throwawaymath

I can't tell if the author has done a real analysis course before, but if they haven't that's the one they should choose next. If they already have that under their belt they should go to probability or complex variables. I don't see the utility of re-doing calculus or linear algebra if the author is already strong in both.

I appreciate the enthusiasm for math that's evident here, that's great! But if you're learning math the "right" way - by actively engaging with the material - there's only so much of it you can learn at once.

Speaking as someone who was in a gifted program in my youth (and who knew others in more advanced programs), I would like to caution against this perspective. My achievements are not notable, and I would not use that as a heuristic for determining whether or not a particular program/standard is successful or useful. Yet I found my experience to be very positive. Despite the fact that not all programs are created equal, I would generally recommend a suitable one to any parent with a gifted child.

I understand MENSA is a bit loaded since it can come across as pretentious, so let me reframe the example for you. Take a look at past winners of the Putnam exam. Most of them are not nearly as notable as cperciva[1], but they're all demonstrably gifted.

Giftedness is not about being entrepreneurial or about how you apply your intelligence in a notable way. Programs designed for gifted people are not trying to create a class of people who are more impressive. In general, they try to foster natural talent in a way that cannot typically be accommodated in the modal classroom setting.

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1. For those unaware I'm referring to Colin Percival, an HN user who designed scrypt and developed Tarsnap. He won the Putnam.

A friend of mine was in a much more rigorous gifted program (through Stanford) and I think it was the best thing that ever happened to him. He had a very poor home life but he's phenomenally brilliant (he's one of a specific handful of people I've personally met who I use that term for). All he really enjoyed doing from the time he was 12 was reading math and physics books. Going through the gifted program put him on a path that exercised his talents in a way that he found personally very fulfilling. He ended up finishing undergrad at Harvard before the age most kids become sophomores, and then completed a PhD from Harvard before the age most people even begin one.

Then he went on to work for the NSA and, later, a hedge fund. Those things probably look soulless to a lot of people, but he's very happy.

> There are people who want to invest in ICOs, Ponzi schemes

Ponzi schemes are fraudulent operations intrinsically designed to extract money from investors by misleading them. Slack is a software company with a cogent, well-defined plan to leverage unprofitability now into significantly greater profitability later. Not only are these things meaningfully different, they're categorically incomparable.

Even if Slack ultimately fails it has a reasonable plan of action that distinguishes it from fraud. The SEC does not exist to eliminate all risk from investor portfolios, that's impossible (and suboptimal!). It exists to (among other things) ensure the risk to investor capital occurs through the normal procedure of markets instead of outright fraud. For similar reasons most ICOs (and particularly fraudulent ones) are categorically dissimilar from a tech company going public at a net loss.

> others want to rape, murder...why should anyone’s opinions be forced on these people?

I can't believe I have to say this, but the two things you described are violent acts with directly violate the rights of other human beings. I'm not sure how much else needs to be said here. We started off at people being allowed to engage in risky but fundamentally legal and mundane investments, and ended up at fraud, rape and murder. Somewhere along the way we've gotten quite lost.

The point is the Siemens sensor is a superfluous correlation with turbine failure, because the underlying dataset is biased towards Siemens sensors. The scenario suggested by the author is one in which your turbine failure dataset does not match reality.

No amount of sample enlargement will correct sample bias. You have a variable which is disproportionately represented in your underlying dataset despite being independent from a collection of variables correlated to failure, and the algorithm is learning that one instead.

Real world ways this is plausible and cannot be corrected by increased sampling:

1. Your telemetry data is accurate, but your logging service providing that data is faulty and only consumes data from a subset of meaningful publishers.

2. Whoever provided this dataset fat fingered a SQL query which joined too few tables including the sensor vendors, but correctly returned only the failing turbines.

3. Your data has (unnormalized) duplicates, because more than one system is providing telemetry data for Siemens sensors without the older systems being retired.

4. You use mostly Siemens sensors, and simply didn't correct for this in your sample.

I'm not sure what the parent means by the metric being the identity function, however. The Euclidean metric is basically the hypotenuse of a triangle parameterized by two vectors. The adjacent and opposite sides of the triangle are measured to be the Euclidean norm of each vector (their length), and the hypotenuse is the shortest distance between them.

The Euclidean metric is not the only metric - you can define distance however you'd like as long as it's consistent. But I'm not sure how the identity function works as a metric, because that would map a vector to another vector, not a scalar.

It would ordinarily be weird to represent shear transformations using rotations and scalings because shear matrices are elementary. But it checks out.

The reason determinants are hard to teach (in my opinion) is because a rigorous derivation of their formula isn't possible without first teaching multilinear algebra and constructing the exterior algebra. Once you do those things, the natural geometric interpretation of the determinant basically falls onto your lap. But it's still very useful for e.g. computing eigenvalues and using the characteristic polynomial, so it's taught before that context can be formalized.

Professors shouldn't teach determinants in the context of matrices, at least not at first. That's heavily computation-focused, and the symbol pushing looks really unmotivated and strange to students. Instead they should teach the basis-free definition of determinants (i.e. focus on the linear map, not the matrix transformation representing the linear map for some basis). Then the determinant is "only" the volume of the image of the unit hypercube under the linear transformation, which is where the parallelepiped comes in. If the linear transformation is invertible, the unit hypercube is transformed from an n-dimensional cube into an n-dimensional parallelogram, from which you can geometrically see the way the linear map transforms the entire vector space it's defined over.

3Blue1Brown has a very good video on the geometry underlying the determinant[2]. For a more rigorous presentation which constructs the exterior algebra and derives the determinant formula using the wedge product, Noam Elkies has notes[3][4] for when he teaches Math 55A at Harvard. Incidentally Noam Elkies uses Axler's book, and while he obviously approves of it he's pretty upfront in asserting that the determinant should be taught anyway[5].

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1. http://mathb.in/33068

2. https://www.youtube.com/watch?v=Ip3X9LOh2dk

3. http://www.math.harvard.edu/~elkies/M55a.10/p8.pdf

4. http://www.math.harvard.edu/~elkies/M55a.10/p9.pdf

5. http://www.math.harvard.edu/~elkies/M55a.10/index.html

1. All math and CS prerequisites to an intro computational complexity course.

2. An intro computational complexity course.

3. All math and CS prerequisites to an advanced, graduate-level computational complexity course. Let’s say by this point you have worked through two linear algebra courses, three calculus courses, one or two discrete mathematics courses, some graph theory, some combinatorics, some logic, and a couple of algorithms courses.

4. The advanced, graduate-level computational complexity course. Hopefully you work on probabilistic computation and, in particular, quantum computation.

5. Now read and understand the relevant papers advancing the field (not just this one problem) for the last two decades. That one paper from 1990 about reducing the permanent to the determinant? You should know about that.

6. Finally, to reach the temple you must walk the path littered with the bodies of would-be explorers before you. Read the papers of failed proofs, starting with the easily refutable ones. The most sophisticated failed proofs have errors so subtle that they are useful for the research community in their own right as an exercise in peer review.

As a rule, a grad student near their PhD in this area should know of everything Aaronson mentioned here: https://www.scottaaronson.com/talks/pvsnp.ppt. None of that should be unfamiliar or unknown. A postdoc and beyond should actively have ideas percolating on how to chip away at some small aspect of a lesser problem featured therein.

Here’s the reality: at almost every stage of a researcher’s career, “taking a stab at” a famous outstanding problem is the wrong way forward. Usually a problem is still outstanding because it actually needs a new theory - this is the practical utility in solving theoretical problems in the first place. Therefore the best bet for solving this problem is actually to chip away at it over a long period of time.

Think hacking through a rainforest to reach a goldmine, not managing to somehow parachute to the goldmine directly when it’s hidden beneath the trees.

In the early to mid 20th century, it was feasible for someone to pick up a book on number theory and apply relative genius to an open problem to quickly solve it. The bottom line is that prerequisites for doing so were often just basic calculus and high school algebra. An understanding of sequences and series went a long way.

This isn't the case in the 21st century. We're firmly out of that territory. You can't offer a profound new perspective on a thing which you can't understand, and the barrier to understanding research mathematics continually rises. The forefront of modern mathematics is so far removed from even graduate level mathematics course material that it's not going to just be intuited through untrained brilliance. You have to actively learn it, which is (unfortunately) vanishingly unlikely outside of academia.

If the tools available to you are basic real analysis and linear algebra, P/NP is beyond your reach - full stop. At this point we actually have proofs that a valid proof of P/NP (and similar problems) cannot be achieved through large swathes of elementary techniques.

We don't live in a world like Good Will Hunting where an amateur can succeed by being a genius. That's useful in the long term but not enough on its own.

The field simply has far less low hanging fruit than it did a century, or even a half-century ago. It's not reasonable to expect that a lone genius with little to no academic background in mathematics could resolve a famous open problem through sheer creativity. For literally generations the field's most intelligent researchers have spent their careers chipping away at this problem.

The short answer is yes. We haven't had a Ramanujan in decades, and they were more or less always extraordinary. Insofar as you can generalize all of its subfields together, it's reasonable to say mathematics is an extremely mature discipline now.

What I mean by that is there isn't much low hanging fruit around. Even when Ramanujan was dazzling Hardy with his results, many of them were already known. That was over a century ago. There just isn't much fertile ground left where a single brilliant mind can make significant headway based on raw talent or intuition, without first working through significant education in prerequisites. It's more and more commonplace for authors to collaborate on their work, because compelling problems at the forefront of mathematics research are increasingly requiring cross-disciplinary knowledge to resolve.

To focus on Ramanujan a bit in particular - one of the reasons Ramanujan was so talented was because he not only came up with novel results in complete isolation; rather, he also came up with original perspectives and theorems. That means he was capable of posing interesting questions, which is usually much more interesting and inspiring for further research. That's very inspiring, but the reason I point it out is because Ramanujan wasn't solving open problems in the West left and right so much as he was exercising ingenuity to pose - and resolve - open problems in fertile ground he could reach. But a century later, the amount of existing mathematics has increased so much in both breadth and depth that a burst of creativity is almost certainly not going to get an undergrad to find something that e.g. Rudin hasn't written down somewhere already. But more importantly, it's vanishingly unlikely they will resolve a famous open problem which is the subject of intense research interest. All the stories of untrained amateurs doing that are from the mid 20th century or earlier.

To turn your question on its head a bit: consider how astonishingly brilliant an untrained individual would have to be find a nontrivial result that the rest of the mathematical community hasn't found. Either they're inventing their own definitions and discovering entirely new mathematics in isolation (wow!), or they've somehow found a way to use the tools accessible to them (university analysis and algebra, at best) in a way every other trained mathematician has not. It's rare even for senior math undergrads to publish nontrivial research on their own. It's significantly rarer for that research to improve progress towards an open problem. The only example I can think of off the top of my head where undergrads actually resolved a well known open problem is AKS, and even then it wasn't a solo author.

In other words that shouldn't be the issue, because the correct "setting" for the stochastic matrix still allows for that possibility. It's not something you introduce by using real entries.

Great example! Do you have an outline of this proof? I can see the error in defining a unitary matrix over R instead of C, but I'm not immediately seeing how you can exploit that error to bypass the halting problem.

I'm guessing the concrete error would be introduced by overlooking that the real-valued matrix won't preserve the correct probability amplitude? If so, what's the next step to (falsely) deciding that a given algorithm will complete?

Yes, implementing that is the difficult part. I'm not saying it's infeasible - I'm saying it's very difficult. Writing provably correct software is also very difficult. This kind of theorem checking is becoming more common for theorems in e.g. combinatorics, but it's still very uncommon due to the added effort.

As for what I meant by that last point - there is a huge amount of mathematics to encode, almost all of which has not been formally specified into types, and much of which is structurally redundant. Not only is it difficult to encode necessary prerequisite mathematics for a theorem, but that requires its own effort and care, which multiplies the effort required for a given nontrivial proof to be checked.

Also note that I'm distinguishing between the general infeasibility of automated theorem proving and the otherwise great (but ordinarily feasible) difficulty of automated proof checking.

The steps to construct a proof of a theorem are as follows:

1. Formally define a language specification which will ostensibly contain the theorem statement and a proof.

2. Search the language space (the set of all strings of the language) for a proof of the theorem statement.

3. If step 2 fails, change the language specification to include prerequisite mathematics or an alternative statement of the theorem.

Even when you exploit clever structural properties of the theorem statement and the language definition, it's extremely easy to encounter a combinatorial explosion when you search for a proof.

On the other hand, suppose you already have a proof and simply want to verify it using a proof assistant. In order to do so, the language you define must be able to automatically verify all prerequisite mathematics which your proof depends on. Encoding that mathematics into a language is difficult and tedious, and dependency checking is itself vulnerable to combinatorial explosion.

It's a very active area of research in its own right, but it can't obviate this problem for the foreseeable future.

Aaronson’s rules are more technical than the ones given in this article, the latter of which can be generally applied to most claimed (dis)proofs of famous open problems. In particular, poorly typeset papers from a solo, unknown authors are highly unlikely to be correct when they’re claiming to resolve well known open problems.

In regard to the P/NP problem specifically, Aaronson’s examples of weaker open problems which would likely be resolved first - or at least accounted for in some nontrivial way - are analogous to this article’s point about proving weaker theorems first. It would be very suspicious for a proof that P and NP are separable (or not separable) to work at such a high level that it doesn’t prove also new results about the lower bounds of a litany of other problems.

Your geometric series is not a summation of the steps or positions, but rather the time required to complete each step. Therefore your example is characterized by an identical geometric series to the model I used in my previous comment.

More generally, Zeno’s paradox can be succinctly resolved by citing the monotone convergence theorem. Every bounded, monotonically decreasing function converges. The time required to complete the infinite series of half steps converges, because (again, with the definition of a metric) the time required to complete each individual step decreases commensurate with the change in distance.

One really active area of research is type theory. Another easy one is differential geometry and linear algebra, which are fundamental to machine learning. Try searching for resources about manifolds and dimensionality reduction. While we're at it, any kind of vectorized computation can also be modeled through linear algebra. There's probably material somewhere that formalizes a lot of Intel CPU architecture algebraically.

Actually, we can basically just take linear algebra and talk about its applications to nearly any area of computing, since it's one of the most ubiquitous areas of mathematical theory. What are you interested in? Graphics? Algorithm design? Data analysis? Parallelism? Statistics?

Combinatorics is also a big one. Knuth's The Art of Computer Programming has a wealth of material on combinatorics and its applications to computing (and in particular, algorithms thereof). You also might like learning about the way complex numbers and quaternions are used to computing spatial rotations.

I'm not personally a fan of category theory, but if abstractions are your thing you might enjoy Bartosz Milewski's Category Theory for Programmers. Along similar lines (and circling back to type theory), there's CMU's Software Foundations series of books[1].

If you are interested in cryptography, you might really enjoy reading Chris Peikert's A Decade of Lattice Cryptography[2]. The author is a prominent researcher in post-quantum cryptography, and walks through the last decade or so of research in cryptography based on a type of structure in abstract algebra called a lattice. This is very technical material, since it's more like a survey than a book.

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1. https://softwarefoundations.cis.upenn.edu/

2. https://web.eecs.umich.edu/~cpeikert/pubs/lattice-survey.pdf