Computer programs exist that can tell you what the next digit is. That means it’s deterministic, and running the program will give you a prediction for each digit (within the memory constraints of your computer).
The fact that it’s deterministic is exactly why pi is interesting. If it was random it would typically be much easier to prove properties about it’s digits.
There’s no way to predict what the next unsolved pi digit will be just by looking at what came before it. It’s neither predictable nor deterministic. The very existence of calculations to get the next digit supports that.
Note: I’m not saying Pi is random. Again, the calculations support the general non-randomness of it. It is possible to be unpredictable, undeterministic, and completely logical.
Note Note: I don’t know everything. For all I know, we’re in a simulation and we’ll eventually hit the floating point limit of pi and underflow the universe. I just wanted to point out that your example doesn’t quite fit with pi.
As I said, you can’t predict the next number simply based upon the set of numbers that came before. You have to calculate it, and that calculation can be so complex that it takes insane amounts of energy to do it.
Also, I think I was thinking of the philisophical definition of “deterministic” when I was using it earlier. That doesn’t really apply to pi… unless we really do live in a simulation.
This might just be my computer-focused life talking, but I’ve never heard of deterministic meaning anything but non-random. At best philosophic determinism is about free will and the existence of true randomness, but that just seems like sacred consciousness.
I also don’t know why predictability would be solely based on the numbers that came before. Election predictions are heavily based on polling data, and any good CEO will prepare for coming policy changes, so why ignore context here? If that’s a specific definition in math then fair enough, but that’s not a good argument for or against the existence of arbitrary strings in some numbers. Difficult is a far cry from impossible.
This might just be my computer-focused life talking
I’m a software eng too, but I have broad interests. Like I said, the philosophic use doesn’t really have a place in this discussion and I messed up by bringing it in. The only way it would be relevant is if the universe is a simulation because, as you guessed, then free will itself becomes part of the equation.
I also don’t know why predictability would be solely based on the numbers that came before
There’s a miscommunication happening here, and I’m wondering if I’m not explaining myself well. Election predictions use polling as their dataset, and there are no calculations that really go into predicting the results other than comparing the numbers within those sets. That’s why they’re notoriously garbage (every single pollster had Hillary winning in late October 2016, for example). Also, there aren’t any calculations that go into a CEO/Boardroom’s intuitions on how shareholders will react to policy changes, so I’m not sure about the relevance here. In the case of pi, there is no dataset that you can use that tells you what the next unknown number in pi is. The only way to get that number is to run a very complex calculation. Calculations are not predictions.
If a new government makes it known that they’ll increase tariffs, any company dealing with international trade can prepare for increased costs sooner. Or even better, if there’s a movement for emissions regulations, they can get lobbyists and lawyers to find or add loopholes nice and early, long before cars would actually need to be more efficient.
Anyway, the miscommunication here seems to be what you mean by nondeterministic and unpredictable. We’ve been through deterministic, and that doesn’t perclude unpredictable.
For example, cryptographic hashes are completely deterministic yet impossible to predict. The determinism allows easily checking for the correct string, but the unpredictability makes guessing the correct string impossible beyond brute force. Yet if a security protocol used π to seed it’s hashes, it would be way more predictable than most methods. Even if your psudorandom table is indistinguishable from noise, if the table is known the whole system can be cracked relatively easily. Thus π would make that method predictable.
Now you could mean that each character or string says very little about each other character or string, but that’s a different claim; that you can’t predict one part of the number using another part. For example, if you say the code to your luggage is a five digit string starting at the 49702nd digit of π, that’s easy to lookup. But no amount of digits will help you figure out that this string really is from π and not something else. I’d call that chaotic rather than unpredictable, as unpredictable makes me think of probability more than calculability. π is found in so many places that many sci-fi stories use it in first contact scenarios, alongside e, the hydrogen line, Fibonacci numbers, and c or sometimes hbar. Dependable is hardly unpredictable.
If we go back to your original reason for describing the predictability of some numbers, we find a simple nonrepeating number (101100111000… let’s call it b for binary) and π. With b, any string can at least narrow down it’s location in the number, and if a string contains both 10 and 01 we can positively fix it’s location, even to a place that no one has calculated before. This is impossible with π, we can positively rule out many positions, but no position can be confirmed for any string, and any string may appear further in the digits as well, giving multiple possible positions for any string.
However we can still compute π, and thus can know (even better than predict!) any arbitrarily precise digit in finite time. There are numbers where that’s not possible, so-called non-computable numbers (For example). This number cannot to computed in finite time, only approximated. This sounds more unpredictable.
Predictability could be seen as a function of the ease of calculability, especially when time is a limited resource, but why not just say that π is more complex to predict than b, or that the existence of b doesn’t disprove that π can contain all finite strings? That was the original issue after all.
Sorry, TL;DR, I don’t think unpredictable is a good word to use outside of probability, and even so an easily predictable number is enough to prove that not all irrational numbers are normal numbers (not all numbers with infinite digits contain all finite strings).
Yeah, but your number doesn’t fit pi. It may not have a pattern, but it’s predictable and deterministic.
Pi is predictable and deterministic.
Computer programs exist that can tell you what the next digit is. That means it’s deterministic, and running the program will give you a prediction for each digit (within the memory constraints of your computer).
The fact that it’s deterministic is exactly why pi is interesting. If it was random it would typically be much easier to prove properties about it’s digits.
There’s no way to predict what the next unsolved pi digit will be just by looking at what came before it. It’s neither predictable nor deterministic. The very existence of calculations to get the next digit supports that.
Note: I’m not saying Pi is random. Again, the calculations support the general non-randomness of it. It is possible to be unpredictable, undeterministic, and completely logical.
Note Note: I don’t know everything. For all I know, we’re in a simulation and we’ll eventually hit the floating point limit of pi and underflow the universe. I just wanted to point out that your example doesn’t quite fit with pi.
π isn’t deterministic? How do you figure that? If two people calculate π they get different answers?
What π is, is fully determined by it’s definition and the geometry of a circle.
Also, unpredictable? Difficult to predict, sure. Unpredictable by simple methods, sure. But fully impossible to predict at all?
As I said, you can’t predict the next number simply based upon the set of numbers that came before. You have to calculate it, and that calculation can be so complex that it takes insane amounts of energy to do it.
Also, I think I was thinking of the philisophical definition of “deterministic” when I was using it earlier. That doesn’t really apply to pi… unless we really do live in a simulation.
This might just be my computer-focused life talking, but I’ve never heard of deterministic meaning anything but non-random. At best philosophic determinism is about free will and the existence of true randomness, but that just seems like sacred consciousness.
I also don’t know why predictability would be solely based on the numbers that came before. Election predictions are heavily based on polling data, and any good CEO will prepare for coming policy changes, so why ignore context here? If that’s a specific definition in math then fair enough, but that’s not a good argument for or against the existence of arbitrary strings in some numbers. Difficult is a far cry from impossible.
I’m a software eng too, but I have broad interests. Like I said, the philosophic use doesn’t really have a place in this discussion and I messed up by bringing it in. The only way it would be relevant is if the universe is a simulation because, as you guessed, then free will itself becomes part of the equation.
There’s a miscommunication happening here, and I’m wondering if I’m not explaining myself well. Election predictions use polling as their dataset, and there are no calculations that really go into predicting the results other than comparing the numbers within those sets. That’s why they’re notoriously garbage (every single pollster had Hillary winning in late October 2016, for example). Also, there aren’t any calculations that go into a CEO/Boardroom’s intuitions on how shareholders will react to policy changes, so I’m not sure about the relevance here. In the case of pi, there is no dataset that you can use that tells you what the next unknown number in pi is. The only way to get that number is to run a very complex calculation. Calculations are not predictions.
If a new government makes it known that they’ll increase tariffs, any company dealing with international trade can prepare for increased costs sooner. Or even better, if there’s a movement for emissions regulations, they can get lobbyists and lawyers to find or add loopholes nice and early, long before cars would actually need to be more efficient.
Anyway, the miscommunication here seems to be what you mean by nondeterministic and unpredictable. We’ve been through deterministic, and that doesn’t perclude unpredictable.
For example, cryptographic hashes are completely deterministic yet impossible to predict. The determinism allows easily checking for the correct string, but the unpredictability makes guessing the correct string impossible beyond brute force. Yet if a security protocol used π to seed it’s hashes, it would be way more predictable than most methods. Even if your psudorandom table is indistinguishable from noise, if the table is known the whole system can be cracked relatively easily. Thus π would make that method predictable.
Now you could mean that each character or string says very little about each other character or string, but that’s a different claim; that you can’t predict one part of the number using another part. For example, if you say the code to your luggage is a five digit string starting at the 49702nd digit of π, that’s easy to lookup. But no amount of digits will help you figure out that this string really is from π and not something else. I’d call that chaotic rather than unpredictable, as unpredictable makes me think of probability more than calculability. π is found in so many places that many sci-fi stories use it in first contact scenarios, alongside e, the hydrogen line, Fibonacci numbers, and c or sometimes hbar. Dependable is hardly unpredictable.
If we go back to your original reason for describing the predictability of some numbers, we find a simple nonrepeating number (101100111000… let’s call it b for binary) and π. With b, any string can at least narrow down it’s location in the number, and if a string contains both 10 and 01 we can positively fix it’s location, even to a place that no one has calculated before. This is impossible with π, we can positively rule out many positions, but no position can be confirmed for any string, and any string may appear further in the digits as well, giving multiple possible positions for any string.
However we can still compute π, and thus can know (even better than predict!) any arbitrarily precise digit in finite time. There are numbers where that’s not possible, so-called non-computable numbers (For example). This number cannot to computed in finite time, only approximated. This sounds more unpredictable.
Predictability could be seen as a function of the ease of calculability, especially when time is a limited resource, but why not just say that π is more complex to predict than b, or that the existence of b doesn’t disprove that π can contain all finite strings? That was the original issue after all.
Sorry, TL;DR, I don’t think unpredictable is a good word to use outside of probability, and even so an easily predictable number is enough to prove that not all irrational numbers are normal numbers (not all numbers with infinite digits contain all finite strings).