The Map of Mathematics [video] | Hacker News Reader

The Map of Mathematics [video]

openculture.com

249 pointsghosh38 comments

I found this to be a nice summary of all the themes. I don't understand why comments here are so critical—it's a youtube video not a PhD thesis!

Here is my take on a concept map of math topics: https://minireference.com/static/tutorials/conceptmap.pdf (covers only high school math + calculus + linear algebra)

You have to be critical so that the real good stuff gets the attention it deserves. This just makes too many glaring errors to justify its claim to be "the map of mathematics".

I think http://www.math-atlas.org was a way better attempt and hope it will come back.

The last copy I could find on archive,org is http://web.archive.org/web/20150616152045/http://www.math-at...

A good presentation style doesn't make something correct. The correctness is what folks are critical of, not the medium.

... And then someone more familiar with, y'know, mathematics went and made one: http://nada.kth.se/~axelhu/mapthematics.pdf

I've always like this tree [0] because it shows clearly which subjects are prerequisite for learning others.

0. http://space.mit.edu/home/tegmark/toe.gif

This is great -- I have been looking for different dependency models, and I hadn't seen this one.

It's not a tree ;)

Whoops, you're right. Just a DAG.

I registered here to share this image exactly. Saw it over on /r/math and while any such endeavor is doomed from the start to be unrealistic or incomplete, this one didn't anger me quite as much.

A 2d-map of arxiv papers: http://paperscape.org/

I'd never seen this before. This is fucking great. Much better than the parent post. Thanks for sharing it!

A two dimensional projection of the space might imply too much distance between fields that are otherwise incredibly close; like vectors/versors/quaternions or combinatorics and with CS and optimization.

Maybe one could do an interactive version where the nodes can move in different dimensions like historic timeline, field of use, mathematical area.

On some online learning sites there are interactive knowledge graph mapping prerequisites such as:

https://www.khanacademy.org/exercisedashboard

https://www.expii.com/map/0

The presenter does explicitly call out the problem with 2D presentation.

Or computer science, which also deals with sets, logic, coplexity and groups.

Related - map of Physics: https://www.youtube.com/watch?v=ZihywtixUYo

Any chance of an international treaty whereby all parties agree to stop saying either "maths" or "math"?

No criticism here, I'll agree to either one.

"called Godel's Incompleteness theorems, which for most people means that mathematics does not have a complete and consistent set of axioms. Which means that it's all kind of made up by us humans"

Uhh... that's not the interpretation of the incompleteness theorems...

See also the Halting Problem

This is a great visualisation. And quite instructive to find myself mostly operating on the Pure Math side. Maybe that explains the gravitational pull I feel in the direction of FP and the like.

It's a followup to the Map of Physics https://www.youtube.com/watch?v=ZihywtixUYo

Wikipedia's version: http://i.imgur.com/zGcdMVl.png

https://en.wikipedia.org/wiki/Areas_of_mathematics

I watched the video and it definitely does not paint an accurate picture of mathematics. Additionally, there's a heap of misinformation (e.g., "fractals are scale invariant", "group theory is about groups [of things]", "Gödel's incompleteness theorem leads to a mystery of why math is even useful", all of which is not true whatsoever).

The most beautiful part of math wasn't explained at all, which is how the fields relate! How do geometry and algebra come together? How about algebra and topology? How about prime number theory and complex numbers? Many of the most influential, important, deep, and illuminating theorems of mathematics are precisely those that make such bridges.

Instead, the video gave extremely high-level mathematical "buzzword soup" with artificial boundaries and an explanation that seems to be derived after the fact.

I'm all for educating the masses on the magnificent landscape of higher mathematics, but I think it's a disservice to do it non-factually.

The video does mention that "how the field relate" can't be drawn properly on the same 2d map.

So maybe the next step is now to make other maps using different projections to show those relations ? Using the same pictograms would help people visualize better, and it would make an interesting collection of maps.

Indeed, it would have been better if they had one or more mathematicians in their team. I'm pretty sure that most mathematicians would have loved to support them, at least to proofread their script and to review their animations.

These are the things I found I would have known to be wrong at the end of high school:

The complex plane doesn't usually have the imaginary on the x axis.

Real numbers are not the only ones that have infinite digits, think 1/3.

e is not called 'the exponential'.

Also, why the hell is probability applied math?

1/3 is a real number, did you mean irrationals?

The video described real numbers (beyond rationals, hence irrationals) as ones with non-terminating decimal expansions.

I have not watched the video, but for people reading only comments let me clarify.

Numbers go naturals < integers < rationals < reals. Reals are the union of rationals (quotient of integers) with irrationals.

Rationals may have an infinite decimal expansion, like 1/3 has, but it has a repeating pattern at some point. Irrationals have an infinite decimal expansion and has no repetition of that kind.

This characteristic of irrationals does not depend on the base, it is always the same way. The finitude or infinitude of the representation of a rational depends on the base, but if infinite, there is a repeating pattern.

The video says the difference between reals and rationals is that reals can have infinite decimal expansions. But I concede you're right (if a bit pedantic), the complement of reals and rationals is irrationals.

Do you have a better source? I would love to see something that focuses on the relations, such as what Langland tried to do, in video form.

The book "Elements of Mathematics: From Euclid to Gödel" by Stillwell does this in terms of identifying what is considered advanced in a survey of elementary math topics and showing how they fit together or don't, like how Groups drop commutativity of multiplication from the ring http://press.princeton.edu/titles/10697.html

Great video, one mistake I spotted: he did include the number 1 among the prime numbers.

Because 1 is not a prime: "A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself." https://www.wikiwand.com/en/Prime_number

Funner definition is "positive number with exactly two divisors". This reduces the amount of special casing

The reason:

"The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization"

Here's the map in case you don't want to watch the entire 11 min video:

https://www.flickr.com/photos/95869671@N08/32264483720/in/da...

I wonder if there are a set of features and distance metric that could describe each field well enough to do hierarchical cluster analysis -- maybe through scraping keywords from enough mathematics journals, etc?

Tangential, but what tools do people use to make such videos?

It really depends on how much effort you want to put in, and your existing skills. I personally use Final Cut Pro. YouTube also has built-in editing features for stitching, cutting, and possibly overlaying videos.

Final Cut Pro is quite high end so I only use 5-10% of the features to make this video: https://vimeo.com/73754523

It took a few hours of storyboarding and editing once I had the footage.

Reply on news.ycombinator.com