5 ms·

This reflects basic properties of the real numbers, e.g., that almost all of them are irrational, almost all of them are transcendental, almost all of them are

by pash 10y ago

This reflects basic properties of the real numbers, e.g., that almost all of them are irrational, almost all of them are transcendental, almost all of them are uncomputable, etc. The familiar real numbers, those that people (including mathematicians) actually use, and that our machines use, are atypical of the set of real numbers as a whole in many important respects. The real numbers are a formal abstraction to an extent that few people realize.

api 10y ago
IMHO this is a special case of something much more profound. I studied biologically inspired and evolutionary computing very intensely in college, and I reached the conclusion fairly early that the way we do compute is very unlike the natural world. Our computers are designed to compute finite values using instruction sets that are explicitly designed to be precise, which has the side effect of these instruction sets and encodings being highly brittle and un-evolvable. This is what makes things like neural nets and genetic programming hard-- making these systems work involves creating virtual anti-computers within our anti-biological existing ones that are fuzzy, mutable, evolvable, etc. I speculate that this is because we've built computers explicitly to do things we are not good at: crunching massive amounts of data very precisely. We are pretty good at more "organic" forms of cognition so there was little evolutionary "market need" for us to augment ourselves in that way.
- j1vms 10y ago
  Though for this specific discussion (which illustrates the uncountably infinite nature of the set of real numbers), it is useful to note that the class of realizable computers includes both analog and digital variants, though usually we are more familiar with the digital in the form of our desktop/laptop computers or servers. Analog computers can be indeed be used for these types of problems, and their usage in the field predates their digital siblings. Sampling of the result of an analog calculation is of course limited by analog noise (e.g. thermal noise), which we must contend with in this universe. On the other hand, processing in the analog domain gets us further than quantization noise and heat dissipation (due to clock signal & switching) would in the digital domain. Edit: If you are interested further, here is a discussion of the application of analog computing in solving differential equations: http://chalkdustmagazine.com/features/analogue-computing-fun-differential-equations/ http://chalkdustmagazine.com/features/analogue-computing-fun... Also, someone on here once posted a link to a recent dissertation comparing the efficiency of solution finding for a problem tackled analog vs. digital, wherein given the problem's nature, the analog computer outperformed its digital counterpart both in terms of performance and energy consumption. Edit2: Here is the dissertation http://www.cisl.columbia.edu/grads/gcowan/vlsianalog.pdf http://www.cisl.columbia.edu/grads/gcowan/vlsianalog.pdf And the comment it was from: https://news.ycombinator.com/item?id=11728186 https://news.ycombinator.com/item?id=11728186
- wyager 10y ago
  Being a messy computer doesn't make one better at messy computations. The reason humans are decent at certain tasks is usually because we have a lot of (low-quality) hardware directed at the task and we've had many years to evolve a workable hack-job. There is a common pattern in AI where a problem that humans do easily seems intractable, right up until we find a cheap trick that gets the job done and seems to have similar quirks to the biological equivalent. If we're lucky, we find an even better way than what evolution has stumbled upon. For example, computers are way better than humans at landmark recognition (using SIFT-inspired algorithms), but it took us a while to find the tricks to do that efficiently.
chriswarbo 10y ago
> The real numbers are a formal abstraction to an extent that few people realize. This is what makes me dislike the name "real" so much; I long ago grew tired of mathematicians making non-intuitive claims about the world which turn out, in fact, to be claims about the real numbers. As Kronecker said, "God made the natural numbers; all else is the work of man".
- wyager 10y ago
  Tons of stuff in physics is real-valued (or complex-real-valued). Phase difference between eigenstates is real. The energy, momentum, etc. of an unbound particle has an uncountably infinite number of eigenstates, so you need to index them with a real number.
  - gryn 10y ago
    thats because real numbers are a convenient abstraction that allow you to have an infinite range of numbers between any two real numbers. that doesn't meaning that there is anything inherently real about them. an analogy would be using a software abstraction that ignores the memory limits of computers, its useful to simplify your model but not something realizable in real life.
    - wyager 10y ago
      Do you have any evidence that the abstraction is divorced from reality?
  - Retra 10y ago
    Most physics is not accurate to arbitrary precision, so you would do just as well making predictions and models with a small subset of the rational numbers.
    - wyager 10y ago
      >Most physics is not accurate to arbitrary precision, According to whom? Our measurements aren't arbitrarily accurate, but as far as we know, physics itself doesn't suffer from accuracy loss at some point.
noobermin 10y ago
I've thought a little bit about this as a physicist. Given that the numbers we really need for science are countable and they are dense in the real numbers, why don't people imagine how math would look like restricted to this set? My best example: epsilon delta proofs...well, usually when I run physics simulations, I have an explicit error I can't overcome, so I don't really need to know whether a limit converges for all epsilon, just some larger than the largest error in my problem.
- danking00 10y ago
  People do, it's called computable analysis. https://en.m.wikipedia.org/wiki/Computable_analysis https://en.m.wikipedia.org/wiki/Computable_analysis The first thing on that page are the computable real numbers. They have this annoying property that the set itself is not computable enumerable: there a real numbers that, if we only knew which ones they were, we could know all their digits; but there is no algorithm that will list all such numbers. https://en.m.wikipedia.org/wiki/Computable_number https://en.m.wikipedia.org/wiki/Computable_number
- Retra 10y ago
  People do imagine that. It's actually a huge area of research, across many different fields. It's umbrella'd under 'Constructive Mathematics', and it is big in logic, computing, analysis, and type theories.

by pash 10y ago

api 10y ago

IMHO this is a special case of something much more profound. I studied biologically inspired and evolutionary computing very intensely in college, and I reached the conclusion fairly early that the way we do compute is very unlike the natural world. Our computers are designed to compute finite values using instruction sets that are explicitly designed to be precise, which has the side effect of these instruction sets and encodings being highly brittle and un-evolvable. This is what makes things like neural nets and genetic programming hard-- making these systems work involves creating virtual anti-computers within our anti-biological existing ones that are fuzzy, mutable, evolvable, etc. I speculate that this is because we've built computers explicitly to do things we are not good at: crunching massive amounts of data very precisely. We are pretty good at more "organic" forms of cognition so there was little evolutionary "market need" for us to augment ourselves in that way.

j1vms 10y ago
Though for this specific discussion (which illustrates the uncountably infinite nature of the set of real numbers), it is useful to note that the class of realizable computers includes both analog and digital variants, though usually we are more familiar with the digital in the form of our desktop/laptop computers or servers. Analog computers can be indeed be used for these types of problems, and their usage in the field predates their digital siblings. Sampling of the result of an analog calculation is of course limited by analog noise (e.g. thermal noise), which we must contend with in this universe. On the other hand, processing in the analog domain gets us further than quantization noise and heat dissipation (due to clock signal & switching) would in the digital domain. Edit: If you are interested further, here is a discussion of the application of analog computing in solving differential equations: http://chalkdustmagazine.com/features/analogue-computing-fun-differential-equations/ http://chalkdustmagazine.com/features/analogue-computing-fun... Also, someone on here once posted a link to a recent dissertation comparing the efficiency of solution finding for a problem tackled analog vs. digital, wherein given the problem's nature, the analog computer outperformed its digital counterpart both in terms of performance and energy consumption. Edit2: Here is the dissertation http://www.cisl.columbia.edu/grads/gcowan/vlsianalog.pdf http://www.cisl.columbia.edu/grads/gcowan/vlsianalog.pdf And the comment it was from: https://news.ycombinator.com/item?id=11728186 https://news.ycombinator.com/item?id=11728186
wyager 10y ago
Being a messy computer doesn't make one better at messy computations. The reason humans are decent at certain tasks is usually because we have a lot of (low-quality) hardware directed at the task and we've had many years to evolve a workable hack-job. There is a common pattern in AI where a problem that humans do easily seems intractable, right up until we find a cheap trick that gets the job done and seems to have similar quirks to the biological equivalent. If we're lucky, we find an even better way than what evolution has stumbled upon. For example, computers are way better than humans at landmark recognition (using SIFT-inspired algorithms), but it took us a while to find the tricks to do that efficiently.

chriswarbo 10y ago

> The real numbers are a formal abstraction to an extent that few people realize. This is what makes me dislike the name "real" so much; I long ago grew tired of mathematicians making non-intuitive claims about the world which turn out, in fact, to be claims about the real numbers. As Kronecker said, "God made the natural numbers; all else is the work of man".

wyager 10y ago
Tons of stuff in physics is real-valued (or complex-real-valued). Phase difference between eigenstates is real. The energy, momentum, etc. of an unbound particle has an uncountably infinite number of eigenstates, so you need to index them with a real number.
- gryn 10y ago
  thats because real numbers are a convenient abstraction that allow you to have an infinite range of numbers between any two real numbers. that doesn't meaning that there is anything inherently real about them. an analogy would be using a software abstraction that ignores the memory limits of computers, its useful to simplify your model but not something realizable in real life.
  - wyager 10y ago
    Do you have any evidence that the abstraction is divorced from reality?
- Retra 10y ago
  Most physics is not accurate to arbitrary precision, so you would do just as well making predictions and models with a small subset of the rational numbers.
  - wyager 10y ago
    >Most physics is not accurate to arbitrary precision, According to whom? Our measurements aren't arbitrarily accurate, but as far as we know, physics itself doesn't suffer from accuracy loss at some point.

noobermin 10y ago

I've thought a little bit about this as a physicist. Given that the numbers we really need for science are countable and they are dense in the real numbers, why don't people imagine how math would look like restricted to this set? My best example: epsilon delta proofs...well, usually when I run physics simulations, I have an explicit error I can't overcome, so I don't really need to know whether a limit converges for all epsilon, just some larger than the largest error in my problem.

danking00 10y ago
People do, it's called computable analysis. https://en.m.wikipedia.org/wiki/Computable_analysis https://en.m.wikipedia.org/wiki/Computable_analysis The first thing on that page are the computable real numbers. They have this annoying property that the set itself is not computable enumerable: there a real numbers that, if we only knew which ones they were, we could know all their digits; but there is no algorithm that will list all such numbers. https://en.m.wikipedia.org/wiki/Computable_number https://en.m.wikipedia.org/wiki/Computable_number
Retra 10y ago
People do imagine that. It's actually a huge area of research, across many different fields. It's umbrella'd under 'Constructive Mathematics', and it is big in logic, computing, analysis, and type theories.