8 ms·

I'm not trying to be flippant, although it may come off that way: why does any of this matter?

deleted 5y ago
[deleted]
sillysaurusx 5y ago
It took many years to understand that! It’s not a flippant question at all. The answer is, it doesn’t matter. And that’s the joy of it. It wasn’t until I got into ML that I learned the value of doing unimportant work. When you’re free to think about inconsequential matters very seriously, you end up discovering so many useful things. It was how I independently rediscovered what is apparently called the Discrete Hartley Transform. You have a vague notion that what you’re doing might be important, but it’s not the focus. It matters because it’s fun in a way that nothing else can be. Of course, the more serious folk won’t admit it’s fun. It’s serious work. And they’re not wrong; nobody’s fooling around. Our time is every bit as precious as an executive’s. But their goal is to change the world. Ours — or mine, at least — is to know the truth of a thing that most people don’t know.
- fighterpilot 5y ago
  > it doesn’t matter It doesn't matter yet. Applications of pure math happen downstream decades or centuries later. It's hard to predict the impact.
  - l33t2328 5y ago
    I don’t like this framing because it implies any piece of pure math becoming useful is just a matter of time, and I see no reason to suspect this is the case.
    - bmitc 5y ago
      > I see no reason to suspect this is the case Isn’t history enough reason?
      - erichahn 5y ago
        No history actually tells you that only 1% of produced maths is useful. You only see the good stuff not the ugly stuff.
        bmitc 5y ago
        I don't know why people are throwing around percentages so much in these replies. Percentage of what? Plus, the body of work of mathematics continues to grow all the time. The core of the original comment that started this chain was: > Applications of pure math happen downstream decades or centuries later. My argument was that history shows that a surprising amount of mathematics does eventually trickle down into applications. I don't think anyone is arguing all of mathematics eventually sees an application. Again, my point was that history does indeed seem to show that "applications of pure math happen downstream decades or centuries later". As the original commenter said, it's hard to predict what will be the next thing to be applied.
      - l33t2328 5y ago
        In addition to what the other commenter said, it’s important to understand the scale. Cutting edge research 600 years ago was solving cubic equations. Today, cutting edge research requires potentially years of advanced study to understand the objects involved. There are more mathematicians alive today than at any point in human history, and their work has become specialized to a tremendous degree. Gone are the days where one mathematician could do substantial new work in a dozen diverse areas. So, in short, the fact that historically lots of pure math has not found use, the specialized nature of modern research math, the volume of work being out out, and an irreverence for applications(again, this is _pure_ math) leads me to be doubtful that even a moderate amount of modern pure math research will ever be useful in any practical sense.
      - jonnycomputer 5y ago
        I've read lots of math papers that were a complete waste of time because they were banal. The only "innovation" was to use different words to describe the same thing. New mathematics is actually pretty damn hard I suspect.
    - hawkice 5y ago
      I think it's more about being open minded to strange uses. Maybe 3% of strange pure math ends up being used for something bigger? But since nobody knows what it'll be, it's good to be curious about all sorts of nonsense. Riemannian geometry was OBVIOUSLY absurd useless pure math sophistry, but then it turned out we lived in it. I think a lot of the work on modular forms and elliptic curves has come across that way too, despite the crypto intersection. The general idea is that, even if it's been a century or two, it's too soon to tell if something is useless, that's solid. And formalization of infinity has been useful in some practical exploration of proof systems used by humans. It might be the future of programming, who knows?
  - dfdz 5y ago
    To follow up on this comment, I think that math should be viewed as a giant tree structure or pyramid At the bottom are all sorts of applications of linear algebra and optimization (calculus) are related to machine learning, deep learning (AI), physics, engineering etc. One level up there are all sorts of ideas in algebra and mathematical analysis that are leading to new ideas in linear algebra and optimization (calculus) Going one level higher, there are ideas in mathematical logic and set theory that are improving our understanding our algebra and analysis. If you look at a specific result at the very top of the pyramid as ask why does it matter? It is difficult to give a good answer. Clearly we want to base of the pyramid, but do we need to keep building it up? Should we stop at a certain level?
- just_one_time_ 5y ago
  Lololol. U gay.
- mr_gibbins 5y ago
  I agree entirely. Industrially, I work a role within an IT team. My work is, I hope, useful and strives towards a purpose - my employer's purpose/business model. Do I 'go home' - leave my home office - and think about it? No. Do I muse on how to solve a work-related issue when I'm showering? Also no. Will I forget almost everything about this job as or when I move onto the next? Definitely yes. But academically, I'm interested, and actively publish albeit modest, low-impact research, in relational databases. I love the set-based approach to all matters SQL and after many years in the field still fool around trying to embellish, attack, improve and invent the core ideas. Why bother? Even if I come up with some magical new improvement to RDBMSs, it doesn't matter. If I fork MySQL and try out my ideas, no one cares. I'll never get the traction or the FOSS community support, I'll spend more time playing politics and managing collaborators than I'd like (zero) and I enjoy the independent thought that my interests bring. So my academic work is pointless. But it's meaningful to me. And I think that's what matters. I don't understand one-tenth of the ideas in this article, but that doesn't matter. What matters is that it's interesting to somebody.
Y_Y 5y ago
I think you'd have to say exactly what you mean by "matter". At one point in my life I felt like the continuum hypothesis was the most important question in the world. Now I'm mostly concerned about finite material issues, e.g. my health. I think a good argument can be made why either of them matter much more than the other.
soperj 5y ago
This is kind of the equivalent of asking why does your life matter?
l33t2328 5y ago
It’s interesting, and fun. Most modern pure math is done completely independent of and without regard to applications(if it were, it would be ‘applied math’).
bmitc 5y ago
We know that the cardinality of the natural numbers is less than the cardinality of the real numbers. The Continuum Hypothesis, which is a long unsolved problem, states that there are no sets with cardinality between the two. The posted article states that this new result strengthens the case against the hypothesis, that is that’s it’s probably false. All of this is nuanced but is important to mathematics and philosophy.
- thom 5y ago
  Why does “the set containing the natural numbers and a sandwich” not have cardinality between the two?
  - hawkice 5y ago
    Because it's the same size. I can show it's the same size because I can just change the labels on the sets (label change of elements doesn't change group size) and go back and forth. In this case, 0 is the sandwich, and all the nonnegative numbers get relabelled down by one. Going back, sandwiches become 0 and you add one to nonnegative numbers. So they've got to be the same size, if I can go back and forth and everything just gets a new label and I don't miss anything.
  - helen___keller 5y ago
    You can create a bijection between that set and the set of natural numbers, defined by the function f: 0 <-> sandwich 1 <-> 0 ... n+1 <-> n By definition, if you can biject two sets, they have the same cardinality.
  - magicalist 5y ago
    One way to say two sets of things are of equal size is if we can pair up all the elements of one set with all the elements of the other. If we're ok with extending that to sets of infinite things (we can still pair elements of each set, we'd just never be able to finish listing all the pairs), then we can say that the natural numbers and "the set containing the natural numbers and a sandwich" are of equal size because we could pair 1 from the first set with the sandwich from the second set, pair 2 from the first set with 1 from the second set, 3 from the first set with 2 from the second set, etc etc. There's no element of either set without a match in the other set, so they have the same cardinality as the natural numbers, with or without the sandwich.
pantulis 5y ago
Because this advances math. Same could be said for lambda-calculus and yet here we are with FP.
helen___keller 5y ago
Others have addressed why it matters (or doesn't matter) when viewed from outside mathematics. But within mathematics, unsolved problems usually matter for two reasons: (1) When lots of really smart people spend lots of time trying to solve something, and fail to do so, it becomes even more interesting for other smart people. A well-known example is the Collatz Conjecture[0] which, by most accounts is a meaningless problem, but endlessly fascinates many for its renowned difficulty. (2) In mathematics, you often stumble upon problems that are equivalent to other problems. If X is difficult to solve so you give up and go work on Y, only to find out that Y as a problem is basically the same to X, or they have some other intimate relationship, and it makes X that much more tantalizing. A common example for computer scientists is the whole class of NP complete problems[1] for which we don't know if there exists a polytime algorithm to decide. Every time we discover a new problem that lives in the realm of NP-complete, the P vs NP problem gets a little more tantalizing. [0] https://en.wikipedia.org/wiki/Collatz_conjecture https://en.wikipedia.org/wiki/Collatz_conjecture [1] https://en.wikipedia.org/wiki/NP-completeness https://en.wikipedia.org/wiki/NP-completeness
- sillysaurusx 5y ago
  This is an excellent answer. I have a mostly unrelated question. In the case of the Collatz conjecture, it seems all but proved: > If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. There are lots of histograms and empirical data supporting the conjecture. My question is, why is this empirical data not "good enough" for mathematicians? As something closer to a physicist, I do wonder what the fascination is with trying to prove conclusively that 3n+1 xor n/2 will eventually encounter 1. It seems a bit like trying to prove conclusively that the gravitational constant is so-and-so, when it seems the best you can do is to measure it as precisely as possible: > Jeffrey Lagarias stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics." As someone who loved mathematics but was never much good at it, what's the fascination here? (I'm only trying to understand the motivations of people smarter than I am.)
gorgoiler 5y ago
We don’t know but looking into the past hints at the future. Cantor, Hilbert and Gödel gave us Church who gave us Turing. Turing and Flowers gave us the machines as well as the theory. All of them put together gave us type systems and types are how you formally prove that your 747 software is free of, if not all bugs, then at least certain large classes of error. There is a clear line of connections from Cantor (1890s) to jumbo jet fly-by-wire (1990s.) Countability of sets and Cantor’s diagonal argument — the subject of the first part of this article — are some of the first topics teenagers learn about in high school CS (if you’re lucky and on a very modern course) or CS101 (if you’re at a University.) Types are sets. Who knows what today’s mathematics will bring us in the year 2121?
- mmastrac 5y ago
  I'd love to see James Burke tackle this line.
- lupire 5y ago
  Eh, this argument kind of falls apart with higher infinities, where the numbers you are studying are bigger than the power set of all interactions among all particles in all possible universes.
ssivark 5y ago
Let me venture a possible application, since there are already many responses celebrating pure mathematics. As far as we understand, the natural numbers are not sufficient for modeling physical phenomena. The reals/complex while immensely useful also occasionally turn out to be “too complicated” to give theoretical guarantees/proofs of models working well. On the practical side, that means that these models/algorithms can’t be guaranteed to not give junk results, while working on the domain of real numbers. Now, purely speculatively, since the naturals/rationals are aleph0 in size and the reals are aleph2, that means there likely exists a set of numbers of size aleph1 sitting in between the two. What if we could use that set of numbers to construct our physical models? Could we somehow guarantee better behavior… Eg: in quantum mechanics, or field theory, or chaos, etc?! :-)
- drdeca 5y ago
  Whether the cardinality of the reals is aleph 1, aleph 2, or something larger, is independent of ZFC. In a model of ZFC in which the set of reals has cardinality greater than aleph_1 , I wouldn’t be surprised if there is a subfield of the reals of cardinality aleph_1 , but I would be surprised if such a field was useful for things like that. Such a field would, of course, not be complete with respect to the usual metric on the rationals, so we wouldn’t have the desired convergence properties. We wouldn’t really be able to do infinite sums in it? (Well, perhaps some other sense of infinite summation could be done, but it wouldn’t be the usual sense.) In addition, because such a subfield would only exist as an uncountable proper subfield in some models of ZFC, I find it hard to imagine that it would allow computations that wouldn’t otherwise work? I suppose it could motivate some computations which would then also work regardless of what model of ZFC is being used / is true ?
pjfin123 5y ago
I understand the diagnol proof but not a lot of the rest. If you assume only computable numbers exist though a lot of this seems to really not matter.
- lupire 5y ago
  Even if only computable numbers exist, it's inconvenient, to do math on them in bulk, so it helps to have a set of virtual/potential numbers (misleadingly named the "real" numbers, even though almost all of them (100%) will never be realized in any way).
shockeychap 5y ago
Why spend time on it? To borrow from George Mallory: Because it’s there.
Hamlet42 5y ago
Some math problems seem uselees when first encountered until they become useful some dacades later. An example of this is knot theory and medicine[1] [1] https://science.sciencemag.org/content/255/5043/403 https://science.sciencemag.org/content/255/5043/403
- db48x 5y ago
  Also certain of the more recondite aspects of number theory that deal with prime numbers: completely useless right up until RSA was invented; now we use them to protect every webpage we view, every email we send, and every financial transaction we make.

deleted 5y ago

[deleted]

It took many years to understand that! It’s not a flippant question at all. The answer is, it doesn’t matter. And that’s the joy of it. It wasn’t until I got into ML that I learned the value of doing unimportant work. When you’re free to think about inconsequential matters very seriously, you end up discovering so many useful things. It was how I independently rediscovered what is apparently called the Discrete Hartley Transform. You have a vague notion that what you’re doing might be important, but it’s not the focus. It matters because it’s fun in a way that nothing else can be. Of course, the more serious folk won’t admit it’s fun. It’s serious work. And they’re not wrong; nobody’s fooling around. Our time is every bit as precious as an executive’s. But their goal is to change the world. Ours — or mine, at least — is to know the truth of a thing that most people don’t know.

fighterpilot 5y ago
> it doesn’t matter It doesn't matter yet. Applications of pure math happen downstream decades or centuries later. It's hard to predict the impact.
- l33t2328 5y ago
  I don’t like this framing because it implies any piece of pure math becoming useful is just a matter of time, and I see no reason to suspect this is the case.
  - bmitc 5y ago
    > I see no reason to suspect this is the case Isn’t history enough reason?
    - erichahn 5y ago
      No history actually tells you that only 1% of produced maths is useful. You only see the good stuff not the ugly stuff.
      - bmitc 5y ago
        I don't know why people are throwing around percentages so much in these replies. Percentage of what? Plus, the body of work of mathematics continues to grow all the time. The core of the original comment that started this chain was: > Applications of pure math happen downstream decades or centuries later. My argument was that history shows that a surprising amount of mathematics does eventually trickle down into applications. I don't think anyone is arguing all of mathematics eventually sees an application. Again, my point was that history does indeed seem to show that "applications of pure math happen downstream decades or centuries later". As the original commenter said, it's hard to predict what will be the next thing to be applied.
    - l33t2328 5y ago
      In addition to what the other commenter said, it’s important to understand the scale. Cutting edge research 600 years ago was solving cubic equations. Today, cutting edge research requires potentially years of advanced study to understand the objects involved. There are more mathematicians alive today than at any point in human history, and their work has become specialized to a tremendous degree. Gone are the days where one mathematician could do substantial new work in a dozen diverse areas. So, in short, the fact that historically lots of pure math has not found use, the specialized nature of modern research math, the volume of work being out out, and an irreverence for applications(again, this is _pure_ math) leads me to be doubtful that even a moderate amount of modern pure math research will ever be useful in any practical sense.
    - jonnycomputer 5y ago
      I've read lots of math papers that were a complete waste of time because they were banal. The only "innovation" was to use different words to describe the same thing. New mathematics is actually pretty damn hard I suspect.
  - hawkice 5y ago
    I think it's more about being open minded to strange uses. Maybe 3% of strange pure math ends up being used for something bigger? But since nobody knows what it'll be, it's good to be curious about all sorts of nonsense. Riemannian geometry was OBVIOUSLY absurd useless pure math sophistry, but then it turned out we lived in it. I think a lot of the work on modular forms and elliptic curves has come across that way too, despite the crypto intersection. The general idea is that, even if it's been a century or two, it's too soon to tell if something is useless, that's solid. And formalization of infinity has been useful in some practical exploration of proof systems used by humans. It might be the future of programming, who knows?
- dfdz 5y ago
  To follow up on this comment, I think that math should be viewed as a giant tree structure or pyramid At the bottom are all sorts of applications of linear algebra and optimization (calculus) are related to machine learning, deep learning (AI), physics, engineering etc. One level up there are all sorts of ideas in algebra and mathematical analysis that are leading to new ideas in linear algebra and optimization (calculus) Going one level higher, there are ideas in mathematical logic and set theory that are improving our understanding our algebra and analysis. If you look at a specific result at the very top of the pyramid as ask why does it matter? It is difficult to give a good answer. Clearly we want to base of the pyramid, but do we need to keep building it up? Should we stop at a certain level?
just_one_time_ 5y ago
Lololol. U gay.
mr_gibbins 5y ago
I agree entirely. Industrially, I work a role within an IT team. My work is, I hope, useful and strives towards a purpose - my employer's purpose/business model. Do I 'go home' - leave my home office - and think about it? No. Do I muse on how to solve a work-related issue when I'm showering? Also no. Will I forget almost everything about this job as or when I move onto the next? Definitely yes. But academically, I'm interested, and actively publish albeit modest, low-impact research, in relational databases. I love the set-based approach to all matters SQL and after many years in the field still fool around trying to embellish, attack, improve and invent the core ideas. Why bother? Even if I come up with some magical new improvement to RDBMSs, it doesn't matter. If I fork MySQL and try out my ideas, no one cares. I'll never get the traction or the FOSS community support, I'll spend more time playing politics and managing collaborators than I'd like (zero) and I enjoy the independent thought that my interests bring. So my academic work is pointless. But it's meaningful to me. And I think that's what matters. I don't understand one-tenth of the ideas in this article, but that doesn't matter. What matters is that it's interesting to somebody.

Y_Y 5y ago

I think you'd have to say exactly what you mean by "matter". At one point in my life I felt like the continuum hypothesis was the most important question in the world. Now I'm mostly concerned about finite material issues, e.g. my health. I think a good argument can be made why either of them matter much more than the other.

soperj 5y ago

This is kind of the equivalent of asking why does your life matter?

l33t2328 5y ago

It’s interesting, and fun. Most modern pure math is done completely independent of and without regard to applications(if it were, it would be ‘applied math’).

bmitc 5y ago

We know that the cardinality of the natural numbers is less than the cardinality of the real numbers. The Continuum Hypothesis, which is a long unsolved problem, states that there are no sets with cardinality between the two. The posted article states that this new result strengthens the case against the hypothesis, that is that’s it’s probably false. All of this is nuanced but is important to mathematics and philosophy.

thom 5y ago
Why does “the set containing the natural numbers and a sandwich” not have cardinality between the two?
- hawkice 5y ago
  Because it's the same size. I can show it's the same size because I can just change the labels on the sets (label change of elements doesn't change group size) and go back and forth. In this case, 0 is the sandwich, and all the nonnegative numbers get relabelled down by one. Going back, sandwiches become 0 and you add one to nonnegative numbers. So they've got to be the same size, if I can go back and forth and everything just gets a new label and I don't miss anything.
- helen___keller 5y ago
  You can create a bijection between that set and the set of natural numbers, defined by the function f: 0 <-> sandwich 1 <-> 0 ... n+1 <-> n By definition, if you can biject two sets, they have the same cardinality.
- magicalist 5y ago
  One way to say two sets of things are of equal size is if we can pair up all the elements of one set with all the elements of the other. If we're ok with extending that to sets of infinite things (we can still pair elements of each set, we'd just never be able to finish listing all the pairs), then we can say that the natural numbers and "the set containing the natural numbers and a sandwich" are of equal size because we could pair 1 from the first set with the sandwich from the second set, pair 2 from the first set with 1 from the second set, 3 from the first set with 2 from the second set, etc etc. There's no element of either set without a match in the other set, so they have the same cardinality as the natural numbers, with or without the sandwich.

pantulis 5y ago

Because this advances math. Same could be said for lambda-calculus and yet here we are with FP.

helen___keller 5y ago

Others have addressed why it matters (or doesn't matter) when viewed from outside mathematics. But within mathematics, unsolved problems usually matter for two reasons: (1) When lots of really smart people spend lots of time trying to solve something, and fail to do so, it becomes even more interesting for other smart people. A well-known example is the Collatz Conjecture[0] which, by most accounts is a meaningless problem, but endlessly fascinates many for its renowned difficulty. (2) In mathematics, you often stumble upon problems that are equivalent to other problems. If X is difficult to solve so you give up and go work on Y, only to find out that Y as a problem is basically the same to X, or they have some other intimate relationship, and it makes X that much more tantalizing. A common example for computer scientists is the whole class of NP complete problems[1] for which we don't know if there exists a polytime algorithm to decide. Every time we discover a new problem that lives in the realm of NP-complete, the P vs NP problem gets a little more tantalizing. [0] https://en.wikipedia.org/wiki/Collatz_conjecture https://en.wikipedia.org/wiki/Collatz_conjecture [1] https://en.wikipedia.org/wiki/NP-completeness https://en.wikipedia.org/wiki/NP-completeness

sillysaurusx 5y ago
This is an excellent answer. I have a mostly unrelated question. In the case of the Collatz conjecture, it seems all but proved: > If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. There are lots of histograms and empirical data supporting the conjecture. My question is, why is this empirical data not "good enough" for mathematicians? As something closer to a physicist, I do wonder what the fascination is with trying to prove conclusively that 3n+1 xor n/2 will eventually encounter 1. It seems a bit like trying to prove conclusively that the gravitational constant is so-and-so, when it seems the best you can do is to measure it as precisely as possible: > Jeffrey Lagarias stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics." As someone who loved mathematics but was never much good at it, what's the fascination here? (I'm only trying to understand the motivations of people smarter than I am.)

gorgoiler 5y ago

We don’t know but looking into the past hints at the future. Cantor, Hilbert and Gödel gave us Church who gave us Turing. Turing and Flowers gave us the machines as well as the theory. All of them put together gave us type systems and types are how you formally prove that your 747 software is free of, if not all bugs, then at least certain large classes of error. There is a clear line of connections from Cantor (1890s) to jumbo jet fly-by-wire (1990s.) Countability of sets and Cantor’s diagonal argument — the subject of the first part of this article — are some of the first topics teenagers learn about in high school CS (if you’re lucky and on a very modern course) or CS101 (if you’re at a University.) Types are sets. Who knows what today’s mathematics will bring us in the year 2121?

mmastrac 5y ago
I'd love to see James Burke tackle this line.
lupire 5y ago
Eh, this argument kind of falls apart with higher infinities, where the numbers you are studying are bigger than the power set of all interactions among all particles in all possible universes.

ssivark 5y ago

Let me venture a possible application, since there are already many responses celebrating pure mathematics. As far as we understand, the natural numbers are not sufficient for modeling physical phenomena. The reals/complex while immensely useful also occasionally turn out to be “too complicated” to give theoretical guarantees/proofs of models working well. On the practical side, that means that these models/algorithms can’t be guaranteed to not give junk results, while working on the domain of real numbers. Now, purely speculatively, since the naturals/rationals are aleph0 in size and the reals are aleph2, that means there likely exists a set of numbers of size aleph1 sitting in between the two. What if we could use that set of numbers to construct our physical models? Could we somehow guarantee better behavior… Eg: in quantum mechanics, or field theory, or chaos, etc?! :-)

drdeca 5y ago
Whether the cardinality of the reals is aleph 1, aleph 2, or something larger, is independent of ZFC. In a model of ZFC in which the set of reals has cardinality greater than aleph_1 , I wouldn’t be surprised if there is a subfield of the reals of cardinality aleph_1 , but I would be surprised if such a field was useful for things like that. Such a field would, of course, not be complete with respect to the usual metric on the rationals, so we wouldn’t have the desired convergence properties. We wouldn’t really be able to do infinite sums in it? (Well, perhaps some other sense of infinite summation could be done, but it wouldn’t be the usual sense.) In addition, because such a subfield would only exist as an uncountable proper subfield in some models of ZFC, I find it hard to imagine that it would allow computations that wouldn’t otherwise work? I suppose it could motivate some computations which would then also work regardless of what model of ZFC is being used / is true ?

pjfin123 5y ago

I understand the diagnol proof but not a lot of the rest. If you assume only computable numbers exist though a lot of this seems to really not matter.

lupire 5y ago
Even if only computable numbers exist, it's inconvenient, to do math on them in bulk, so it helps to have a set of virtual/potential numbers (misleadingly named the "real" numbers, even though almost all of them (100%) will never be realized in any way).

shockeychap 5y ago

Why spend time on it? To borrow from George Mallory: Because it’s there.

Hamlet42 5y ago

Some math problems seem uselees when first encountered until they become useful some dacades later. An example of this is knot theory and medicine[1] [1] https://science.sciencemag.org/content/255/5043/403 https://science.sciencemag.org/content/255/5043/403

db48x 5y ago
Also certain of the more recondite aspects of number theory that deal with prime numbers: completely useless right up until RSA was invented; now we use them to protect every webpage we view, every email we send, and every financial transaction we make.