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- Before starting everything, a disclaimer that all these are learned from Stanford Philosophy Encyclopedia page. So a huge credit. At the front.
- First let’s look at the properties that are measured. According to Euclid, there is this property with lines, surface or solid called magnitude. If one magnitude is a multiple of another, we can use the second one as a measurement; if two are both multiples of some third magnitude, then the third is a measure of the other two. Alright, but not all magnitudes have this whole number relation, then how can we quantify the relation between them then? Euclid used ratio of magnitudes, which uses both rational and irrational numbers, thus can describe numerical relations in a broader spectrum. *my take on this: measurement essentially is about comparison, or relation. Comparison is only possible between magnitudes that are about the same property, e.g. we cannot compare the mass of an object with the volume of another. And when the comparison can be expressed with multiples of some value, it is called measurement; otherwise there is a ratio concept to solve the problem. Bottom line is that the magnitudes have to be of the same properties. I also conceptualize magnitude as property of properties.
- Now comes Aristotle. What does he have to say about this? Well, he seperates properties as quantities and qualities. Quantities are those that can be compared among different objects with a standardized scale, and the results are repeatable; while qualities are the properties that after being compared by different people, the outcomes cannot be guaranteed to be the same. Aristotle thinks quantities of one property can only be perceived as an amount and that’s it, there isn’t a more or less degree of that amount; but qualities have degrees. But as for one quality, he doesn’t say whether the different degrees of that quality are all still considered that quality, or are considered different qualities. Some guy (Duns Scotus) believes that qualities can also be operated on! Like adding or subtracting degrees.
- Let’s welcome Leibniz to the stage. As usual, he got something to say about this topic. His idea basically is that, look, Euclid, I know it’s pretty cool to measure extended magnitudes in your geometrical thingies, but see, actually the intensities of sensations we perceive can also be measured, and thanks to me, we an now unify all natural changes with degrees, and let’s just agree that this has a cool name of “principle of continuity.”
- Here comes Kant. He pretty much agrees on Leibniz, but he thinks Leibniz’s idea can look more elegent. So adding on that, he says that we can now classify magnitudes with being extensive or intensive. The former ones are those extend in time and space, those “in which the representation of the parts makes possible the representation of the whole.” And the latter is the properties that are preceived by us immediately, like color and warmth, instead of being perceived part by part like the boring spatial ones. How to represent the latter? He thinks it should be represented by how far it goes from that point to negation.
- Alright, those are not absolute. New science came, and there are now more ideas on what magnitudes are, but I refrain from digression.
- Let’s get right into measurement, after all the digression, yay! There are multiple schools, and let’s tackle one by one.
- First, mathematical measurements. But before that, let’s get something straight. So the fact that we often represent reality with numbers are actually not that sound. Imagine temperature. 20 degree celcius is not twice hot as 10, because the 0 point on that thermometer is not the absence of temperature, but an arbitrary point just for the purpose of convenience. Intervals don’t always carry meaning, too. Imagine our opinions on something, scaled from 1-10, like a hot guy. Between a 7 and an 8 doesn’t mean they have the interval of 1 as between 5 and 6. There isn’t an exact 1-worth of handsomeness. Now let’s dive into mathematical measurement. These people are all over how to better represent reality in terms of numbers and number relations. This person Hermann von Helmholtz has this beautiful quote, “What is the objective meaning of expressing through denominate numbers the relations of real objects as magnitudes, and under what conditions can we do this?” Basically, he asks 1) what have we assumed before using this mathematical measurement and 2) how adequate or limited are we when we use them? Following these, a general rule of thumb is that we use mathematical structure to measure the relations in reality that mirrors math relations. Like bigger than in numbers and lengths in reality. In other words, we map relations in reality to relations in maths. And people would assume this kind of mapping as isomorphisom or homomorphism. Actually, people have a lotta different views apart from the aforementioned. Firstly, they have different views on what actually is this object in real world. The object being measured, the sophisticated philosophers call “relata,” can be among the following concepts (or more): it is the thing that’s there; it is our perception; it should be an idealized object of that thing; it is a universal property that can belong to a whole lotta things. Based on these differences, the measurability is debatable because they don’t even agree on what is being measured. Now we can introduce some different ideas on that. A consensus, though, is that measurement is assigning numbers to magnitudes with a unit. Unite is important because it is 1, wheras others are all compared to it. Then the question is what is the right way of asigning? Some people says you gotta make sure when you make algebraic operations like adding, multiplying, etc, the reality stays true. Basically, after manipulating the numbers for an equation, the reality must stay good as well. Some believes that the two expectations from objects are comparison and addition, so they two suffice as a additive numerical representation. And if you can measure 1 thing directly then you are measuring fundamental magnitudes. Other magnitudes have to be calculated and they are called derivative magnitudes.
- There are also 4 different scales: objects in the same class without orders, nominal; ordered but interval has no standard meaning, ordinal; ordered and interval has meaning, interval; values that can be calculated as their porportion because they have an absolute zero point, ratio. This is according to a guy with a cool name of S.S. Stevens (is that really not a stutter?). Anyways, he later refined this theory, adding linear and logrithmatic into the interval kind, cause some intervals have equal magnitudes while others have exponential ones, and thus only their logrithmatic value is taken onto the scale. For ratio scales, he catagorized them into those with natural units and those without. Why would our Stevens wanna catagorize that? Well, by catagorization, he can make different operations on different catagories of scales without losing their empirical truthfulness, and that in turn could mark the scales’ catagory. (Actually I don’t really understand why would he wanna do that. Isn’t that circular?)
- Now comes the measurement of sensation. There’s a guy named Gustav Fechner (I just realized how this whole area is a boy’s club, but I won’t digress to that now) realized that we can actually measure it with experiments, basically taking down the scales of stimuli with “just noticeable differences.” Turns out when sensations go up linearly, the magnitudes of stimuli go logarithmically. With this law, people started to measure sensation with the measurement of stimuli. But some people beg to differ, e.g. Campell, who thinks as long as the numbers cannot be concantenated and then represent a new reality bearing that number, that doesn’t count as a foundamental measurement! But Stevens argues back cuz he thinks they are consistent and non-random assignments, so they are fine.
- I didn’t understand how RTM works with triplets.
