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Dimensions or units are a type of idea or concept. Two or more of them can be integrated together into a single conceptual unit. This follows the general rules of integration of ideas.
Any group of ideas can be integrated together, but most groupings and combination methods result in nonsense. Also, the usefulness of concepts is contextual. So context determines which ideas are useful to combine in what ways. You need to start with, and conclude with, concepts that fit your context.
A concept that seems like nonsense could make sense in some other logically possible context. That works for all possible finite results of integration. Speaking more broadly, the result of an integration is information (and it’s logically possible to get the same information by another method without integrating it). So the question is whether any arbitrary information would be meaningful in the right context. The answer is “yes” for finite information because you can imagine e.g. a context where that information is the password to decrypt a computer file.
With integration, we’re always considering which results are relevant to our goals. We try combining many ideas in many ways to find some combinations that are useful for us (that are adapted to our goals and context). You can integrate any ideas, but the large majority of results are junk.
Combining dimensions is the same way. You can combine any dimensions but most of the results are junk – they don’t mean something meaningful to you.
Integration is a generic idea about combining multiple parts into a greater whole. It doesn’t specify a specific way of combining. There are many ways to combine ideas and we don’t know what they all are. When we integrate ideas, we often don’t consciously know what method we used. We’re bad at explaining in words how ideas combine.
Dimensions, being ideas, could be integrated with any general purpose method. But we generally focus on a few combination methods which are not general purpose: they only apply to numbers. As long as we only define dimensions for things we know how to put a number on, then we can combine dimensions using numerical operations (math), like multiplication or division (addition doesn’t work well).
Can we put numbers on all dimensions? Not very well. Some dimensions, like length or mass, are measurable physical quantities. Others, like cuteness or deliciousness, are things we don’t know how to measure accurately. We don’t know, clearly enough, what they are. And we don’t know the right units. However, people make up very loose, approximate numbers for dimensions like those, often on scales like 1-100 (similar scales include 1-10, 1-5, percentages, and real numbers from 0-1). Scales like those are often used in a comparative way. You imagine some super cute or uncute stuff, and some middle of the road stuff, and then consider how close the thing you’re judging seems to those points of reference. These comparisons are not made in a reliable, accurate way like measuring, so doing math with these dimensions, at all, is questionable. Happily, Critical Fallibilism (CF) does not do math with those dimensions.
CF uses yes-or-no questions to convert to related binary dimensions. You can still use 1-100 scales for informational purposes when you find them useful, but no math on problematic numbers is needed. By switching to a 0 to 1 scale, with no decimals allowed, CF allows for clear, accurate answers instead of intuitively trying to guess the right number (and never getting it perfect, and not having any good way to figure out how much you’re wrong by or how consistent your evaluations are for different things – e.g. did you really estimate the cuteness of three different cats on the same scale so that the numbers are all directly comparable?).
CF also allows for non-binary numbers when they represent distinct categories. It’s problematic to use a 1-5 scale for how good something is. But it’s OK to have 5 discrete, qualitatively-different categories with breakpoints differentiating them, and to rank them by how good they are and number them. Not all categories have clear rankings (in your context, for your goals) but some do. The reason this works is that category membership is a clear, decisive thing – whether something is a member of a category is a binary judgment (it is or isn’t a member). So you aren’t trying to measure an intellectual quantity. The numbers do not represent different amounts of the same thing. Only physical quantities can really be measured; using measurement of amounts intellectually is only an approximation. (BTW, some people don’t realize it, but counting is a type of measurement, even though we can count without a tool like a ruler or scale.)
With binary judgments, the numbers are not different amounts of the same thing. When you say something like “how good is this on a 1-10 scale?” you’re looking at an amount or degree of one thing. When you ask, “Is X the case, yes or no?” you are not considering an amount or degree. It’s not a spectrum. In binary thinking, we can use the numbers 0 and 1, but they represent qualitative differences (like yes/no, true/false or pass/fail) not quantitative differences (different amounts or degrees of something).
Suppose you’re considering pets and use numbers for different species of animal. You rank them by length and assign dog=4, cat=3, hamster=2, goldfish=1. These numbers are not measurements of anything, including length. They are not degrees or quantities. Quantities would be: dog = 36 inches, cat = 20 inches, hamster = 8 inches, goldfish = 4 inches. Numbers are more normally useful for quantities/amounts/degrees/spectrums, but they can be used for qualities/categories (like using 1 and 0 for true and false is common).
In the example, I ranked the animals from 1 to 4. It’s customary for the lowest number to be either 0 or 1, and for the numbers to be consecutive integers. But you could use other numbers, e.g. dog=5743, cat=994, hamster=22, goldfish=-5. And you don’t have to rank things and put the numbers in any particular ordering. You could just have five things and give them five different numbers in no particular order. Then the numbers are essentially just new names for the things, which can be useful if they didn’t have names already or the names were long. When using numbers that way, small positive numbers are good because they are short to say or write.
There’s a related technique to help figure out if numbers are being used in a quantitative or qualitative way. You can try changing the numbers around. Can you add 10 to all the numbers, or triple them, without screwing things up? If the specific numbers used seem to matter and changing them causes problems, that’s an indication that you’re dealing with quantities. If other numbers would work fine, that’s an indication you’re dealing with qualities/categories. If the numbers are measurements in inches, changing the numbers won’t work – they will no longer correspond to reality.
People sometimes talk about cardinal numbers (counting numbers – one, two, three) and ordinal numbers (rankings in a sequence like first, second, third). Cardinal numbers are normally used in a quantitative way – to say how many of one thing there are. Whenever there is more than one number for different amounts of one thing, it’s quantitative. When each number represents a different thing, then it’s qualitative and suitable for some important uses in Critical Fallibilism. Ordinal numbers are technically qualitative. First, second and third are different categories. Third is not some amount of firstness or fifthness. You can’t do math with ordinal numbers in the regular ways that you do with cardinal numbers. And ordinal numbers do not directly represent measurements. However, people general think of ordinal numbers in connection with quantities – e.g. people run a race and are ranked by their time (a quantity), and first place corresponds to a particular measured time.
I’ve related integrating ideas with combining dimensions, discussed numbers and measurements, and talked about qualitative and quantitative differences. Being able to distinguish qualitative and quantitative differences, and spotting when they come up in any part of life, is crucial to understanding and using CF well.