51–60 of 106 posts
Earlier quoted context omitted.
The very notion of discreteness depends on subjective definitions of "objects". We take concepts of objects for granted because they make interacting with the world tractable, but it's really hard to define them outside of minds.
As far as we know, the universe is made up of discrete units and any other type of math is an abstraction over discrete math.
Earlier quoted context omitted.
As far as we know, the universe is made up of discrete units and any other type of math is an abstraction over discrete math.
As far as we know, the universe is a single unity, and any discrete units and any other type of math are human distinctions overlaid upon that unity.
Earlier quoted context omitted.
Discrete math is the single most "observed property of reality", and nothing else comes even close.
The very notion of discreteness depends on subjective definitions of "objects". We take concepts of objects for granted because they make interacting with the world tractable, but it's really hard to define them outside of minds.
Earlier quoted context omitted.
Discrete math is the single most "observed property of reality", and nothing else comes even close.
The very notion of discreteness depends on subjective definitions of "objects". We take concepts of objects for granted because they make interacting with the world tractable, but it's really hard to define them outside of minds.
that's why I already got the double twin prime conjecture ready:
there exists an infinite number of consecutive twin primes. 3 examples: 11,13; 17,19. 101,103;107,109, AND 191,193;197,199... I know of another example near the 800s
there's also the dubious, or trivial, or dunno (gotta generalize this pattern as well) of the first "consecutive" twin prime but they overlap which is 3,5 and 5,7.... which reminds me of how only 2 and 3 are both primes off by one; again, I need to generalize this pattern of "last time ever primes did that"
I'm a little confused at the significance here. Before I read the definition of the M_a, this seemed crazy, but on actually reading it, M_1 is just the sum-of-divisors function (usually denoted sigma). So, n is prime iff M_1(n)=n+1. That's much simpler than the first equation listed there! Indeed, looking things up, it seems that in general the functions M_a can be written as a linear combination (note: with polynomi…
The M functions are the MacMahon’s partition functions (see the paper [1]). They were not known to relate to the sum of divisors. The M_a function counts partitions in a parts but weighing multiplicities in the partion. [1]: https://arxiv.org/abs/2405.06451
M_1(n) = sum(
m
for m in range(1, n+1)
for s in range(1, n+1)
if m*s = n
)https://news.virginia.edu/content/faculty-spotlight-math-pro...
Earlier quoted context omitted.
As far as we know, the universe is made up of discrete units and any other type of math is an abstraction over discrete math.
I thought it was a smooth continuous manifold
(3n^3 − 13n^2 + 18n − 8)M_1(n) + (12n^2 − 120n + 212)M_2(n) − 960M_3(n) = 0
is equivalent to the statement that n is prime. The result is that there are infinitely many such characterizing equations.