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Measurement-omission in mathematical models - also my take on dimensions

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Measurement-omission in mathematical models.

Mathematical models are special kind of models and they too involve particular measurement omissions and the preservation of generalities of special-case events/objects of reality.

These measurement omissions in the physical context must be dictated essentially by the scale of time and change-magnitude:
1) The lower bound of the energies that can be measured. Eg: The fine structure of hydrogen-spectrum can be omitted.
2) The lower bound to the frequency of recurrence of events that can be measured. Eg: The rotation of the earth about its axis is too slow we may consider our laboratories to be inertial frames.
3) The upper bound to the frequency of recurrence of events that can be measured. Eg: The motion of gas molecules is too fast, the precise knowledge of momentum-impinges on a surface can be sacrificed for one measurement: pressure.

Any given mathematical model has a precise number of parameters required to describe a particular instance of it. This is its dimension. (One can also use a model described by a function over the real numbers - in that case the dimension is infinite. An example for that is the temperature distribution of a rod modeled as a straight line-segment. But in this case we are constraining our system more than we have information about it. After all one cannot measure upto arbitrary precision.)

A model is only an abstracted essence as a tuple of numbers can only describe one level of information of the actual object. For example, an actual physical triangle is more than just the side lengths, it has material properties, etc.

According to the physical context, a new model is constructed by including new parameters (to make finer distinctions), for example, in expressing the state of a molecule we may either choose to only keep track of the velocity coordinates of its center of mass, or we may include coordinates describing its vibrational modes or its rotational motion, etc increasing the dimension of the model as we include more complexity. The new model may even omit the previous parameters (in the sense that the parameters do have particular values but may have any of the permissible values) if they are assumed to have no bearing in the working of the considered details. For example, in the wave-function description of the molecule one is unconcerned with the motion of the molecule as a whole. Prior to Einstein, it was believed that one can use purely spatial models for dynamics plugging in time independently.

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This is not quite a parallel or symmetrical analogy with Rand's central general idea of "measurement omission" and its relationship with conceptualization and abstraction, although you have presented an isolated concrete example to which Rand's measurement omission may be applied.  Your identification of limits and granularity and their "omission" is not the same sort of thing as Rand's "measurement omission".

The concept of "numerical measurement" I will call " quantification" in order to avoid confusing Rand's idea of measurement omission with the context of its application here, which is numerical quantification.

Numerical quantifiability as a concept applies to physical and mathematical quantities and states that objects which are subject to such quantification can be quantified at various resolutions and within limits.  The general idea of quantification which spans various applications, i. e. an idea that subsumes quantifiabilty is a property of a class of entities physical and mathematical, omits the specific quantity in question and its unitary bases (charge, momentum, etc) and associated granularity or limits of quantification as specifics i.e. as "measurements" characterizing the particular quantification as simply being quantifiable.

So the concept of quantifiability itself may be abstracted away from its particular concretes using "measurement omission" in the same way any other concept.

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Let me further elaborate my position:

The same physical entity can be classified into the different categorical concepts of ... -> furniture -> chair -> rocking chair ... according to the level of measurements you choose to retain/omit.

Similarly, a physical entity can be treated, in science, as a member of different categorical concepts/mathematical models depending on what level of coarseness of information we are interested in. ...-> ideal-gas-molecule -> molecule-with-internal-properties-like-magnetic-moment -> many-body-quantum-system-of-n-dimensions -> ...

Dimension then becomes the number of different measurements/quantifiers associated with that model.

Now, to your reply (as far as I could follow it):

The key idea is that objects are quantifiable to varying degrees and the degree determines the model by which it would be treated. Such that in the ideal gas model, all molecules of the gas are equivalent no matter what their energetic state may be.

It is exactly the same as making statements such as:

"My furniture is ugly." or "The tables are not that flat."

When a particular existent is being treated as furniture as opposed to table there is only so much that you can say about it.

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Thanks, Abhijeet, for sharing these resemblances and overlaps between quantitative characterizations in physics and in Rand's theory of concepts.

Another aspect of physics and Rand's theory I'd say lies in her remarks on commensurability (which has some relation to Aristotle) and their connection to dimensional analysis. Then too, there are straight examples of Rand's idea of measurement omission in the character of concepts in very explicit form in physics. Such would be the concept of a solid as material having some resistance, however much not zero, to shearing stresses. Or we speak of an electrified body as having some amount, however much not zero, of net electrical charge. Another nice example from the exact sciences is the measurement of shape. Unfortunately for Rand, she did not understand how shape is measured and gave an erroneous account of it in trying to use that as an example in her theory. The correct method is a splendid example for her theory and is given here. Search also on the word football here.

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