This might be a dumb question but are there any current or foreseeable practical applications of this kind of result (like in the realm of distributed computing) or is this just pure mathematics for its own sake?
Not a dumb question. The links to mesh networking etc seem interesting. It sounds like the insights from descriptive set theory could yield new hardness/impossibility results in computational complexity, distributed algorithms etc.
are there any current or foreseeable practical applications of those results?
the math of infinity isn't very relevant to the finite programs that humans use. Even today's astronomically large computing systems have size approximately equal to 3 compared to infinity.
> All of modern mathematics is built on the foundation of set theory
That's ignoring most of formalized mathematics, which is progressing rapidly and definitely modern. Lean and Rocq for example are founded on type theory, not set theory.
usually you're more interested in better ergonomics: can you do X with less work?
it's like picking a programming language - depending on what you're attempting, some will be more helpful.
and ZFC is a lot more low level than what day-to-day mathematics usually bothers with. So most mathematians actually work in an informally understood higher-order wrapper, hoping that what they write sufficiently explains the actual "machine code"
the idea then behind adopting alternative foundations is that these come with "batteries included" and map more directly to the domain language.
Isabelle/HOL is still types. The underlying type theory of Isabelle/HOL is not theory of dependent types, but theory of simple types.
Isabelle/ZF would be a better example as it encodes Zermelo–Fraenkel set theory.
And formalised mathematics are ignored in mathematical practice by most mathematicians, and even the ones that know of it often don't use that as a foundational language due to relative inconvenience.
I think at this stage, most mathematicians recognize that formal proof verification is a real and interesting thing. We have extremely prominent mathematicians like Scholze & Tao making a point of using these tools.
But in many cases, it's extra effort for not much reward. The patterns which most mathemematicians are interested in are (generally) independent of the particular foundations used to realize them. Whether one invests the effort into formal verification depends on how hard the argument is and how crucial the theorem.
This is enormously off topic but if one person sees this and ends up not missing the show then it was worth mentioning. I think there’s a reasonable crossover between math, discrete math, hacking, and mathcore :)
If you wanted to dive in, here is a Jazz version of one of their singles. The real version is 2 guitars, 1 bass, 1 drums, and someone screaming into the microphone -- its real rad.
I think this is pretty common for Quanta, and it might be sticking out more because it's a field we're familiar with.
I'm really torn about this, because I think they're providing a valuable service. But their general formula doesn't diverge a whole lot from run-of-the-mill pop-science books: a vague, clickbaity title and then an article that focuses on personalities and implications of discoveries while glancing over a lot of important details (and not teaching much).
I agree with our assessment of Quanta. I used to enjoy reading their articles, but the clickbait title formula has put me off. Also their status as a mouthpiece of the Simons foundation grantees.
I feel like I’m being a bit curmudgeonly, but I don’t read them much any more.
Initially I too thought - but we try to approximate infinity in CS all the time.
But I have come to think, well actually, approximate is doing some heavy lifting there AND I have never used infinity for anything except to say "look I don't know how high this should go, so go as far as you can go, and double that, which is really saying, you are bound by finite boundaries, you'll have to work within them, and the uncountable thing that I was thinking about is really finite.
Edit: Think of it like this
We know that it's most likely that the universe is infinite, but we can only determine how big it is by how far we can see, which is bounded by the speed of light, and the fact that we can only see matter emitting light (I'm being careful here, if the big bang theory is right, and I am understanding it correctly, there is a finite amount of matter in the universe, but the universe itself is infinite)
Computer science isn't concerned with the uncountable infinite though, which is what measure theory is mostly concerned with as countable sets all have measure zero.
What? You can find tons of textbook examples when dealing with the uncountable sets from non-deterministic finite automata. Those frequently deal with power sets (like when you have an infinite alphabet) which are quintessentially uncountable
Every non halting program with partial results correlates to a real number, and this sort of thing is talked about in theoretical C's all the time. Broadly speaking program analysis is akin to statements on the reals. The fields are the same. Even most day to day computer programming is ultimately reliant on the same kind of reasoning
I studied math for a long time.
I’m convinced math would be better without infinity. It doesn’t exist. I also think we don’t need numbers too big . But we can leave those
I haven't studied math beyond what was needed for my engineering courses.
However, I also am starting to believe that infinity doesn't exist.
Or more specifically, I want to argue that infinity is not a number, it is a process. When you say {1, 2, 3, ... } the "..." represents a process of extending the set without a halting condition.
There is no infinity at the end of a number line. There is a process that says how to extend that number line ever further.
There is no infinity'th prime number. There is a process by which you can show that a bigger primer number must always exist.
The statement and proof of the theorem are quite accessible and eye-opening. I think the number line with ordinals is way cooler than the one without them.
Whether you think infinity exists or not is up to you, but transfinite mathematics is very useful, it's used to prove theorems like Goodstein's sequence in a surprisingly elegant way. This sequence doesn't really have anything to do with infinity as first glance.
Actually, all numbers are functions in Peano arithmetic. :)
For example, S(0) is 1, S(S(0)) is 2, S(S(S(0))) is 3, and so on.
There is no end of a number line. There are lines, and line segments. Only line segments are finite.
> There is no infinity'th prime number. There is a process by which you can show that a bigger primer number must always exist.
You misunderstand the concept of infinity. Cantor's diagonal argument proves that such a bigger number must always exist. "Infinity'th" is not a place in a number line; Infinity is a set that may be countable or uncountable, depending on what kind of infinity you're working with.
There are infinities with higher cardinality than others. Infinity relates to set theory, and if you try to simply imagine it as a "position" in a line of real numbers, you'll understandably have an inconsistent mental model.
I highly recommend checking out Cantor's diagonal argument. Mathematicians didn't invent infinity as a curiosity; it solves real problems and implies real constraints. https://en.wikipedia.org/wiki/Cantor's_diagonal_argument
Numbers don't exist, either. Not really. Any 'one' thing you can point to, it's not just one thing in reality. There's always something leaking, something fuzzy, something not accounted for in one. One universe, even? We can't be sure other universes aren't leaking into our one universe. One planet? What about all the meteorites banging into the one, so it's not one. So, numbers don't exist in the real world, any more than infinity does. Mathematics is thus proved to be nothing but a figment of the human imagination. And no, the frequency a supercooled isolated atom vibrates at in an atomic clock isn't a number either, there's always more bits to add to that number, always an error bar on the measurement, no matter how small. Numbers aren't real.
Why is it that when I have a stack of business cards, each with a picture of a different finger on my left hand, then when I arrange them in a grid, there’s only one way to do it,
but when I instead have each have a picture of either a different finger from either of my left or right hand, there is now two different arrangements of the cards in a grid?
I claim the reason is that 5 is prime, while 10 is composite (10 = 5 times 2).
Wouldn't it follow that those "things" we're pointing to aren't really "things" because they're all leaking and fuzzy? Begging the question, what ends up on a list of things that do exist?
It turns out rather ok without actual infinity, by limiting oneself to potential infinity. Think a Turing Machine where every time it reaches the end of its tape, an operator ("tape ape") will come and put in another reel.
> I’m convinced math would be better without infinity. It doesn’t exist.
I agree finitism and ultrafinitism are worth more development. Infinity is a hack in the sense that it effectively represents a program that doesn't halt, and when you start thinking of it this way, then all sorts of classical mathematical arguments start looking a little fishy, at least when applying math to the real world. For instance, I do think this is at the heart of some confusions and difficulties in physics.
Math isn't concerned with what exists or not, circles doesn't exist in reality either. And infinity is useful, if we didn't have it we would need to reinvent it in a clumsier form, like "the limit if x goes to a very, very big number, bigger than anything you could name".
Infinity is simply the name for a process that does not end. Some of these processes increase faster than others (hence larger infinities). Personally, I also like the Riemann sphere model of infinity.
I don't think this is a good description of mathematicians using infinite objects. It typically doesn't involving something being given the label "infinity". That's used in like, taking limits and such, but there it is just a notation.
When mathematicians are working with infinite objects, it is not by plugging in "infinity" somewhere a number should go, in order to imagine that the rule that would construct an object if that were a number, constructs an object. No. Rather, (in a ZF-like foundations) the axiom of infinity assumes that there is a set whose elements are exactly the finite ordinals. (Or, assumes something equivalent to that.) From this, various sets are constructed such that the set is e.g. in bijection with a proper subset of itself (there are a handful of different definitions of a set being "infinite", which under the axiom of choice, are equivalent, but without AoC there are a few different senses of a set being "infinite", which is why I say "e.g.").
In various contexts in mathematics, there are properties relevant to that specific context which correspond to this notion, and which are therefore also given the name "infinite". For example, in the context of von Neumann algebras, a projection is called a "finite projection" if there is no strict subprojection that is Morray-von Neumann equivalent to it. Or, in the context of ordered fields, an element may be called "infinite" if it is greater than every natural number.
Usually, the thing that is said is that some object is "infinite", with "infinite" an adjective, not saying that some object "is infinity" with "infinity" a noun. One exception I can think of is in the context of the Surreal Numbers, in which the Gap between finite Surreal Numbers and (positive) infinite Surreal Numbers, is given the name "infinity". But usually objects are not given the name "infinity", except as like, a label for an index, but this is just a label.
I suppose that in the Riemann sphere, and other one-point compactifications, one calls the added point "the point at infinity". But, this kind of construction isn't more "make believe" than other things; one can do the same "add in a point at infinity" for finite fields, as in, one can take the projective line for a finite field, which adds a point that is doing the same thing as the "point at infinity" in the complex projective line (i.e. the Riemann sphere).
Math is a tool for structured reasoning. We often want to reason about things that physically exist. Why should we use a tool that assumes the existence of something that literally cannot physically exist? That introduces the danger that of admitting all sorts of unphysical possibilities into our theories.
I think Baez's paper, Struggles with the Continuum, shows a lot of past difficulties we've had that resulted from this:
Is this a joke or are you deeply interested in some ZFC variant that im unaware of? We absolutely need infinity to make a ton of everyday tools work, its like saying we dont need negative numbers because those dont exist either.
A ZFC variant without infinity is basically just PA. (Because you can encode finite sets as natural numbers.) Which in practice is plenty enough to do a whole lot of interesting mathematics. OTOH by the same token, the axiom of infinity is genuinely of interest even in pure finitary terms, because it may provide much simpler proofs of at least some statements that can then be asserted to also be valid in a finitary context due to known conservation results.
In a way, the axiom of infinity seems to behave much like other axioms that assert the existence of even larger mathematical "universes": it's worth being aware of what parts of a mathematical development are inherently dependent on it as an assumption, which is ultimately a question of so-called reverse mathematics.
There’s tons of variants of ZFC without the “infinity”. Constructivism has a long and deep history in mathematics and it’s probably going to become dominant in the future.
To echo the sibling comment, that answer is specifically referring to the type theory behind Lean (which I’ve heard is pretty weird in a lot of ways, albeit usually in service of usability). Many type theories are weaker than ZFC, or even ZF, at least if I correctly skimmed https://proofassistants.stackexchange.com/a/1210/7.
"the study of how to organize abstract collections of objects" is not really a great explanation of set theory. But it is true that the usual way (surely not the only way) to formalize mathematics is starting with set-theoretic axioms and then defining everything in terms of sets.
Sure, but is discarding Type Theory and Category Theory really fair with a phrase like "All of modern mathematics"? Especially in terms of a connection with computer science.
Arguably, type theory is more influential, as it seems to me all the attempts to actually formalize the hand-wavy woo mathematicians tend to engage in are in lean, coq, or the like. We've pretty much given up on set theory except to prove things to ourselves. However, these methods are notoriously unreliable.
Regardless of the name, descriptive set theory does not seem to have all that much to do with "set theory" in a foundational sense; it can be recast in terms of types and spaces with comparative ease, and this can be quite advantageous. The article is a bit confusing in many ways; among other things, it seems quite obviously wrong to suggest that recasting a concept that seems to be conventionally related to mathematical "infinities" in more tangible computational terms is something deeply original to this particular work; if anything, it happens literally all the time when trying to understand existing math in type-theoretic or constructive terms.
Most modern mathematicians are not set theorists. There are certain specialists in metamathematics and the foundations of mathematics who hold that set theory is the proper foundation -- thus that most mathematical structures are rooted in set theory, and can be expressed as extensions of set theory -- but this is by no means a unanimous view! It's quite new, and quite heavily contested.
Confused why the article author believes this is a surprise. The foundations of mathematics and computer science are basically the same subject (imho) and dualities between representations in both fields have been known for decades.
If something people dedicated multiple years of their lives studying and proving seems trivial to you, you're either a genius or you didn't understand the problem.
There is a world of difference between the obvious and the trivial. The post you are replying to only implied it was obvious, the retort is unnecessary.
The notion of algorithms and computer science were invented to discuss the behavior of infinite sequences: examining if there’s descriptions of real numbers. This was extended by connecting complicated infinite structures like the hyperreals with decisions theory problems.
That descriptions of other infinite sets also corresponds to some kind of algorithm seems like a natural progression.
That it happens to be network theory algorithms rather than (eg) decision theory algorithms is worth noting — but hardly surprising. Particularly because the sets examined arose from graph problems, ie, a network.
One, in theory, can construct number sets (fields) with holes in them - that's truly discrete numbers. such number sets are at most countably infinite, but need not be.
One useful such set might have Planck's (length) constant as its smallest number beyond which there is a hole. The problem with using such number sets is that ordinary rules of arithmetic breakdown, ie division has to be defined as modulus Planck constant
You can define an additive group $\frac{1}{n}\mathbb{Z}$ if you like. However (for $n > 1$) it would not even be a ring, because it would not be closed under multiplication. (It's closure under multiplication would be $\mathbb{Z}[\frac{1}{n}]$, which would not have a smallest positive element, contrary to your design criterion.)
(Of course, you could define a partial multiplication on it. I don't think there's a good name for such a thing. I guess you could just call it "a subgroup of the rational numbers under addition, equipped with a partial multiplication operation that is defined and agrees with the usual multiplication on rational numbers when the result would still be in the subgroup")
The field Qp of p-adic numbers is complete with respect to the p-adic norm, but is not ordered in the same sense as field of real numbers. It's still uncountable infinite. If there is a sense in which "gaps" or Holes can be introduced without breaking its completeness, that would make it very useful for modeling reality
When you talk about infinity, you are no longer talking about numbers. Mix it with numbers, you get all sorts of perplexing theories and paradoxes.
The reason is simple - numbers are cuts in the continuum while infinity isn't. It should not even be a symbolic notion of very large number. This is not to say infinity doesn't exist. It doesn't exist as a number and doesn't mix with numbers.
The limits could have been defined saying "as x increases without bound" instead of "as x approaches infinity". There is no target called infinity to approach.
Cantor's stuff can easily be trashed. The very notion of "larger than" belongs to finite numbers. This comparitive notion doesn't apply to concepts that can not be quantified using numbers. Hence one can't say some kind of infinity is larger than the other kinds.
Similarly, his diagonal argument about 1-to-1 mapping can not be extended to infinities, as there is no 1-to-1 mapping that can make sense for those which are not numbers or uniquely identifiable elements. The mapping is broken. No surprise you get weird results and multiple infinities or whatever that came to his mind when he was going through stressful personal situations.
I don't think you can just call Cantor's diagonal argument trash without providing a very strong, self-consistent replacement framework that explains the result without invoking infinity.
This sounds like ranting from someone who doesn't deeply understand the implications of set theory or infinite sets. Cardinality is a real thing, with real consequences, such as Gödel's incompleteness theorems. What "weird results" are tripping you up?
> when he was going through stressful personal situations
Ad hominem. Let's stick to arguing the subject material and not personally attacking Cantor.
The Löwenheim-Skolem theorem implies that a countable model of set theory has to exist. "Cardinality" as implied by Cantor's diagonal argument (which happens to be a straightforward special case of Lawvere's fixed point theorem) is thus not an absolute property: it's relative to a particular model. It's internally true that there is no bijection as defined within the model between the naturals and the reals, as shown by Cantor's argument; but externally there are models where all sets can nonetheless be seen as countable.
I just gave the reason - The notion of comparison and 1-to-1 mapping has an underlying assumption about the subjects being quantifiable and identifiable. This assumption doesn't apply to something inherently neither quantifiable nor is a cut in the continuum, similar to a number. What argument are you offering against this?
The difference between infinities is I can write every possible fraction on a piece of A4 paper, if the font gets smaller and smaller. I can say where to zoom in for any fraction.
You can't enumerate the real numbers, but you can grab them all in one go - just draw a line!
The more I learn about this stuff, the more I come to understand how the quantitative difference between cardinalities is a red herring (e.g. CH independent from ZFC). It's the qualitative difference between these two sets that matter. The real numbers are richer, denser, smoother, etc. than the natural numbers, and those are the qualities we care about.
That doesn't make one set "larger" than the other. You need to define "larger". And you need to make that definition as weird as needed to justify that comparison.
.. because that is where you are allowed to challenge some biblical stories of the math without the fear of expulsion from the elite clubs.
Most of math history is stellar, studded with great works of geniuses, but some results were sanctified and prohibited for questioning due to various forces that were active during the times.
Application of regular logic such as comparison, mapping, listing, diagonals, uniqueness - all are the rules that were bred in the realms of finiteness and physical world. You can't use these things to prove some theories about things are not finite.
…that existed in the world of descriptive set theory — about the number of colors required to color certain infinite graphs in a measurable way.
To Bernshteyn, it felt like more than a coincidence. It wasn’t just that computer scientists are like librarians too, shelving problems based on how efficiently their algorithms work. It wasn’t just that these problems could also be written in terms of graphs and colorings.
Perhaps, he thought, the two bookshelves had more in common than that. Perhaps the connection between these two fields went much, much deeper.
Perhaps all the books, and their shelves, were identical, just written in different languages — and in need of a translator.
This might be a dumb question but are there any current or foreseeable practical applications of this kind of result (like in the realm of distributed computing) or is this just pure mathematics for its own sake?
Not a dumb question. The links to mesh networking etc seem interesting. It sounds like the insights from descriptive set theory could yield new hardness/impossibility results in computational complexity, distributed algorithms etc.
are there any current or foreseeable practical applications of those results?
the math of infinity isn't very relevant to the finite programs that humans use. Even today's astronomically large computing systems have size approximately equal to 3 compared to infinity.
First sentence:
> All of modern mathematics is built on the foundation of set theory
That's ignoring most of formalized mathematics, which is progressing rapidly and definitely modern. Lean and Rocq for example are founded on type theory, not set theory.
I think what is more accurate to say is that the broad crisis in foundations of mathematics was initially resolved by formalizing set theory.
Not a mathematician, but AFAIK ZFC is a valid foundation. Dependent types helps a lot with bookkeeping, but cannot prove more theorems.
Lawrence Paulson is a great person to clarify those topics (Isabelle/HOL is not based on types yet it can proof most maths).
> but cannot prove more theorems
usually you're more interested in better ergonomics: can you do X with less work?
it's like picking a programming language - depending on what you're attempting, some will be more helpful.
and ZFC is a lot more low level than what day-to-day mathematics usually bothers with. So most mathematians actually work in an informally understood higher-order wrapper, hoping that what they write sufficiently explains the actual "machine code"
the idea then behind adopting alternative foundations is that these come with "batteries included" and map more directly to the domain language.
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Isabelle/HOL is still types. The underlying type theory of Isabelle/HOL is not theory of dependent types, but theory of simple types. Isabelle/ZF would be a better example as it encodes Zermelo–Fraenkel set theory.
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It’s not really. It leads to paradoxes. There are alternative formulations which avoid these.
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And formalised mathematics are ignored in mathematical practice by most mathematicians, and even the ones that know of it often don't use that as a foundational language due to relative inconvenience.
I think at this stage, most mathematicians recognize that formal proof verification is a real and interesting thing. We have extremely prominent mathematicians like Scholze & Tao making a point of using these tools.
But in many cases, it's extra effort for not much reward. The patterns which most mathemematicians are interested in are (generally) independent of the particular foundations used to realize them. Whether one invests the effort into formal verification depends on how hard the argument is and how crucial the theorem.
You are conflating two things - the language of math, and the language of proving facts about the language of math.
The former is almost entirely sets, while the later is necessarily types.
It's cons'es all the way down.
Finally - we can calculate infinity.
Been a long way towards it. \o/
If you’re in the Bay Area next month then The Dillinger Escape Plan are bringing the Calculating Infinity circus to town(s):
https://www.theregencyballroom.com/events/detail/?event_id=1...
This is enormously off topic but if one person sees this and ends up not missing the show then it was worth mentioning. I think there’s a reasonable crossover between math, discrete math, hacking, and mathcore :)
Took me an embarrassingly long time to realize you're talking about a music event not a math lecture
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If you wanted to dive in, here is a Jazz version of one of their singles. The real version is 2 guitars, 1 bass, 1 drums, and someone screaming into the microphone -- its real rad.
https://www.youtube.com/watch?v=D4-erceTpc8
Fairly trivial, in haskell:
let x = x in x
Completely encapsulates a countable infinity.
>Finally - we can calculate infinity.
And Beyond!
∞/1 + 1/∞
One infinitesimal step for the numerical, one giant step for mathkind.
Chuck Norris counted from one to infinity. Twice.
Come on, now you know better than that. It was infinity that counted to Chuck Norris
This has indeed taken forever /s
> computer science with the finite
um... no... computer science is very concerned with the infinite. I'm surprised quanta published this. I always think highly of their reporting.
I think this is pretty common for Quanta, and it might be sticking out more because it's a field we're familiar with.
I'm really torn about this, because I think they're providing a valuable service. But their general formula doesn't diverge a whole lot from run-of-the-mill pop-science books: a vague, clickbaity title and then an article that focuses on personalities and implications of discoveries while glancing over a lot of important details (and not teaching much).
I agree with our assessment of Quanta. I used to enjoy reading their articles, but the clickbait title formula has put me off. Also their status as a mouthpiece of the Simons foundation grantees.
I feel like I’m being a bit curmudgeonly, but I don’t read them much any more.
Initially I too thought - but we try to approximate infinity in CS all the time.
But I have come to think, well actually, approximate is doing some heavy lifting there AND I have never used infinity for anything except to say "look I don't know how high this should go, so go as far as you can go, and double that, which is really saying, you are bound by finite boundaries, you'll have to work within them, and the uncountable thing that I was thinking about is really finite.
Edit: Think of it like this
We know that it's most likely that the universe is infinite, but we can only determine how big it is by how far we can see, which is bounded by the speed of light, and the fact that we can only see matter emitting light (I'm being careful here, if the big bang theory is right, and I am understanding it correctly, there is a finite amount of matter in the universe, but the universe itself is infinite)
Asymptotics is a good example of CS using infinity practically.
Also, a small aside: there’s a finite amount of matter in the visible universe. We could have infinite matter in an infinite universe.
Computer science isn't concerned with the uncountable infinite though, which is what measure theory is mostly concerned with as countable sets all have measure zero.
What? You can find tons of textbook examples when dealing with the uncountable sets from non-deterministic finite automata. Those frequently deal with power sets (like when you have an infinite alphabet) which are quintessentially uncountable
Every non halting program with partial results correlates to a real number, and this sort of thing is talked about in theoretical C's all the time. Broadly speaking program analysis is akin to statements on the reals. The fields are the same. Even most day to day computer programming is ultimately reliant on the same kind of reasoning
Another glaring example:
> Set theorists use the language of logic, computer scientists the language of algorithms.
Computer science doesn’t use logic? Hello, Booleans.
So lazy, especially when you can ask an AI to tell you if you’re saying something stupid.
[dead]
That’s nothing, ‘node_modules’ has been linking the math of infinity to my filesystem for years.
I studied math for a long time. I’m convinced math would be better without infinity. It doesn’t exist. I also think we don’t need numbers too big . But we can leave those
I haven't studied math beyond what was needed for my engineering courses.
However, I also am starting to believe that infinity doesn't exist.
Or more specifically, I want to argue that infinity is not a number, it is a process. When you say {1, 2, 3, ... } the "..." represents a process of extending the set without a halting condition.
There is no infinity at the end of a number line. There is a process that says how to extend that number line ever further.
There is no infinity'th prime number. There is a process by which you can show that a bigger primer number must always exist.
> There is no infinity at the end of a number line. There is a process that says how to extend that number line ever further.
Sure, but ordinal numbers exist and are useful. It's impossible to prove Goodstein's theorem without them.
https://en.wikipedia.org/wiki/Ordinal_number
https://en.wikipedia.org/wiki/Goodstein%27s_theorem
The statement and proof of the theorem are quite accessible and eye-opening. I think the number line with ordinals is way cooler than the one without them.
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Whether you think infinity exists or not is up to you, but transfinite mathematics is very useful, it's used to prove theorems like Goodstein's sequence in a surprisingly elegant way. This sequence doesn't really have anything to do with infinity as first glance.
Actually, all numbers are functions in Peano arithmetic. :)
For example, S(0) is 1, S(S(0)) is 2, S(S(S(0))) is 3, and so on.
There is no end of a number line. There are lines, and line segments. Only line segments are finite.
> There is no infinity'th prime number. There is a process by which you can show that a bigger primer number must always exist.
You misunderstand the concept of infinity. Cantor's diagonal argument proves that such a bigger number must always exist. "Infinity'th" is not a place in a number line; Infinity is a set that may be countable or uncountable, depending on what kind of infinity you're working with.
There are infinities with higher cardinality than others. Infinity relates to set theory, and if you try to simply imagine it as a "position" in a line of real numbers, you'll understandably have an inconsistent mental model.
I highly recommend checking out Cantor's diagonal argument. Mathematicians didn't invent infinity as a curiosity; it solves real problems and implies real constraints. https://en.wikipedia.org/wiki/Cantor's_diagonal_argument
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Numbers don't exist, either. Not really. Any 'one' thing you can point to, it's not just one thing in reality. There's always something leaking, something fuzzy, something not accounted for in one. One universe, even? We can't be sure other universes aren't leaking into our one universe. One planet? What about all the meteorites banging into the one, so it's not one. So, numbers don't exist in the real world, any more than infinity does. Mathematics is thus proved to be nothing but a figment of the human imagination. And no, the frequency a supercooled isolated atom vibrates at in an atomic clock isn't a number either, there's always more bits to add to that number, always an error bar on the measurement, no matter how small. Numbers aren't real.
Why is it that when I have a stack of business cards, each with a picture of a different finger on my left hand, then when I arrange them in a grid, there’s only one way to do it, but when I instead have each have a picture of either a different finger from either of my left or right hand, there is now two different arrangements of the cards in a grid?
I claim the reason is that 5 is prime, while 10 is composite (10 = 5 times 2).
Therefore, 5 and 10, and 2, exist.
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Wouldn't it follow that those "things" we're pointing to aren't really "things" because they're all leaking and fuzzy? Begging the question, what ends up on a list of things that do exist?
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Infinity doesn't exist, and neither does 4, or even a triangle. Everything is a concept or an approximation.
A triangle exists in physics.
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Doing math without it is pretty ugly.
(Granted, there are other objects that may seem ugly which can only be constructed by reference to infinite things.)
You can do a lot of things without assuming that the natural numbers keep going, but it is plain awful to work with.
It turns out rather ok without actual infinity, by limiting oneself to potential infinity. Think a Turing Machine where every time it reaches the end of its tape, an operator ("tape ape") will come and put in another reel.
I think we can stop at 8.
7, if the Extremely Strong Goldbach Conjecture holds. [1]
[1] https://xkcd.com/1310/
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> I’m convinced math would be better without infinity. It doesn’t exist.
I agree finitism and ultrafinitism are worth more development. Infinity is a hack in the sense that it effectively represents a program that doesn't halt, and when you start thinking of it this way, then all sorts of classical mathematical arguments start looking a little fishy, at least when applying math to the real world. For instance, I do think this is at the heart of some confusions and difficulties in physics.
Math isn't concerned with what exists or not, circles doesn't exist in reality either. And infinity is useful, if we didn't have it we would need to reinvent it in a clumsier form, like "the limit if x goes to a very, very big number, bigger than anything you could name".
Infinity is simply the name for a process that does not end. Some of these processes increase faster than others (hence larger infinities). Personally, I also like the Riemann sphere model of infinity.
How would you handle things like probability distributions with infinite support?
Infinity is a hack.
It is very tempting to use it in place of a number, and so mathematicians (being humans) did that.
Yes, mathematicians hack too. Maybe they even invented it.
I don't think this is a good description of mathematicians using infinite objects. It typically doesn't involving something being given the label "infinity". That's used in like, taking limits and such, but there it is just a notation.
When mathematicians are working with infinite objects, it is not by plugging in "infinity" somewhere a number should go, in order to imagine that the rule that would construct an object if that were a number, constructs an object. No. Rather, (in a ZF-like foundations) the axiom of infinity assumes that there is a set whose elements are exactly the finite ordinals. (Or, assumes something equivalent to that.) From this, various sets are constructed such that the set is e.g. in bijection with a proper subset of itself (there are a handful of different definitions of a set being "infinite", which under the axiom of choice, are equivalent, but without AoC there are a few different senses of a set being "infinite", which is why I say "e.g.").
In various contexts in mathematics, there are properties relevant to that specific context which correspond to this notion, and which are therefore also given the name "infinite". For example, in the context of von Neumann algebras, a projection is called a "finite projection" if there is no strict subprojection that is Morray-von Neumann equivalent to it. Or, in the context of ordered fields, an element may be called "infinite" if it is greater than every natural number.
Usually, the thing that is said is that some object is "infinite", with "infinite" an adjective, not saying that some object "is infinity" with "infinity" a noun. One exception I can think of is in the context of the Surreal Numbers, in which the Gap between finite Surreal Numbers and (positive) infinite Surreal Numbers, is given the name "infinity". But usually objects are not given the name "infinity", except as like, a label for an index, but this is just a label.
I suppose that in the Riemann sphere, and other one-point compactifications, one calls the added point "the point at infinity". But, this kind of construction isn't more "make believe" than other things; one can do the same "add in a point at infinity" for finite fields, as in, one can take the projective line for a finite field, which adds a point that is doing the same thing as the "point at infinity" in the complex projective line (i.e. the Riemann sphere).
Some people have this view, but I don't get why.
Math is a tool for structured reasoning. We often want to reason about things that physically exist. Why should we use a tool that assumes the existence of something that literally cannot physically exist? That introduces the danger that of admitting all sorts of unphysical possibilities into our theories.
I think Baez's paper, Struggles with the Continuum, shows a lot of past difficulties we've had that resulted from this:
https://arxiv.org/abs/1609.01421
Me neither. David Deutsch had some interesting thoughts on why finitists are wrong in TBOI but I never fully understood it.
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Is this a joke or are you deeply interested in some ZFC variant that im unaware of? We absolutely need infinity to make a ton of everyday tools work, its like saying we dont need negative numbers because those dont exist either.
A ZFC variant without infinity is basically just PA. (Because you can encode finite sets as natural numbers.) Which in practice is plenty enough to do a whole lot of interesting mathematics. OTOH by the same token, the axiom of infinity is genuinely of interest even in pure finitary terms, because it may provide much simpler proofs of at least some statements that can then be asserted to also be valid in a finitary context due to known conservation results.
In a way, the axiom of infinity seems to behave much like other axioms that assert the existence of even larger mathematical "universes": it's worth being aware of what parts of a mathematical development are inherently dependent on it as an assumption, which is ultimately a question of so-called reverse mathematics.
There are a couple philosophies in that vein, like finitism or constructivism. Not exactly mainstream but they’ve proven more than you’d expect
https://en.wikipedia.org/wiki/Finitism
https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...
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There’s tons of variants of ZFC without the “infinity”. Constructivism has a long and deep history in mathematics and it’s probably going to become dominant in the future.
> All of modern mathematics is built on the foundation of set theory, the study of how to organize abstract collections of objects
What the hell. What about Type Theory?
Type theory is actually a stronger axiomatic system than ZFC, and is equiconsistent with ZFC+ a stronger condition.
See this mathoverflow response here https://mathoverflow.net/a/437200/477593
To echo the sibling comment, that answer is specifically referring to the type theory behind Lean (which I’ve heard is pretty weird in a lot of ways, albeit usually in service of usability). Many type theories are weaker than ZFC, or even ZF, at least if I correctly skimmed https://proofassistants.stackexchange.com/a/1210/7.
"the study of how to organize abstract collections of objects" is not really a great explanation of set theory. But it is true that the usual way (surely not the only way) to formalize mathematics is starting with set-theoretic axioms and then defining everything in terms of sets.
"Usual", "most common by far" etc. are all great phrases, but not "all of mathematics", esp when we talk about math related to computer science
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Is there a collection of type theory axioms anywhere near as influential as ZF or ZFC?
Sure, but is discarding Type Theory and Category Theory really fair with a phrase like "All of modern mathematics"? Especially in terms of a connection with computer science.
Arguably, type theory is more influential, as it seems to me all the attempts to actually formalize the hand-wavy woo mathematicians tend to engage in are in lean, coq, or the like. We've pretty much given up on set theory except to prove things to ourselves. However, these methods are notoriously unreliable.
Regardless of the name, descriptive set theory does not seem to have all that much to do with "set theory" in a foundational sense; it can be recast in terms of types and spaces with comparative ease, and this can be quite advantageous. The article is a bit confusing in many ways; among other things, it seems quite obviously wrong to suggest that recasting a concept that seems to be conventionally related to mathematical "infinities" in more tangible computational terms is something deeply original to this particular work; if anything, it happens literally all the time when trying to understand existing math in type-theoretic or constructive terms.
the empirical modern mathematics are build on set theory, type and category theory are just other possible foundations
Most modern mathematicians are not set theorists. There are certain specialists in metamathematics and the foundations of mathematics who hold that set theory is the proper foundation -- thus that most mathematical structures are rooted in set theory, and can be expressed as extensions of set theory -- but this is by no means a unanimous view! It's quite new, and quite heavily contested.
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That’s nothing wait til you see my math.
Confused why the article author believes this is a surprise. The foundations of mathematics and computer science are basically the same subject (imho) and dualities between representations in both fields have been known for decades.
If something people dedicated multiple years of their lives studying and proving seems trivial to you, you're either a genius or you didn't understand the problem.
There is a world of difference between the obvious and the trivial. The post you are replying to only implied it was obvious, the retort is unnecessary.
It's not a surprise that math and CS are related. It's a surprise that this particular subject in math and CS are so intimately related.
Why? Both are basically about how to handle information with a priori indefinitely many cases, types, instances, species
Is it?
The notion of algorithms and computer science were invented to discuss the behavior of infinite sequences: examining if there’s descriptions of real numbers. This was extended by connecting complicated infinite structures like the hyperreals with decisions theory problems.
That descriptions of other infinite sets also corresponds to some kind of algorithm seems like a natural progression.
That it happens to be network theory algorithms rather than (eg) decision theory algorithms is worth noting — but hardly surprising. Particularly because the sets examined arose from graph problems, ie, a network.
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Reactions to your statement seem to hinge on whether or not the person replying has a CS degree from a math focused program!
It makes perfect sense to me.
True. After Turing and Co, mathematical logic became almost like applied theoretical computer science.
One, in theory, can construct number sets (fields) with holes in them - that's truly discrete numbers. such number sets are at most countably infinite, but need not be. One useful such set might have Planck's (length) constant as its smallest number beyond which there is a hole. The problem with using such number sets is that ordinary rules of arithmetic breakdown, ie division has to be defined as modulus Planck constant
Such a thing would not be a field.
You can define an additive group $\frac{1}{n}\mathbb{Z}$ if you like. However (for $n > 1$) it would not even be a ring, because it would not be closed under multiplication. (It's closure under multiplication would be $\mathbb{Z}[\frac{1}{n}]$, which would not have a smallest positive element, contrary to your design criterion.)
(Of course, you could define a partial multiplication on it. I don't think there's a good name for such a thing. I guess you could just call it "a subgroup of the rational numbers under addition, equipped with a partial multiplication operation that is defined and agrees with the usual multiplication on rational numbers when the result would still be in the subgroup")
The field Qp of p-adic numbers is complete with respect to the p-adic norm, but is not ordered in the same sense as field of real numbers. It's still uncountable infinite. If there is a sense in which "gaps" or Holes can be introduced without breaking its completeness, that would make it very useful for modeling reality
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When you talk about infinity, you are no longer talking about numbers. Mix it with numbers, you get all sorts of perplexing theories and paradoxes.
The reason is simple - numbers are cuts in the continuum while infinity isn't. It should not even be a symbolic notion of very large number. This is not to say infinity doesn't exist. It doesn't exist as a number and doesn't mix with numbers.
The limits could have been defined saying "as x increases without bound" instead of "as x approaches infinity". There is no target called infinity to approach.
Cantor's stuff can easily be trashed. The very notion of "larger than" belongs to finite numbers. This comparitive notion doesn't apply to concepts that can not be quantified using numbers. Hence one can't say some kind of infinity is larger than the other kinds.
Similarly, his diagonal argument about 1-to-1 mapping can not be extended to infinities, as there is no 1-to-1 mapping that can make sense for those which are not numbers or uniquely identifiable elements. The mapping is broken. No surprise you get weird results and multiple infinities or whatever that came to his mind when he was going through stressful personal situations.
I don't think you can just call Cantor's diagonal argument trash without providing a very strong, self-consistent replacement framework that explains the result without invoking infinity.
This sounds like ranting from someone who doesn't deeply understand the implications of set theory or infinite sets. Cardinality is a real thing, with real consequences, such as Gödel's incompleteness theorems. What "weird results" are tripping you up?
> when he was going through stressful personal situations
Ad hominem. Let's stick to arguing the subject material and not personally attacking Cantor.
https://en.wikipedia.org/wiki/Cantor's_diagonal_argument#Con...
The Löwenheim-Skolem theorem implies that a countable model of set theory has to exist. "Cardinality" as implied by Cantor's diagonal argument (which happens to be a straightforward special case of Lawvere's fixed point theorem) is thus not an absolute property: it's relative to a particular model. It's internally true that there is no bijection as defined within the model between the naturals and the reals, as shown by Cantor's argument; but externally there are models where all sets can nonetheless be seen as countable.
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I just gave the reason - The notion of comparison and 1-to-1 mapping has an underlying assumption about the subjects being quantifiable and identifiable. This assumption doesn't apply to something inherently neither quantifiable nor is a cut in the continuum, similar to a number. What argument are you offering against this?
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The difference between infinities is I can write every possible fraction on a piece of A4 paper, if the font gets smaller and smaller. I can say where to zoom in for any fraction.
I can't do that for real numbers.
You can't enumerate the real numbers, but you can grab them all in one go - just draw a line!
The more I learn about this stuff, the more I come to understand how the quantitative difference between cardinalities is a red herring (e.g. CH independent from ZFC). It's the qualitative difference between these two sets that matter. The real numbers are richer, denser, smoother, etc. than the natural numbers, and those are the qualities we care about.
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That doesn't make one set "larger" than the other. You need to define "larger". And you need to make that definition as weird as needed to justify that comparison.
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All very well but I used to say to my sister "I hate you infinity plus one", so how do you account for that?
> Cantor's stuff can easily be trashed.
Only on hackernews.
.. because that is where you are allowed to challenge some biblical stories of the math without the fear of expulsion from the elite clubs.
Most of math history is stellar, studded with great works of geniuses, but some results were sanctified and prohibited for questioning due to various forces that were active during the times.
Application of regular logic such as comparison, mapping, listing, diagonals, uniqueness - all are the rules that were bred in the realms of finiteness and physical world. You can't use these things to prove some theories about things are not finite.
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Emdash all over the article.
It wasn’t just that.. all over the article..
I am getting really strong LLM article vibes.
…that existed in the world of descriptive set theory — about the number of colors required to color certain infinite graphs in a measurable way.
To Bernshteyn, it felt like more than a coincidence. It wasn’t just that computer scientists are like librarians too, shelving problems based on how efficiently their algorithms work. It wasn’t just that these problems could also be written in terms of graphs and colorings. Perhaps, he thought, the two bookshelves had more in common than that. Perhaps the connection between these two fields went much, much deeper. Perhaps all the books, and their shelves, were identical, just written in different languages — and in need of a translator.