They don’t exist as concepts. A rational number whose square is 2 is (convenient prose for) a formal symbol describing some object. It happens that it does not describe any object. I am claiming that many objects described after the explosion of mathematics while putting calculus on a firmer foundation to resolve infinitesimals do not exist.
List functions like that need to be handled carefully to ensure termination. Summations of infinite series deal are a better example, consider adding up a geometric series. You need to add “all” the terms to get the correct result.
Of course you don’t actually add all the terms, you use algebra to determine a value.
I hold that the discovery of computation was as significant as the set theory paradoxes and should have produced a similar shift in practice. No one does naive set theory anymore. The same should have happened with classical mathematics but no one wanted to give up excluded middle, leading to the current situation. Computable reals are the ones that actually exist. Non-computable reals (or any other non-computable mathematical object) exist in the same way Russel’s paradoxical set exists, as a string of formal symbols.
Formal reasoning is so powerful you can pretend these things actually exist, but they don’t!
I see you are already familiar with subcountability so you know the rest.
This is really just the constructive/classical argument but I want to be specific.
You just named a function and specified a property you want it to have. However no function with this property meaningfully exists. We can manipulate the symbol just fine, but we can never look inside it because it’s not real. Classical mathematics was developed before computation was relevant and the question of decidability arose fairly late, it makes more sense to consider it an early attempt at what is now called intuitionistic mathematics. The halting problem disproved excluded middle.
This document plays at least two shell games, declaring that “homosexuality” as its own concept is recent (within 200 years) but then smoothly omitting this when discussing scripture, instead of analyzing scripture and then inserting the modern concept. No wonder it doesn’t find any condemnation of a concept it excluded from consideration!
It then does a similar trick where the authors of the New Testament are acknowledged to have poor Greek in many cases but then using specific word choice to claim they meant an extremely forced reading, relying on the previous trick a bit too.
There’s even a discussion of how nitpicking word choice is bad practice earlier in the same document!
Papal infallibility is not invoked that often. Here’s an example, in section 4 (wherefore…) [0]
In particular papal infallibility was not involved in the Protestants’ complaints, and the response to their complaints (Trent) was a council and again has nothing to do with papal infallibility.
The pope was also an absolute monarch at the time, but protestants didn’t care about that aspect.
This is just a polite fiction and no one is ever punished for giving illegal orders or for following them. Bloody Sunday resulted in absolutely no consequences for 1 Para. Tiger Force had illegal orders as their priority, were investigated and found guilty, and then absolutely nothing happened. The My Lai massacre resulted in a slap on the wrist for one guy.
He was “just following orders”, which has been an airtight defense for the members of Tiger Force. If they’re going to be rounded up along with all the other Americans known to be guilty of war crimes, fine prosecute nazi soldiers. But if not their immunity should be equally as strong.
AZ average students per class is 18. $21/hr for 2000 hours is $42k. Double it to overaccount for total cost of employee. Where's the $96k remaining going?
I'm not going to analyze further, I don't live in AZ. But you should check out the actual answers. Teachers could make more without spending more on education. Or rather spending more on education and less on "education"
So you claim it was a total coincidence he went after Soviet agents? The witch analogy here is that there are real witches but they are so powerful/well hidden witch hunters cannot find them and are diverted to patsies. This wouldn't mean witch hunting is bad, just that we needed better witch hunters.
Look up the total expenditure on education per student in your state, then do a rough calculation of how much of it is actually going to teaching students (facilities, teacher salaries, school supplies). It's usually not that much and school districts spend it on nonsense instead. There's probably enough money already so adding more won't help.
Just say witch hunt. McCarthy was entirely correct that there were a lot of communists and communist sympathizers, so many that many of the people he thought were helping him were themselves communists or communist sympathizers. Witches on the other hand are not real.
The history of these discoveries is fascinating and shows that Kuhn’s scientific revolutions idea is wrong but it’s always rounded off to “Copernicus and Galileo” and doesn’t even get them right
And in the P=?NP case Aaronson uses, the answer wouldn't be "P=NP" (a classical answer -- totally useless) but the actual function NP->P.
People just instinctively know that you need to know which side of the disjunction you're on, and they haven't been trained in classical logic to forget it.
Your information is outdated. Cada is at parity with lego now, they have even introduced brick designs that lego later copied (flip flop technic beams). The rest of the Chinese brands are similar, and even aliexpress mystery bricks are serviceable. Sometimes their color matches are not exact, but lego had the reddish brown issue so that's also at parity.
List functions like that need to be handled carefully to ensure termination. Summations of infinite series deal are a better example, consider adding up a geometric series. You need to add “all” the terms to get the correct result.
Of course you don’t actually add all the terms, you use algebra to determine a value.