the early weaklings decided to take up their own interest in body building, they may have reached a similar strength to that of their descendants (though even that is debatable since that specific individual wouldn’t have access to all the training techniques and diets developed over the course of its species’ future); however, it seems like an awkward interpretation to say therefore the strength of the species has not changed.

This is similar to the situation we find ourselves regarding intelligence in the human species. Humans gain intelligence by exercising their brains and engaging in mental activity, and humans today are far more occupied by these activities than our ancestors were. This, in my view, makes it accurate to claim that human intelligence has changed significantly since the advent of religion. Individual capacity for intelligence may not have changed much, but the intelligence of humans as a whole has changed.

Note that my argument does not conclude that human knowledge or understanding has changed over time. These attributes certainly have changed - I’m sure not many would doubt that. It also doesn’t conclude that every modern human is more intelligent than every ancient human. Instead, it concludes that human intelligence as a whole has changed as a result of changes in our culture that influence us to spend more time training our intelligence than our ancestors.

You have the spirit of things right, but the details are far more interesting than you might expect.

For example, there are numbers past infinity. The best way (imo) to interpret the symbol ∞ is as the gap in the surreal numbers that separates all infinite surreal numbers from all finite surreal numbers. If we use this definition of ∞, then there are numbers greater than ∞. For example, every infinite surreal number is greater than ∞ by the definition of ∞. Furthermore, ω > ∞, where ω is the first infinite ordinal number. This ordering is derived from the embedding of the ordinal numbers within the surreal numbers.

Additionally, as a classical ordinal number, ω doesn’t behave the way you’d expect it to. For example, we have that 1+ω=ω, but ω+1>ω. This of course implies that 1+ω≠ω+1, which isn’t how finite numbers behave, but it isn’t a contradiction - it’s an observation that addition of classical ordinals isn’t always commutative. It can be made commutative by redefining the sum of two ordinals, a and b, to be the max of a+b and b+a. This definition is required to produce the embedding of the ordinals in the surreal numbers mentioned above (there is a similar adjustment to the definition of ordinal multiplication that is also required).

Note that infinite cardinal numbers do behave the way you expect. The smallest infinite cardinal number, ℵ₀, has the property that ℵ₀+1=ℵ₀=1+ℵ₀. For completeness sake, returning to the realm of surreal numbers, addition behaves differently than both the cardinal numbers and the ordinal numbers. As a surreal number, we have ω+1=1+ω>ω, which is the familiar way that finite numbers behave.

What’s interesting about the convention of using ∞ to represent the gap between finite and infinite surreal numbers is that it renders expressions like ∞+1, 2∞, and ∞² completely meaningless as ∞ isn’t itself a surreal number - it’s a gap. I think this is a good convention since we have seen that the meaning of an addition involving infinite numbers depends on what type of infinity is under consideration. It also lends truth to the statement, “∞ is not a number - it is a concept,” while simultaneously allowing us to make true expressions involving ∞ such as ω>∞. Lastly, it also meshes well with the standard notation of taking limits at infinity.

My superiority complex is stronger than your inferiority complex.

I don’t recall any socialized courier or food delivery services.

This is just a continuous extension of the discrete case, which is usually proven in an advanced calculus course. It says that given any finite sequence of non-negative real numbers x,

lim_n(Sum_i(x_i^n ))^(1/n)=max_i(x_i).

The proof in this case is simple. Indeed, we know that the limit is always greater than or equal to the max since each term in the sequence is greater or equal to the max. Thus, we only need an upper bound for each term in the sequence that converges to the max as well, and the proof will be completed via the squeeze theorem (sandwich theorem).

Set M=max_i(x_i) and k=dim(x). Since we know that each x_i is less than M, we have that the term in the limit is always less than (kM^n )^(1/n). The limit of this upper bound is easy to compute since if it exists (which it does by bounded monotonicity), then the limit must be equal to the limit of k^(1/n)M. This new limit is clearly M, since the limit of k^(1/n) is equal to 1. Since we have found an upper bound that converges to max_i(x_i), we have completed the proof.

Can you extend this proof to the continuous case?

For fun, prove the related theorem:

lim_n(Sum_i(x_i^(-n) ))^(-1/n)=min_i(x_i).

OCB Virgin Slims are my go-to. They’re not quite as thin as Raw Blacks, but they have less of a taste in my experience. They also don’t tear as easily and stick more reliably than Raws. They’re not nearly as common as Raw Blacks, but they’re a far superior product in my opinion. It’s sad they aren’t more popular.

2 may be the only even prime - that is it’s the only prime divisible by 2 - but 3 is the only prime divisible by 3 and 5 is the only prime divisible by 5, so I fail to see how this is unique.