I had never heard of Carl Gustav Jacob Jacobi until this week. I have only known of Charlie Munger for a couple of years. I mention that up front because it is the whole point of what follows.
When I tell people how I work a problem — start at the point of failure and trace backward, rather than start with the goal and build forward — a fair number of them scratch their heads. It sounds backward because it is backward, deliberately. And the pushback is usually some version of: that’s not how you’re supposed to solve things. You define success, then you build toward it.
It turns out I am in good company holding the opposite view — company I did not know I was keeping. So let me do the honest thing first and draw the distinctions, because these are three different methods, and I am not going to claim they are the same. Then I will show you the principle underneath that makes them kin.
First, the differences — because they are real
Jacobi inverted equations. The 19th-century German mathematician Carl Gustav Jacob Jacobi is remembered for a maxim — man muss immer umkehren, “invert, always invert.” He found that certain hard problems in mathematics could not be solved by pushing forward through the standard formulas. So he flipped them. Faced with elliptic integrals that resisted a direct assault, he inverted them — and in doing so opened an entire new field. His inversion was mathematical: restate the problem in its inverse form, and the path that was invisible from the front becomes obvious from the back.
Munger inverted to avoid failure. Charlie Munger, Warren Buffett’s late partner at Berkshire Hathaway, took Jacobi’s maxim out of mathematics and turned it into a decision tool. His version is preventive. Instead of asking “how do I succeed?”, he asked “how would I fail?” — then made a list of the ways, and avoided them. As he put it, all he wanted to know was where he was going to die, so he’d never go there. Munger inverts toward a hypothetical failure: imagine every way this could go wrong, and steer clear.
My method starts from a failure that is already happening. Here is where I diverge from both. I do not invert an equation, and I do not imagine a hypothetical failure to avoid. I start from an actual, observed, present failure — something that is breaking right now — and I trace its mechanism backward, step by step, to the real cause. When a defence maintenance operation was delivering parts on time only 51 percent of the time against a 90 percent requirement, I did not start from the target and build toward it. I started from the breakage and asked why, and followed the answer out of procurement, into the service department, back through the suppliers and the border, and finally into finance — where the real cause was hiding. Munger’s inversion is preventive; mine is diagnostic. He inverts to avoid failure. I trace into a failure that has already occurred, to find what caused it.
Three different operations. Jacobi flips an equation. Munger imagines a failure to dodge. I trace a failure that is live. I want that clear before I claim any kinship, because the kinship is not that we do the same thing.
Now, the principle — because it is the same
Underneath the three different operations sits one idea, and it is Jacobi’s: some problems cannot be solved forward.
That is the whole of it. Most people, most of the time, think in one direction — forward, from where they are toward where they want to be. And for a great many problems, that works. But there is a category of problem where the forward path actively hides the answer. In mathematics, Jacobi found it in the integrals. In investing and life, Munger found it in the human tendency to chase success while walking straight into avoidable stupidity. And in enterprise systems — the world I have spent nearly three decades in — I found it in failure diagnosis: you cannot find the cause of a failure by starting from the outcome you wanted, because starting from the desired outcome makes you build a case for it, and the real cause is precisely the thing that case is designed not to see.
This is what I have called the difference between an equation-based approach and an agent-based one. The equation-based approach starts from the successful goal and builds forward toward it — and in doing so, it commits you, structurally, to being right about the path you chose. Every fact gets sorted into “helps me reach the goal” or “obstacle.” You are no longer investigating; you are defending a destination. The agent-based approach starts from where the thing is failing and traces backward — and because you began at the breakage rather than the goal, you have nothing to defend. You are free to follow the evidence wherever it actually leads, even out of your own department, even to a cause you never suspected.
Jacobi would recognize that immediately. It is his maxim, applied to systems instead of integrals. The forward path obscures; the backward path reveals. He said it about elliptic functions. Munger said it about avoiding ruin. I say it about diagnosing why an operation is failing. None of us learned it from the others — I genuinely did not know these two men’s work when I developed and proved my method in 1998, with funding from the Government of Canada’s Scientific Research & Experimental Development (SR&ED) Program — and that, I think, is the most persuasive part. A principle that surfaces independently in pure mathematics, in investing, and in enterprise diagnosis, across two centuries and three people who never compared notes, is not a preference. It is closer to a law.
Why I am telling you this
Not to borrow the pedigree of a great mathematician and a great investor. I cannot borrow what I never had — I did not stand on their shoulders, because I did not know they were there. I am telling you because if you are one of the people who scratches their head when I say start at the failure and work backward, it may help to know that the instinct behind it is not idiosyncratic. It is the same instinct that let Jacobi crack problems that would not yield to the forward formulas, and the same one that kept Munger out of the traps that took down cleverer investors.
Invert, always invert. Jacobi meant it about equations. Munger meant it about avoiding failure. I mean it about tracing failure to its root. Three roads, no map shared between them, arriving at the same place — which is the surest sign the place is real.
If you want to see the inversion applied — the forward map and the backward trace set side by side, with the full 1998 case that first proved it and the diagram that makes the difference visible — I have laid it out here: The Map You Design and the Map You Trace.
-30-
Truth Is Believing. Accuracy Is Knowing. Outcome Is Proof.™
Related
The Hansen Models™ Approach Shares a Principle With a 19th-Century Mathematician and a 20th-Century Titan of Business — Arrived at Independently
Posted on August 2, 2026
0
I had never heard of Carl Gustav Jacob Jacobi until this week. I have only known of Charlie Munger for a couple of years. I mention that up front because it is the whole point of what follows.
When I tell people how I work a problem — start at the point of failure and trace backward, rather than start with the goal and build forward — a fair number of them scratch their heads. It sounds backward because it is backward, deliberately. And the pushback is usually some version of: that’s not how you’re supposed to solve things. You define success, then you build toward it.
It turns out I am in good company holding the opposite view — company I did not know I was keeping. So let me do the honest thing first and draw the distinctions, because these are three different methods, and I am not going to claim they are the same. Then I will show you the principle underneath that makes them kin.
First, the differences — because they are real
Jacobi inverted equations. The 19th-century German mathematician Carl Gustav Jacob Jacobi is remembered for a maxim — man muss immer umkehren, “invert, always invert.” He found that certain hard problems in mathematics could not be solved by pushing forward through the standard formulas. So he flipped them. Faced with elliptic integrals that resisted a direct assault, he inverted them — and in doing so opened an entire new field. His inversion was mathematical: restate the problem in its inverse form, and the path that was invisible from the front becomes obvious from the back.
Munger inverted to avoid failure. Charlie Munger, Warren Buffett’s late partner at Berkshire Hathaway, took Jacobi’s maxim out of mathematics and turned it into a decision tool. His version is preventive. Instead of asking “how do I succeed?”, he asked “how would I fail?” — then made a list of the ways, and avoided them. As he put it, all he wanted to know was where he was going to die, so he’d never go there. Munger inverts toward a hypothetical failure: imagine every way this could go wrong, and steer clear.
My method starts from a failure that is already happening. Here is where I diverge from both. I do not invert an equation, and I do not imagine a hypothetical failure to avoid. I start from an actual, observed, present failure — something that is breaking right now — and I trace its mechanism backward, step by step, to the real cause. When a defence maintenance operation was delivering parts on time only 51 percent of the time against a 90 percent requirement, I did not start from the target and build toward it. I started from the breakage and asked why, and followed the answer out of procurement, into the service department, back through the suppliers and the border, and finally into finance — where the real cause was hiding. Munger’s inversion is preventive; mine is diagnostic. He inverts to avoid failure. I trace into a failure that has already occurred, to find what caused it.
Three different operations. Jacobi flips an equation. Munger imagines a failure to dodge. I trace a failure that is live. I want that clear before I claim any kinship, because the kinship is not that we do the same thing.
Now, the principle — because it is the same
Underneath the three different operations sits one idea, and it is Jacobi’s: some problems cannot be solved forward.
That is the whole of it. Most people, most of the time, think in one direction — forward, from where they are toward where they want to be. And for a great many problems, that works. But there is a category of problem where the forward path actively hides the answer. In mathematics, Jacobi found it in the integrals. In investing and life, Munger found it in the human tendency to chase success while walking straight into avoidable stupidity. And in enterprise systems — the world I have spent nearly three decades in — I found it in failure diagnosis: you cannot find the cause of a failure by starting from the outcome you wanted, because starting from the desired outcome makes you build a case for it, and the real cause is precisely the thing that case is designed not to see.
This is what I have called the difference between an equation-based approach and an agent-based one. The equation-based approach starts from the successful goal and builds forward toward it — and in doing so, it commits you, structurally, to being right about the path you chose. Every fact gets sorted into “helps me reach the goal” or “obstacle.” You are no longer investigating; you are defending a destination. The agent-based approach starts from where the thing is failing and traces backward — and because you began at the breakage rather than the goal, you have nothing to defend. You are free to follow the evidence wherever it actually leads, even out of your own department, even to a cause you never suspected.
Jacobi would recognize that immediately. It is his maxim, applied to systems instead of integrals. The forward path obscures; the backward path reveals. He said it about elliptic functions. Munger said it about avoiding ruin. I say it about diagnosing why an operation is failing. None of us learned it from the others — I genuinely did not know these two men’s work when I developed and proved my method in 1998, with funding from the Government of Canada’s Scientific Research & Experimental Development (SR&ED) Program — and that, I think, is the most persuasive part. A principle that surfaces independently in pure mathematics, in investing, and in enterprise diagnosis, across two centuries and three people who never compared notes, is not a preference. It is closer to a law.
Why I am telling you this
Not to borrow the pedigree of a great mathematician and a great investor. I cannot borrow what I never had — I did not stand on their shoulders, because I did not know they were there. I am telling you because if you are one of the people who scratches their head when I say start at the failure and work backward, it may help to know that the instinct behind it is not idiosyncratic. It is the same instinct that let Jacobi crack problems that would not yield to the forward formulas, and the same one that kept Munger out of the traps that took down cleverer investors.
Invert, always invert. Jacobi meant it about equations. Munger meant it about avoiding failure. I mean it about tracing failure to its root. Three roads, no map shared between them, arriving at the same place — which is the surest sign the place is real.
If you want to see the inversion applied — the forward map and the backward trace set side by side, with the full 1998 case that first proved it and the diagram that makes the difference visible — I have laid it out here: The Map You Design and the Map You Trace.
-30-
Truth Is Believing. Accuracy Is Knowing. Outcome Is Proof.™
Share this:
Like this:
Related