Felix Breuer's Blog

The Geometry of Restricted Partitions

Slide from Talk

Last week, I attended the IMA Workshop on Geometric and Enumerative Combinatorics which was, hands down, the best conference I have ever attended. The speaker lineup was simply amazing and I return brimming with ideas for future projects and collaborations. I felt, therefore, particularly honored to be invited to speak in front of this audience, given that so many of my personal “academic heroes”, who I have cited so often, were present.

Given these high stakes, I am delighted to say that the talk went very well and that the reception was overwhelmingly positive! I spoke on joint work with Brandt Kronholm and Dennis Eichhorn on a geometric approach to witnessing congruences of restricted partition functions. The great thing about this subject is that it allowed me to do what I love the most about mathematics: using geometric insight to visualize combinatorial objects in an unexpected way. In this case, this approach allowed us to - quite literally - look at partitions from a new point of view and derive new theorems as a result of this shift in perspective.

Accordingly, the talk is very rich in visuals. I always think talks about geometry should put pictures before text. This time around, I went even further than I usually do and created interactive 3d graphics to illustrate the ideas. (See more on the technology below.) This turned out to be a big amount of work, but it was, I feel, well worth the effort. Especially in complex mathematical illustrations, interactivity, animation and 3d can transport much more information in a short amount of time than a static picture (or a bunch of formulas) ever could.

Slides and Video

  • Slides. You can view the slides online. They require a modern browser and were designed to work on my system on the day of presentation. They may or may not work on your system today. Recommended window size for viewing: 1024x768 pixels. Use arrow keys to navigate. Also, the IMA hosts a ZIP archive of the slides for offline viewing - beware, though, that it may take some fiddling to run (see README.txt).
  • Video. Happily, the IMA made a video recording of my talk in Minnesota. It turned out very well, except for the fact that the colors are very much off. But you should still be able to make sense of everything.

Title and Abstract

Oh yes, title and abstract, I almost forgot. Who needs those, anyway, when you got slides and video? Nonetheless, here are title and abstract from the Minnesota-version of the talk:

Combinatorial Witnesses for Congruences of Restricted Partition Functions via Ehrhart Theory

Abstract. The restricted partition function $p ( n,d )$ which counts the number of partitions of $n$ into parts of size at most $d$ is one of the most classic objects in combinatorics. From the point of view of Ehrhart theory, $p ( n,d )$ counts integer points in dilates of a $( d-1 )$-dimensional simplex.

In this talk we use this geometric point of view to study arithmetic progressions of congruences of the form $$ p ( s \cdot k+r,d ) \equiv 0 \mathrm{mod} m \forall k \geq 0. $$ Motivated by the work of Dyson, Andrews, Garvan, Kim, Stanton and others on the general partition function, we are not interested in arithmetic proofs of such congruences, but instead ask for combinatorial witnesses: To show divisibility we want to organize the set of partitions into disjoint cycles of length $m$.

It turns out that geometry is an excellent tool for constructing such combinatorial witnesses. Ehrhart theory induces a natural tiling of the partition simplex that can be used to construct natural cycles in several different ways. Following this approach we obtain combinatorial witnesses for several infinite families of arithmetic progressions of congruences. Moreover, these cycles have a direct interpretation on the level of combinatorics, which leads us to a new type of decomposition of partitions with great potential for further applications.

Finally, one of the great benefits of the application of geometry to combinatorial problems is that one can draw pictures. Instead of using Ferrers diagrams to visualize one partition at a time, we can use the theory of lattice points in polytopes to visualize all partitions of a given number simultaneously and gain insight from their spatial relationship. In this talk, we will therefore take a very visual approach to the subject and present a new way of “looking at partitions” – literally.

This talk is about joint work with Dennis Eichhorn and Brandt Kronholm.

Past Shows

I presented some version of these slides at several occasions:

  • Workshop Geometric and Enumerative Combinatorics, IMA, University of Minnesota, Nov 13th, 2014.
  • Seminar, University at Albany, Nov 5th, 2014. (presented together w/ Brandt Kronholm)
  • Partition Seminar, Penn State, Nov 4th, 2014. (w/ Brandt Kronholm)
  • 73rd Séminaire Lotharingien de Combinatoire, Strobl, Sep 9th, 2014. (w/ Brandt Kronholm)


The slides are made using HTML and JavaScript (or, more precisely, CoffeeScript). The basic slide infrastructure is provided by deck.js and the mathematical typesetting is courtesy of MathJax.

The most important part are obviously the graphics. They are all hand-written using Three.js, which is a terrific library that I have blogged about before. The graphics are all implemented on a very low level and could use a couple of layers of abstraction. I am looking forward to trying out MathBox in the future, which unfortunately is between versions right now and so was not an option.

Let me emphasize that the slides should absolutely not be used as a template. They are hacked together on a very tight schedule using copious amounts of copy-and-paste and an excessive neglect of refactoring. I provide these slides so that you 1) can learn about my research, 2) are inspired to communicate your research visually and 3) start writing your own presentations from scratch, using an appropriate library. I might also be willing to contribute to a library for making math presentations – if you are interested in creating such a library, feel free to get the ball rolling and send me a message!

Using Three.js to Create Vector-Graphics from 3D-Visualizations Right in Your Browser

When you want to prepare your 3D visualizations for publication in print/PDF you typically want to convert them into a vector graphics format. Three.js is a JavaScript library for creating 3D graphics that can be easily used to create vector graphics images in SVG format from 3D visualizations. Here is an example of how to achieve this – and you can modify the example right in your browser!

The Story

Last year, I created a 3D-visualization of the greatest commond divisor function. Then, a few weeks ago, I wanted to take snapshots of the visualization for inclusion in a research paper I was writing. However, as the research papers was intended for print/PDF publication, I was not satisfied with rasterizing the image (i.e., “taking a screenshot”). I wanted to have a vector graphics version of the image, especially since my visualization consisted entirely of lines. Ideally, I wanted to have the image in SVG format so that I could edit the result in Inkscape, my favorite program for creating mathematical illustrations. Unfortunately, Sage, the computer algebra system in which I had originally prepared the visualization, does (at the time of this writing) not support exporting 3D plots in a vector graphics format. So I had to find a different tool.

Three.js came to the rescue. Three.js is a fantastic JavaScript library for creating 3D graphics in the browser. It is mature, easy to use, has a large community and a creator, Ricardo Cabello, who is extremely helpful. Moreover, I think scientific publishing needs to move away from the PDF as its main medium and start creating web-first publications for the web browser – very much in the spirit of, e.g., MathJax and Substance. So, getting my feet wet with three.js was certainly a worthwhile investment.

In my opinion, the three.js version turned out much nicer than the original. Just as with the original version, you can modify the code producing the visualization right in the browser. However, in contrast to the Sage-based version, there is no need for clunky Java applets to do the rendering and there is no dependency on a Sage Cell server that benevolently does all the computation for the reader of the blog post – now, everything happens right inside the reader’s browser, which makes these kind of interactive documents far easier to host. And of course, you can now take SVG snapshots of the visualization, which was the motivation for the whole exercise.

So, let’s move on to the graphics and code.

The 3D Visualization

Below is a graph of the greates common divisor function. See this blog post and this expository paper for the math behind this picture. Use the mouse to zoom and rotate.

To take an SVG snapshot of this picture, click here: SVG snapshot.

The SVG Snapshot

This is a static SVG snapshot of the 3D image above. The SVG source is shown below. Paste it into an .svg file and open that in Inkscape for editing. This post-processing by hand is extremely useful!

The Code

Here is the code for the visualization. The language is CoffeeScript. Feel free to experiment and change the code. To reload the visualization with the updated code, click: Run!

Naturally, when working on a project of your own, you will want to have the code on your own system. If you want to use this example as a template to start from, feel free to clone my GitHub repository fbreuer/threejs-examples.

The above code uses both the SVGRenderer and OrbitControls which are not part of the standard three.js library but can instead be found in the examples directory of official three.js repository. This also means that they have some limitations. For example, the SVGRenderer does not support all features of the standard three.js renderers and some fiddling may be required.

Next Steps

Of course this is not the end of the story. Inspired by this success, I already have big plans for a next visualization project based on research papers I am currently writing with Dennis Eichhorn and Brandt Kronholm. I also plan to take a closer look at MathBox, which is a wonderful library for math visualization based on three.js.

Thoughts on QED+20

The potential of the comprehensive formalization of mathematics has fascinated me for quite some time - even though I am a working mathematician whose research is not part of the area conventionally known as “formal methods”. Last week, I took the opportunity to attend the QED+20 workshop which was part of the Vienna Summer of Logic and which celebrates anniversary of the original QED workshops and manifesto.

20 years ago, the QED project set out to create “a computer system that effectively represents all important mathematical knowledge and techniques”. Today, as even a cursory glance at the field will reveal, we are still a long way from achieving that goal. Nevertheless, as the excellent talks at QED+20 showed, there has been a tremendous amount of progress. I also enjoyed the workshop because the illustrious range of speakers at the event gave a wonderful overview of the many different perspectives of the researchers in the area.

However, for most of the workshop, I was struck by an odd incongruence between what researchers in the formalization of mathematics are doing and what I, as a working mathematician, would be interested in.

The potential of QED

From my perspective, there are many benefits that a wide-spread machine-processable formalization of mathematics could have. These potential benefits include:

  1. Ascertaining correctness of mathematical theorems.
  2. Revolutionizing mathematical publishing.
  3. Automation of large-scale case analyses and computations in mathematical proofs.
  4. Semantic search of mathematical methods and results.
  5. Computerized discovery of new mathematical ideas.

The incongruence that I perceived is this:

To me, correctness is the most boring and least important of these benefits. Yet, it appears to be the one thing that the QED community focuses on almost exclusively.

By contrast, the other items are all very interesting and appear crucial to the advancement of mathematics. However, with some exceptions they do not seem to be what current research in QED is all about. Let me explain what I mean by that by addressing these items one by one.


Mathematical research has two different modes: On the one hand, there is the “hot phase” of research (as Bruno Buchberger likes to call it) where you creatively explore new ideas to gain insight into the problem. On the other hand, there is the “cold phase” where you iron out the details and make your ideas precise in a semi-formal way. Research requires both of these processes. And yet, it is the hot phase that excites me and where actual discovery takes place. The cold phase, on the other hand, is due diligence – hard boring work that is necessary.

Now, just as John Harrison mentioned at one point, it would be fantastic if QED could make it easier for me to get the cold phase research over and done with. But instead, QED appears to be all about turning cold phase research into an art form all by itself. Beautiful ideas are ground out to a level of detail, formality and technicality that not only appears excessive and unintelligible (even to mathematicians) but that – as several talks at the workshop made evident – turn the project of formalizing a major proof into an epic undertaking, requiring quite literally man-decades of work.

The main outcome from these huge formalization projects is that a given proof is certified to be correct. For proofs of long-standing conjectures – such as Thomas Hales’ proof of the Kepler conjecture – this certification is worthwhile. But in my every-day work as a mathematician, correctness is not an issue. The interesting thing about a proof of a theorem in a research paper is not that it provides a certificate of correctness for a theorem. A proof is interesting when it conveys an intuitive idea for why a theorem should hold and when it reveals a new method for solving a particular type of problem. Whether the authors got all the details right is beside the point. The interesting question is whether the key idea works, and for this the human semi-formal process of mathematical reasoning suffices. Moreover, to paraphrase an argument of Michael Kohlhase, even if a proof idea turns out to be wrong: Either the idea is irrelevant anyway and then it does not matter that it is wrong. Or the idea is interesting, and then people will look at it, work with it and figure out how to get it right by way of a social process.


The fact that formalism, technicalities and correctness are secondary becomes particularly evident when writing math papers. From my own experience I know the more technical a paper is, the less it gets looked at. By contrast, papers that convey the key ideas clearly get read, even if that entails a very informal presentation of theorems and proofs. This makes writing math papers a frustrating process, as one has to strike a fine balance between between technical correctness and conveying the intuitive ideas – often even within a single sentence. Here, innovation in mathematical publishing could be a huge help. Separating the concerns of conveying intuitive ideas and their technical implementation – possibly by going beyond the static PDF as the delivery format of mathematical articles – would make life a lot easier for authors, readers and computer alike.

Computations in Proofs

Another area where computers can contribute a lot to mathematical research is carrying out large-scale case analyses and large-scale computations as part of a mathematical proof. Both the proof of the Kepler conjecture and the 4-color theorem are prime examples of this, but they are far from the only ones. Researchers at RISC routinely do such computer proofs in many different areas of mathematics. However, even as far as doing computer proofs is concerned, there seems to be a mismatch between the interests of mathematicians and those of the formal methods community. On the one hand, Thomas Hales reported at QED+20 that a major challenge in formalizing the proof of the Kepler conjecture was getting the “human proof” and “computer proof” parts to play well with each other within the formalization. Apparently, ITP systems are not tailored towards executing formally verified algorithms efficiently and accepting their computation as part of a proof. On the other hand, the mathematical community interested in computer proofs is happy to accept the computations of computer algebra systems or custom-written code as proof, even though none of these systems are formally verified! So while there certainly is interest in computer proofs, I see hardly any interest in formally verified computer proofs.

Semantic Search

As I have written previously, I view the increasing fragmentation of mathematics as a huge challenge. More and more, researchers are unaware of results in other areas of mathematics that would be relevant to their own work. This hampers scientific progress, makes research more redundant and decreases the impact of new results. Here, an efficient semantic search over a comprehensive database of known results could be a tremendous help. If researchers could simply ask a mathematical question to an automated system and be pointed to the relevant literature – especially if the relevant literature is phrased in a vernacular of a different area of mathematics, which they may not be familiar with – research productivity would be drastically increased.

New Ideas

Finally, there is of course the utopian hope that one days computers can discover genuinely new mathematical ideas that also provide new intuitive insight to human mathematicians. In a limited form, there are already isolated examples of such discoveries. But it seems that in general this utopian scenario is still far far away.


In view of this apparent mismatch between the focus of the QED community and my own interest in the subject, I was feeling somewhat out-of-place during most of the workshop. Thus, I was very happy with Michael Kohlhase’s talk towards the end of the day.

Kohlhase argued that when comparing the number of formalized theorems and the number of math research papers published, QED is losing the race by orders of magnitude (which reminded me of a I similar remark I made some time ago). He went on to say that correctness is over-rated (mirroring my sentiment exactly) and that to be able to bring the widespread formalization of mathematics about, we need to relax what we mean by “formal”. He argued that there is a whole spectrum of “flexiformal” mathematical articles, between the informal math paper and the formal proof document in use today. Moreover, he argued that based on flexiformal articles we can achieve a whole range of objectives, such as semantic mathematical search, as long as we do not focus on certifying correctness - which is a non-issue anyway.

These comments were not only very well placed - their were also particularly delightful to me, as I had spent the afternoon brainstorming and scribbling notes (under the vague headline “zero foundations” ) on what I would expect from a useful “flexiformal” mathematical proof document.

What is a proof anyway?

I think, if we want to make any progress towards QED, then we have to radically rethink what we mean by a human-written mathematical proof document. Currently, a formal mathematical proof is taken to be a text written in a formal language with a precisely defined semantics which then is compiled – using a fixed ITP system with a deterministic behavior – to a low-level proof in terms of the axioms and inference rules of whatever logic the ITP is based on.

This, however, is not what a human-written proof in mathematical research paper is. A proof in a math paper has no well-defined semantics. It is not based on any clear-cut foundations of mathematics whatsoever. It is not “bound” to any fixed library of definitions and theorems or any fixed dictionary of terms. Notation and terminology is inconsistent. Citations are vague. But these are not bugs – these are features.

The purpose of a mathematical article is not to prove a theorem, it is to convey an idea.

For this purpose, lack of any precise foundations is an essential feature of mathematical writing. The informality makes the writing far more concise and it makes the ideas stand out more clearly. It makes the content far more resilient to changes in the way definitions and theorems are stated and used in the contemporary literature (truly formal proofs “bit-rot” alarmingly fast). And it makes the content accessible to a wider range of readers from a variety of different backgrounds.

Despite all this ambiguity, I still think it is perfectly feasible to define what a mathematical article really is:

A mathematical article is an advice string intended to help the reader solve the in general undecidable problem of proving a theorem.

Most importantly, the obligation of proof lies with the reader of the article – no matter whether the reader is a human or a computer: It is up to the reader to pick the foundations on which they want to base the proof. It is up to the reader which background knowledge they are going to use. It is up to the reader to pick the theorem that they want to prove in the first place (which may well be a different theorem than the author of the article had in mind). And in the end it is also up to the reader to decide whether the ideas contained in the article will be at all useful for the task the reader has set themselves. In particular, the advice string is in no way assumed to be trustworthy – it does not certify anything.

This “advice string interpretation” of a mathematical article lies somewhere in the middle of the flexiformal spectrum. How such a flexiformal proof sketch might look like in detail I do not know, even though I have speculated about this before. The main objectives would be to produce an article format that is

  • vastly easier for humans to read,
  • vastly easier for humans to write,
  • vastly more flexible in its relation to previous knowledge,
  • significantly more versatile in its applicability to different problems,
  • not necessarily endowed with a precise formal meaning,
  • not necessarily correct, but still
  • useful to computers as advice for producing proofs.

Of course such a format would be vastly more difficult for a machine to handle than today’s formal proofs. Which brings me to the last topic of the workshop.

Artificial Intelligence

Fortunately, artificial intelligence is (finally) making huge progress in QED as well. Cezary Kaliszyk and Josef Urban have done amazing work integrating existing provers with machine learning methods for premise selection and turning the whole thing into a service that people can actually use. To mention just one highlight, the hybrid system they constructed can prove automatically almost all of the individual steps that appear in today’s large libraries of declarative proofs.

This is not quite as impressive as it looks at first glance: As I found in my own experiments with declarative proofs, a large part of the work involved in formalizing an existing mathematical proof goes into choosing the intermediate steps in just the right way. Nonetheless, it is already a huge help when systems can justify individual proof steps automatically.

Of course, we still have a long ways to go before such technology could automatically interpret flexiformal proof sketches in order to produce a fully formal proof acceptable to one of today’s ITP systems. And, given that I called into doubt the importance of formal correctness checks, the question arises why we should strive to have machines translate flexiformal proof sketches into low-level formal proofs at all. Perhaps Kohlhase is right, and we can achieve all the objectives that we care about without getting computers to reason at a more formal level than humans would.

However, most interesting applications, for example semantic mathematical search, arise when computers can in fact reason at a deep level about the mathematical content. Precisely because mathematicians in different areas phrase the same ideas in different language, the most interesting synergies will arise when computers can help us draw the non-obvious connections. Of course, nothing says that this reasoning has to be formal. But, in order to make sure that humans and computers do in fact talk about the same thing when reasoning about mathematics, a formal foundation certainly does seem like the most natural common ground.


The QED community’s focus on correctness may well be a necessary first step that we need to take before we can attack the more exciting applications. However, I think the time is right to start talking about what else QED can do, especially if the QED community want to attract the attention of the wider mathematical community. I am glad that projects like MathHub.info are taking first steps in this direction, and place a particular focus on exploring new kinds of mathematical documents. No matter whether we start from an entirely formal level and build on top of that, or whether we start from an informal level and dig down: Getting working mathematicians interested in QED will require new kinds of flexiformal mathematical articles – in between the extremes of complete formal correctness and informal prose stuck in a PDF file.

Time Flies at RISC

It is amazing how quickly time can pass: I am already half a year at the Research Institute for Symbolic Computation in Linz, Austria. (Actually, RISC is in Hagenberg, but I am living in Linz.) I am working in Peter Paule’s Partition Analysis project, which is part of the SFB Algorithmic and Enumeratice Combinatorics, a joint special research program of Johannes Kepler University Linz, the University of Vienna and the Technical University of Vienna. The people at RISC are fantastic and I enjoy the lively - and very international - research environment.

I have been positively surprised by both the quality and the quantity of events that are taking place around here. In addition to the regular Algorithmic Combinatorics Seminar and Theorema Seminar, there are always interesting workshops and conferences to go to. Mentioning just those I attended, there were:

in addition to several internal meetings of our SFB, both at RISC and in Vienna. Unrelated to all these local events, I was also happy to travel to Geneva to participate in the Open Knowledge Conference, back in September, which was a great experience.

Between all these goings-on and the general excitement of a transatlantic move to a new country, I also got a lot of research done. Primarily, I have been working with my good friend Zafeirakis Zafeirakopoulos on the joint project we started back in San Francisco and I have been exploring new territory with Brandt Kronholm, my fellow postdoc in the partition analysis project, and Dennis Eichhorn. In addition, I managed to finally get a couple of papers out the door which have been in the pipeline for a long time, and I have enjoyed many stimulating conversations with the people at RISC and in the SFB, leading up to exciting future projects.

Last but not least, I met a couple of very nice coders in the coworking space Quasipartikel right around the corner from my apartment. Michael Aufreiter and Oliver Buchtala are working on the fantastic editing and publishing platform Substance and I have had many in-depth discussions about Fund I/O with Martin Gamsjäger and Milan Zoufal.

All of the items above, would deserve their own blog post, but that is (obviously) not going to happen. However, judging by the experience of the last few years, the spring is always a good time to start blogging again. So, look forward to further updates soon!


I want to close this post by sharing a couple of pictures of the beautiful scenery that comes with doing research at RISC. The institute itself is housed in the venerable Castle of Hagenberg.

Castle of Hagenberg, home of RISC

The castle’s tower offers a nice view of Wartberg and the valley.

View from tower

My own office is in a very nice new extension building overlooking the pond.

RISC extension building

RISC extension building in the evening

I myself do not live in Hagenberg but the city of Linz. Here you see the view over Linz from the Pöstlingberg.

Linz skyline

As you can see from the list above, Strobl is a favorite location for status seminars of the different working groups here at JKU. Unfortunately, these usually go from early morning to late in the evening, so that there is usually no time to enjoy the scenery. But if you can manage to squeeze in a walk between sessions and the weather plays nice, the view on the Wolfgangsee can be truly breathtaking.

Wolfgangsee, seen from Strobl

From Open Science to Open Mathematics

“Science is based on building on, reusing and openly criticizing the published body of scientific knowledge. For science to effectively function, and for society to reap the full benefits from scientific endeavors, it is crucial that science data be made open.” Panton Principles

“A piece of data or content is open if anyone is free to use, reuse, and redistribute it — subject only, at most, to the requirement to attribute and/or share-alike.” Open Definition

I could not agree more. However, what do open science and open data mean for mathematics?

As exciting as the open science and open data movements are, they appear at first glance to be largely unrelated to the world of pure mathematics, which revolves around theorems and proofs instead of experimental data. And theorems and proofs are “open” the moment they are published, right? Does this mean that mathematics is already “open”?

Of course, the word “published” is loaded in this context: The debate around open access publishing in academia is ongoing and far from settled. My personal view is that the key challenge economic: We need new funding models for open access publishing - a subject I have written a lot about recently. However, in this blog post I want to talk about something else:

What does open mathematics mean beyond math papers being freely available to anyone, under an open license?

The goal is to make mathematics more useful to everyone. This includes:

  • Discovery. How can we make relevant mathematical research easier to find?
  • Understanding. How can we make math papers easier to comprehend for a wider range of people?
  • Exploration. How can we allow readers to interact with our work?
  • Application. How can we make mathematical knowledge easier to use in many different contexts?
  • Modification. How can we make it easier to build upon mathematical research?

We can open up new possibilities in each of these areas by reimagining what it means to publish mathematical research.

Mathematics as data

Examples, definitions, theorems, proofs, algorithms - these are the staples of mathematical research and constitute the main body of tangible mathematical knowledge. Traditionally we view these “items” of mathematical knowledge as prose. What if we start to view examples, definitions, theorems, proofs and algorithms as data?

Examples have always been the foundation of any mathematical theory and the discovery of new examples has been a key driver of research. As systematic search for examples (with computers and without) is becoming increasingly important in many fields, experimental mathematics have flourished in recent years. However, while many researchers publish the results of their experiments, and some great open databases exist, experimental results often remain stuck in a tarball on a personal website. Moreover, the highly structured nature of the mathematical objects encoded has led to a profusion of special purpose file formats, which makes data hard to reuse or even parse. Finally, there is a wealth of examples created with pen and paper that either are never published at all, or remain stuck in the prose of a math paper. To make example easier to discover, explore and reuse, we should:

  • Create decentralized databases of examples. Think both OEIS and “github for examples”.
  • Promote the use standard formats to represent structured data, such as YAML or JSON.
  • Acquire the parsing skills to deal with special-purpose file formats where necessary.
  • Complement LaTeX papers with data sets of examples in machine readable form.
  • Make uploading data sets on the arXiv common practice.
  • Publish examples even if they don’t make it in a paper.

The rise of experimental mathematics goes hand in hand with the rise of algorithms in pure mathematics. Even in areas that were solidly the domain of pen-and-paper mathematics, theoretical algorithms and their practical implementation play an increasingly important role. We are now in the great position where many papers could be accompanied by working code - where papers could be run instead of read. Unfortunately, few math papers actually come with working code; and even if they do the experiments presented therein are typically not reproducible (or modifiable) at the push of a button. Many important math software packages remain notoriously hard to compile and use. Moreover, a majority of mathematicians remains firmly attached to low-level languages, choosing small constant-factor improvements in speed over the usability, composability and readability afforded by higher-level languages. While Sage has done wonders to improve interoperability and usability of mathematical software, the mathematical community is still far away from having a vibrant and open ecosystem as available in statistics. (There is a reason why package managers or a corner stone of any programming language that successfully fosters a community.) In order to make papers about algorithms actually usable and to achieve the goal of reproducible research in experimental mathematics, we should:

  • Publish the software we write. This includes publishing the scripts we use to run our experiments in order to make the easily reproducible.
  • Write software to be used - even outside of our own office. Invest the time to polish and document code.
  • Use common package repositories to publish software, not just the personal home page.
  • Prefer high-level languages over low-level languages to make our libraries easier to reuse and our code easier to read and modify.
  • Make software easy to install.
  • Make coding part of the pure math curriculum, not just part of applied math.

Theorems and proofs are the main subject of the vast majority of pure math papers - and we do not consider them as data. However, opening up theorems and proofs to automatic processing by making their semantic content accessible to computers has vast potential. This is not just about using AI to discover new theorems a couple of decades in the future. More immediate applications (in teaching as well as research) include using computers to discover theorems in the existing literature that are relevant to the question at hand, to explore where a proof breaks when modifying assumptions, to get feedback while writing a proof about the soundness of our arguments or to verify correctness after a proof is done. The automatic and interactive theorem proving communities have made tremendous progress over the last decades, and their tools are commonly used in software verification. To be able to apply these methods in everyday mathematics, we should:

  • Develop formal tools suitable for everyday use by working mathematicians (as opposed to experts in software verification or formal methods).
  • Start formalizing parts of the mathematical articles we write.
  • Create the infrastructure to publish and integrate formal content with prose articles.
  • Explore the use of formal methods in teaching mathematics.

Mathematics for people

The points mentioned so far focus on making mathematical knowledge more accessible for computers. How can we make mathematical knowledge more usable for humans?

First of all, there is of course the issue of accessibility. From screen readers to Braille displays and beyond, there is a wealth of assistive technologies that can benefit from mathematics published in modern formats. For example, MathML provides richer information to assistive technologies than do PDF documents. Adopting modern formats and publishing technology can do a world of good here and have many positive side-effects, such as making math content more readable on mobile devices as well. But even assuming a readers are comfortably viewing math content on a desktop screen, there is a lot of room for improving the way mathematical articles are presented.

Communication depends on the audience. Math papers are generally written for other experts in the same field of mathematics, and as such, their style is usually terse and assumes familiarity with facts and conventions well-known to this core audience. However, a paper can also be useful to other readers who would prefer a different writing style: Researchers from other fields might prefer a summary that briefly lays out the main results and their context without assuming specific prior knowledge. Students would appreciate a wealth of detail in the proofs to learn the arguments a senior researcher takes for granted. Newcomers could benefit from links to relevant introductory material elsewhere. And everyone appreciates richly illustrated examples.

A single static PDF document is not the best tool for achieving all of the above objectives at the same time. By experimenting with dynamic, interactive documents, we can create articles that are more useful to a wider range of audiences. Documents could be “folded” by default, giving readers an overview first and allowing them to drill down for details where needed, possibly all the way to a formal proof. Examples could be presented side-by-side with the results they illustrate instead of the two being interleaved in a linear text. Software can enable readers to dynamically rearrange the text, for example by “pinning” definitions from the preliminaries to the screen while working through the proofs. Procedurally generated figures can be modified and explored interactively. Algorithms can be run and their execution observed - and articles could even be used as libraries from other software. Social annotation frameworks can allow readers everywhere to engage in a dialogue.

As soon as we leave the printed page behind us, the possibilities are endless. However, for these visions to fulfill their potential, openness is key. In particular:

  • File formats have to be open and not proprietary. Everyone has to be able to create their own software for reading and writing these files.
  • File formats have to be easily extensible, so that everyone can experiment with what a “document” can be.
  • It should be possible to inspect a document to learn how it was written. (Think “show source” on a web page.) This way, authors can learn from each other by default.
  • There is no reason why there should be separate software programs for “reading” and “writing” a document. The transition from “reading” to “working with” to “building on” can and should be seamless.
  • Finally, licenses have to allow all of the above.


Open data matters for pure mathematics. Taking open principles seriously can transform mathematical research and make it more useful and relevant both within academia and in the real world.

To conclude, I want to add three more thoughts:

  • Intuition is a form of mathematical knowledge that I have not mentioned so far. In my view, it is the most important one, but at the same time it is by far the most difficult one to convey in writing. The best way to communicate intuition is through dialogue. Intuitive explanations on a printed page can confuse more than they help, which is why they are often excluded from published papers. Dynamic documents can offer new room for experimenting with intuitive explanations - this cannot replace interaction in person, but which can be very valuable for anyone without access to an expert in the subject area.
  • Open science and open mathematics have to be at least as much about education as they are about research. Open data that requires a team of data scientists and a compute cluster to make use of may create huge value for industrial applications, but excludes a large part of society. One of the key tenets of open knowledge is that it should be open to everyone. Being open to everyone is not just about licenses and price, however. It also means giving everyone the means to benefit from these resources. Open mathematics should empower not just experts, but learners at all levels.
  • Making even a tiny fraction of these ideas happen will require a huge amount of work, and this work does not come for free. Making one math paper open means that a second paper is not going to get written. I think this is a worthy investment and creates far more value for society. However, as long as the academic job market values publication counts above everything else, this may not be a wise choice for many, career-wise. The transition to open mathematics will require both young researchers who are willing to take that risk and academic search committees who value innovation.

Here is how to fund Open Textbooks and MOOCs

Open textbooks and MOOCs are the way of the future. Yet, traditional business models, from donations to sales, do not suffice to make the full breadth of educational content open to everyone. Fund I/O is the business model to fill this gap.

It is simple, really: the more people have access to an educational resource, the more can benefit from it. In an age where copying has almost zero cost and education becomes increasingly important, open textbooks and open online courses seem like the obvious way to go. However, the question how open educational resources can be funded remains unanswered.

The classical approach is to sell educational resources for profit. However, the price of textbooks is skyrocketing, instead of going down. Reasons are publishers’ profit interests, market failure (professors choose textbooks, but students pay), high production costs (both for content and for printing hardcopies) and the increase in the number of students’ buying used textbooks, renting textbooks and downloading unlicensed digital copies (aka “piracy”) to avoid paying the full price. These trends lead to publishers pricing their textbook out of the market and are thus self-reinforcing. Clearly, the traditional business model is not sustainable.

The alternative way to fund educational resources is through donations. This can take many forms. Professors may choose to devote their time to write a book (or produce an online course) for free. Governmental agencies or private foundations may provide grant money. Companies may sponsor a course (in exchange for some form of advertising). Or a fundraising campaign could gather donations from the interested public. If donations suffice to fund an open textbook, that is great. However, except for a few high-profile examples, donations won’t suffice to support high-quality educational publishing in its full breadth. (In fact, there is a theorem which, roughly, says that, if people behave rationally, the amount of money that can be raised for open content through donations is just the square root of the amount that can be raised through sales, i.e., \$1.000 instead of \$1.000.000.)

Crowdfunding is a new funding model that is working spectacularly well in the entertainment industry. While crowdfunding has yet to be tried at scale for funding open textbooks, there are two reasons why current reward-based crowdfunding models will not be a viable option for educational publishing in general. First, most successful crowdfunding projects do not promise the creation of open content. Instead, their success is critically tied to the exclusivity of rewards. Second, crowdfunding projects are typically carried by high-tier backers who pay exorbitant amounts for token rewards. While it stands to reason that the hard-core fans of an artist will happily donate large amounts of money in exchange for a casual meet-and-greet, it is hard to imagine students paying huge sums to meet their textbook authors.

Enter Fund I/O. Fund I/O is a new kind of business model that can provide access to textbooks to as many people as possible while at the same time covering the cost of production. The model is independent of donations and maximizes access instead of publisher profits. Moreover, Fund I/O provides a smooth transition towards making the textbooks open to everyone, according to a transparent mechanism.

An illustrated introduction to the Fund I/O model is given here. In a nutshell, the idea is this:

  • The more students buy the textbook, the lower the price.
  • As the price drops, early buyers receive a full refund for the price difference. So nobody has an incentive to wait with their purchase.
  • When the price reaches a minimum value, the textbook is made free for everyone.

In particular, students (or universities, or libraries) can pledge how much they are able to pay for the content. If enough students pledge that amount, the price will drop automatically. In this way, students can determine the price of educational content.

From the publishers’ point of view, the deal looks like this: Publishers limit their profits to a fixed maximum amount, right from the outset. In return, they greatly reduce their financial risks and give their customers a rational incentive to reveal how much they are truly willing to pay for the content. By giving publishers and their readers a mechanism to communicate rationally about price, Fund I/O resolves the vicious cycle of publishers charging ever more and customers finding new ways to avoid paying.

In short, whenever donations or ads do not suffice to cover costs, Fund I/O is the business model to use for funding content that serves a common good.

A different way to fund websites, web services and software development

In the age of the Internet, it is difficult for websites, web services and software developers to monetize the value they create. Most customers are extremely price sensitive and try to pay as little as possible for software or web services provided through software.

The prevalent business model is to provide a basic, ad-supported service for free and to charge for the full-featured service. Rates of conversion from the free to the premium service are generally low, and ads do not pay much per impression. As a result, such services are only sustainable if they reach massive scale quickly.

These economic constraints shape the software services we use - often not in a good way: Software that could run perfectly well as a standalone application is turned into a web service to funnel traffic past advertisements. Services that would work perfectly well for a small group of users are turned into yet-another social network, in order to advance scale through network effects. Companies put the interests of advertisers before the interests of their users, because advertisers are where the money comes from. User data offers additional revenue streams at the expense of user privacy. And software that caters to niche markets cannot be financed, even if the value created for its users would cover the costs of development.

Many of these constraints are not facts of life, but simply a feature of current business models. In this post I describe a radically different business model, dubbed Fund I/O for web services, that provides incentives for users to finance the development of software products and web services directly. This model is based on the Fund I/O mechanism, and provides one key feature: it gives developers and users a rational mechanism to communicate about price.

Key Benefits

Fund I/O for web services offers a number of advantages for both users and developers.

Advantages for users

  • Users get access for the lowest possible price, that covers the cost of providing the service.
  • Users can pledge how much they are willing to pay. If enough users pledge, the price of the service will go down.
  • Early adopters receive a refund as the price goes down. Everybody pays the same price, nobody overpays.
  • Users pay only for what they use. If a user ends up not using the service after they sign up, they won’t pay anything.
  • Users have full cost control. They never pay more than their pledge for a subscription period, no matter how much they use the service.
  • By supporting developers directly, users know that the company has the economic incentives to serve their interests over the interests of advertisers or investors.

Advantages for developers

  • Costs are covered through revenues. No need to finance operations through equity or debt.
  • Fund I/O is optimized for growth. Prices are set automatically to maximize the number of active users, while costs are covered.
  • Customers reveal how much they are willing to pay for the service, providing invaluable market data.
  • Fund I/O provides incentives for price-sensitive customers to support the service financially and stop free-riding.
  • Developers can focus on creating value for their customers, instead of worrying about placing advertisements or maximizing profit.

Fund I/O for web services creates a playing field that is radically different from the current model centered around VC capital, ads, freemium services and massive scale. As such, it caters primarily to web services for which the current model is not a good fit. Fund I/O is a great choice for web services that:

  • do not have the scale neccessary to support itself through ads or sales of premium features,
  • cannot be supported through donations alone, or
  • do not have the venture capital to support years of operations at a deficit.

How it works

Fund I/O for web services is around the following three assumptions about the behavior of the majority of users with regard to web services and software:

Users want to pay as little as possible. Here, the Fund I/O model subscriptions can be of tremendous value, as it gives users rational incentives to reveal their true valuation for the product or service. This is in contrast to a classical sales context, where users would understate their true valuation and, e.g., opt for the free tier of a service, even if the value they obtain from the service exceeds the price of the premium tier.

Users do not want to buy something they do not know. Most software products and web services are not commodities. They cannot be perfectly substituted. Instead it depends on the particular characteristics of a given software product whether a user will want to work with that software at all. Thus, users avoid making a significant up-front investment without trying the product first. However, even after a trial is over, relatively few users make a purchase. In short, discontinuities in spending turn away users. A simple solution is to adopt a simple rule: Charge users in proportion to their actual usage.

Users do not want to constantly monitor their usage. Hence the popularity of flat rate offers for all kinds of services. Therefore, users should be charged in proportion to their actual usage, but only up to a fixed total amount. This way, users have peace of mind, even if they do not pay attention to how much they are using the product.

Here is an example how this could work in practice. Suppose a company wants to develop an HTML5 web app. It needs \$10,000 per month for developer salaries. To keep things simple, let us assume that on top of that, additional hosting costs are \$1 per user per month of non-stop usage. However, the average user would use the app for about 30 hours per month, which amounts to average hosting costs of about 5 cents per user per month.

The pricing now happens as follows:

  • Development cost are distributed equally among all users, according to the Fund I/O for subscriptions model.
  • A user counts only as a “full user” when they used the web app for 20 hours or more in the current month. A user who just tried the web app for 2 hours would just have to pay for “one tenth” of the average development costs.
  • Hosting costs are passed on to the user directly. This boils down to a rate of less than 0.2 cent an hour, with an absolute maximum of \$1 per month.

Suppose the web app has a small but consistent user base of 1,000 “full” users, 500 occasional users at 10 hours per month and another 1,000 users who just stop by and look at the app for 1 hour. Then the app would have a total of 1,000 + 500 * 0.5 + 1000 * 0.05 users, which amounts to a total of 1,300 “full” users. Distributed evenly among them, the development costs amount to \$7.70 per user. So, the 1,000 full users are charged \$7.75. The 500 “half” users are charged \$3.87. And the 1,000 “window shoppers” have to pay 77 cents each. Just opening the app and looking at it for a couple of minutes would cost less than 10 cents.

Now, suppose over time the service grows by a factor of ten, so that we have 10,000 full users, 5,000 half users and 10,000 window shoppers. Then the respective price would drop to \$0.77 + \$0.05 = \$0.82 for full users, \$0.385 + \$0.02 = \$0.41 for half users and 8 cents for window shoppers. Just looking at the app for a couple of minutes would cost just 1 cent. The tremendous economies of scale inherent in web apps are passed on entirely to users. Moreover, the Fund I/O concept of average cost pricing and refunds as prices drop mean two things:

  • Users can pledge how much they would be willing to pay for the service, even if their pledge is below the current price. When enough users pledge, the price will go down.
  • To compensate early adopters for their investment in the service, a part of the revenues can be passed on to them as a refund for the premium they paid early on. The golden rule is that within one subscription period, everyone pay the same price for the service, no matter when they join or how much they pledge. There can also be transfers across subscription periods, for example if version 1.1 of a web service builds on version 1.0, for which early adopters paid a premium.

Of course this is not the end of the story. There are many variations that can be used to tailor this protocol to particular use cases. (See this list of posts for more details on these variations.)

  • To price-discriminate low-valuation from high-valuation customers, web services can offer tiered subscriptions where the upper tiers are tied to a higher base cost. The higher tiers can include dontations made in exchange for special rewards. The lower tiers can include an ad-supported free tier; however, as a free tier decreases the incentive for users to participate in the pricing mechanism, this is not recommended.
  • Producers can choose the cost function such that they make a limited profit if the service becomes popular. * Stretch goals can be used to achieve specific growth targets and provide funding for additional features.
  • The fact that prices drop over time can be used to create a smooth transition to a release of the software developed under an open source license. (See here and here for more on Fund I/O for free software.)


Fund I/O for web services is a business model that breaks with many of the conventions we currently take for granted in the internet economy. At the heart of the model is a transparent mechanism for users and developers to communicate rationally about price. Users actively shape the pricing of the web service through the very payments they make. Developers cover their development costs while reducing their dependence on advertisers and venture capital. Fund I/O for web services makes web services for niche audiences sustainable and at the same time provides them with a smooth transition to serving mass audiences if they become popular. Finally, Fund I/O offers a flexible toolbox that can be adapted to many different use cases.

Fund I/O offers a felxible toolbox of concepts for designing innovative business model, and it is just the beginning: There is plenty of room for disrupting the internet economy.

A Business Model for Crowdfunding Subscription Services

Fund I/O is a model for funding the production of any type of good with massive economies of scale. In its basic version, Fund I/O is intended to finance fixed costs that only apply once, such as the one-off cost of writing a book. (See here for an introduction.) However, many important goods and services incur fixed costs on a recurring basis. Examples include ongoing software development, research or journalism. These services require financing a staff of people to produce new software releases, white papers or articles on a continuing basis. However, once a given release/paper/article is produced, it can be distributed to an arbitrary number of people at a vanishingly small marginal cost. Thus the basic Fund I/O concept still applies.

A variant of Fund I/O that finances ongoing fixed costs is the following subscription model. Of course, each particular use-case will require further tweaks, but the basic model is this.

  • Each period creates new fixed costs. A period can be a time interval or a new release. An example would be \$5,000 every month.
  • Each customer submits a pledge for how much they would be willing to pay each period for access to the subscription. Let’s assume that on June 1st there are 100 people who submitted a pledge of \$50 or more, and a few more who pledged less than \$20.
  • The price is chosen as low as possible such that the costs are covered. Costs are distributed evenly among all subscribers who get access. In the example, the price would be set \$50 and the 100 people who pledged at least as much receive access for the current month of June.
  • As more subscribers sign up the costs drop. Say, another 100 people pledge at least \$25 by June 5th, then the price drops to \$25 and these 100 people receive access to the subscription for the month of June.
  • Previous subscribers are refunded the price difference. Each of the original 100 subscribers receive \$25. This can either be a cash refund or a credit that can be applied to the next months’ subscription fees.
  • Pledges remain active across periods. So if nobody changes their pledge, the subscription will cost exactly the same amount every month.

Just as in the case of one-off payments, the Fund I/O model has a number of advantages over classic subscription models.

  • Customers have an incentive to reveal their true valuation for the product.
  • The subscription is provided at the lowest possible price to the largest possible audience, while costs are covered.
  • Customers determine the price of the subscription through a transparent mechanism.

The subscription model described above can serve as a foundation for many different variants.

  • Rolling subscriptions can be incorporated into the pricing scheme. One customer can subscribe for June 1st till July 1st while another subscribes from June 5th till July 5th.
  • “Stretch-goals” can allow customers to determine the scope of the subscription through their pledges. If pledges are high enough to provide \$10,000 per month, another staff member can be hired, for example.
  • Periods can build on each other. Subscribers to version 1.0 of a software package can get refunds from sales of version 1.1, as the development work put into 1.0 clearly benefits 1.1.
  • And, of course, a share of the revenues can go to the producers as profit beyond the fixed costs for each month. These profits can serve as capital to provide funding for months with insufficient subscriptions.

Bottom line: Fund I/O is well-suited to subscription services. In the next post, I will go into detail on how this can be used to fund web services.

Funding Vaccine Production

Fund I/O can be extremely useful in funding projects with a social or environmental impact: producing vaccines, for example. One challenge of vaccine production is getting the market for a new vaccine to scale to the point where the vaccine becomes affordable. Because prospective buyers expect prices to drop, many will wait with their purchase, which may even prevent the market from reaching the required size. Fund I/O offers an original solution to this problem, by giving buyers incentives to buy now. This can provide a viable alternative to difficult-to-negotiate advance market commitments.

Fund I/O can be used not only for digital goods, but for anything that has large fixed costs and low marginal costs. This includes physical goods that have a large social impact, such as vaccines. Here is how this might work.

Suppose a vaccine for a certain disease has already been developed. Now it needs to be produced at scale to reach millions of people world wide. The problem is vaccine production requires a substantial investment to get going. This means, if a supplier were to produce just 100,000 doses of the vaccine, each dose might be prohibitively expensive. But at a scale of 10 million doses, each dose could be very cheap. As long as the market is small, the price for the vaccine will be very high so that only few people will be able to afford it. But once the market grows, the price will drop, and many more people will be able to afford the vaccine.

This creates a chicken and egg problem: To get the price down, many people would need to buy the vaccine. But for many people to buy the vaccine, the price would need to drop. So how can we get the market started?

One approach to getting the market started are advance market commitments. A couple of large buyers of the vaccine, such as governments or charitable donors, make a joint commitment to buying a large number of doses at a specified price. The idea is that by guaranteeing a certain demand for the vaccine in advance, the market can be jump-started at a volume attractive to suppliers and at a price attractive to buyers.

There is a catch, however: Because, the price for the vaccine will drop over time, buyers have an incentive to wait. Those who start buying early will pay a premium. It you wait, you pay less, and the later you get in, the more you save. Even if everyone wants to buy the vaccine, those who manage to start buying last will pay the least. Early adopters effectively subsidize late entrants.

You can imagine that this leads to quite a lot of maneuvering over who gets to be the last. Especially because the market for governments buying vaccines has several features that exacerbate the problem: First, very large sums of money are at stake. Second, buyers (such a governments) are often frugal, by necessity. Third, implicitly subsidizing other parties may be unpopular, even if it is for a good cause such as getting more vaccines to more people. And finally, the only benefits of entering early are measured in terms of impact (saving lives) and not profit (return on investment), which often makes spending money harder instead of easier - unfortunately. Advance market commitments can help a lot in this setting: by forming a coalition of first movers, the financial disadvantage to buying early is reduced considerably. But the incentive to get in later remains, and this makes it often very hard to establish such a coalition.

Here Fund I/O can help. In fact, Fund I/O was designed to solve exactly this chicken-and-egg problem. Using Fund I/O, buyers do not have an incentive to wait. To achieve this, all you need to do is apply the refund mechanism used in the second phase of the Fund I/O model. Buyers can purchase immediately and still benefit from economies of scale as the market matures. Through the refund mechanism, they can rest assured they will benefit from decreasing prices in just the same way as a late entrant.

This solution makes jump-starting markets much easier, by offering a smooth transition from a low volume/high price market to a high volume/low price market. Moreover, it does not require complicated negotiations aimed at getting buyers to commit to making purchases now, even though it would be in their financial interest to wait. The Fund I/O mechanism guarantees the same financial benefits as an advance market commitment involving all potential buyers, present and future, but it can be set in motion by just a few interested parties right away. Agreements are not made at a negotiating table but on a platform of exchange that incentivises all participants to reveal their true valuation for the vaccine.

Of course there are many more details to be addressed here, both as far as the theoretical model and practical issues are concerned. But these general considerations already make clear the Fund I/O shines at resolving difficult negotiations of this type by intermediating price differences across time.

A Crowdfunding Model for Open Source Software?

I think we need more crowdfunding projects for open source software! And here is one, for example: Yorba, the guys behind Shotwell, are currently running an Indigogo campaign to fund the development of their open source email client Geary. As of this writing, they have raised about \$50,000 of their \$100,000 goal and the rush of the last 24 hours has just begun. So, please head over there now and consider making a dontation. You can always come back here to read on!

Why Funding Open Source Software is Difficult on Kickstarter

Crowdfunding has taken off, for physical goods as well as for software-related products like games. So why do we see so few open source project take advantage of crowdfunding? In fact, Kickstarter is based on the Street Performer Protocol (SPP), a method for raising funds that was originally intended as a model for financing open source software, and that had some early successes such as making Blender open source. Could it be that rewards-based crowdfunding as on Kickstarter simply does not work as well for open source software?

There is one subtle difference between open source and closed source projects on Kickstarter, and this difference may be crucial: The successful game projects on Kickstarter have no intention of giving away their core product for free. This makes pledges pre-sales instead of donations. Open source projects on the other hand announce right away that their product is going to be free. Does this influence backer behavior?

It is certainly true that many crowdfunding campaigns are successful because some altruistic backers pledge huge amounts of money to the project - much more than they would need to pledge to get a copy of the product for themselves. But projects rarely can be funded through such altruistic contributions alone. Take a look at Shrouds of the Avatar, for example: the large majority of backers pledged the minimum amount required to receive a copy of the game. These backers are obviously price conscious and don’t pledge more than they have to. I will call these backers rational, in the economic sense. This means that they are self-interested and act to maximize their own welfare.

This leads me to the following working hypothesis:

Crowdfunding projects need contributions from both rational and altruistic backers to raise large amounts of money.

This would explain why open source projects appear (so far) to be less successful than other creative projects at raising money via crowdfunding: Because open source software is (going to be) free, rational backers don’t pledge anything.

In fact, there is research in the economics literature on the optimal behavior of an entirely rational backer who is faced with the decision of backing an open source project. The amount he or she is willing to pledge is determined by the probability of their pledge being pivotal and getting the project funded. Backers correctly observe that this probability decreases with the total number of backers interested in a project. A consequence is that - if all backers were rational - the amount of money open source projects can expect to raise grows only proportional to the square root of the number of people interested in the project. In contrast, the amount a closed source project can raise grows linearly in the number of people interested in the project. This can quickly lead to a difference of several orders of magnitude!

(Now, before anyone gets a wrong impression: I do not think that people behave entirely rationally or that they should behave rationally. It is vital for our society that people act out of altruistic motives. And it is great that crowdfunding has found a way to get altruism to scale, so that we can make great things happen. I think we should look for more ways in which we can integrate altruism into our economic activities. All I am saying is that there is a reason that our entire economic system is built on the assumption that people tend to behave rationally. Our goal should be to find economic models that incorporate both altruistic and rational motives.)

Fund I/O as a Model for Funding Open Source Software

Fund I/O is a new crowdfunding model based on the Average Cost Threshold Protocol (ACTP). (If you are not yet familiar with Fund I/O, you may want to check out this example before continuing here.) When I developed the ACTP, one of my goals was to find a crowdfunding model that would work for open source software. And in many ways Fund I/O achieves precisely that:

  • Most importantly, Fund I/O gets rational backers to reveal how much they would truly be willing to pay for the software.
  • This allows open source project to raise as much money as closed source projects.
  • Moreover a project can raise more money on Fund I/O than using the Kickstarter model, because it lets the market determine the optimal price of the product.
  • At the same time, Fund I/O still allows altruistic backers to dontate as much as they want to the project. (This requires Fund I/O to be used with the option of allowing each backer to set their own minimum price level.)
  • Finally, Fund I/O provides a clear path towards the relase of the software product under an open source license.

Now, you may have noticed that I did not quite say Fund I/O “works” for open source software. In fact, I do not think the Fund I/O model gives me everything I would personally want from an open source funding model. However, I do think it works much better than anything that is currently out there. So let me first explain in what way it does work for open source software. In the next section I will then go on to explain the limitations I see.

Fund I/O works great for open source projects if your project meets the following assumptions:

  • Your goal is to finance the creation of version 1.0 - no more.
  • You are willing to release your software under a closed license at first.

Then you just run through the three phases of the Fund I/O mechanism (with individual minimal price levels) and at the end:

  • you have completed the software you wanted to write,
  • you have released the software and its source code under an open source license, free for everyone to use and modify,
  • your costs were covered right from the start and
  • you even made a certain profit that you can determine in advance.

The release under an open source license happens if and only if there is enough altruism among your customers to finance the costs of production through voluntary donations. However, this absolutely not the same as simply using the SPP. First of all, your project gets made if there is enough rational interest in your software. It just won’t be open sourced unless there is enough altruistic interest. Second, Fund I/O achieves a smooth transition from the state where there are few users who would need to donate very large amounts to the state where there are many users who need to donate just a couple of dollars. This lessens the burden on any individual donor considerably. Moreover, the donation is a sunk cost for the customer: donating means forgoing future refunds instead of paying money now. All of these factors make it much easier to gather the required donations, than using the SPP, even if Fund I/O is used with individual minimal price levels. If a fixed minimal price level is used for everyone, then gathering the required donations gets easier still.


Now, I believe the above could work great for developing, say, an open source game. But for open source software I see a couple of caveats.

First of all, the user base of the software has to be willing to use closed source software during the second phase of the protocol. If a substantial number of users will never run anything but free software, then you are of course out of luck. I do not think this will be a problem; it was not a problem for Light Table during its crowdfunding campaign.

Second, it has to be reasonable to make the software non-free during the second phase. This means that the software is for sale and according to the license users are not allowed to pass copies of the software along to their friends. This is a stronger restriction than just not releasing the source code and may turn off more of the free software crowd. More importantly, this limits viral marketing effects during the crowdfunding and sales phases. People have to buy the product to try it out. (Even though a demo could certainly be offered.)

Of course the whole point of Fund I/O is that because of the refund mechanism, there is less of a disincentive to buying than in a classical sales context. In particular, if a customer sets their minimal price level to zero, then they get a full refund in case the software is released as open source. But in a world where many people expect application software (as opposed to games) to be free, this may be a hurdle. I’d be optimistic and say that the compromise of making software non-free for a limited amount of time, according to a transparent mechanism, in order to cover the costs of development is convincing enough to assuage these concerns.

Finally, and most importantly from my point of view, software requires maintenance. Application software needs to be developed continually. A game may be “done” after the 7th patch or so, but application software never is. If the only goal is to fund the creation of a version 1.0, then this is not a problem. But what if you are looking for a way to finance the development of open source software throughout its entire lifecycle?

An obvious modification to Fund I/O for financing several consecutive versions would be to run a new campaign for each major version. The mechanism could easily be adjusted to make a part of the revenues from the version 2.0 sales refunds for version 1.0. And once 1.0 is free, buyers/backers of subsequent versions could stop paying refunds to version 1.0 owners. The problems I see with this are therefore not theoretical, but practical in nature. The company developing the software would have to maintain several versions of the software side-by-side, they would have to backport security patches and provide support, and they would have to finance these activities from their version 2.0 revenues. While all of these concerns appear in commercial (and open source) software development anyway, they are exacerbated by keeping different versions under different licenses around - just because of the funding scheme.

Ultimately, it will depend on the particular project which of these considerations are relevant, and which variant of Fund I/O makes the best fit. Personally, I view Fund I/O as a toolbox that can be tailored to fit many different needs.


I hope to have convinced you that crowdfunding open source software is possible at much bigger scale than we have seen so far. What we have to do to make this happen is to find a crowdfunding model that takes the backers’ diverse patterns of economic behavior into account. I think Fund I/O is a step in the right direction, but can certainly be improved upon with regard to funding open source software.

What do you think would be the challenges when applying Fund I/O to open source software? How would you modify Fund I/O to make it more applicable to open source software? Comments and questions are welcome!