[xnor]

  Everyone here is using a different meaning of "model" so it's a bit confused. I'm using the sense described in the Wikipedia entry on "Scientific Modeling" -- the essence of it is some kind of construct (mathematical or not) that represents some part of reality (or some part of a more complex system).

On that very page it says that models are used as a substitute for measurements and experimentation. A FR measurement is a measurement.

 

[manbear]

The frequency of a signal is not a model. Its a naturally occurring phenomenon which is assigned a number.
The definition of a second is also tied to natural phenomenon (cesium atoms).
Hence measuring an FR is not a model, unless you can prove that a fourier analysis is a model.

LOL. I'm not switching to the OP's side here, but it seems pretty obvious to me that Fourier analysis is a model. It relies on a host of mathematical assumptions. Just for starters, http://cnx.org/content/m11003/latest/

 

[xnor]

Relying on axioms doesn't make it a model.
 
Otherwise even simple addition of two numbers is a model...

 

[manbear]

  Relying on axioms doesn't make it a model.
 
Otherwise even simple addition of two numbers is a model...

Hmm... Would you say the act of curve fitting (say, linear regression) is not modelling? I'm just talking about the mathematical process, not the interpretation applied once the process is completed. 

All of math is built up from ZFC axioms. I would say that all of math is a model, as ZFC requires viewing (modelling) mathematical entities as sets. The properties and theorems of math are extensions of set-theoretic logic. In any case, it seems like all mathematics is either a model or not. 

I also dispute your claim that Fourier analysis relies purely on axioms. There are some non-axiomatic assumptions in there as well, at least in practical application (which is true for most discrete computational processes that rely on sampling). For example, on Fast Fourier Transforms:

"When the FFT is used, attention should be paid to leakage, which is caused by the FFT's assumption that the input signal repeats periodically and that the periodic length is equal to the length of the actual input. However, if the true signal is not periodic or if the assumed periodic length is not correct, leakage will occur. This will cause both the amplitude and position of a frequency measurement to be inaccurate."

http://www.originlab.com/www/helponline/origin/en/Category/Fast_Fourier_Transform_(FFT).html

EDIT -- Of course, all of this depends on how define the word model. This isn't a trivial task, and our choice of definition will decide most of these issues. Here's some food for thought: http://plato.stanford.edu/entries/models-science/#OntWhaMod

Without getting too deep into the semantics, I'm inclined to say that modelling is any act of representation. 

Some interesting points: 

"Another kind of representational models are so-called ‘models of data’ (Suppes 1962). A model of data is a corrected, rectified, regimented, and in many instances idealized version of the data we gain from immediate observation, the so-called raw data. Characteristically, one first eliminates errors (e.g. removes points from the record that are due to faulty observation) and then present the data in a ‘neat’ way, for instance by drawing a smooth curve through a set of points." 

This is essentially what happens when we take the raw electrical output of a microphone and turn it into a picture we can look at or a chart we can understand (e.g. frequency response). 

"In modern logic, a model is a structure that makes all sentences of a theory true, where a theory is taken to be a (usually deductively closed) set of sentences in a formal language (see Bell and Machover 1977 or Hodges 1997 for details). The structure is a ‘model’ in the sense that it is what the theory represents. As a simple example consider Euclidean geometry, which consists of axioms—e.g. ‘any two points can be joined by a straight line’—and the theorems that can be derived from these axioms. Any structure of which all these statements are true is a model of Euclidean geometry."

This is similar to my point about math earlier. I would have no problem calling "1+1=2" a model. We're taking sentences of a formal language (set-theoretic axioms), using symbols to represent objects (numbers), relations (equality), and functions (addition), and we're saying that this thing we've written models reality (the act of putting one pencil next to another gives us two pencils, for example). 

I'm discussing this because I find it interesting. I don't think this has much practical relevance in terms of how we should react to headphone measurements. 
 

 

[raddle]

The reason I think it's important to understand measurements as models is this: I think there are certain phenomena that the current state of sound science fails to explain. I think that sound science is consistent-- it's a set of observations and theories that reinforce each other. I think it has holes, however.
 
But first I'm finding it hard to believe that anyone would think FR is not a model. I'm interested in the reality of how it is used and measured. Apart from that, it's just meaningless numbers.
 
However, I think that I've been a little too rigid about defining "models" and indicating how they are used. I admit there are a lot of ways that models are used and defined, but this doesn't change my point about FR.
 
I would be interested in how people here would answer the question: "Why do we measure FR?"
 
Besides that I'll note that FR models a device as linear and can be used to predict its response to future signals that have not been previously measured, although not with perfect accuracy.

 

[manbear]

  The reason I think it's important to understand measurements as models is this: I think there are certain phenomena that the current state of sound science fails to explain. I think that sound science is consistent-- it's a set of observations and theories that reinforce each other. I think it has holes, however.
 
But first I'm finding it hard to believe that anyone would think FR is not a model. I'm interested in the reality of how it is used and measured. Apart from that, it's just meaningless numbers.
 
However, I think that I've been a little too rigid about defining "models" and indicating how they are used. I admit there are a lot of ways that models are used and defined, but this doesn't change my point about FR.
 
I would be interested in how people here would answer the question: "Why do we measure FR?"
 
Besides that I'll note that FR models a device as linear and can be used to predict its response to future signals that have not been previously measured, although not with perfect accuracy.

You need to actually state your points, instead of hinting at them. I don't understand why you think we should care about whether measurements are models or not. 

What does sound science fail to explain? 

What is your point about FR? That it is a model? --- Why does that matter? 

 

[proton007]

Hmm... Would you say the act of curve fitting (say, linear regression) is not modelling? I'm just talking about the mathematical process, not the interpretation applied once the process is completed. 


All of math is built up from ZFC axioms. I would say that all of math is a model, as ZFC requires viewing (modelling) mathematical entities as sets. The properties and theorems of math are extensions of set-theoretic logic. In any case, it seems like all mathematics is either a model or not. 


I also dispute your claim that Fourier analysis relies purely on axioms. There are some non-axiomatic assumptions in there as well, at least in practical application (which is true for most discrete computational processes that rely on sampling). For example, on Fast Fourier Transforms:


"When the FFT is used, attention should be paid to leakage, which is caused by the FFT's assumption that the input signal repeats periodically and that the periodic length is equal to the length of the actual input. However, if the true signal is not periodic or if the assumed periodic length is not correct, leakage will occur. This will cause both the amplitude and position of a frequency measurement to be inaccurate."

http://www.originlab.com/www/helponline/origin/en/Category/Fast_Fourier_Transform_(FFT).html


EDIT -- Of course, all of this depends on how define the word model. This isn't a trivial task, and our choice of definition will decide most of these issues. Here's some food for thought: http://plato.stanford.edu/entries/models-science/#OntWhaMod


Without getting too deep into the semantics, I'm inclined to say that modelling is any act of representation. 


Some interesting points: 


"Another kind of representational models are so-called ‘models of data’ (Suppes 1962). A model of data is a corrected, rectified, regimented, and in many instances idealized version of the data we gain from immediate observation, the so-called raw data. Characteristically, one first eliminates errors (e.g. removes points from the record that are due to faulty observation) and then present the data in a ‘neat’ way, for instance by drawing a smooth curve through a set of points." 


This is essentially what happens when we take the raw electrical output of a microphone and turn it into a picture we can look at or a chart we can understand (e.g. frequency response). 


"In modern logic, a model is a structure that makes all sentences of a theory true, where a theory is taken to be a (usually deductively closed) set of sentences in a formal language (see Bell and Machover 1977 or Hodges 1997 for details). The structure is a ‘model’ in the sense that it is what the theory represents. As a simple example consider Euclidean geometry, which consists of axioms—e.g. ‘any two points can be joined by a straight line’—and the theorems that can be derived from these axioms. Any structure of which all these statements are true is a model of Euclidean geometry."


This is similar to my point about math earlier. I would have no problem calling "1+1=2" a model. We're taking sentences of a formal language (set-theoretic axioms), using symbols to represent objects (numbers), relations (equality), and functions (addition), and we're saying that this thing we've written models reality (the act of putting one pencil next to another gives us two pencils, for example). 


I'm discussing this because I find it interesting. I don't think this has much practical relevance in terms of how we should react to headphone measurements. 

 

Mathematics is not a model, because it doesn't rely on assumptions, but certain universal truths.
1+1=2 has a logical proof, one that won't fit in here.

The other way is to go by the skeptics way (Sextus Empericus).

Those who claim for themselves to judge the truth are bound to possess a criterion of truth. This criterion, then, either is without a judge's approval or has been approved. But if it is without approval, whence comes it that it is truthworthy? For no matter of dispute is to be trusted without judging. And, if it has been approved, that which approves it, in turn, either has been approved or has not been approved, and so on ad infinitum.

 

[manbear]

Such a crude answer makes the question less interesting. If you actually wrote out the proof that 1+1=2, you might come closer to seeing my point....

Is the axiom of choice an assumption or a universal truth? What about the law of the excluded middle?

Or a less abstract example: it is not possible to prove the parallel postulate using the other axioms of Euclidean geometry. It must be assumed.

You might want to read up on incompleteness and decidability, if axioms look more like universal truths than assumptions to you.

No criterion of truth other than what is shared by mathematicians is required.

Sigh.

 

[higbvuyb]

Mathematics itself is somewhat different to a scientific 'model'. A scientific model has to be related in some way to reality, which pure mathematics doesn't do. Mathematics can be used to construct a model, but this is extending beyond mathematics. For example 1+1=2 does not model anything; 1 apple + 1 apple = 2 apples does.
 
More subtly, simply applying mathematics to a data set is not modelling. However as soon as an interpretation is made based on the manipulated data, this is modelling.
It seems like applying the FFT is a model, but only because basically every time someone does so, they are doing it to interpret the data in a certain way.
 
 

  The reason I think it's important to understand measurements as models is this: I think there are certain phenomena that the current state of sound science fails to explain. I think that sound science is consistent-- it's a set of observations and theories that reinforce each other. I think it has holes, however.

Can you link to evidence suggesting that a significant number of people believe our understanding of sound and hearing is 100% complete?
 
I don't think I've seen anyone express that opinion before.
 

  but this doesn't change my point about FR.

What is your point? If you can post a dozen times in one thread yet nobody else understands what you're trying to actually claim, there's a bit of a problem.

 

[proton007]

Such a crude answer makes the question less interesting. If you actually wrote out the proof that 1+1=2, you might come closer to seeing my point....

Is the axiom of choice an assumption or a universal truth? What about the law of the excluded middle?

Or a less abstract example: it is not possible to prove the parallel postulate using the other axioms of Euclidean geometry. It must be assumed.

You might want to read up on incompleteness and decidability, if axioms look more like universal truths than assumptions to you.

No criterion of truth other than what is shared by mathematicians is required.

Sigh.

 
Its a crude answer because the whole topic itself is muddled. The proof of 1+1=2 is present in many forms, some of which like the Principia Mathematica takes about 300 pages because it starts from scratch.
 
The axiom of choice can be an assumption, or a logical axiom which can be universally true (x=x).
Since you gave the example of Euclidean geometry, it only defines the relationship between abstract geometrical objects.
What makes it a 'concrete' set of axioms is that the relations are present in nature rather than relying on another set of axioms. Hence its usefulness. You can draw lines with varying widths, but they all follow the same postulates.
 
In terms of FFT, the Fourier Transform is a 'transformation'. Taken as it is, its not a model until you start applying it to human hearing. What is modeled is the prediction that a certain FR will be perceived in a certain manner. This is something which can only be tested experimentally.
 
Again, how does this go back to the OP's original point?

 

[manbear]

  Mathematics itself is somewhat different to a scientific 'model'. A scientific model has to be related in some way to reality, which pure mathematics doesn't do. Mathematics can be used to construct a model, but this is extending beyond mathematics. For example 1+1=2 does not model anything; 1 apple + 1 apple = 2 apples does.
 
More subtly, simply applying mathematics to a data set is not modelling. However as soon as an interpretation is made based on the manipulated data, this is modelling.
It seems like applying the FFT is a model, but only because basically every time someone does so, they are doing it to interpret the data in a certain way.

It comes down to how we choose to define a model. If we want to say that a model must apply to measurable properties of the physical world, then you would be right. I'm not really committed to a particular definition of model, but the idea that a model merely has to represent some other object (even mental objects) makes the most intuitive sense to me. So for the sake of argument, I might say that the symbols "1+1=2" model the behavior of the successor function, as applied to the natural number 1. Or we could dig in deeper and say that it's a model of the construction {{}} U {{{}}} = {{},{{}}}, whatever that is. 

Then again, this definition of model is pretty broad. I'm not sure what the use of it is. Though I'm not really sure what the use of any definition of model is. Because I don't see why it matters if something is a model or not. I'm waiting for the OP's explanation on that one. 

IDK. I'm mostly just bored  

 

 

[proton007]

 
It comes down to how we choose to define a model. If we want to say that a model must apply to measurable properties of the physical world, then you would be right. I'm not really committed to a particular definition of model, but the idea that a model merely has to represent some other object (even mental objects) makes the most intuitive sense to me. So for the sake of argument, I might say that the symbols "1+1=2" model the behavior of the successor function, as applied to the natural number 1. Or we could dig in deeper and say that it's a model of the construction {{}} U {{{}}} = {{},{{}}}, whatever that is. 

Then again, this definition of model is pretty broad. I'm not sure what the use of it is. Though I'm not really sure what the use of any definition of model is. Because I don't see why it matters if something is a model or not. I'm waiting for the OP's explanation on that one. 

IDK. I'm mostly just bored  

 

 
Funny how we've ended up here. Going back to the OP's point, is measurement a model?  If I assign '1' to an apple, does that make it a model? 1+1= 2 apples, but you cannot assign this number to a liquid for instance. There, 1+1=2 doesn't hold physically, but the addition on one side and the relationship of doubling still holds.

 

[manbear]

  Funny how we've ended up here. Going back to the OP's point, is measurement a model?  If I assign '1' to an apple, does that make it a model? 1+1= 2 apples, but you cannot assign this number to a liquid for instance. There, 1+1=2 doesn't hold physically, but the addition on one side and the relationship of doubling still holds.

Funny indeed. This thread got so dumb that it imploded and started becoming smart. Or did it. 

1 liquid + 1 liquid  =  2 liquids. Sounds legit. LOL. Maybe the OP can clear that one up for us. 

 

[proton007]

 
Funny indeed. This thread got so dumb that it imploded and started becoming smart. Or did it. 

1 liquid + 1 liquid  =  2 liquids. Sounds legit. LOL. Maybe the OP can clear that one up for us. 

 
Well, it seems this is one of those threads that has a multiple personality disorder.

 

[KraftD1]

It is models of tortoises all the way down.