# Thought Experiments, Minus the Thought

My second-favorite Newton fact is that, despite inventing calculus, he refused to use it for his most famous work of physics, the Principia. Instead, he used geometrical proofs, tweaked to smuggle in calculus without admitting it.

Essentially, these proofs were thought experiments. Newton would start with a standard geometry argument, one that would have been acceptable to mathematicians centuries earlier. Then, he’d imagine taking it further, pushing a line or angle to some infinite point. He’d argue that, if the proof worked for every finite choice, then it should work in the infinite limit as well.

These thought experiments let Newton argue on the basis of something that looked more rigorous than calculus. However, they also held science back. At the time, only a few people in the world could understand what Newton was doing. It was only later, when Newton’s laws were reformulated in calculus terms, that a wider group of researchers could start doing serious physics.

What changed? If Newton could describe his physics with geometrical thought experiments, why couldn’t everyone else?

The trouble with thought experiments is that they require careful setup, setup that has to be thought through for each new thought experiment. Calculus took Newton’s geometrical thought experiments, and took out the need for thought: the setup was automatically a part of calculus, and each new researcher could build on their predecessors without having to set everything up again.

This sort of thing happens a lot in science. An example from my field is the scattering matrix, or S-matrix.

The S-matrix, deep down, is a thought experiment. Take some particles, and put them infinitely far away from each other, off in the infinite past. Then, let them approach, close enough to collide. If they do, new particles can form, and these new particles will travel out again, infinite far away in the infinite future. The S-matrix then is a metaphorical matrix that tells you, for each possible set of incoming particles, what the probability is to get each possible set of outgoing particles.

In a real collider, the particles don’t come from infinitely far away, and they don’t travel infinitely far before they’re stopped. But the distances are long enough, compared to the sizes relevant for particle physics, that the S-matrix is the right idea for the job.

Like calculus, the S-matrix is a thought experiment minus the thought. When we want to calculate the probability of particles scattering, we don’t need to set up the whole thought experiment all over again. Instead, we can start by calculating, and over time we’ve gotten very good at it.

In general, sub-fields in physics can be divided into those that have found their S-matrices, their thought experiments minus thought, and those that have not. When a topic has to rely on thought experiments, progress is much slower: people argue over the details of each setup, and it’s difficult to build something that can last. It’s only when a field turns the corner, removing the thought from its thought experiments, that people can start making real collaborative progress.

## 3 thoughts on “Thought Experiments, Minus the Thought”

1. pete1187

Great entry (though as a mathematical realist I’d push back on Newton having “invented” Calculus rather than having “discovered” it)

I wanted to ask you, and I know this is tangential to the above post, but what are your thoughts on the recent searches for Dark Matter coming up empty? I check into Not Even Wrong every now and then and saw this today: http://www.math.columbia.edu/~woit/wordpress/?p=8654

Towards the end Woit writes: “With SUSY and the “WIMP miracle” now dead ideas, perhaps that will lead to focus on more promising ones,” and he also notes in one of the comments from a 2001 paper on SUSY predictions:

“When CDMS is moved to the Soudan mine, its sensitivity will drop to between 10^-8 and 10^-7 pb and GENIUS claims to be able to reach 10^-9. At those levels, direct detection experiments will either discover supersymmetric dark matter or impose serious constraints on supersymmetric models.

The LUX result is below 10^{-9} pb over a very large range, down to 2.2 x10^-10 at 50 GeV. So, it is very much “imposing serious constraints” on SUSY, but of course there will always be SUSY models with smaller cross sections. I don’t think there’s a sensible way to put a measure on such things.”

Your my go-to as far as questions on ST/SUSY goes, so I wanted a possibly slightly more nuanced version of what this might mean than the opinion of a (very intelligent) guy with a vicious hatred of all things String Theory and the bias that entails. Is the space excluded really that detrimental to SUSY’s prospects? Could an axion/sterile neutrino that he hints to as possibly better solutions be incorporated in a supersymmetric model? And what are your thoughts on the whole thing?

Any input is greatly appreciated, as always.

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