Mathematicians and startup founders

Throughout history, mathematicians and physicists, like entrepreneurs, have craved for a theory of everything. They’ve been looking for a set of principles that can predict and explain all phenomena that we observe. In case of mathematics, this search led to mathematicians Bertrand Russel and Alferd Whitehead write a three volume (thick book) Principia Mathematica. Among other results in the book, they dedicated several hundred pages to derive 1+1=2 from even more basic “self-evident truths” (called axioms).

In the footsteps of Bertrand and Alferd, another famous mathematician David Hilbert in 1920s set out to derive all of mathematics from as few axioms as possible. The aim of Hilbert’s program was to formalize all of mathematics so that any given mathematical statement could be proven true or false using only initial axioms. A related expectation from this program was that if such axioms are found, then doing mathematics would be equivalent of mechanically combining these axioms. In other words, Hilbert wanted to transform mathematics from a creative enterprise to a mechanistic one.

Then Gödel dropped a bomb on this project in 1931.

His incompleteness theorems shook the foundation of mathematics. What Gödel essentially did was to formalize this silly little sentence (“This statement is false”) into mathematics and ask whether it was true.

Since we’re talking about mathematics here, it’s easy to get the details wrong so I’m going to quote someone else on this topic to explain incompleteness theorem.

if we have a list of axioms which we can enumerate with a computer, and these axioms are sufficient to develop the basic laws of arithmetics, then our list of axioms cannot be both consistent and complete.

In other words, if our axioms are consistent then in every model of the axioms there is a statement which is true but not provable.

The proof of incompleteness theorem is very interesting and I encourage you to go through it. However, what I find more interesting is what it implies. Incompleteness means that all systems of basic principles are insufficient to enumerate all true statements that can come out of it. There will always exist mathematical facts that you cannot prove or disprove. You can observe them to be working but there’s no step-by-step explanation of why they work.

For a long time, most mathematicians suspected that Fermat’s last theorem is true but nobody could prove that. Similarly, currently, there are many unproven guesses (called conjectures) in mathematics that are suspected to be true but nobody knows how to go from what we know to deriving these unproven results. For example, mathematicians strongly suspect that P is not equal to NP but nobody has found a proof for it. What incompleteness theorem says is that we can’t even say whether we will ever find a proof for it.

What all of this has to do with entrepreneurship?

Gödel’s incompleteness theorem is precisely defined and it has been abused to even “prove” God. Mathematicians would rightly cringe at the application of something as specific as a mathematical theorem to the vague world of humans.

Knowing fully that I’m stretching the limits of logic here, I find it interesting that the process of doing mathematics and starting a business is quite similar. To succeed, entrepreneurs, like mathematicians, must spend a lot of time gawking at their respective playing fields and intuit a truth that’s invisible to others. The gut feeling that Steve Jobs had for iPhone was not very dissimilar to what Ramanujan had for mathematical results that he attributed as God’s thoughts.

There are two types of entrepreneurs and mathematicians in the world. The first type – let’s call them safe ones – take what’s already proven to work and either add something incremental on top of it or extend it to another area. These are mom and pop shops that generate small but consistent cash flow or tenured professors who publish low impact results but do that consistently.

The second type – let’s call them risky ones – venture out into dark and unknown, emerging either dead or victorious. They’re either spectacularly unsuccessful or they end up changing the world. These are entrepreneurs and mathematicians who try to go from zero to one by getting lucky with a completely novel insight. They have a completely new way of looking at the world that’s either completely correct or completely bogus – there’s no inbetween. Most of such attempts fail but a few produce breakthrough results. They usually have a convincing answer to Peter Thiel’s question: “What important truth do very few people agree with you on”.

Good mathematicians internalize Gödel’s incompleteness theorem, so they know that following the rules will not get them closer to breakthrough results. Similarly, entrepreneurs know that while generating revenue is predictable, making profit requires creativity.

I’m excited about this connection because it implies that there’s no formula for creating a successful startup. Yes, like existing mathematical principles, you can rely on existing business principles, but breakthrough results only come when someone breaks the mold. Successful startup, like a successful science experiment or mathematical result, is a knowledge generating event in the world.

If there was a formula for doing mathematics, the field of mathematics will disappear. Similarly, if there was a formula for creating successful startups, existing companies would already be applying it and the act of entrepreneurship will disappear.

Doing mathematics and starting companies is fun because it cannot be taught, only be trained for.

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