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Cake day: July 11th, 2023

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  • 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.










  • 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.




  • 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.



  • 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.


  • 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.