from Hacker News

Is mathematics mostly chaos or mostly order?

by baruchel on 6/20/25, 3:21 PM with 88 comments

• by danwills on 6/24/25, 9:45 AM

  > the mathematical universe, like our physical one, may be made up mostly of dark matter. “It seems now that most of the universe somehow consists of things that we can’t see,”

  Not heaps fond of relating invisible things in the mathematical universe to dark matter! Although maybe both might turn out to be imaginary/purely-abstract? Imaginary things can absolutely influence real things in the universe, it's just that they are not usually external to the thing they are influencing. If I imagine making a cake say, and then I go ahead and make the one I imagined, the 'virtual' cake was already inside me to begin with, and wasn't 'plucked' from a virtual universe of possible cakes somewhere outside my knowledge of cake-making.

  Something nags at the back of my mind around this about maths though, as if to suggest that as soon as there was one-of-anything that was kinda an 'instantiation' of the most abstract "one" object from the mathematical universe.. (irrespective of what axioms are used as long as they support something like one) But I doubt there's never been exactly-PI-of-anything in the real universe, just a whole bunch of systems that behave as if they know (or are perhaps in the process of computing) a more exact value! (spherical planets, natural sine waves etc!)

  Very interesting article, I wish my math was stronger! I can just skirt the edges of what they're actually talking about and it's tantalizing! Would love to know more about these new types of cardinal numbers they've developed/discovered.

• by Sniffnoy on 6/24/25, 4:10 AM

  > But add a smaller cardinal to one of the new infinities, and “they kind of blow up,” Bagaria said. “This is a phenomenon that had never appeared before.”

  I have to wonder just what is meant by this, because in ZFC, a sum of just two (or any finite number) of cardinals can't "blow up" like this; you need an infinite sum. I mean, presumably they're referring to such an infinite sum, but they don't really explain, and they make it sound like it's just adding two even though that can't be what is meant.

  (In ZFC, if you add two cardinals, of which at least one is infinite, the sum will always be equal to the maximum of the two. Indeed, the same is true for multiplication, as long as neither of the cardinals is zero. And of course both of these extend to any finite sum. To get interesting sums or products that involve infinite cardinals, you need infinitely many summands or factors.)

• by jerf on 6/24/25, 2:10 PM

  Math is entirely chaos. In a slangy sense I can't prove the set of math that we would call "ordered" is of measure 0 against all the mathematical structures that "exist", without getting into exactly what that means.

  That's also the interesting math, so it is worthy of study. But the math that is interesting is the exception.

  A "randomly" chosen function from the set of all possible functions is a function with some infinite input that maps it to an infinite output (with any of the infinite ordinals in play you like) where there is no meaning to any of the outputs at all, indistinguishable from random. (The difficulties of putting distributions on infinite things is not relevant here; that's a statement of our limitations, it doesn't make these structures that we can't reach not "exist".)

  It's not amazing that if we take a "wrong" turn down the interesting math we end up in increasing levels of chaos. What's impressive is how interesting the not-pure-chaos subset manages to be, and how well it holds together.

• by zzo38computer on 6/25/25, 3:02 AM

  Principia Discordia says some things about chaos and order. Below are some quotations (which do not appear immediately next to each other in the original text, but are within one section of the original text):

  > Both order and disorder are man made concepts and are artificial divisions of PURE CHAOS, which is a level deeper that is the level of distinction making.

  > We look at the world through windows on which have been drawn grids (concepts). Different philosophies use different grids. A culture is a group of people with rather similar grids. Through a window we view chaos, and relate it to the points on our grid, and thereby understand it. The ORDER is in the GRID. That is the Aneristic Principle.

  > The point is that (little-t) truth is a matter of definition relative to the grid one is using at the moment, and that (capital-T) Truth, metaphysical reality, is irrelevant to grids entirely. Pick a grid, and through it some chaos appears ordered and some appears disordered. Pick another grid, and the same chaos will appear differently ordered and disordered.

  The mathematical ideas described in the article are interesting and mathematicians might be able to try to figure out these things, but it is only about ZFC and the variants with the additional axioms, not about "all of mathematics" (for one thing, there are other kind of set theory too; but there are other things too), which cannot be answered.

• by kazinator on 6/25/25, 5:35 AM

  Mathematics is dominated by chaos. Let's consider the real numbers. Most of them can't be written down. Most real-valued functions of just one variable cannot be written as a symbolic formula of that variable. Consider the n-body problem. The underlying rules of motion are easy to write down symbolically, but the resulting trajectories have no symbolic solution and are chaotic. Even in discrete math or number theory there is chaos. Like there is no pattern as to where on the integer number line the prime numbers land.

• by scrubs on 6/24/25, 5:36 AM

  If I had a semester or two of free time I'd love to hit this subject again. I once told my math prof (logician) who made a comment about transfinite cardinals: careful it's powerful but it's power from the devil. I half regret that comment in retrospect.

  I've never made peace with Cantor's diagonaliztion argument because listing real numbers on the right side (natural number lhs for the mapping) is giving a real number including transedentals that pre-bakes in a kind of undefined infinite.

  Maybe it's the idea of a completed infinity that's my problem; maybe it's the fact I don't understand how to define (or forgot cauchy sequences in detail) an arbitrary real.

  In short, if reals are a confusing you can only tie yourself up in knots using confusing.

  Sigh - wish I could do better!

• by b0a04gl on 6/24/25, 3:50 AM

  how much of modern set theory is reverse engineered from axioms rather than discovered. we're always building highways through a forest we haven't mapped, assuming every tree will fall in line. and suddenly these new large cardinals show up that don't even sit neatly in the ladder. it's maynot be failure of math,but failure of narrative. we thought the infinite was climbable, now it's folding sideways. maybe the math we're building is just a subset of what's possible, shaped by what's provable under our current tools. lot of deep shit probably hiding in the unprovable.

• by dgfitz on 6/24/25, 6:04 AM

  I’ve always considered math is something that is discovered, neither chaotic or orderly, it just… is. Really brilliant people make new discoveries, but they were there the whole time waiting to be found.

  This article seems to kind of dance around yet agree with the discovery thing, but in an indirect way.

  Math is just math. Music is just music. Even seemingly-random musical notes played in a “song” has a rational explanation relative to the instrument. It isn’t the fault of music that a song might sound chaotic, it’s just music. Bad music maybe. This analogy can break down quickly, but in my head it makes sense.

  Disclaimer - the most advanced math classes I’ve taken: calc3/linear/diffeq.

• by charlieyu1 on 6/24/25, 2:18 PM

  The orderly stuff are already well studied

• by jibal on 6/24/25, 9:15 PM

  Gregory Chaitin already proved that there are mathematical truths that are completely random.

• by anthk on 6/24/25, 6:36 PM

  Measuring. Movement it's applied calculus. Pi it's not magical, it's all the distances from a point related to the distance itself.

• by revskill on 6/24/25, 7:09 AM

  We need a word-less world of math where all meaning is derived from figures. WOrds are confusing.

  "If you can't describe the meaning using only pencils and compass, you don't mean it"

• by metalman on 6/24/25, 3:13 PM

  mathematics is a human construct, one among many others, such as order and chaos. One of the characteristics of human constructs is the never ending battles to redefine them as is evident in any investigation of the historical uses and definitions of these ideas. while we can point to any number of ordery things that are admired, no one can point to an orderly framework that shapes our universe, well no one who does not have that flash fryed all seeing thousand yard stare that will stay with you, which is the little game we are blithely toying with in the title of the article..........as in be carefull what questions you ask, as you just might get an answer

• by lordfrito on 6/24/25, 1:11 PM

  Isn't that a bit like asking if computing is mostly ones or mostly zeroes?

  It's the relationship between order and chaos that matters. Everything interesting always happens on the boundary between the two.