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Promoting the accessibility and appreciation of mathematics and its history#

We all believe that mathematics is an art. … We all have experienced on some rare occasions the feeling of elation in realizing that we have enabled our listeners to see at a moment’s glance the whole architecture and all its ramifications. How can this be achieved?

Emil Artin

What We’re Doing#

Alpine Mathematics is a non-profit organization that exists to promote the accessibility and appreciation of mathematics and its history:

  • Accessibility: What does it mean to be able to “access” mathematics and its history? It means being able to read a proof, understand it, and know where it came from. This is a challenge for everyone, from beginner to expert. Newcomers can use help in getting started, established students in going farther. Accessibility is about depth.

  • Appreciation: People who don’t paint or write songs still enjoy paintings and music. Even if you aren’t advancing mathematics, you should know that you can understand and enjoy the work of mathematicians. Meanwhile, those who already do enjoy math can branch out into new areas, and learn more about the subject’s history. Appreciation is about breadth.

We want to make mathematical proofs and their history more understandable and more accessible, both to beginners and to established students. As a result, the number of people enjoying and appreciating these great products of the human mind will naturally grow.

We believe that:

Please read on in our blog posts to learn more about our ideas and projects, and about how you can get involved.

Who this is for#

If you are already “in math,” this is for you. If you feel like an outsider, but are curious about math, this is for you. This is for everyone with an interest or involvement in math, at any and every level.

Everyone who studies mathematics, even the most accomplished professor, knows that accessibility is always an issue. You can be a world expert in one area of math, and still be like a first-year student in another area you haven’t had the time to delve into yet.

We are not about “dumbing anything down.” We are about providing tools to make your studies more fun and more effective, and even more inviting for those who haven’t really gotten into math yet. We like to put it this way: You can’t lower the mountain, but you can improve the trail.

How we’re doing it#

Our approach involves two key ideas:

  • A focus on the history of mathematics

  • New ways of presenting and communicating proofs

Understanding math through its history#

One of our primary goals is to deliver in new forms the classic literature in which modern mathematics emerged. In the HistArch project, we’re encoding proofs from the 18th, 19th, and 20th centuries, in order to bring them to students’ fingertips. We want students to be ready to delve into the subject’s history, and develop a familiarity with its origins.

Why is this so important? Because a big part of understanding a subject is understanding where it came from. Mathematics didn’t just appear one day, it was developed over thousands of years. And yet students are routinely asked to understand in a day an idea that might have taken many years to develop. We want students to know where, when, and how these ideas emerged.

From a logical point of view only the answers are needed, but from a psychological point of view, learning the answers without knowing the questions is so difficult that it is almost impossible. That, at any rate, is my own experience. I have found that the best way to overcome the difficulty of learning an abstract mathematical theory is to follow Toeplitz’s advice and to ignore the modern treatises until I have studied the genesis in order to learn the questions.

Harold M. Edwards

How we communicate proofs#

Proofs are what mathematicians produce. Proofs are a hidden art form. Proofs are the stories that take you, miraculously, from not knowing why a theorem is true, to knowing.

Sadly, proofs are hard to read, and most people never enjoy the great stories they have to tell. In one of our blog posts, we talk about why this is so.

In further blog posts, we start sketching our ideas for what we can do about this situtation. We talk about making annotated, electronically browsable proofs. We talk about restructuring studies to provide a top-down approach. We even talk about how to make math: the video game.

What to expect#

We’re calling on the mathematical community to help us build something big. It will take years, and the involvement of many. There are many ways to contribute.

So far, we’ve built some software which we think is a pretty good start, but also has enormous room to grow. We’ve also just begun to build some of the content – the proofs, the commentary, the annotations, the lessons – that the software is designed to display to the student. In our blog posts, we’ve sketched some visions of where this can all go.

If you have never studied math, will you open up our software today and be handed an instantaneous and effortless understanding of the proof of Fermat’s Last Theorem? No. Please don’t expect that!

As Euclid said, there is no royal road to mathematics. But as we have said already, and will say again, we can’t lower the mountain, but we can make the trail better. We can provide maps; we can remove obstacles; we can build bridges and ladders. We can even invite you to go skiing. We can make the trails more and more fun, but you still have to put your own energy into it, and spend time learning, practicing, and exercising.

Looking for PISE Online?#

PISE is the Proofscape Integrated Study Environment, and it’s free and open-source software.

You can run it on your own machine, or you can access it online.

The online version is called PISE Online, and it’s a place to publish and study proofs, proof expansions, and proof annotations.

If you already know what you’re doing, you can access PISE Online by clicking here: