sykh
Joined 8 years ago
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Comment · 8 years ago
Comment · 8 years ago
Let S be an encoding of the works of Shakespeare in binary. A normal number contains S in its binary expansion. We can do this for all information. In this sense is it correct to say that a normal number contains infinite information in it?
Comment · 8 years ago
The author's paper is about normal numbers (numbers that contain infinite amount of information). If a normal number isn't considered to have random digits then what notion would you use?
Comment · 8 years ago
The set theory that most working mathematicians deal with is ZFC. In ZFC it is not known what cardinal the continuum is. Hence the statement that I was responding to is incorrect. The person I responded to said that they do know how many reals there are.
The cardinality of the reals is called c. It is known to the be the same as the cardinality of the power set of the naturals. It is not known, in ZFC, which aleph this is. We just know that it is the same as the size of another set.
If you want to add an axiom and say that c is aleph1 then you are free to do so. But if you don't have this axiom then you don't know which aleph it is. So in what sense can you say that you know how many reals there are? You only know it if you add an axiom that says, "It is aleph1."
If I have a jar of pennies and I know it has the same number of pennies as the number of quarters in another jar that I have, does this mean I know how many pennies are in the jar?
Comment · 8 years ago
If sqrt(2) is a normal number then its decimal expansion would contain infinite information.
Comment · 8 years ago
A normal number can be computable. It is not known if sqrt(2) is normal.
Comment · 8 years ago
The cardinality of the continuum is a cardinal number. It’s one of the alephs. It is not known which aleph it is. So it’s not known what the cardinality of the powerset of the naturals is. It’s just known that it is the same cardinal number of the reals. Basically, we have two jars of marbles that contain the same number of marbles but it’s not known how many marbles that is.
Comment · 8 years ago
A computable number can be a normal number.
Comment · 8 years ago
The complement of the set of normal numbers has measure zero. A normal number is what the author of the paper is talking about when referring to random numbers. I think most mathematicians, if they had to bet, would bet that sqrt(2) is normal. It is not known though. If it is normal then it’s digits are random.
I haven’t read the paper but I think the author is arguing that most real numbers don’t make sense physically. If sqrt(2) were shown to be normal and since it’s the hypotenuse of a right triangle of legs with length 1, I wonder what the author's response to this would be. Perhaps he’d say that such a triangle doesn’t exist physically.
Comment · 8 years ago
I didn’t make any self referential call out to authority. I provided no authority whatsoever. I posited beliefs and spoke in such terms. I just stated how I see things. As I said, I believe the problems are emergent phenomena. Of course I could be wrong. It could be an organized effort as you say. It seems reasonable that it would become organized even if the origin isn’t.
Comment · 8 years ago
Since we're discussing nothing short of the US government acting as an enemy to the citizens it is supposed to serve....
Your premise is incorrect. The "sort of" part of my statement is important. There are lots of forces at play that cause this. I see it as sort of an emergent phenomenon and not the design of some clever genius or powerful organization.
Comment · 8 years ago
It’s a sign of how bad things are in the U.S. State something obvious, like you did, that jives with someone’s attachement to a political party and you get irrational responses. The polarization is palpable. People don’t seem to rationally discuss policy. It’s about defending/justifying political parties. It’s the party that matters.
Comment · 8 years ago
It’s sort of the divide and conquer strategy. Pit groups against each other whilst the looting occurs. While people fight over who is the true snowflake and that kind of stuff all sorts of shitty policies get enacted. Things didn’t just suddenly get unbearable because <person of opposite party> just got elected. If one thinks this then the problem was already there and one's lack of awareness of it is the real problem.
Comment · 8 years ago
...pensions create a long term liability in a way that putting cash in employee 401k doesn’t.
Assuming we don’t want to live in a society in which vast numbers of elderly live in penury this isn’t true. It’s well known that when retirement savings are the responsibility of the emoloyee then they don’t save enough.
What you say about bad governance is certainly true and incompetent or badly incentivized leadership will exacerbate problems. I don’t know what the solution is for the U.S. but the current system is going to make us end up with a society with large numbers of destitute elderly. Shifting the onus to the employee via 401Ks is just kicking the can down the road so to speak. It also allows politicians and voters off the hook because the liabilities aren’t on the books. On paper it looks good but the reckoning will occur.
Comment · 8 years ago
My wife is a psychiatrist and has dealt with people who survived suicide attempts. Mostly people who tried to kill themselves by shooting themself in the head. She says that her experience is that people do not immediately regret the decision just the consequence of having survived severe damage to the head.
It seems, at first thought, unreasonable that people trying to kill themselves by jumping off a bridge would be more likely to regret the decision than people who shoot themselves in the head. I wonder if this is mostly a selective choosing of the survivors of jumping off the bridge to fit a narrative.
But a normal number has "random" digits in the sense that all finite sequences of digits (in a given base) occurs uniformly in the expansion. What other notion of random can one meaningfully give for an expansion of a number's digits? Without getting into too much philosophy.
From [1]: We call a real number b-normal if, qualitatively speaking, its base-b digits are “truly random.
My expertise is in commutative algebra so I'm outside of my comfort zone.
[1]: https://www.davidhbailey.com/dhbpapers/bcnormal.pdf