Originally Posted by manbear
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.