The Mathematics of Product

“When you reach the end of what you should know, you will be at the beginning of what you should sense”

— Kahlil Gibran, Sand and Foam

In some sense, abstract mathematics is the art and science of discovering genuine insights about things that don’t exist. When I say “things that don’t exist,” I mean things that really only exist within the contours of your thoughts.

Consider prime numbers as an example. They don’t “exist” in the way your chair, or I do. You’ll never bump into 37 on the street. And yet, mathematicians have discovered all sorts of truths about the primes.

So to do abstract mathematics well, you need to learn how to reliably and repeatedly discover truths about things that don’t exist. It’s not an easy task. You are essentially in the business of pulling symbolic truth from the ether, like some kind of ex-psychic turned book-nerd.

Yet, mathematicians do it on a regular basis. Their livelihoods depend on it. So how do they manage?

I don’t claim to know all their methods. Certainly they are vast, and some methods work better than others. Consider as an example Srinivas Ramanujan, widely regarded as one of the greatest mathematicians in world history. Ramanujan stated that his family goddess gave him all his formulas and that he was not the least bit interested in an equation unless it was a thought of God. If you’re skeptical of this — note that it worked quite well for him. When he died at the tender age of 32, he left behind some 3,900 mathematical results, but left out the proofs for most of them. Since his death mathematicians have looked and found that nearly all of them are true. Some of his work is now used to describe the physics of black holes, even though the very concept of a black hole didn’t exist during Ramanujan’s life.

For those of us without access to Ramanujan’s seemingly divine prowess in pulling truth from the ether, there are more mundane and accessible methods. One such method is to learn how to systematically produce intuition.

Step 1: Gather and clarify relevant facts.

Step 2: Stop trying.

The thing about problem solving in any domain, not just math, is that it is really tempting to just sit down and “think really hard.” But as anybody who has sat and “thought hard” for 12 hours to no avail will tell you — the real juice in problem solving doesn’t seem to come about when you’re thinking real hard. It seems to come when you’ve stopped thinking entirely. It comes at you as a sort of feeling, as an intuition.

Alas, once you’re aware that intuition is the key to genuine insight, frustration awaits. Unlike your intellect, intuition is not push-to-start. It is rather like a pretty woman perennially playing hard to get. You never stop needing to earn her trust and presence in your life.

And so the question is how? How do you get intuition to show up and give you the juice you need to arrive at genuine insight?

The essence of the skill is to transcend the desire to “think real hard” and instead understand how to use what you can access on demand (your intellect) to prime something you need but can not access on demand (your intuition).

Systematic production of intuition is straightforward.

Step 1: Gather and clarify relevant facts.

Step 2: Stop trying.

In math, gathering and clarifying relevant facts is straightforward. You clarify the definition of your mathematical object. You remind yourself of the results that other mathematicians have proven about it. You study up on related mathematical structures that you feel might prove useful. And you just keep going, continuously trying to learn more relevant facts while making sure that you understand them as clearly as possible.

And then you stop trying. You play foosball, or dance, or sleep, or go swimming. You totally and completely forget that mathematics ever existed in the first place.

And then when you’re in the middle of your 15th lap in the pool, pssst. Intuition strokes you gently and whispers “try this.” And you get out of the pool, and you try it and lo and behold, it leads you to a fundamentally new approach to the problem and then you’re there. You’ve pulled truth out of the ether. You are a mathematical wizard.

To use an analogy, the whole thing is rather like cooking a meal with an A-list chef who won’t tell you what he’s going to make. All you know is that this chef is going to make a meal if you give him a bunch of ingredients and then leave the kitchen. But you don’t know which ingredients he needs and when he wants you to leave the kitchen. So you just go on gathering and prepping ingredients. You buy some broccoli and some onions and some chicken and prep it all and then leave the kitchen. But you come back the next day and the chef didn’t make anything. So you go back to the store and you buy some tomatoes and shallots and sweet potatoes and a little Parmesan. Then you come home and prep it all and leave the kitchen again. And then you come back the next day and the chef has left you a note with a recipe and nothing else. He didn’t even bother to say hello and he certainly did not cook the meal because that’s just how he is. But he tells you “do this, then that and then this.” So you follow the instructions and before you know it you’ve made a broccoli/chicken/sweet potato quiche with shredded Parmesan as an accoutrement. It’s delicious, and you’re at once satiated and confused. You never even knew that such a concoction was possible.

Contrast this with sitting in the kitchen and “trying real hard” to make something tasty, and you see why there is a subtle but important difference between the discipline of reliably inviting your intuition to come and play, and the hard work approach of thinking your way through problems.

The curious thing I’ve found early in my career as a product manager and entrepreneur is that this line of problem solving translates quite well, specifically for one core aspect of building product: figuring out what to build in the first place.

Oddly enough, figuring out what to build is a lot like discovering something new about the prime numbers (or any other mathematical object, for that matter). Like prime numbers, the future version of your product really exists only in your mind. Yet there are aspects of your product that you can grapple with right now.

And so the systematic process ports over almost precisely. Rather than gathering and clarifying mathematical facts, you research your user, you check usage rates on your current features, and you clarify how the product fits into your user’s daily life. You learn anything and everything you can about your product, your user, the business, the vision, and the market. And you keep distilling your understanding of all those things.

And then you go for a walk with your director of engineering, or you sit for dinner with the family. You completely forget that this whole “product” thing exists in the first place.

And when you least expect it, intuition whispers a cool “build that.”

Now at this point it bears stating that your intuition does not always lead you right. (Part of this is due to the undeveloped mind’s inability to discern between mental chatter and intuition, but that’s a story for another time). That is why in math, you must produce an intellectually rigorous proof to justify your intuitive leaps of faith. In product, the equivalent is to create an MVP (or some other kind of low-work test case) and see how users respond to it.

If they respond well, you’ve “proven” your product insights. (I use quotations here because truth in the world of product is a rather fuzzy thing. It’s more a matter of what works than an accurate description of reality, although the two are intertwined).

If they don’t, you just go back to the grocery store and hope the chef visits you again tomorrow.