Cauchy Principal Value - Real or Not Real?

[J_B_]

As I worked through my engineering degrees, I often found myself using math & science I didn't really understand. As is often the case, no one wants to be the first to speak up and admit a weakness, but I suspect it is the same for many who work their way through the sciences. Having a philosophical nature, working that way created a cognitive dissonance I was very much aware of. The justification I settled on, was that if, at the end of the day, the machine worked, the means justified the end. Further, no machine is ever perfect (just as no person is ever perfect). So, if we're working to improve ourselves and the machine, that's the best we can do.

All of this is a preamble to my question, but feel free to comment on it if you like.

Over my decades-long career I've continued to knock down the things I don't understand one at a time. The latest hill to conquer was the Kalman Filter, which depends on the Cauchy Principal Value. Once the light bulb of understanding went off, my immediate reaction was, "Well, nuts." A common topic in the Philosophy of Science is the meaning of the correlation between mathematics and reality. In this case, there is none. The Cauchy Principal Value is something we can pretty easily say is not real (though I'm sure someone is going to disagree with that). The math is valid - I don't question that - but it has no correlation to anything in reality. All it does is help us make a better guess.

It's part of being an engineer that you are constantly making estimates (guesses). So, knowing a Kalman Filter is based on a guess of a function's value rather than the function itself isn't going to slow down engineering. But what about science? Of course people work on improvements to the Kalman Filter, but I don't see any signs that the goal is to find the "real" value. It's more just to make a better guess.

Finally, the question. If it were known (e.g. widely accepted) that a scientific model isn't real, but merely our best guess, should we continue to build on that, pushing the extrapolation farther and farther? Or should our efforts be focused on a better model of reality? In other words, engineering is essentially saying, "As long as machine performance continues to improve, it's not worth the cost, even though we know our model isn't based on reality." Is it OK for science to also adopt that attitude?

 

[essentialsaltes]

The Cauchy Principal Value is something we can pretty easily say is not real

Only because (IIRC) it can involve teeny tiny detours into the complex (ergo not real) plane to avoid the poles.

But to the larger question, I think science is always/often looking at results and methods being better and better approximations to reality. If a better model arises, the old one gets abandoned (in the fulness of time).

 

[J_B_]

Only because (IIRC) it can involve teeny tiny detours into the complex (ergo not real) plane to avoid the poles.

But to the larger question, I think science is always/often looking at results and methods being better and better approximations to reality. If a better model arises, the old one gets abandoned (in the fulness of time).

My understanding of this topic is recent, so I will assume you have a better grasp of it than I. Therefore, my response is meant only as a clarification of my perspective, not a challenge of yours.

"A teeny tiny detour" - yes. The word would be "infinitesimal", as many matters of calculus tend to be. The size of the detour is not the issue, nor the use of the complex plane. The label given to imaginary numbers is an artifact of times past, and not a definition of their real nature - or lack thereof.

Nor is this about Kalman filters, which are by definition state estimators. It goes deeper than that. The Cauchy Principal Value (CPV) is not a value of the function in question. It replaces an undefined value. To use it in a state estimator, therefore, isn't really a problem, as it's acknowledged from the outset that the result is an estimate. A Kalman filter will never lead to the true value of the state (except by a happy accident), and therefore there is no claim that the model represents reality. The underlying function that is being estimated, however. That is a claim to represent reality.

And CPV is used in other places. It's used in QM ... though I'm not sure that's a problem either, given all of QM is probabilistic. It's always been a head scratcher to me that some physicists appear to be claiming the quantum level IS probabilistic rather than just saying it's represented as such ... Anyway, I've not fully investigated it's uses in Thermodynamics and Aerodynamics, but if any of these uses are standing on a claim that reality IS the CPV ... I'm not ready to accept that.

 

[essentialsaltes]

My understanding of this topic is recent, so I will assume you have a better grasp of it than I.

My understanding is old and foggy, so you have no reason to be deferential. Honestly, I know nothing at all about Kalman filters.

 

[Ophiolite]

I know nothing at all about Kalman filters

I don't even drink coffee!

More seriously, two points:

• Isn't it an informal principle of engineering that "No model is real, some models are useful" ? Which is what I think @J_B_ is saying.
• You mention abandoning old and less effective models in science, "in the fulness of time". The clique who bemoan the openess of conventional science to new ideas, does find some support in this cautious delay - the fullness of time can appear to take too long. Thoughts?

 

[Stopped_lurking]

As I worked through my engineering degrees, I often found myself using math & science I didn't really understand. As is often the case, no one wants to be the first to speak up and admit a weakness, but I suspect it is the same for many who work their way through the sciences. Having a philosophical nature, working that way created a cognitive dissonance I was very much aware of. The justification I settled on, was that if, at the end of the day, the machine worked, the means justified the end. Further, no machine is ever perfect (just as no person is ever perfect). So, if we're working to improve ourselves and the machine, that's the best we can do.

All of this is a preamble to my question, but feel free to comment on it if you like.

Over my decades-long career I've continued to knock down the things I don't understand one at a time. The latest hill to conquer was the Kalman Filter, which depends on the Cauchy Principal Value. Once the light bulb of understanding went off, my immediate reaction was, "Well, nuts." A common topic in the Philosophy of Science is the meaning of the correlation between mathematics and reality. In this case, there is none. The Cauchy Principal Value is something we can pretty easily say is not real (though I'm sure someone is going to disagree with that). The math is valid - I don't question that - but it has no correlation to anything in reality. All it does is help us make a better guess.

It's part of being an engineer that you are constantly making estimates (guesses). So, knowing a Kalman Filter is based on a guess of a function's value rather than the function itself isn't going to slow down engineering. But what about science? Of course people work on improvements to the Kalman Filter, but I don't see any signs that the goal is to find the "real" value. It's more just to make a better guess.

Finally, the question. If it were known (e.g. widely accepted) that a scientific model isn't real, but merely our best guess, should we continue to build on that, pushing the extrapolation farther and farther? Or should our efforts be focused on a better model of reality? In other words, engineering is essentially saying, "As long as machine performance continues to improve, it's not worth the cost, even though we know our model isn't based on reality." Is it OK for science to also adopt that attitude?

First an answer to your question. I believe all scientific models only represent out best explanation at the moment. From the models we can hopefully make predictions which we then can compare with observations. If there is a discrepancy, we check our assumptions, we check our observations and if it still remains we update our understanding of reality and the model. Rinse and repeat, there's no guarantee that we ever are working with some metaphysical truth.

Some questions about the preamble. Do Kalman filters depend on CPV even if they use a gaussian kernel (when the covariance matrix is well defined)? Isn't the CPV used to calculate the covariance matrix when using a Cauchy-distribution kernel? It is more than 20 years ago I read about Kalman filters, and nowadays all math I do is applied statistics.