The Nontriviality Argument
Every so often, I make an argument that makes me sound like the least cool kid in school. It goes like this: "Yes, from A follows B, but it is nontrivial." Sometimes this "follows" is a formal mathematical proof. Other times, it means that A provides enough empirical evidence that B has to be true. But neither of those explanations accounts for the nontriviality. Either A makes B come true, or it doesn't.
There are variations on the argument that I also employ. One useful exercise I do is to count the discrete logic leaps. At the top level, indeed, from A follows B. If you start unpacking, it might turn out that A implies C, C leads to D, D requires E, and from E finally follows B - that is four leaps we had to make. If the proof at each leap isn't ironclad and has a chance of failure, then a longer chain is harder to check and believe than a shorter one. Apart from truthfulness of logic, there is also an explanatory or pedagogical hurdle if you want anyone else to believe your claim. Why did you pick this particular train of thought from all other possible ones? Where is your chain leading? The longer the chain, the harder it is to explain.
Sometimes people ask me hard, loaded questions immediately before I have to depart. I can only say "Sorry, I cannot answer this question in 1 sentence/1 minute". Distilling tl;dr is hard. The difficulty of writing short and to the point communications was acknowledged by Blaise Pascal in the 1600s if not before. Even if your conclusion is valid and sound (as they preach in intro philosophy classes), it is still hard to convince anyone else.
Communication is not just my problem, it's everyone's problem, with huge implications. Every time we communicate, we experience a little piece of weight, or friction, or cost, and these pieces add up. In the book Why Information Grows, the antidisciplinarian César Hidalgo accounts for these pieces across a dizzying range of scales, from atoms to national economies (even says so on the jacket). Summarizing the whole arc of his argument in a few words would be contrary to my message here, so I focus just on Chapter 7, in the middle of the book, that weaves in the transaction cost theory.
In economics, transaction cost theory discusses the non-market cost of employee interactions within organizations. In the market, independent agents can buy and sell goods and services at posted prices. But apart from directly exchanging money for what you need, you need to negotiate such an exchange. The negotiation is even more obvious when there is no money changing hands. A typical scenario is working on a project with your colleague who reports to the same boss: you don't pay each other money but need to get along. As an organization grows, its workers pay more and more transaction costs in form of meetings, audits, reports, approvals, coordination (read: bureaucracy) - and thus spend less time on the primary job. Existence of such transaction costs puts a limit on how large an organization can get before it has to fragment in order to keep communication costs bearable. Hidalgo calls this limit of organizational knowledge a firmbyte. While communication advances lower the transaction costs, they can't get rid of the costs completely, the limit is always there.
I am an academic scientist. While I get paid a salary in money and use it to support my livelihood, it is not my primary academic interest. I am instead interested in science communities and cracks between them. Anyone who tried to read a book or present at a conference from a different discipline is aware of the translation difficulties. What is an acceptable object of study, what is a typical method, what is the assumed knowledge, what is the common jargon all differ between fields. Explaining ourselves to a different audience incurs a transaction cost. With experience, the cost decreases, but never goes away completely. If there was no communication barrier, we would all be doing the same blob of "science", rather than separate physics or biology or sociology. The exact lines along which science fractured into pieces were driven by historical contingencies. However, since there is a cost to explaining things, as soon as we reached a critical number of scientists, some sort of fracture was inevitable.
Of course, I seem to be jumping across these fractures quite easily in my scientific career. I recently gave a popular talk introducing the field of complex systems, following the taxonomy of this booklet. Complex systems, I said, are loosely defined by several features - from multiscale emergence to unpredictable dynamics to interdisciplinarity of the practitioners. During the Q&A one audience member was very upset that my classification mixed up objective and subjective features, math and people. However, the very reason why the "complex systems" field can exist is because of the simultaneous search for commonality of behavior and respect to domain knowledge. The complex systems conferences and research groups have a higher respect for the weird stuff at the margins and in the cracks between the disciplines, and a willingness to listen. In this atmosphere, the communication costs are lower, and thus more actual work can get done. The Nobel Prize in Physics a few weeks ago placed complex systems front and center in its citation, and the community rejoiced with this new legitimacy. Whatever the laureates accomplished did not become any less or any more true or important, but life for complex systems scientists got a tad easier.
Why can't we just be objective about science? Why all these games about acceptance?Mathematical theorems might be formally, innately true, but making claims about them, whether correct or wrong, takes effort by scientists. Which theorems do we choose to prove? Which experiments do we make? Which science gets done? These questions don't have singular answers, and they are not even objective in the sense of being divorced from particular human points of view. These questions are solved by the politics of inclusion and exclusion of people, by allocation of resources, by rewarding and punishing behaviors. The innocuous, uncool accounting of nontriviality leads us to a whole labor-centric view of science.
I had the pleasure of meeting César Hidalgo in person on several occasions. One of the more recent ones was at the PhD defense of his last MIT student. The post-defense reception was at the research group space which was being packed up for the move out. In César's office there was a collection of translations of Why Information Grows into various languages, and a stack of author's copies of the original US print, so I asked for a signed copy. He graciously agreed and wrote "Dear Andrei, Never Stop Computing!" Today, this phrase strikes me as particularly appropriate. Computing, processing information, making nontrivial claims, laboring is how we make stuff happen.