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mathgeniuson Matrices as Tensor Network Diagrams

Indeed, the "einstein summation convention" is a slightly more primitive version of this graphical tensor notation.

mathgeniuson Matrices as Tensor Network Diagrams

Yes it really is just like lego, you can make all kinds of bizarro shapes.

mathgeniuson Matrices as Tensor Network Diagrams

Yeah, and this works for any "algebra" (ie. vector space with a way of multiplying vectors.) The three legged thing is what they call the "structure constants" for the algebra.

mathgeniuson The Six-Fold Nature of Snow

John Baez has a nice post about ice crystals here: https://johncarlosbaez.wordpress.com/2012/04/15/ice/

mathgeniuson The value of friendships in academic success

Do they separate causation from correlation, or is that not the point?

mathgeniuson My Billion Dollar Mistake

These people* think that you build a plan, or set a direction, by assembling many details together. Imagine just memorizing facts until you had enough facts in your brain that you could formulate a plan. It's ridiculous, but surprisingly difficult to criticize, because they have their facts to back them up!

The other way is this: use your intuition, and then go to the facts (details) to verify or disqualify the intuition. It's top-down rather than bottom-up. You don't get intuition by memorizing stuff. Beware the clean desk. I could go on and on ...

* I hate to categorize people, but there it is.

mathgeniuson My Billion Dollar Mistake

Thankyou dang. This is not personal at all. I wish this person well, I'm sure he's a nice guy!

I wanted to point out a phenomena, that I happen to be passionate about. I have thought alot about this. It was not meant to be a random twitter comment. But I am very serious about this, and hope that people notice this pathology. This guy is clearly lost in his "left-brain" [1]: full of details, trees with no forrest. It's all about plans, manipulation, linearity, verbiage, control. No wider context, no magic. Probably he has an impressive mastery of details, but basically clueless, as I said. Very likely he is a positive energetic person, full of charisma, nice to be around. Yet too often we put these charismats in charge! So that's my Dr. House run-down that I was reluctant to give. In summary, good with spreadsheets, hopeless at finding a direction.

[1] See "The Master and His Emissary" by Iain Mcgilchrist. Awesome book.

mathgeniuson My Billion Dollar Mistake

Ugh. I've worked for, and with, people like this. Never again. It's clear from this article that he still has no idea why he "failed". Apparently his failure was too many ideas, and yet later on in the article he says they failed because execution slowed. Finally he admits it was magic lost, ie. he has no idea.

The best solution he comes up with amounts to better planning and "Decide where to focus limited resources." Duh.

This guy is ernest, but clueless. If you meet someone like this, run for the hills. They should not be in charge of anything apart from the excel spreadsheet in front of them.

mathgeniuson Why have so many physicists shrugged off the paradoxes of quantum mechanics?

There have been several books recently that describe the saga of twentieth century quantum physics, culminating in the realisation that entangelement is a real thing that has consequences. My favourite is "The Age of Entanglement: When Quantum Physics Was Reborn" by Louisa Gilder. It's a fun book.

The history is fascinating. I'm waiting for a book that describes the deeper sociological aspect. A book with attitude. For instance, several of the founders of quantum physics had mystico-philosophical things to say about the quantum. But all of this was thrown out the window with the advent of the second world war. Physics became overrun by calculations. It took quite a while for people to get around to thinking about entanglement again. John Bell struggled for decades to get people to notice his work on this. Even the guy that invented the laser was met with stern disbelief when he presented his results to the "elders". Partly this is science just doing its thing, but there also does seem to be a history of science being dragged, kicking and screaming. Especially with quantum physics.

mathgeniuson Applied Category Theory

For a first introduction to category theory, from the masters thereof, I would highly recommend the book by Lawvere and Shanual "Conceptual Mathematics: A First Introduction to Categories". It almost doesn't look like a mathematics book, but believe me, these guys know what they are talking about. They distill some very high-brow ideas down to something us mere mortals can grok.

mathgeniuson Applied Category Theory

Have you seen Robert Ghrist's book, "Elementary applied topology" ? There's plenty of category theory in there, even though it is mostly implicit. (And applications to ML and robotics.)

https://www.math.upenn.edu/~ghrist/notes.html

mathgeniuson Applied Category Theory

Category theory is like putting types around mathematics. Like how dimensional analysis in physics points the way toward what operation to perform, category theory helps to show what mathematics "makes sense". Another example: I've heard that Haskell's type system is so good that you can often find the function you need just by searching for the function signature (it's type). Category theory does that for mathematics.

mathgeniuson How the Boeing 737 Max disaster looks to a software Developer

This was not buggy software. This was a bad specification. The software did what it was supposed to do.

mathgeniuson Pure Maths in Crisis?

> Proof is the essence of mathematics.

Proof may be essential, but it is not the essence of mathematics. Like many things, the essence of mathematics is beauty.

In this mad world, of which academia is a part, we only seek to measure things. And how can you measure beauty? You cannot, or, at best, it is subjective. So, in place of beauty we have big complicated proofs, that no-one understands.

John Conway, later on in life, decided to quit being a "real mathematician" and focus instead on trying to understand mathematics. His student Borcherds came up with a proof of Conway & Norton's moonshine conjecture, but Conway himself never accepted it. Why? Because it was too complicated, it couldn't possibly be the right explanation for moonshine. What a dude.

mathgeniuson Homeless, living in a tent and employed: The changing face of homelessness

I was totally interested in watching this, but this youtube is complete garbage, and does not belong on HN.

mathgeniuson ‘A toxic culture of overwork’: The graduate student mental health crisis

> machine learning of logic programs from examples.

This sounds like it would have use in industry. You are really saying that it does not?

mathgeniuson ‘A toxic culture of overwork’: The graduate student mental health crisis

Not a stupid question at all.

History is littered with examples of major scientific discoveries that went completely ignored because the discoverer was not associated with a prestigious institution, or some other reputable platform.

> quality work

So what? There is no shortage of quality work.

If someone really wants to make it as an outsider (or even an insider) I would suggest spending some time on youtube watching videos about how to make it in hollywood. It is much the same: fashion-driven, full of bean-counters, whats the next hot thing, who do you know, are you fun to work with, etc. etc.

People have this idea that science is the objective search for truth. But this is just a small part of what is going on.

mathgeniuson Ask HN: Startup failed after years of work – Can I even get a job now?

You have been working for someone: your customers. Hopefully you got some good reviews along the way.

mathgeniuson The Gyllenhaal Experiment

Where is the journalism here? Where is the story? I don't see any. "People can't spell" isn't a story...

mathgeniuson Ask HN: What VPN service are you currently using?

Not if you use one of their servers in sweden.

mathgeniuson Impossible Mathematics of the Real World (2017)

> The precise explanation is complicated, but hinges on the fact that 163 is what is called a Heegner number.

These are the nine Heegner numbers: 1, 2, 3, 7, 11, 19, 43, 67, 163

Here is a python session:

  >>> import gmpy2
  >>> gmpy2.get_context().precision=1000
  >>> pi = 2*gmpy2.acos(0.)
  >>> f = lambda i : gmpy2.exp(pi * gmpy2.sqrt(i))
  >>> f(163)
  mpfr('262537412640768743.99999999999925007259719818568...

> Or take the mathematical relationship fancifully known as “Monstrous Moonshine.”

I wouldn't qualify this as a near miss... This relationship has been shown to hold exactly. I guess in this equation "196,884 = 196,883 + 1", the +1 looks like a glitch, but if you look at the larger pattern it is not a glitch.

mathgeniuson A monad is a monoid in the category of endofunctors, what’s the problem?

For me category theory is more concrete. The basic reason is simple: categories provide type information. If all you want to do is grind out some calculations to pass an exam, then yeah, memorize the ring axioms and have at it. Anything more than that, and you need direction, or context, Ie. types, categories. Once you de-categorify that's when things become mysterious and confusing, and the impressive calculations abound. But it's just because you forgot the concrete thing that you are calculating.

It's all a bit paradoxical, and I'm not explaining it very well. (And I have no clue about the Haskell stuff.)

mathgeniuson A monad is a monoid in the category of endofunctors, what’s the problem?

I would highly recommend the book by Lawvere and Schanuel: "Conceptual Mathematics: A First Introduction to Categories". It is a very down-to-earth introduction to category theory, written by two masters of the field.

mathgeniuson In the foundations of physics we haven't seen progress for the past four decades

There is a large overlap of mathematical techniques between cond-mat and high energy physics. This has led some to speculate that particles are just excitations in some deeper "condensed matter" substrate.

Overall I find Hossenfelder's attitude to be far too negative. I'm fine with canning the big particle accelerators, but apart from that, there is a huge amount of exciting research going on in physics, theoretical physics (not just high energy stuff!), and mathematics.

mathgeniuson Visual Programming Doesn’t Suck

Totally agree. For me any such system would have to come with vim bindings. This pretty much rules out "wires". All wires would need to come from some kind of geometric locality. For example, I have often dreamed of a programming language that uses horizontal and vertical compositions of tiles. It would be easy enough to layer this on top of a language like python, assigning semantics to the horizontal and vertical directions. But it would probably be a bit arbitrary.

mathgeniuson Visual Programming Doesn’t Suck

Thanks for linking to graphicallinearalgebra.net . I finally sat down and read it today, and (loveheart-emoji). This stuff looks more complicated than just writing a bunch of matrices, but it really does go to the deep of linear algebra.

mathgeniuson Visual Programming Doesn’t Suck

The canonical example of this phenomenon is the spreadsheet.

Otoh, it does often lead to job security.

mathgeniuson Brexit Deal Fails in Parliament

Politics should be utterly boring, instead we get this endless theatrics. Politics should not be entertaining at all. Same goes for finance & banking. These jobs should be the most boring jobs in the world. Just get on with it, and forget about all the posturing and game-of-thrones crap.