Posted by Jason Polak on 01. December 2017 · Write a comment · Categories: opinion

Three things I would like to see happen with the practice of mathematics in the next ten years are:

Computer proof assistants…

…that are easy to use for the typical mathematician. Why? Mathematics is expanding tremendously and some areas are getting quite abstract. Some areas are so complex that I even wonder if anyone in some of these fields has time to verify important details. Proof assistants will probably become necessary, not just for the creation of new mathematics but to help us organize the amazing amount of existing mathematics.

Actually, there are several proof assistants that exist are and currently maintained. I’ve tried ones like Isabelle, Mizar, and Coq, but have not found them easy to use. Unfortunately making it possible to input typical mathematics into a computer so that it can be verified mechanically is a pretty difficult problem.

Blockchain Paper Publishing

This isn’t just for mathematics, but I think if anyone could adopt blockchain technology for journals, it would be mathematicians. Actually compared to other fields I think math has done pretty well with creating independent, open-access journals. However, we could go further and adopt blockchain technology.

If mathematicians used a single blockchain system (or maybe a few), every peer-review process could be recorded on the blockchain ledger. Traditional journals could still operate on this system, and in this case the name of the journal could be recorded in the blockchain ledger. In fact, this would be a benefit to journals because the book-keeping records would be in the blockchain and external hosting would not be necessary.

But one wouldn’t need a traditional ‘journal unit’; someone who writes a paper could send it off to a willing independent editor and that editor could just send the paper to be reviewed as usual. The result of this independent transaction would be recorded on the blockchain.

The upshot is that all peer-reviewed transactions would be recorded on a distributed blockchain and all paper-reviewing activities would receive a quantitative measure of credit.

De-emphasis of proving new results in mathematics

This relates back to the idea that mathematics today is progressing so quickly that theorems are being proved more quickly than people can understand them. So today you have hundreds of fields, some of them with only a few people inside them knowing what’s going on. That’s kind of a shame because mathematics would be enriched if there were a little more cross-disciplinary understanding.

Thus it would be great if more emphasis and credit were placed upon simplifying existing research and making it more accessible. This would help the next generation of mathematicians enter the field. But it would also alleviate the pressure that exists for many existing research to output theorem after theorem, which is probably not all that useful for developing the understanding of mathematics as a whole anyway.

I still think we should prove theorems, and there are many cool things that haven’t been done yet. But with a more varied approach to math, we could concentrate on fewer, more quality theorems rather than pure quantity.

Posted by Jason Polak on 23. September 2017 · Write a comment · Categories: opinion

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Posted by Jason Polak on 06. August 2017 · Write a comment · Categories: math, opinion

A senior mathematician who will remain nameless recently said in a talk, “there is nothing left to prove”. In context, he was referring to the possibility that we are running out of math problems. People who heard laughed, and first-year calculus students might disagree. Was it said as a joke?

Because of the infinite nature of mathematics, there will always be new problems. On the other hand, there are only finitely many theorems we’ll ever know; only finitely many that we’ll ever be interested in. Are we close to knowing all the interesting theorems? Is the increasing specialisation of the literature a sign of a future with a thousand subfields each with only one or two devotees?

Truthfully, I don’t think math is running out of problems at all. I think it’s more like good, nonspecialist exposition isn’t really keeping up with the rapid development of mathematics and so we know less and less about what our colleagues are doing. So we should attempt to prevent the future where every person is their own research field. Here are some ways we could do that:

  1. Make part of your introduction in your paper understandable to a much wider range of mathematicians. This will encourage more collaboration and cross-disciplinary understanding. For example, once I was actually told by a journal to cut out a couple of pages from a paper because it was well-known to (probably ten) experts, even though that material was literally not written down anywhere else! Journals should actually encourage good exposition and not a wall of definition-theorem-proof.
  2. Have the first twenty minutes of your talk understandable by undergraduates. Because frankly, this is the only way mathematicians (especially young ones) in other fields will actually understand the motivation of your work. How are we supposed to ask good questions when we can’t figure out where our research fits in with the research of others?
  3. Use new avenues of mathematical exposition like blogs and nontechnical articles. Other fields like physics and biology appear in magazines like Scientific American and have an army of people working to make specialised work understandable to the nonspecialist.
  4. Encourage new, simplified proofs or explanations of existing results. And by ‘encourage’, I mean count high-quality, expository papers on the level of original results in determining things like tenure and jobs! There are already journals that publish these types of papers. Chances are, any expository paper will actually help at least as many people as an original result, perhaps more. And there are still hundreds of important papers that are very difficult if not impossible to read (even by many experts), with no superior alternative exposition available.

I think it’s been a long-lived fashion in mathematics to hide the easy stuff in favour of appearing slick ever since one dude tried to hide how he solved the cubic from another dude, and it’s probably something we can give up now.