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In the world we deal with concrete 'unit' quantities, the 'units' are very concrete, ten pencils, for example we can tell the difference between one pencil and another and we can also trace and identify one of them and see where it goes and differentiate it from other pencils.

If we measure '10 meters' can we do the same? A meter is a certain measurement on a ruler, it does not exist like a pencil, we cannot differentiate a meter unit from another, only a two things that are a meter in length, the same for any 'measured' quantities. If I have 5 pencils and then get more, such that I have '10 pencils' I can trace which pencil is which, if I have an object of mass 2kg and gain 3kg to have mass 5kg I can trace each kg worth of matter, but not the 'kilograms' themselves.

Is this part of how we should view 'units of measurement' or simply as values we would expect to find on measurement, can an object 'possess' units at one time and have two objects each with 'one unit' of length have two different 'units', with one each? What is the best way to deal with these abstract 'objects'?

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  • We count physical objects: pencils. They are "concrete" not their numbers. We count other "objects": money. The 20 written on the twenty dollars banknotes does not count physical objects: there are no twenty "things" in the piece of papers. We call them "abstract objects". Commented Apr 29, 2022 at 9:54
  • the question is: are units of measurement like money, where you and me can each have an 'abstract object' or is the 'amount' just a relative value that gives the property of the scale, so there's no 'you have 5 units and I have 3 units' just you have such a property state that would measure 5 units and I have a such a property state that measures 3 units'
    – Confused
    Commented Apr 29, 2022 at 10:25

4 Answers 4

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You "pencils" example appear to use pencil itself as the unit. This is somewhat workable in English because English omits the unit for countable objects, and as a result, leaves some open area of interpretation on what the actual unit could/should be.

However, some other languages, specifically Chinese that I know of, do use identity-less (abstract) units for countable objects (e.g. "30 rounds (rd) of bullets").

So I think the solution to your problem is to avoid using concrete objects as units for countable objects, and consider there is an abstract unit like "count" (ct) that is omitted when expressed in English.

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A measurement with units is not a counting of units, but a ratio to units. That is: 10 meters is that length which, divided by ten, is the length we call 1 meter. This for example lets us have things like nanometers: that length which, divided by 1/1000000000, is the length we call 1 meter, or negative meters (which just measure displacements in the negative direction, but good luck finding a negative meter stick!)

On the other hand, there's no such thing as a nanoiPhone. If you chop an iPhone into a billion pieces, you can't take one of those pieces, multiply it by a billion, and arrange it into an iPhone. You'd just have a pile of dust. And there's certainly no such thing as a negative iPhone.

Even when using units that are attached to countable values, we use ratios to units, not the countable value itself. For example, electric charge has a fundamental unit of the charge of a single electron. The standard unit for charge is the Coulomb, approximately equal to -6.24 * 10^18 electron charges. But the Coulomb is emphatically not equal to -6.24 * 10^18 electrons. The actual countable object - the electron - is abstracted out, leaving just the the charge associated with an electron. This turns out to be more useful than anyone might have imagined at the turn of the 20th century. Quarks - the building blocks of protons and neutrons which were discovered around 1960, turn out to have charges of absolute value 2/3 or 1/3 electron charge.

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Scales of measurement are always conventional: arbitrary, but systematic and functional. They are usually organized into a few types:

  • Categorical: individual, indivisible objects sorted into groups, without any necessary hierarchical evaluation: e.g., sorting cards into four suits (clubs, hearts, diamonds, spades)
    • This is sometimes called Binary when there are only two categories
  • Ordinal: individual, indivisible objects sorted into groups, with a necessary but ill-defined hierarchical evaluation: e.g., 'low', 'middle', 'high'
  • Interval: measurements that might relate to individual, indivisible objects, but are themselves infinitely divisible: e.g.' measurements of temperature or length.
    • This is called Ratio when the measurement has defined and absolute zero-point (making ratio comparisons meaningful). Thus temperature Fahrenheit is only interval (since the zero-point is arbitrarily chosen, so that 4°F is not twice as hot as 2°F), but length is ratio (since a length of zero is meaningful, so that 4" is twice as long as 2"

The question points at the distinction between counting and measurement, which is a bit esoteric, but not as firm as one might think. One of the assumptions in counting is that the things being counted (for practical purposes) are indistinguishable and interchangeable: members of a 'class' of objects. So yes, even though we can track each individual pencil in a box of ten pencils, doing so violates the spirit of counting. If we do that, we no longer have a collection of ten pencils, we have a collection of ten unique objects: ten categories with one object each, not one category (pencil) with ten objects. Likewise, if we take our ten pencils and cut them all in half, we now (technically) have twenty pencils which we must treat as indistinguishable and interchangeable (despite the fact that half of them lack eraser ends). The notion of 'a pencil' is a convention that ignores individual differences between objects to categorize them along abstract functional/structural grounds.

Measurement, by contrast, generally involves things that are infinitely (uncountably) divisible. We can think of a 'length' measurement (like a meter) as a collection of infinitely many infinitely small 'objects' chained together: this is the intuition behind things like Zeno's paradoxes or Newton's calculus. The actual unit — like 'meter' or 'inch' — is purely conventional, of course. Someone at some point said it was so, and the rest of us consented (because... why not?). Once that convention is established, an inch or a meter becomes an abstract object-category, in much the same way that 'pencil' is an abstract object-category: we call things a 'meter' or an 'inch' (or for that matter a 'pencil') because it fits a restrictive set of conditions for inclusion.

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  • Surely, being able to track individual pencils makes no difference? Being able to differ between them makes no difference they can be very different and still be 'pencils' to be 'a pencil' is to be a member of a class. I can say I have 10 pencils, Pencil A, Pencil B etc and it makes no difference, there are 10 pencils, but only one pencil A. Knowing what happens to Pencil A makes no difference as I still have only 10 members of a class called 'Pencil'? Being members of a class does not stop us from distinguishing them?
    – Confused
    Commented Apr 30, 2022 at 11:14
  • Perhaps I've misunderstood you slightly, do you mean we create a sort of set of 'identical' units called 'Pencil'?
    – Confused
    Commented Apr 30, 2022 at 11:19
  • We can still have a 'unique' pencil
    – Confused
    Commented Apr 30, 2022 at 12:13
  • @user1007028: Of course we can have unique pencils; each of the myriad pencils in the world is unique. But in counting we have to treat objects as interchangeable, otherwise counting (and math) falls apart.consider... Say we have nine yellow pencils and one green one. Do we say we have ten pencils? That obviates the uniqueness of the green one. Do we say we have nine yellow and one green? That preserves uniqueness, but creates two distinct categories. Do we say we have ten yellow-or-green pencils? That preserves the color difference, but not uniqueness (we no longer know which are green). Commented Apr 30, 2022 at 14:48
  • @user1007028: The point is that a 'pencil' isn't exactly a thing in the world. A 'pencil' is any object that we have assigned to the mental category 'pencils'. Its uniqueness as a real-world object is disregarded; all that matters is that it fits the purposes and characteristics that are intrinsic to the mental category. I mean really, when we say "give me a pencil" we mean something like "give me one of those objects that has been standardized for the purpose of making clean marks on sheets of paper". It doesn't matter which of those objects we get. Commented Apr 30, 2022 at 14:55
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The point of measurements is to make an objective judgement on a property of an object avoiding subjectivity (roughly). This might explain why comparing them is complicated.

For example, imagine a group of biologist trying to understand what different species of trees there were based on their leaf sizes. 1 biologist would take a ruler and announce this oak tree has 5cm wide leaves. But having only 1 measurement is not enough proof, so biologist 2 takes the ruler and measures the leaves and announces that they indeed 5cm wide which creates a stronger argument.

The measure is important because now they both agree on a measurement that is not subjective to either of them.

Taking the alternative where there is no measure, one biologist would say "the leaf is quite large" and the other might say "the leaf is very large", but do these individual statements supply the same evidence for the proof that the tree is a certain species?

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