I heard it was supposed to be human body temperature, but they used horse body temperature instead because it was close to human body temperature but more… stable.

A vector space is a collection of vectors in which you can scale vectors and add vectors together such that the scaling and addition operations satisfy some nice relationships. The 2D and 3D vectors that we are used to are common examples. A less common example is polynomials. It’s hard to think of a polynomial as having a direction and a magnitude, but it’s easy to think of polynomials as elements of the vector space of polynomials.

Vote splitting is not a myth. It’s just math. Let me explain with an example:

1000 people at a conference are deciding where to order catering and hold a vote:

• 490 people want Mexican and do not want Asian
• 510 people want Asian:
  • 480 people want Vietnamese, would be satisfied with Thai, and do not want Mexican
  • 30 people want Thai, would be satisfied with Vietnamese, and do not want Mexican

The restaurants on the ballot are:

1. A Mexican restaurant,
2. A Vietnamese restaurant, and
3. A Thai restaurant.

If the people who want Asian recognize the strength of their combined numbers, then they can tip the scales by all voting for the favorite between Vietnamese and Thai. In this situation, we get 490 votes Mexican, 510 votes Vietnamese, and 0 votes Thai. This time Vietnamese wins and the majority of people, the 510 who prefer Asian, are either happy or satisfied with the result while only 490 are disappointed.

If everyone votes for their favorite, then we get 490 votes Mexican, 480 votes Vietnamese, and 30 votes Thai. In this case, Mexican wins and the majority of people, the 510 who prefer Asian, are left disappointed while only 490 people are happy with the result. The vote has been split and the result is that the entire conference is worse off for it.

By the way, the ratio of 480 Vietnamese to 30 Thai is irrelevant as long as neither value is 0. That ratio can be fixed to any positive value and a situation can be described in which vote splitting occurs with that specific ratio of Vietnamese supporters to Thai supporters. That’s why vote splitting isn’t too uncommon - any number of people voting Thai has the potential to split the vote. The one caveat is if literally every Vietnamese supporter decides to vote Thai as well; in that scenario, no vote splitting can occur. Unfortunately, that doesn’t happen in practice because it’s easier to convert the Thai supporters who are smaller in number than it is to convert the Vietnamese supporters who have greater numbers.

If you want examples from history, there are plenty. Our electoral college amplifies the effect since it breaks one federal election down into a large number of state elections, any of which can exhibit vote splitting. Other people have linked to them in this discussion and you can find more elsewhere online.

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

Nobody is arguing that a grocery stocker requires less skill and training than brain surgery. Literally nobody. And yet you people repeat this idea over and over.

We know you aren’t arguing that every job requires the exact same degree of skill. All that we want to do is say that there are jobs whose required skills are quick to acquire and are therefore easily replaceable. Meanwhile, there are other jobs whose skills take a long time to acquire and are not easily replaceable. We use the term “unskilled labor” to refer to the former group and “skilled labor” to refer to the latter group as a point of convention. When people claim that unskilled labor doesn’t exist, they imply that every single job requires skills that are slow to obtain and therefore every worker is difficult to replace, which is clearly false.

I mean this not as an attack on you but a chance to expand your worldview. Cognitive dissonance hurts, and it’s important to recognize when it’s happening so you can ask further questions.

Where is the cognitive dissonance? Where is the contradiction in distinguishing between jobs that require trained applicants and jobs that don’t require trained applicants?

There is no such thing as an “unskilled worker” because all jobs require skill. It’s called human skill, and it’s what enables us to build societies greater than the sum of its citizens.

If you decide to use “skilled worker” to mean a worker who has a skill, then you are correct that “unskilled workers” do not exist. Unfortunately, that’s not what the phrase “skilled worker” means. If that’s how you use the term, then you’re talking about something different to everyone else.

The logical conclusion you are suggesting is that because some humans are less capable, they don’t deserve basic needs such as a home, reliable transportation, internet, food, utilities, etc.

The logical conclusion of “unskilled labor exists” is simply that unskilled labor exists. You cannot jump from the observation that “unskilled labor exists” to the claim that “some people don’t deserve their basic needs.” It’s a non sequitur, and it’s not a position anyone in this thread would support.

And if your basic premise starts with the notion that society should not be meeting the basic needs of its people, then there’s only one thing that would convince you anyway.

This is a straw man. No one here has expressed the position that society shouldn’t meet the basic needs of its people. The position you are arguing against is the position that some jobs require training before hiring and others don’t. Again, that’s just what people mean when they refer to skilled labor and unskilled labor.

Yeah, you’re close. You seem to be suggesting that any measurement causes the interference pattern to disappear implying that we can’t actually observe the interference pattern. I’m not sure if that’s what you truly meant, but that isn’t the case. Disclaimer: I’m not an expert - I could be mistaken.

The particle is actually being measured in both experiments, but it’s measured twice in the second experiment. That’s because both experiments measure the particle’s position at the screen while the second one also measures if the particle passes through one of the slits. It’s the measurement at the slit that disrupts the interference pattern; however, both patterns are physically observable. Placing a detector at the slit destroys the interference pattern, and removing the detector from the slit reintroduces the interference pattern.

Binary supremacy!!!

This problem doesn’t involve cardinal numbers.

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 know the reason. I think not having the extra blank lines would be better, but it works just fine as is - even the post admits this much. That’s why it’s an enhancement. It’s possible for software to be functional and consistent and still have room for improvement - that doesn’t mean there is a bug.

My point is that someone made the decision for it to do that and that the software works just fine as is. It’s not a bug, it’s just a weird quirk. The fact that they made the enhancement you requested doesn’t make the old behavior buggy. Your post title said “it’s not a bug, it’s a feature!”, but the behavior you reported is not accurately classified as a bug.

It’s not a bug just because the software doesn’t conform to your personal preferences. You’re asking for what would be considered an enhancement - not a bug fix.

It depends. If the variable names are arbitrary, then a map is best. If the variable names are just x_1, x_2, x_3, …, x_n, then a list or dynamic array would be more natural. If n is constant, then a vector or static array is even better.

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

I see what you’re saying about moving away from smarmy pop culture videos, but the claim that all the “random smart things” ultimately don’t matter is weak when you’re comparing them to topics that literally do not matter at all. I’m not invalidating your example, just pointing something out.

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.