what the author means to allude to, through some nonsensical rambling, are the incomputable numbers 
the cardinality of all real numbers that can be described by a terminating computer program to some accuracy is a countable set (since the number of such programs is countable) however, the cardinality of the reals is uncountable.
hence most real numbers cannot be computed beyond a certain accuracy.
Edit (additionally): the set of computable numbers forms a field (if a,b are computable, so is their sum, etc). and there are several movements in "constructive" mathematics, to work exclusively in this field, instead of the field of real numbers. however, many cornerstone theorems in analysis fail in this context, such as, the least upper bound of a bounded increasing computable sequence of computable numbers need not be a computable number .
Yeah, but you can describe many incomputable numbers. Indescribable numbers are not the incomputable numbers- Chaitin's constant's a nice one, it's the proportion of Turing machines that halt. Described. Now, compute it...
There's actually no adding. Just multiplying by two. That's what makes the result so surprising.
The key is the metric on the bi-infinite sequence space. Differing in the digit closest to the decimal point on either side gives a distance of 1/2, differing one digit further out is 1/4, then 1/8, 1/16, and so on (the total distance is the sum; the maximum distance between two sequences is 2 if they differ in every digit.) So two sequences are "close" to one another if their middle digits are all the same.
This gives you the key attributes of chaos:
(1) Sensitive Dependence on Initial Conditions. Two sequences can be identical for BIG_NUM digits on each side of the decimal point, and then completely different outside of that area. This means their initial distance apart is roughly 1/2^BIG_NUM. But after BIG_NUM multiply-by-two operations, the trajectories have diverged to a distance of around 2.
(2) Topological Mixing. A similar construction to above -- select the first N digits of a sequence to make it fall into a particular neighborhood, and the next M digits to make it fall into a different neighborhood after the appropriate number of bitshifts. This can be done for any pair of neighborhoods.
(3) Density of periodic orbits. Pick any point X in the space, and a distance epsilon. Construct a periodic orbit that gets within epsilon of X simply by taking the central digits of X (using log_2(epsilon) to select how many digits) and repeating those digits.
There you have it -- chaos as a result of a single bitshift operation on bi-infinite sequences, with no addition or other operations.
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The reason why this problem is difficult is because primes have everything to do with multiplication (every number is the product of primes), and summing numbers have everything to do with...well, addition. To give additive properties of numbers in terms of primes is like putting a round peg in a square hole.
The reason why this problem is important, is that in the grand scheme of things, it'a more about the math and less about the applications. People have already assumed it to be true, and still not applied it to anything practical like cryptography, as far as I know. Please correct if not.
Easily, act as if you proved it. We are generally very hesistent to do this (with good reason), but sometimes practicality takes over.
For example, we have not proven that are current construction of mathematics contains no contradictions (indeed, we have proved that such a proof is impossible), however, we continue to act as if it does.
We have also not proved that factoring numbers is hard, or that NP/=P, but we still act as if these are true.
Can you explain your question? It seems obvious that you can take something that is not proven as an axiom and try to use it in some practical applications. Maybe to disastrous results if it ends up being wrong, but no worse than what is already a risk with bugs in implementation.
> Maybe to disastrous results if it ends up being wrong
This is what I meant. Bugs are never introduced intentionally in this manner. Though I suppose we can say that most encryption is used based on an unproven assumption, so I see your and dmvaldman's point!
Also, Golbach's conjecture has been verified up from 5 to 4000000000000000000 computationally.
But in general, it's common for mathematicians to assume a conjecture is true and go on from there, because if they arrive at a contradiction, then they can conclude the conjecture false (or that another interesting fact may be logically equivalent to the conjecture).
Pure mathematics consists entirely of such assertions as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true…. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
- Bertrand Russell
>He discredits one of Sandberg's scientifically verified claims that women take criticism worse than men,
I don't know about the "scientifically verified claim". In general, scientific claims made in scientific journals are nuanced, and context sensitive. I would be wary of a lay person with a particular position weaving attributions to journal papers indiscriminately in their essays. Furthermore, Humanities research is notoriously error-prone and lacking in rigor. In this respect, I'd say Greenspun is probably close to the truth.
> saying left-handed people make 20% more than right-handed people is equally (if not much more) dubious.
That was his point! Sheryl Sandberg has a particular thesis and cherry picked some numbers from some study to support her assertion. He mocked this by cherry picking another. I'm pretty sure he doesn't particularly know (or care) if right-handed people are more successful in their careers.
This may be fair or it may be unfair criticism, but the fact that Sandberg threw a whole bunch of references in doesn't automatically mean her assertions are supported by science.
>if you ever find yourself arguing that someone's references are likely wrong because they are in the humanities... well I just want to punch you in the face really..
Your reading comprehension is suspect. I certainly didn't argue this, and neither did Greenspun.
> that's not the point. saying left-handed people make 20% more than right-handed people is equally (if not much more) dubious.
The sentence before that is "Hardly anybody knows anything about why some workers succeed more than others." His citation of the US News report is meant as an example that there is a lot of confusion on the subject. He is using it specifically to illustrate that dubious claims are flying around.
Sandberg confirms that “A is Average” at Harvard. Her brother David, a neurosurgeon whom Sandberg admires because currently “he splits child care duties with his wife fifty-fifty”, was also a Harvard undergrad. He takes “a class in European intellectual history”, skips all but two lectures and all but one book, gets tutored for three hours and receives an A for the semester (p32-33). The guy’s success is attributed to the general confidence of men. Sandberg does not consider how likely it is that her brother’s confidence would have resulted in an A in a physics class at Caltech.
Yes it is. Because Sandberg never says she admires her brother because of that fact. She says she admires her brother, and then states that fact about the brother. Greenspun is the one who makes the false connection. Here is the full quote:
I should have understood that this kind of self-doubt was more common from females from growing up with my brother. David is two years younger than I am and one of the people in the world whom I respect and love the most. At home, he splits child care duties with his wife fifty-fifty; at work, he's a pediatric neurosurgeon whose days are filled with heart wrenching life-and-death decisions…
With Greenspun's logic, you could equivalently claim Sandberg admires her brother because he is a neurosurgeon.
This is why I unsubscribed from Greenspun's blog a long time ago (that and him deleting my comments); he makes some good points (Sandberg is indeed extremely exceptional and much of her views are ludicrous), but he buries them in a sea of bias and partisanship.
Easy exercise for readers: go through this post and a few previous posts, and count every use of 'meta' points like Ioannidis's paper against a claim. Notice any patterns?
And then there are the gratuitous digs...
> Sandberg identifies the same tendencies for underlings in a bureaucracy to hold their tongues that Max Weber noticed 100+ years ago (p85; no reference to Weber).
And why should there be any reference at all, Phil? Most people never read a page of Weber, and it's not an observation which requires uniquely keen insight or experience...
In the quote about right-handed men, I think you missed the following parenthetical phrase: "(and as noted below (Ioannidis), this result is itself probably false)"? It's not (A and ~A), it's the same thing in both statements (i.e. published studies aren't reliable).