The point of mathematics is not to prove results. It is to build conceptual thinking about mathematics. Important problems are important because in order to solve them we have to build concepts tying different things together.
We're not searching for answers. We're searching for insights. Trying to understand the problem causes us to draw the connections and find those insights.
AI gives us answers. But it doesn't help us build those insights. AI has a complete mastery of existing human insights. But doesn't build new ones from its own experience. In a real way, it does not find the opportunity to really learn.
So it tackles problems and either solves them or not. If solved, we now have an answer. If not, it's too hard for humans.
That's the viewpoint of everyone sensible outside of Alzheimer's research.
Those in it are still throwing billions per year at the idea.
Meanwhile, back in reality, no amyloid-beta drug has had any clinical effect in humans, other than reducing the plaques. But both the shingles and RSV vaccines are proven to reduce Alzheimer's risk.
Which did not stop the FDA from approving a useless anti-amyloid drug, leading to the resignation of several experts, one of whom called it "probably the worst drug approval decision in recent U.S. history" in his resignation letter. [0]
Bad example. You can do all of this with constructivism. Any constructable Cauchy sequence converges to a constructable member of the space.
What you get for the formalism around computable numbers is this. Every mathematical object in the theory is something that can be, at least in principle, actually written down. When we say that it exists, this existence is of the most tangible form that any mathematical thing could have.
Having constructible Cauchy sequences doesn't guarantee that we can construct unbounded operators. I'm no expert, but the little searching I've done suggests this is an open research question.
I don't see the benefit of being able to write something down "in principle." A number can only ever be computed to a finite number of digits in practice. If we're talking about finite approximations, then the standard approach using numerical solutions to the Schrödinger equation handles this just fine, no alternative mathematics needed. If we're talking about theories, then we should choose whatever abstraction is most convenient for expressing the theory.
Personally, I don't believe numbers "exist." The physical universe exists, and numbers are abstractions that we invent to describe it. In that sense, uncomputable numbers are just as "real" as computable ones.
But there are numbers in constructivism for which it is unknown whether they are zero. Some of which must remain unknown, if mathematics is consistent. This is a rather important and weird edge case.
No. Tor is for the CIA. It won't work for them unless we use it as well. Criminals also find it useful.
It's easy to verify this. Tor was originally written by Paul Syverson, Michael G. Reed, and David Goldschlag. While all three were working at the U.S. Naval Research Laboratory.
These are the origins of Tor yes. The same technology that protects the spy, protects the journalist, or the citizen whose government blocked them, or placed a wall of ID verification checks.
That depends entirely on their perceived vulnerability.
Actors who feel they have little vulnerability --- that they can act with immunity or impunity --- regardless of the validity of that assessment, can and will act overtly.
Those who feel they are in a hostile or vulnerable environment will seek to conceal their actions and communications, and practice strong tradecraft.
Epstein (incorrectly as it happens) presumed impunity. His (mostly) unidicted co-conspirators have thus far been shown correct in their similar presumptions.
No appeal process.
Luckily she didn't use Facebook much.
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