Undeterred after three decades of looking, and with some assistance from a supercomputer, mathematicians have finally discovered a new example of a special integer called a Dedekind number.
In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. The Dedekind number M(n) is the number of monotone boolean functions of n variables. Equivalently, it is the number of antichains of subsets of an n-element set, the number of elements in a free distributive lattice with n generators, and one more than the number of abstract simplicial complexes on a set with n elements.
Pretty simple to understand. I mean, I understand it, for sure. Totally.
Good work everyone. I stay more with the stereo boolean variables, but the news about those lattices being free now is really great stuff. We really did something here
Imagine you have a puzzle with a set of rules about how you can put the pieces together. This puzzle isn’t made of typical jigsaw pieces, but instead uses ideas from math to decide how they fit. A Dedekind number is like counting how many different ways you can complete this puzzle.
In simple terms, a Dedekind number is connected to a concept in mathematics called a “Boolean function.” This is a type of math problem where you only use two things: yes or no, true or false, or in math language, 0 or 1. A “monotone Boolean function” is a special kind of this problem where changing a 0 to a 1 in your problem can only change the answer from 0 to 1, not the other way around.
The big news is that mathematicians and computer scientists just found a new, very large Dedekind number, called D(9). It took them 32 years since the last one was found! To find it, they used a supercomputer that can do lots of calculations at the same time. This was a big deal because Dedekind numbers are really hard to calculate. The numbers involved are so huge that it wasn’t even sure if we could find D(9).
You can think of finding a Dedekind number like playing a game with a cube where you color the corners either red or white, but you can’t put a white corner above a red one. The goal of the game is to count all the different ways you can do this coloring. For small cubes, it’s easy, but as the cube gets bigger (like going from D(8) to D(9)), it becomes super hard.
So, discovering D(9) is a big achievement in mathematics. It’s like solving a super complex puzzle that very few people can understand, let alone solve. It’s significant because it pushes the boundaries of what we know in math and shows how powerful computers can help us solve really tough problems.
That seems more just very resource requiring than hard to do, in a modern world with computers? I get that these were ridiculous to find around 1900 when they were discovered and you had to find them without computers to do the calculations.
“Resource requiring” and “hard to do” are kind of the same in math’s terms. Most unsolved math problems are either because we lack the resources, we lack observation (in case of phisics) or we lack both.
hat useful purpose does these Dedekind numbers have?
Nothing, just like when lasers were first discovered (now we use them for medical and tech purposes)
You can kind of use this as a benchmark for where we are computationally as a society. If you plot these achievements on a graph, maybe we can plot the trajectory of achievement and predict where we will be in 10 years…or something.
🤔 That could matter a lot for chip designers. They’d need to know the ways in which a Boolean function could do such a thing since you use Boolean math to design the chips, and need to understand the math to design the chips in certain ways depending on your needs.
EL5 why this is significant, please.
( Not trying to be any which way.)
I looked it up on Wikipedia.
Pretty simple to understand. I mean, I understand it, for sure. Totally.
Ah, yes, those things, of course.
Glad we cleared that up. In hindsight, it was pretty obvious from the start.
Ah, yes. I know
somenone of these words.I understood most of the words, just the ones that I didn’t made the rest incomprehensible garbledygoop
Good work everyone. I stay more with the stereo boolean variables, but the news about those lattices being free now is really great stuff. We really did something here
Long slaughtering necromancer math
Complements of GPT:
I still don’t understand it, but good job math wizards!
Mathmagicians.
That seems more just very resource requiring than hard to do, in a modern world with computers? I get that these were ridiculous to find around 1900 when they were discovered and you had to find them without computers to do the calculations.
“Resource requiring” and “hard to do” are kind of the same in math’s terms. Most unsolved math problems are either because we lack the resources, we lack observation (in case of phisics) or we lack both.
hat useful purpose does these Dedekind numbers have? Nothing, just like when lasers were first discovered (now we use them for medical and tech purposes)
You can kind of use this as a benchmark for where we are computationally as a society. If you plot these achievements on a graph, maybe we can plot the trajectory of achievement and predict where we will be in 10 years…or something.
🤔 That could matter a lot for chip designers. They’d need to know the ways in which a Boolean function could do such a thing since you use Boolean math to design the chips, and need to understand the math to design the chips in certain ways depending on your needs.