[HN Gopher] Is mathematics mostly chaos or mostly order?
___________________________________________________________________
Is mathematics mostly chaos or mostly order?
Author : baruchel
Score : 106 points
Date : 2025-06-20 15:21 UTC (4 days ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| b0a04gl wrote:
| 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 wrote:
| 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 wrote:
| > 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| Oh, that would make more sense, yeah.
| scrubs wrote:
| 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 wrote:
| 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.
| dmurray wrote:
| > 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].
|
| This argument seems way less convincing to me than the
| diagonalization argument, because as n is going to infinity
| that length does become 1.
| wbl wrote:
| Then we can crank that 2 up to 3 or 4.
| gowld wrote:
| Then let U_i be the interval of length 1/3^i centered on
| f(i), so that the total length is 1/2, far less than 1.
|
| Even though the supposed "surjection" is infinite, it's
| still the case that every x in [0,1] would be in in one of
| the finite U_n and therefore V_n. But every K_n clearly has
| measure > 0 and is therefore non-empty, and since the K_n
| are nested subsets, there is at least one special point
| x_omega that is in all of the K_n.
|
| The "intuitive" problem (not logical problem) with PP's
| proof is that it relies on measure and completeness, which
| is far more technologically complex than the decimal
| diagonizalization argument.
|
| Here is intuitive "rebuttal": the same proof strategy
| seemingly proves that the rationals are uncountable! (This
| is of course technically false, because rational intervals
| are incomplete and all have measure 0 in the first place.
| But understanding this is much more complicated than
| imagining an 2-D infinite spreadsheet of decimal numbers
| between 0 and 1.)
| wbl wrote:
| Are these the same proof?
|
| If I let U_i be the interval of length 1/10^(i) centered
| on f(i), than what I'm saying is pick a different decimal
| digit to avoid this particular real.
| clintonc wrote:
| Then use 1/3 instead of 1/2 for a combined length of 2/3 --
| the total length of the intervals can be as small as you
| like. This hints at the fact that any countable subset of
| the real numbers is Lebesgue measure zero.
|
| Even using 1/2, the set that remains is nonempty due to the
| Cantor intersection theorem. The total length of the
| intervals is 1, which means that the remainder has no
| "interior" (i.e., contains no open interval), but the
| converse is not true: removing intervals whose lengths sum
| to less than one does not mean that the remainder will
| contain any interval. This is the consideration that allows
| you to create what are called "fat Cantor sets" -- the
| middle thirds Cantor set has Lebesgue measure zero, but by
| removing smaller intervals you can get other, homeomorphic
| sets that have positive measure.
| cyborgx7 wrote:
| > 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.
|
| As someone who also has never fully made his peace with the
| diagonality argument, but just chosen to accept it as true, as
| a given, this kind of bumps up against an interesting
| implication of different cardinalities of infinity.
|
| To precisely define an arbitrary real you'd need some kind of
| finite string that uniquely identifies that real number. Finite
| strings can be mapped, 1 to 1, to natural numbers. Therefore
| there can't be a finite string for any real number that
| uniquely identifies it. Otherwise we'd have a mapping between
| natural numbers and real numbers.
|
| In fact, the set of uniquely identifiable real numbers is a
| countable subset of real numbers. [1]
|
| Somehow, this realization has helped me make peace with the
| uncountability of real numbers.
|
| [1] Sorry if use words like "unique", "identify", "define" in
| not quite the right way. I hope the meaning I'm going for comes
| across.
| gowld wrote:
| A way to make peace with the Reals is to understand them as
| "potential numbers". Every where you look, there is Real
| number. Everyone logical agrees about that.
|
| But what about where you _don 't_ look? Either you take the
| orthodox axiomatic view that Real numbers are there too, or
| you take the constuctivist or finitist (or perhaps quantum
| mechanical?) view that nothing is there until you look,
| because the act of looking is the same as the act of
| creation.
| drdeca wrote:
| I wouldn't call it quantum mechanical. The "looking" in
| math is not like the measurement of an observable/operator
| in quantum mechanics. When you consider a thing in math,
| there's no alternative thing that you could have considered
| instead which would correspond to a different operator that
| doesn't commute with the first one.
| scrubs wrote:
| I'm will give this more consideration; thank you for the
| comment.
|
| For now I just want to add you hit a bit closer into the
| slight of hand in Cantors argument (for me) which is alluring
| but hard to surmount in the last 10% of the argument.
|
| The natural numbers are constructible, finite. They are
| finite to write down. It requires a finite amount of code
| (tape) to output one etc. The 1:1 mapping business gets the
| concept of infinity onto the table but without engaging a
| completed infinity. So far, it's solid followable etc ... now
| the next 5% you toss real numbers in rhs ... then produce
| another real off the diagonal for 5% more ... and |Z| /= |R|.
|
| Here real numbers live under the shadow or reflect the light
| of nats, which is misleading. The reals are not well defined
| objects.
|
| Now, the realist (the mathematician) will argue: the point of
| Cantor's argument is not to construct reals as part of the
| solution to |Z| /= |R|. The point is only to establish
| there's no bijection. In truth I agree: the focus is on the
| mapping not getting dragged into the mud of construction.
|
| However, I remain unclear if too much got swept under the rug
| that (practical minded) argument. I will have to re-read
| Chatin/Kolmogorov ... so I need 4 semesters now. This is my
| spooky action at a distance problem.
| AIPedant wrote:
| Cantor's original proof of the uncountability of the reals
| didn't use a diagonalization argument, it used order +
| completeness and in fact applies to any complete poset.
| https://en.wikipedia.org/wiki/Cantor%27s_first_set_theory_ar...
|
| Likewise his proof that there is no surjection from a set to
| its power set uses a more general diagonalization argument that
| doesn't make any uncomfortable assumptions:
| https://en.wikipedia.org/wiki/Cantor%27s_theorem
| bheadmaster wrote:
| > I've never made peace with Cantor's diagonaliztion argument
|
| Maybe you'd prefer a purely set-theoretic one:
|
| ---
|
| Let R be a set. Let S be a set of all subsets of R.
|
| We want to prove that |S| > |R|, by proving that a bijective
| function from R to S cannot exist. We will do that by assuming
| that it _can_ , and then deriving a contradiction.
|
| Assume there is a bijective function f : R -> S. Define D = { r
| [?] R | r [?] f(r) }. Since f is a bijection, there exists some
| r0 [?] R such that f(r0) = D.
|
| However, by the definition of D, we have:
|
| - If r0 [?] D, then r0 [?] f(r0) = D, which is a contradiction.
|
| - If r0 [?] D, then r0 [?] f(r0) = D, which is also a
| contradiction.
|
| Therefore, our assumption that there exists a bijective
| function f : R -> S must be false.
|
| Therefore, |S| > |R|.
| LegionMammal978 wrote:
| If it's the idea of completed infinity that's the objection,
| then it's the first step, constructing the powerset, that
| would be problematic. Various forms of finitism would not
| accept that one can 'take all the subsets, finite or
| infinite, and quantify over them' and obtain a meaningful
| result past the formal level.
| drdeca wrote:
| They should still then accept that there is no surjection
| from a set to a set that is a set of all subsets of that
| set (because of there being no such set).
| jonah-archive wrote:
| > 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.
|
| You're in good company -- from Penelope Maddy's "Believing the
| Axioms"[0]:
|
| ---
|
| Measurable cardinals were introduced by Ulam in [1930], where
| he proved that they are inaccessible. They are now known to be
| much larger than that, larger than all the hyperinaccessibles,
| Mahlos and weakly compacts. Indeed, because of their power,
| they are probably the best known large cardinals of all. The
| voice of caution reminds us that they were invented by the same
| fellow who invented the hydrogen bomb.
|
| ---
|
| 0: https://jwood.faculty.unlv.edu/unlv/Articles/Maddy1.pdf
| dgfitz wrote:
| 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 wrote:
| 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 wrote:
| > all internally consistent
|
| Well, we hope.
| bonoboTP wrote:
| 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
| Godel, 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".
| anthk wrote:
| No, you are wrong. 90% of Philosphy it's bullshit about
| giving a fake truth status depending of WHO said what.
| Meanwhile, Math and Science always put FACTS over personas.
| cheesecompiler wrote:
| The overarching point still stands: our formal systems are
| just models built to describe the patterns we observe. In
| that sense, math "just is." The fact that some models aren't
| compatible with others doesn't undermine that--it just shows
| they're incomplete or context-dependent views into a larger
| structure.
| nexttk wrote:
| About the axioms, not really. Axiom sets is mostly there just
| as a 'short hand' to quickly describe a context we're talking
| about, but ultimately you could just do away with them. E.g.
| if we let A be the set of axioms from some theory (e.g. set
| theory, number theory etc.) and you have a mathematical
| statement of the form X => Y within that theory, you could
| just as well consider the statement "A ^ X => Y" in the
| purely formal system without any axioms at all, then it is
| purely a logical question (essentially, if X => Y is a
| theorem within theory A) and more objectively true than "X =>
| Y" which would be theory-independent.
| cubefox wrote:
| Large cardinal axioms are the paradigmatic example "invented"
| math.
| leftcenterright wrote:
| What makes you consider it a "discovery" instead of a creation
| of us humans?
|
| I am more on the side of seeing maths as a precision language
| we utilize and extend as needed, especially because it can
| describe physically non-existent things e.g. perfect circles.
| esperent wrote:
| I rather think the discovered/invented thing is just
| semantics.
|
| You can say that literally anything was "just discovered".
|
| Thriller by Michael Jackson? Those particular ordering of
| sound waves always theoretically existed, MJ and various
| sound engineers just discovered them, they didn't create
| anything.
|
| The cappucino? It's just a particular orderly collection of
| chemicals, such a collection always theoretically existed.
| Those baristas are explorers, discovering new latte art
| shapes, nothing creative there.
|
| Cantor's diagonal argument? Yep, those numbers where just
| waiting to be discovered and written in that order.
|
| And so on. The entire argument is meaningless, pointless
| philosophizing. Nobody wastes their time saying latte art was
| discovered rather than invented, but somehow when it comes to
| mathematics this is considered a deep and worthy discussion.
| dgfitz wrote:
| To me, the difference is: the way to __make__ music was
| invented, it was already there to be discovered.
|
| Didn't mean to rub you the wrong way.
| bheadmaster wrote:
| > Didn't mean to rub you the wrong way.
|
| To me, it doesn't sound like you did. The parent comment
| of yours just stated, albeit bluntly, that the
| "invention" and "discovery" are fundamentally the same.
| Whether we use one or the other depends on how big the
| size of the space of the possibilities _feels to us_.
| Math has a very rigid and easily enumerable space of
| possibilities (strings of symbols), so we call it
| "discovery", while cooking has an enormous space of
| possibilities (countless pieces of meat and vegetables,
| each unique in its configuration of atoms, etc.), so we
| call it "invention".
|
| When you invent a way to make music, did you really
| invent it? Or did you simply discover a particular
| configuration of atoms that can produce sound when
| handled in a particular way, that was already there in
| some platonic universe of ideals? Either way, the end
| result is the same. Nothing really changes.
| gowld wrote:
| Is TRIZ (and its great-great-grandchildren, LLM GenAI)
| _invention_ , or _discovery_?
| https://en.wikipedia.org/wiki/TRIZ
| esperent wrote:
| > Math has a very rigid and easily enumerable space of
| possibilities (strings of symbols), so we call it
| "discovery", while cooking has an enormous space of
| possibilities (countless pieces of meat and vegetables,
| each unique in its configuration of atoms, etc.), so we
| call it "invention".
|
| I think you've made a good point here.
|
| Although, to nitpick a bit: Both spaces (cooking and
| maths) are infinite, and for most fields of math,
| uncountably infinite. The difference is in the numbers we
| are dealing with. For cooking, it's mixtures of trillions
| of molecules. For maths, it's usually in the order of
| thousands of symbols (although those ellipses do some
| infinitely heavy lifting!).
| esperent wrote:
| > Didn't mean to rub you the wrong way
|
| Not at all, you sparked thought in an area I find
| fascinating (philosophy of maths). Albeit I find this
| specific topic a bit too commonly discussed relative to
| how important it is, but I'm still happy to talk about it
| and share my thoughts.
| frontfor wrote:
| I like to think of axioms as "created" while the
| _consequences_ (i.e. theorems) of said axioms are
| "discovered". You can't create logic consequences
| (conclusions) given a set of axioms, but you can certainly
| create the axioms (premises).
| mjburgess wrote:
| It's not clear to me why people think perfect geometries do
| not exist, they occur all the time in physics.
|
| Of composite matter, sure, because it's composite in a
| certain sort of way, you do not get perfect circles. But the
| structure of macroscopic material does not exhaust the
| physically.
|
| Even here, one could define some process (eg., gravitational)
| which drives matter towards being a perfect circle, because
| perfect circularity is a property of that process. This is,
| as a matter of fact, true of gravity -- if it weren't we'd
| observe violations of lorentz invariance, which we do not.
| IAmBroom wrote:
| Perfect in a single-body universe, perhaps, but the gravity
| field of a particle is perturbed by other nearby particles
| - where "nearby" is relative to precision desired - and
| therefore never a perfect sphere.
|
| Or, to put it another way, so-called "perfect circles"
| exist in a real, 4-D, wibbly-wobbly gravity-distorted
| space, and are no longer perfect Cartesian circles.
|
| They still only exist theoretically; not in practice.
| mjburgess wrote:
| Circularity is still a property of the _process_. One
| requires perfect circles to describe it.
|
| It is also easy enough to construct circular state
| spaces, and the like.
|
| The idea that what's real is simply the geometry of
| macroscopic visible matter, or even of matter alone, is a
| nonesense.
|
| The world is "immanently abstract", and possess
| primeness, circularity, etc. in itself -- not as
| something merely imagined. This is obvious from the
| physical description of its evolution.
|
| Irregularity, of this kind, is _derivative_ of a
| geometrical reality. The irregular doesn 't govern the
| irregular, if it did, there would be no structure
| whatsoever.
| anthk wrote:
| Perfect circles just exist on probabily using 2pi for
| random directions and overall statistics.
| revskill wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| Sure, if you accept reifying concepts into objects as
| valid. But that is a gateway to misery.
| Sharlin wrote:
| 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 wrote:
| Object in OOP ?
| Sharlin wrote:
| Mathematical objects, like numbers, sets, matrices,
| functions, ...
| kbrkbr wrote:
| I can think of N as a process in a sense, because I can keep
| adding a number. But I can't think of R as a process like
| this, specifically because there is no surjective mapping
| from N to R.
|
| How would you think of R as a process?
| vharuck wrote:
| It is amazing what Euclid was able to prove and, frankly, even
| imagine using just geometric figures.
|
| Nowadays, we have symbolic notation. The only research articles
| I read are for epidemiology, so I don't know how much notation
| is used in pure math journals. But I can't remember seeing any
| notation in those articles beyond what one would encounter in
| high school. I guess authors see more value in deceptive
| narrative and descend into strict logical languages only when
| necessary.
| glial wrote:
| Not just Euclid - even Newton's Principia used geometric
| demonstrations, rather than algebraic ones.
| danwills wrote:
| > 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 wrote:
| 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 wrote:
| 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!
| nyrikki wrote:
| _Almost all_ real numbers are normal numbers, which don 't
| even have a finite representation.
|
| Sure you can assign them to an arbitrary set, but you don't
| have access to the value.
|
| It is a hay in the haystack problem, where you really only
| have access to the needles, not the hay.
| gowld wrote:
| It's even more extreme than that!
|
| Take the (uncountable) set of Real numbers. Remove the
| normal numbers, which is _almost all_ of them in the
| sense that the probability that "a uniformly randomly
| chosen real number is normal (and therefore also
| undescribable)" is 1. The remaining set of numbers, which
| has measure 0 in the Real numbers, is _still_
| uncountable, meaning that the proability of randomly
| choosing a describable number in that set is again 0.
|
| I'm not sure how deep this chain can go. Google AI says
| "only 1 steps" but it's not admiting the case described
| in this comment.
| LegionMammal978 wrote:
| > _Almost all_ real numbers are normal numbers, which don
| 't even have a finite representation.
|
| Plenty of normal numbers have a finite representation
| from which digits can be efficiently extracted. E.g.,
| Champernowne's constant (in any base) is normal, and you
| can find its digits with a relatively simple algorithm.
|
| All computable reals can similarly have their digits
| extracted by some algorithm or another, even though it
| may take a long time. I wouldn't call that "not having
| access to the value". Of course, uncomputable numbers are
| a different story, but they have nothing to do with
| normality in any base.
|
| And of course, radix representations are not the only way
| to evaluate real numbers. E.g., you could represent them
| with simple continued fractions (which would still allow
| addition, multiplication, comparison, etc.), and then you
| could write out any quadratic irrational with a periodic
| expansion.
| nyrikki wrote:
| "almost all" or "almost everywhere" is in italics because
| I mean it in the measure theory sense.
|
| Meaning it holds for all elements of a set except for a
| subset that has measure zero.
|
| Yes some normal numbers are in the constructable reals,
| but it is a measure zero subset.
|
| You are putting your hand in the haystack and only
| finding needles, finding the hay in the haystack is the
| problem here.
| dwohnitmok wrote:
| This is subtle and a simple counting argument (definable
| means satisfies a finite formula, there are only countably
| many finite formulas, there are uncountably many reals,
| therefore there must be undefinable reals) doesn't work,
| because "definable in ZFC" is not something that is
| formalizable in ZFC and so the usual set-theoretic counting
| arguments don't work.
|
| So it is in fact possible and consistent with ZFC that all
| reals are definable.
|
| See: https://mathoverflow.net/questions/44102/is-the-
| analysis-as-...
| bubblyworld wrote:
| Thanks, that's a wonderful link and a nice puzzle to think
| about. The best intuition I have for it is that since the
| predicate "isDefinableReal(x)" is not itself definable in
| first-order set theory, there is no way to construct the
| set of all definable reals in the first place. Thus saying
| it's countable is basically meaningless - _what_ , exactly,
| is countable?
| drdeca wrote:
| If you use ZFC+Consistent(ZFC) as your meta-theory, and
| within it consider a model of ZFC, then surely one can
| consider the set (in the meta theory) of sentences which
| pick out a unique real number in the model, and then the
| set of real numbers in the model which are picked out by
| some sentence? It might not be a set that belongs to the
| model, but it's a set in the meta-theory, right?
|
| And, I imagine that the set of real numbers of the meta
| theory could be (in the meta theory) the same set as the
| set of real numbers in the model?
| bubblyworld wrote:
| You can do this, but things get strange in the meta-
| theory. Some models of ZFC are countable according to the
| meta-theory! And some of them have models of the reals
| that are countable according to the meta-theory. There's
| no contradiction here, because what the meta-theory
| thinks "countable" means has nothing to do with what the
| inner model thinks "countable" means.
|
| (for an extreme example of this, by the Lowenheim-Skolem
| theorem there are countable models of ZFC)
|
| So you can do what you are suggesting, and you will of
| course get a countable set of reals (or what are reals
| according to the inner model), but they might not be
| countable according to the inner model. They might not
| even be a set according to the inner model, and there are
| even inner models that think you've got all of the reals!
|
| (see https://mathoverflow.net/questions/351659/set-of-
| definable-r... pretty heavy reading)
|
| So the statement "the set of definable reals is
| countable" is nonsense - you're talking about things that
| live in different universes of meaning.
| gbacon wrote:
| My intuition about the question is related: the set of all
| Turing machines (algorithms) is countable, but the set of all
| languages (problems to solve) is uncountable. If you take
| mathematics to be the bigger, uncountable picture, it's
| mostly chaos, but if you limit consideration to algorithms,
| then it's mostly order.
| Viliam1234 wrote:
| > the set of all languages (problems to solve) is
| uncountable
|
| The set of all problems that can be described by a finite
| description is countable. Why would we care about the rest
| of them?
| jibal wrote:
| Because we're theorists.
| lordfrito wrote:
| 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 wrote:
| 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 wrote:
| The orderly stuff are already well studied
| metalman wrote:
| 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 wrote:
| 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 wrote:
| Gregory Chaitin already proved that there are mathematical truths
| that are _completely random_.
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