I read through most of the paper we are all nominally talking about. He doesn't use normal numbers as you say. He is talking about computable numbers as you say. My apologies.
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.
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?
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?
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?
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.
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.
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.
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.
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.
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.
...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.
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.
This is a famous article in the mathematics community. While her results were known to humans for several hundred years she claimed they were not known to her. If this is correct then her derivation is an impressive feat but certainly not worthy of publication. According to [1] the paper still receives citations and some people refer to "Tai's model" instead of the Trapezoidal Rule.
You can read some comments to her article and her response in [2]. I think she should have acknowledged that her method is the Trapezoidal Rule. Maybe she has in the 20+ years since. I don't know.
The whole saga reminds me of a time that member of the biology department asked me, "Why does the Ti-83 calculate scientific notation wrong?" I asked what she meant. She gave me this example:
Calculate (3.75 x 10^23)/(9.34 x 10^(-5))
She enters the problem into the calculator to show me that it does indeed calculate the wrong value. She entered
3.75 x 10^23/9.34 x 10^(-5)
I had to explain to her that the order of operations was important.
Logically speaking if it is the first subsea roundabout then it is also the first Atlantic subsea roundabout. But due to how we normally communicate and interpret things one naturally might assume as you did. Our normal communication/interpretation does not follow rules of logic. It’s interesting that this is so.
Actually you can and this is a good thing in many cases. Though in some cases you end up in a gray area where it’s not so clear what is right. We force doctors to treat people and in certain circumstances they are forced to treat people even if they can’t pay. We force pharmacists to fill prescriptions. We prevent landlords from discriminating against protected classes.
In the case at hand I think the court got it right. I sympathize with the sentiment that a business owner ought to be able to turn down business but I don’t deny that there are situations where this shouldn’t be the case.
I see your point and would be more sympathetic to it if we had proper sex education in k-12 and made birth control and abortions easier to obtain for poor people. Even so, clearly this lady made some poor choices.
I think it’s clear that parent was equating income taxes to theft. The article we are all nominally commenting on is about theft. Parent says he/she can get behind the movement against said theft and that there is a reason that income taxes were unconstitutional. I think I have enough evidence to base my conclusion on. If you really think that I’m wrong in thinking that parent believes that income taxes are theft then why wasn’t this your objection to begin with?
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