Write Ideas Down Before They’re Formed

What I learned at the intersection of math and fiction

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Sometimes I prove mathematical theorems in my work as an engineer. It’s one of the most intense types of brain work that I do. I also write fiction, and it taught me one of the most important lessons about all my creative work: to write down everything that I know.

Setting the scene

I’m hunched over my desk with grid paper in front of me and a PDF of a textbook with reference material on my computer. I glanced up at the PDF and back at the page full of symbols I’ve written. My brow is furrowed. I’m stuck.

Involuntarily, I stretch my neck a bit and let out the breath I didn’t know I was holding — oblivious to my need for a break — even though my head’s throbbing and my shoulders are stiff.

Almost there. Almost there!

I can’t stop even if I want to. The discomfort is mounting, and so is the promise of success.

This is how I usually feel when I am about to figure something out. I have all the different aspects of the problem stuck in my head and work them simultaneously. I’m switching and comparing approaches, thinking about which ones will probably work, and suddenly become aware of the existent disadvantages! Time to consider borrowing an element of one approach to offset a disadvantage of another.

Prelude to Success, turning Discomfort into Anticipation

Holding so many alternate and competing possibilities in my mind is exhausting. It is also exciting because I have learned to recognize these symptoms as being on the brink of a creative breakthrough.

In the past, sessions like this one, be it over a period of months or even years, usually add up to a proof of a theorem, where every step has been checked from every angle for holes. In one session I build a step, and in the next, I find a flaw in the step. In the next session, I try to fix it. Eventually, I look for flaws and don’t find any. The proof is as bulletproof as I can make it.

And that is the result I’m seeking — the mathematical result I can apply to my engineering problem. And achieving that means I can finally rest my straining, pulsing brain and bask in the glow of success.

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Recently, I learned a new method that I borrowed from one of my other areas of interest: fiction writing. An exercise commonly recommended to fiction writers is free-writing (also known as Morning Pages), that is, stream-of-consciousness writing where you put down on paper whatever is in your mind, without pause, for three pages. You write before you’ve had a chance to complete the train of thought; in fact, the train of thought is altered by the writing of it. The exercise forces your mind to work differently, slows down for your writing and helps increase your ease and fluency.

When I start writing fiction, I don’t have the whole story in my head — at least, not in a form that performs the function of a story. I may have the plot or a loose structure, but I don’t have the elements that make for a memorable reader experience or create an emotional impact.

The emotional complexity of characters and the journey that the reader goes through are too complicated to hold in my head, but I can divide the creative process over several sessions to make it manageable. Before this though, I have to write everything I know about the story at the time and let the story evolve on the page.

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A Mathematical Theorem is Like a Fiction Piece

Mathematical theorems, like fiction pieces, are thought processes too complex to hold in my head. One day, more or less by accident, it occurred to me that I could apply my writing method to my mathematical proving process.

Instead of completing the thought in my mind and then writing it down, I wrote while I was thinking. I started getting my thoughts down while they were still half-formed. By downloading ideas onto the page, I was effectively freeing up my working memory and making space for my next thought. This allowed me to create more complex, layered, and multifaceted ideas because I was not limited by how much I could hold in my mind at a given moment.

For example, while still proving a theorem, I just wrote in my notebook all the half-formed thoughts and possibilities I knew.

Maybe, I can change notation?

Or change parameterizations and have simpler forces?

What about switching parameterizations in between?

I just wrote down my approaches as they appeared and went back to work each of them out in more detail.

Okay. So here’s the sticking point. I think we’re going to get to this result, but unfortunately, this term doesn’t cancel.

I wasn’t able to finish the theorem that day. Thanks to my notebook, though, I could bookmark a Lemma (an intermediate conclusion that will be used in the theorem), and start from where I’d left off, for my next session a few days later.

In the past, I would have wasted my energy carrying around a half-formed idea in my mind for days. I would have felt like I had nothing to show after all my exhausting brain work. Now, when I start, I allow myself not to have every step figured out. I have the statements that I know to be true, guesses about which ones will get me to closer to the conclusion, and an instinctive idea of how the proof will end up. I build from there, just like I would do for a fiction piece.

My New Approach and Why It Works

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The most striking difference in my new approach is how easily-flowing the experience of writing math proofs is compared to my past experience. Instead of feeling like my brain is about to burst and I can’t stop until it releases some of the pressure, I feel relaxed, I pause, flip back in my notebook and enjoy the progress.

I invite you to write down everything you know. Don’t just keep it in your head, write down your current knowledge of the problem, what you expect to happen, and the things you think might be relevant but aren’t sure. You’ll avoid frustration and the unfulfilled expectations of having “The Answer” at the end of your session.

Freeing up the brain-space you were using to think those thoughts will enable you to take unimaginable next steps. Let your notes build on each other on the page and become something that your mind alone could not create. You will be surprised to find the boundaries of your knowledge expanding in ways you could not predict or control.

In my next installment, I will take you through how these ideas can be applied to blogging. In the meantime, write your ideas down and let me know what happens next!

Originally appeared in Evidence Of…

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