Is mathematics mostly chaos or mostly order?
quantamagazine.org
quantamagazine.org
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.)
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.
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.
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!
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.
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.
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.
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.
> Yes some normal numbers are in the constructable reals, but it is a measure zero subset.
Almost all irrational computable reals (in the sense of natural density) will be normal, for any sane enumeration. Just because a real number is computable doesn't mean it's less likely to be normal.
> All computable reals can similarly have their digits extracted by some algorithm or another, even though it may take a long time.
It was an intentional abstraction to avoid a self referencing claim, not to say that they are equivalent to anything.
The nice thing about constructible reals is after the construction you can typically forget how you constructed them, be that through Axioms, Cauchy sequences, Dedekind cuts etc...
The computable reals by definition can be computed to within any desired precision by a finite, terminating algorithm. That is why I said it is begging the question.
> Almost all irrational computable reals (in the sense of natural density) will be normal, for any sane enumeration. Just because a real number is computable doesn't mean it's less likely to be normal.
While some have Conjectured claims close to this, looking into why there have been no proofs for even a single number that was not explicitly created to be normal in any base may be a good lens in to the hay-in-the-haystack problems I was refrencing.
From: "Distribution Modulo One and Diophantine Approximation (Cambridge Tracts in Mathematics, Series Number 193)" Page 81, section 4.1 "Equivalent definitions of normality"
> Lemma 4.3: Let b and r be integers greater than or equal to 2. If a real number is simply normal to base b^r, then it is simply normal to base b.
That there may exclude many of what appear to be simply normal numbers from actual ones. As a graduate level text that book may be a bit expensive for what it is, but a good reference in my experience.
The 'Normal' property that is the full measure set of the reals is far more constrained than natural density. Which is why it is so surprising that is is the property of almost all of them.
That is why it was offered as a lens, specifically one that was worked on before computability as a subject, as an intentional way to gain distance from the almost intractable polysemy problems there.
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-...
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?
(for an extreme example of this, by the Löwenheim–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.
And I think by virtue of this, the statement should have meaning.
As like, a statement in the meta-language that models of ZFC which have as their sets of reals, the (according to the meta-theory) set of reals, that the set of reals definable within ZFC, is a countable set of the meta-theory.
Also, did someone downvote your comment?? I don’t know why if so. It seems a productive comment to me.
By definition an inner model consists of some domain (a set of sets) and some choice of mappings from all the function/relation symbols of ZFC to functions/relations on this domain, satisfying the axioms of ZFC.
You are suggesting to enumerate every formula of ZFC, evaluate them against this inner model, and take the set of all reals that are uniquely picked out by some formula (according to the model).
The trouble is that even though you can make the set of reals the same, your chosen interpretation of all the functions/relations will not match the meta-theory, and in fact cannot match it (i.e. the meta-theory cannot provably construct an inner model like this, by Tarski's undefinability of truth theorem).
So you will get a set of reals, and they will be reals according to the meta-theory too, but the meta-theory cannot relate this set to the definable reals of the meta-theory.
As far as I can see this is the strongest statement you can actually prove: "the set of reals in any inner model of ZFC uniquely definable by a formula (according to the interpretation of the inner model) is countable (according to the interpretation of the meta-theory)".
> Also, did someone downvote your comment??
Someone did, yeah, but I don't mind =) I probably sound like a crackpot to the uninitiated.
The set of all problems that can be described by a finite description is countable. Why would we care about the rest of them?
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.
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.
Well, we hope.
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".
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.
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.
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.
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!).
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.
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.
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.
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.
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.
> 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.
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!
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.
This argument seems way less convincing to me than the diagonalization argument, because as n is going to infinity that length does become 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.)
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.
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.
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.
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.
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.
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
Maybe you'd prefer a purely set-theoretic one:
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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|.
You're in good company -- from Penelope Maddy's "Believing the Axioms"[0]:
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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.
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"If you can't describe the meaning using only pencils and compass, you don't mean it"
How would you think of R as a process?
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.
It's the relationship between order and chaos that matters. Everything interesting always happens on the boundary between the two.