> Definition and equality are really the same thing. [...] So the only difference is [...].
An interesting way of justifying how two things are really the same thing, that is, by acknowledging their difference.
The point is that the pointed ambiguity (the difference) can be quite significant, and these become apparent when studying these sentences in a formal logic. Hence, the significance of the ambiguity.
This (and the article under discussion) just echoes the idea that mathematical sentences and proofs are considered "rigorous" but "informal", where "formal" is the domain of logic.
This was huge in Brazil, too, at the time. Though with some setbacks, when aired in open/FTA networks. For example, the multi-part episodes were cut down (i.e. destroyed) as a single episode -- we only discovered they were actually two episodes a couple of years later, when cable tv got popular and the show aired on Fox Kids without the mutilations.
Also there were some translation mistakes (specially in the first episodes). However, the brazilian voice actors were amazing and extremely memorable -- Brazil had an incredible pool of voice actors in the 80's and 90's. It was so memorable that the same voice actors who did the animated series were used to produce the dubbed versions of the movies a decade+ later, a rare gift given to fans in these lands.
As for the comics, they were also a few years behind US. Uncanny X-Men #262 was published when the show first aired. So as kids coming from the show to the comics, we were also pretty confused about uniforms and team members.
Another example is what happened when Linux moved to 3.0; some programs expected a 2.x version, or even 2.6.x, these programs were clearly buggy, as they should check that the version is greater than 2.x, however, the bugs were already there, and people didn’t want to recompile their binaries, and they might not even be able to do that. It would be stupid for Linux to report 2.6.x, when in fact it’s 3.x, but that’s exactly what they did. They added an option so the kernel would report a 2.6.x version, so the users would have the option to keep running these old buggy binaries. Link here.
> I had no idea where to start interpreting the notation other than maybe attending a class to get the proper starter culture.
More than anything, I think this is the main reason why studying many subfield of mathematics is so difficult, when we spend more time and effort in things like parsing instead of in the actual subject matter.
Even popular subjects may show similar problems. For example, in Sedgewick's Analysis of Algorithms, he gives the following common definition:
> O(f(N)) denotes the set of all g(N)....
In the next page, he presents an exercise:
> Show that f(N) = N lg N + O(N) implies f(N) = ϴ(N log N)
Reader looks at that "N lg N + O(N)" and tries to make sense of the addition of a number to a set of functions. Note, this is in page 5, so many readers unfamiliar with the culture of the area are likely to just abandon the book (and perhaps the study of the subject) as they could not translate to themselves the very first formula presented by the author.
The only clue to anything that could help the newbie to get some answers lies in a couple of references that point to the historical uses of these greek symbols for computational complexity. Within that reference [1], by no other than Knuth, one can find the explanation for such syntax:
> "1+O(n^-1) " can be taken to mean the set of all functions of the form 1+g(n), where |g(n)| < Cn^-1 for some C and all large n.
And then he goes on about the problem of that syntax in respect to one-way equality:
> we write 1+O(n^-1) = O(1) but not O(1) = 1+O(n^-1). The equal sign here really means ⊆ "set inclusion", and this has bothered many people who propose that we not be allowed to use the = sign in this context. My feeling is that we should continue to use one-way equality together with O-notations since it has been common practice of thousands of mathematicians for so many years now, and since we understand the meaning of our existing notation sufficiently well.
The above is not only a reason for logicians to laugh at the pretense that mathematical language is formal, but also an example of things that are likely not to be in books but in an unwritten culture of a field.
--
[1] - Knuth, Big Omicron and big Omega and big Theta
1: the "99%" you're looking at are jobs on a specific niche that uses technology to bounce information from one place to another and display them, mostly known as IT jobs.
2: Hard problems are everywhere. Literally. Even in those "99%" jobs you mentioned -- "doing" such software may not be full of mysteries, but make it evolve without break and respond to constraints are examples of real hard challenges. It's a matter of seeing them and putting it on perspective.
3: there are many ways of framing a problem, making it hard or easier. The harder you frame it, the more instruments you will be required to have to tackle it. Those may involve research skills and fluency in an array of fields that would help understand the phenomena. It mostly depend on where in the spectrum of the problem you want to work, and that's up to you to figure it out.
4: advice to avoid a career working on crud apps: interview the interviewer and find out if the problems of the hiring company are the ones that you value.
5: questions like "where to work on hard/fun stuff" are very hard to give satisfying answers. I suggest you take the time to survey problems you are interested, try talking to people working on them...many problems are tackled by teams, often with people with different background and specialties dealing with different aspects of the problem.
An interesting way of justifying how two things are really the same thing, that is, by acknowledging their difference.
The point is that the pointed ambiguity (the difference) can be quite significant, and these become apparent when studying these sentences in a formal logic. Hence, the significance of the ambiguity.
This (and the article under discussion) just echoes the idea that mathematical sentences and proofs are considered "rigorous" but "informal", where "formal" is the domain of logic.