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.

 

[Larniavc]

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?

Any model is by definition an approximation of reality. For example a sample from a population can be used to model that population but until you use then entire population you’ll never have 100 confidence.

But a model is something more manageable than reality and if the predictions a model makes are accurate we can say the model is accurate to a specified level of accuracy.

That’s why science is tentative. We will never have 100% of the information in the system because that only exists as the system.

We just have to live with that.

 

[essentialsaltes]

the fullness of time can appear to take too long. Thoughts?

Planck was far too pessimistic when he suggested that the old guard have to die before a new scientific paradigm can become established among the experts in a field. On the flip side, a crucial new experiment or idea can hardly change everything overnight. I dunno -- we all know the scientific enterprise (as spiffy as it is as a tool for understanding/explaining natural phenomena) is a human endeavor.

 

[J_B_]

Planck was far too pessimistic when he suggested that the old guard have to die before a new scientific paradigm can become established among the experts in a field. On the flip side, a crucial new experiment or idea can hardly change everything overnight. I dunno -- we all know the scientific enterprise (as spiffy as it is as a tool for understanding/explaining natural phenomena) is a human endeavor.

It's good to hear you say that. But there are many opinions on the spectrum, and some move closer to scientism. It's not surprising that it happens. The pressure to deliver can be considerable - the pressure to express high confidence in a model.

 

[J_B_]

Any model is by definition an approximation of reality. For example a sample from a population can be used to model that population but until you use then entire population you’ll never have 100 confidence.

But a model is something more manageable than reality and if the predictions a model makes are accurate we can say the model is accurate to a specified level of accuracy.

That’s why science is tentative. We will never have 100% of the information in the system because that only exists as the system.

We just have to live with that.

It's one thing to say it. It's another to do something about it. When it comes time to pay the bill, my experience is that the appetite for change quickly disappears. In a few rare instances I've been blessed to work on some very innovative projects, and I'm thankful for it. But that's a small part of my career.

 

[J_B_]

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.

You could be right, but my impression was that a Gaussian kernel would mean it's technically not a Kalman filter. Regardless, if a Gaussian kernel were used, I think my concern would be moot.

 

[sjastro]

The Cauchy Principal Value becomes useless when the integrand cannot be solved directly using integral calculus.

For example:

This has no known solution and functions of complex variables are used where the indefinite integral is converted into a closed contour integral of a complex function where z is the complex variable.

Where

Cos(z) is the real part (Re) of the equation.
Without getting into specifics which are well beyond the scope of this post, the closed contour integral of the complex function can be solved using Cauchy’s residue theorem .

The answer turns out to be.

On the subject of where the Cauchy Principal Value in complex form is relevant and intersects reality is in the field of fluid mechanics which relies heavily on the use of complex variable functions.

Fluids are modelled as incompressible (constant density), inviscid (no viscosity), and irrotational (except at singularities like vortices).
These idealized fluids have been able to make accurate predictions in the real world and are confirmed by experiments.

 

[BeyondET]

Hilbert transform formula

 

[sjastro]

Hilbert transform formula

I assume this response is directed to me.
The equation can be solved using Hilbert transforms as well as Fourier transforms but both are deeply rooted in complex analysis.
The point is the answer to the equation.

Isn't solved using high school integral calculus in the real number domain.

 

[J_B_]

On the subject of where the Cauchy Principal Value in complex form is relevant and intersects reality is in the field of fluid mechanics which relies heavily on the use of complex variable functions.

Fluids are modelled as incompressible (constant density), inviscid (no viscosity), and irrotational (except at singularities like vortices).
These idealized fluids have been able to make accurate predictions in the real world and are confirmed by experiments.

I spent most of my career working with classical mechanics. However, I recently took a job at a new company that requires a lot of fluid mechanics - something I had intentionally avoided earlier in my career.

For me, assuming an incompressible fluid simply means whacking off terms to make the math simpler. I viewed it much the same way in my former job when I did a modal truncation. If pressed, I would have calculated confidence intervals to justify it. I can see, though, why a mathematician would seek formal validation for an approximation. What I hear you saying is that CPV is that validation. Very interesting. Thanks ... Though I can't imagine what would happen if I tried to use CPV to justify incompressibility.

 

[sjastro]

I spent most of my career working with classical mechanics. However, I recently took a job at a new company that requires a lot of fluid mechanics - something I had intentionally avoided earlier in my career.

For me, assuming an incompressible fluid simply means whacking off terms to make the math simpler. I viewed it much the same way in my former job when I did a modal truncation. If pressed, I would have calculated confidence intervals to justify it. I can see, though, why a mathematician would seek formal validation for an approximation. What I hear you saying is that CPV is that validation. Very interesting. Thanks ... Though I can't imagine what would happen if I tried to use CPV to justify incompressibility.

CPV is a computational tool when used in conjunction with Cauchy’s residue theorem it takes singularities out of the equation otherwise as an example normal integration of the potential w(z) = ∫μ/z = μ.log(z) is a problem when the point z=0 is included in the integration range.
In fluid mechanics this is the complex potential for a vortex which has a non-removable singularity at its centre and an infinite velocity.
Physically vortices do not have infinite velocities at their centres so I suppose one can argue CPV with the residue theorem is a validation.

In general a model starts off as an idealized construction because as you state the mathematics is simpler.
Fluid mechanics like many other models such as non-relativistic quantum mechanics is no exception.
Mathematicians and physicists however apply perturbation theory where in the case of fluid mechanics if the complex potential of an ideal incompressible, inviscid and irrotational fluid is w(z) applying a small perturbation ε<<1 , w(z,ε) =wₒ(z)+ϵw₁(z)+(ϵ²/2!)w₂(z)+⋯
where w₁,w₂…..wₙ are the partial derivatives wₙ= ∂ⁿw/∂εⁿ.

These perturbations provide better approximations for real phenomena.