Subnostr

Picture thinkers seem to have that, so some degree. An image that represents a concept is faster to summon than a sentence that describes the same concept.

That said, I'm not too concerned about speed; thought already moves much faster than words, even if it takes non-zero time.

The fastest thought would be pre-symbolic, almost by definition. Spelling out concepts will necessarily take time.

Daniel Wigton 1y ago

I care about speed because I'd rather hear words at the end of a "thought much and said little" process than a stream-of-concious word dump.

Also I like math and logical reasoning. Language is only sorta useful for that. We have to come up with special definitions of words for logical contexts. Programming is an excellent case. By if we mean "if and only if" and or is not exclusive OR. If the symbols, oft times, WERE the conclusion I wouldn't have to hold so much in my head at once.

You'll have to forgive me, I want this to be able to solve the twin prime conjecture, but I run into a hard limit on the complexity that I can reason about at once. I wonder if I could offload some of the work onto a symbolic system that maintains part of the state for me.

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Discussion

MichaelJ 1y ago

Can you explain the twin prime conjecture? I'm not super familiar with it.

Daniel Wigton 1y ago

There are infinite pairs of primes such that P_n+1 - P_n = 2. Examples are (3,5),(5,7),(11,13),(17,19) etc..

Daniel Wigton 1y ago

The best I can do is prove that there are infinitely pairs coprime to any list of primes. Basically you can't give me any list of primes up to but not including infinity for which I couldn't find a pair of number separated by two neither of which have any factors in your list.

MichaelJ 1y ago

So the question is how to generalize from lists of coprimes to *all* the primes, right?

Daniel Wigton 1y ago

Kind of. You don't have to generalize to all the primes if you can show that at least one of the co-prime pairs is "small." small being less than the square of the largest prime in your list.. as long as your list is all the primes up to that prime.

If the co-prime twins are evenly distributed then this shouldn't be a problem there will be an ever increasing number of twin prime candidates less than the square as the list gets longer. But the best I can do there is show that there will be PI[(p_n-1)/(p_n-2)] / 2 less than PI[pn-1] / 2

PI being the product of all the terms. Unfortunately the numbers get so massive compared to p_n^2 that I can't really say anything conclusive even though there should be plenty. I need headspace to think about larger patterns of gaps between gaps of primes. I.e. if the twin primes are far enough apart then they need to spread out enough to have some less than p_n^2.