7 ms·

Is mathematics mostly chaos or mostly order?

117 points by baruchel 1y ago

b0a04gl 1y ago
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.
- QuesnayJr 1y ago
  This isn't really how it went down, historically. They considered lots of different large cardinals, and then they turned out to be linearly orderable by consistency strength. And then it's natural to wonder if it's a general rule.
Sniffnoy 1y ago
> 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.)
- bubblyworld 1y ago
  I suspect they mean "add" in the sense of "add in an axiom asserting the existence of another cardinal". Things like consistency strength of the resulting theory seem to vary wildly depending on what other cardinals you throw into the mix (if I understood the article correctly, haven't read the paper, mea culpa).
  - gowld 1y ago
    I think you are right. Plain English is a terrible language for higher math. It's extremely misleading (especially when talking about infinite objects and probability, where most of the mystery and confusion comes from terms that aren't precisely defined in the reader's mind).
    - jerf 1y ago
      I enjoyed quanta magazine for a while. But I think they ended up pretty much exhausting the set of things that can be explained to an interested audience with a bit more English, but doesn't take actual mathematics. I know enough about this topic to know that reading the article didn't teach me anything useful because I don't know enough to understand what they are saying. And if I did know enough, this article probably still wouldn't effectively teach me because it's too simplified. I haven't seen a quanta article in a while that I found useful. I appreciate the attempt but I don't think it works anymore. And I don't think it's their "fault"... I just think the slice of things this works for was smaller than we might have liked.
  - Sniffnoy 1y ago
    Oh, that would make more sense, yeah.
- whatidunno 1y ago
  [dead]
scrubs 1y ago
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!
- clintonc 1y ago
  There are a couple of strategies for understanding the real numbers. One is to write down a definition of real numbers, for example using rational numbers and Dedekind cuts, hoping that what you're describing is really what you mean. The other is to write down the properties of real numbers as you understand them as "axioms", and go from there. An important property of real numbers that always comes up (either as a consequence of Dedekind cuts or as an axiom itself) is the least upper bound property -- every set which has an upper bound has a least upper bound. That's what gives you the "completeness" of the real numbers, from which you can prove facts like the completeness of the real numbers (i.e., Cauchy sequences always converge), the Heine-Borel theorem (closed and bounded subsets of the reals are "compact", and vice-versa), and Cantor's intersection theorem (that the nested intersection of a sequence of non-empty compact sets is also compact). The diagonalization argument is an intuitive tool, IMHO. It is great if it convinces you, but it's difficult to make rigorous in a way that everyone accepts due to the use of a decimal expansion for every real number. One way to avoid that is to prove a little fact: the union of a finite number of intervals can be written as the finite union of disjoint intervals, and that the total length of those intervals is at most the total length of the original intervals. (Prove it by induction.) THEOREM: [0, 1] is uncountable. Proof: By way of contradiction, let f be the surjection that shows [0, 1] is countable. Let U_i be the interval of length 1/2*i centered on f(i). The union V_n = U_1 + U_2 + ... + U_n has combined length 1 - 1/2*n < 1, so it can't contain [0, 1]. Another way to state that is that K_n = [0, 1] - V_n is non-empty. K_n also compact, as it's closed (complement of V_n) and bounded (subset of [0, 1]). By Cantor's intersection theorem, there is some x in all K_n, which means it's in [0,1] but none of the U_i; in particular, it can't be f(i) for any i. That contradicts our assumption that f is surjective. Through the right lens, this is precisely the idea of the diagonalization argument, with our intervals of length 2*-n (centered at points in the sequence) replacing intervals replacing intervals of length 10*-n (not centered at points in the sequence) implicit in the "diagonal" construction.
dgfitz 1y ago
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.
- isaacfrond 1y ago
  Mathematics isn't monolithic—it depends heavily on the axioms you choose. Change the axioms, and the theorems change. ZFC, ZF¬C, intuitionistic logic, non-Euclidean geometry—each yields a different “math,” all internally consistent. So it’s not right to say math “just is” in some absolute sense. We’re not just discovering math; we’re exploring the consequences of chosen assumptions. For instance: Under Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), every set can be well-ordered, but we do get the Hahn–Banach paradoxes. Under ZF without Choice, analysis as we know it no longer holds. In constructive mathematics, which avoids the law of the excluded middle, many classical theorems lose their usual formulations or proofs. Non-Euclidean geometries arise from altering the parallel postulate. Within their own axioms, they are as internally consistent and "natural" as Euclidean geometry. Do non-intersecting lines exist in this universe? I've no idea.
  - vintermann 1y ago
    > all internally consistent Well, we hope.
  - bonoboTP 1y ago
    This just steps one meta level higher. Yes, you can make your object of analysis the axioms and what they lead to and proof theory etc. But now you've just stepped back one level. What are the axioms that allow you to derive that "ZFC leads to Hahn–Banach paradoxes"? Is this claim True and discovered or is it in itself also simply dependent on some axioms and assumptions? This is part of a broader meta-ization of culture. Philosophers are also much more reluctant to make truth claims in the last century compared to centuries ago. Everything they say is just "To a Hegelian, it is {such and such}. For Descartes, {x, y, z}." If you study theology, they don't teach with conviction that "Statement A". They will teach that Presbyterians believe X while the Anglicans think Y, and the Catholics think it's an irrelevant distinction. Of course when push comes to shove, you do realize that they do have truth claims, and moral claims that are non-negotiable but are shy to come forward with them and explicitly only talk in this "conditional" "if-then" way. In fact many would argue that math is not too far from theology. People who were obsessed with math limits, like Gödel, were also highly interested in theology. I guess physics is the closest to still making actual truth claims about reality, though it's also retreating to "we're just making useful mathematical models, we aren't saying that reality is this way or that way".
revskill 1y ago
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"
- novaRom 1y ago
  And especially when we mix different categories. Like saying about any infinity as about an object is misleading, because it's rather a process
  - Sharlin 1y ago
    Infinities (transfinite cardinals) in the sense used by the article are absolutely objects. We’re not talking about infinite sums or other sequences and their limits. (And limits aren’t really “processes” either – the limit of the sequence 0.9, 0.99, 0.999, … is exactly 1, as a well-known example which nonetheless is controversial among people who don’t know what limits are.)
    - random_ta 1y ago
      Sure, if you accept reifying concepts into objects as valid. But that is a gateway to misery.
      - Sharlin 1y ago
        What's the difference? How is the concept of a transfinite cardinal less of an object than, say, the concept of a set? Or a real number? All are well enough defined that you can do useful math with them, and that's really all that matters.
    - revskill 1y ago
      Object in OOP ?
      - Sharlin 1y ago
danwills 1y ago
> 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.
- cyborgx7 1y ago
  An interesting thing about the quote you highlighted is that it's already true about the set of real numbers itself. The set of real numbers that can be precisely, individually identified is a countable subset of all real numbers. That means the vast majority of real numbers, an uncountable amount of them, can not be individually defined and thought about.
  - danwills 1y ago
    That is very interesting I agree, and certainly any list of descriptions/identifiers must be countable, though I wonder if there's any validity in descriptions that describe things in aggregate? It's certainly a brain-bender that even in the unit interval if we imagine filling in all the the rationals and then adding in the describable-irrationals like PI/4, sqrt(2)/2 and so on.. that this still does not even come close to covering the unit interval - or any interval - of Real numbers! My imagination sees a line with a heck of a lot of dots on it, but still knowing that there clearly still uncountably-more values that are not covered/described! Amazing! The continuum (Real numbers) is such a fascinating concept!
Noelia- 1y ago
[dead]
lordfrito 1y ago
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.
jerf 1y ago
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.
charlieyu1 1y ago
The orderly stuff are already well studied
metalman 1y ago
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
anthk 1y ago
Measuring. Movement it's applied calculus. Pi it's not magical, it's all the distances from a point related to the distance itself.
jibal 1y ago
Gregory Chaitin already proved that there are mathematical truths that are completely random.
zzo38computer 1y ago
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.
kazinator 1y ago
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.

117 points by baruchel 1y ago

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.

QuesnayJr 1y ago
This isn't really how it went down, historically. They considered lots of different large cardinals, and then they turned out to be linearly orderable by consistency strength. And then it's natural to wonder if it's a general rule.

Sniffnoy 1y ago

> 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.)

bubblyworld 1y ago
I suspect they mean "add" in the sense of "add in an axiom asserting the existence of another cardinal". Things like consistency strength of the resulting theory seem to vary wildly depending on what other cardinals you throw into the mix (if I understood the article correctly, haven't read the paper, mea culpa).
- gowld 1y ago
  I think you are right. Plain English is a terrible language for higher math. It's extremely misleading (especially when talking about infinite objects and probability, where most of the mystery and confusion comes from terms that aren't precisely defined in the reader's mind).
  - jerf 1y ago
    I enjoyed quanta magazine for a while. But I think they ended up pretty much exhausting the set of things that can be explained to an interested audience with a bit more English, but doesn't take actual mathematics. I know enough about this topic to know that reading the article didn't teach me anything useful because I don't know enough to understand what they are saying. And if I did know enough, this article probably still wouldn't effectively teach me because it's too simplified. I haven't seen a quanta article in a while that I found useful. I appreciate the attempt but I don't think it works anymore. And I don't think it's their "fault"... I just think the slice of things this works for was smaller than we might have liked.
- Sniffnoy 1y ago
  Oh, that would make more sense, yeah.
whatidunno 1y ago
[dead]

scrubs 1y ago

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!

clintonc 1y ago
There are a couple of strategies for understanding the real numbers. One is to write down a definition of real numbers, for example using rational numbers and Dedekind cuts, hoping that what you're describing is really what you mean. The other is to write down the properties of real numbers as you understand them as "axioms", and go from there. An important property of real numbers that always comes up (either as a consequence of Dedekind cuts or as an axiom itself) is the least upper bound property -- every set which has an upper bound has a least upper bound. That's what gives you the "completeness" of the real numbers, from which you can prove facts like the completeness of the real numbers (i.e., Cauchy sequences always converge), the Heine-Borel theorem (closed and bounded subsets of the reals are "compact", and vice-versa), and Cantor's intersection theorem (that the nested intersection of a sequence of non-empty compact sets is also compact). The diagonalization argument is an intuitive tool, IMHO. It is great if it convinces you, but it's difficult to make rigorous in a way that everyone accepts due to the use of a decimal expansion for every real number. One way to avoid that is to prove a little fact: the union of a finite number of intervals can be written as the finite union of disjoint intervals, and that the total length of those intervals is at most the total length of the original intervals. (Prove it by induction.) THEOREM: [0, 1] is uncountable. Proof: By way of contradiction, let f be the surjection that shows [0, 1] is countable. Let U_i be the interval of length 1/2*i centered on f(i). The union V_n = U_1 + U_2 + ... + U_n has combined length 1 - 1/2*n < 1, so it can't contain [0, 1]. Another way to state that is that K_n = [0, 1] - V_n is non-empty. K_n also compact, as it's closed (complement of V_n) and bounded (subset of [0, 1]). By Cantor's intersection theorem, there is some x in all K_n, which means it's in [0,1] but none of the U_i; in particular, it can't be f(i) for any i. That contradicts our assumption that f is surjective. Through the right lens, this is precisely the idea of the diagonalization argument, with our intervals of length 2*-n (centered at points in the sequence) replacing intervals replacing intervals of length 10*-n (not centered at points in the sequence) implicit in the "diagonal" construction.

dgfitz 1y ago

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.

isaacfrond 1y ago
Mathematics isn't monolithic—it depends heavily on the axioms you choose. Change the axioms, and the theorems change. ZFC, ZF¬C, intuitionistic logic, non-Euclidean geometry—each yields a different “math,” all internally consistent. So it’s not right to say math “just is” in some absolute sense. We’re not just discovering math; we’re exploring the consequences of chosen assumptions. For instance: Under Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), every set can be well-ordered, but we do get the Hahn–Banach paradoxes. Under ZF without Choice, analysis as we know it no longer holds. In constructive mathematics, which avoids the law of the excluded middle, many classical theorems lose their usual formulations or proofs. Non-Euclidean geometries arise from altering the parallel postulate. Within their own axioms, they are as internally consistent and "natural" as Euclidean geometry. Do non-intersecting lines exist in this universe? I've no idea.
- vintermann 1y ago
  > all internally consistent Well, we hope.
- bonoboTP 1y ago
  This just steps one meta level higher. Yes, you can make your object of analysis the axioms and what they lead to and proof theory etc. But now you've just stepped back one level. What are the axioms that allow you to derive that "ZFC leads to Hahn–Banach paradoxes"? Is this claim True and discovered or is it in itself also simply dependent on some axioms and assumptions? This is part of a broader meta-ization of culture. Philosophers are also much more reluctant to make truth claims in the last century compared to centuries ago. Everything they say is just "To a Hegelian, it is {such and such}. For Descartes, {x, y, z}." If you study theology, they don't teach with conviction that "Statement A". They will teach that Presbyterians believe X while the Anglicans think Y, and the Catholics think it's an irrelevant distinction. Of course when push comes to shove, you do realize that they do have truth claims, and moral claims that are non-negotiable but are shy to come forward with them and explicitly only talk in this "conditional" "if-then" way. In fact many would argue that math is not too far from theology. People who were obsessed with math limits, like Gödel, were also highly interested in theology. I guess physics is the closest to still making actual truth claims about reality, though it's also retreating to "we're just making useful mathematical models, we aren't saying that reality is this way or that way".

revskill 1y ago

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"

novaRom 1y ago
And especially when we mix different categories. Like saying about any infinity as about an object is misleading, because it's rather a process
- Sharlin 1y ago
  Infinities (transfinite cardinals) in the sense used by the article are absolutely objects. We’re not talking about infinite sums or other sequences and their limits. (And limits aren’t really “processes” either – the limit of the sequence 0.9, 0.99, 0.999, … is exactly 1, as a well-known example which nonetheless is controversial among people who don’t know what limits are.)
  - random_ta 1y ago
    Sure, if you accept reifying concepts into objects as valid. But that is a gateway to misery.
    - Sharlin 1y ago
      What's the difference? How is the concept of a transfinite cardinal less of an object than, say, the concept of a set? Or a real number? All are well enough defined that you can do useful math with them, and that's really all that matters.
  - revskill 1y ago
    Object in OOP ?
    - Sharlin 1y ago

danwills 1y ago

> 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.

cyborgx7 1y ago
An interesting thing about the quote you highlighted is that it's already true about the set of real numbers itself. The set of real numbers that can be precisely, individually identified is a countable subset of all real numbers. That means the vast majority of real numbers, an uncountable amount of them, can not be individually defined and thought about.
- danwills 1y ago
  That is very interesting I agree, and certainly any list of descriptions/identifiers must be countable, though I wonder if there's any validity in descriptions that describe things in aggregate? It's certainly a brain-bender that even in the unit interval if we imagine filling in all the the rationals and then adding in the describable-irrationals like PI/4, sqrt(2)/2 and so on.. that this still does not even come close to covering the unit interval - or any interval - of Real numbers! My imagination sees a line with a heck of a lot of dots on it, but still knowing that there clearly still uncountably-more values that are not covered/described! Amazing! The continuum (Real numbers) is such a fascinating concept!

Noelia- 1y ago

[dead]

lordfrito 1y ago

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.

jerf 1y ago

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.

charlieyu1 1y ago

The orderly stuff are already well studied

metalman 1y ago

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

anthk 1y ago

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

jibal 1y ago

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

zzo38computer 1y ago

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.

kazinator 1y ago

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.