“Measurement” by Norman Robert Campbell (1880-1949)

“Measurement” by Norman Robert Campbell (1880-1949)

  • Measurement is important because so many sciences are math-based & require it for measurement
    • But explaining what it is is harder – its use, its rationale & this is an attempt at that
  • What is measurement?
    • wasn’t around until a high degree of civilization appeared but today we’re all familiar with it
    • numbers are used to represent properties [time, prices, quantities, etc.]
      • not all properties have numbers [what kind of potatoes are the best for cooking, etc.]
      • we can even refer to them by numbers [e.g. No. 11 potatoes in a catalogue]
      • not all the same though
      • we can count the number of potatoes in a sack but non-measurable qualities are unchanged no matter how many there are
  • Numbers
    • the word “number” can have 2 different meanings
      • it can mean a word or symbol
      • it can be a property of an object
    • When we count the number of potatoes in a sack, we do so by counting – these are NUMERALS
      • without any symbols, I can count the number of potatoes in a sack & compare that number to another sack & its number of potatoes
      • based on these numbers, I can compare the 2 & say which sack has more potatoes in it
  • Rules for Counting
    • the numbers can be used to compare the overall equality, superiority or inferiority of groups being counted
      • seems obvious but it wasn’t always this way
    • starting with a single object & adding to it with other objects, we can build up a series of collections where they can possibly have the same number as one another
      • allows for a standard series of collections to be compared to other collections
      • shows how to make a standard series with least amount of cumbrousness
      • successive members of a standard series are compounded like this – originally with distinguishable objects [fingers & toes]
        • Numerals are distinguishable objects we build our standard series of but adding them to previous members
          • first member of series is 1, then the next 1 – 2, then the next, 3, etc.
          • these are compared to other series done in the same way
        • We quote each series by the number of the last numeral counted
          • Number of the days of the week is referred to as “7”, not using the whole series as in “1, 2, 3, 4, 5, 6, 7”
      • We can combine two or more collections
        • Adding a collection of 2 (1, 2) & a collection of 3 (1, 2, 3) should look like (1, 2, 1, 2, 3) if the numbers assigned to objects counted are fixed on each object
        • But we add the number of objects in a collection by continuing to count beyond the last number of the first group by the number of the second group (1, 2 <–from the first group & then count three more numerals beyond –> 3, 4, 5)
  • What Properties are Measurable?
    • If a property is measurable it must be that
      • 1 – 2 objects which are the same in respect to that property as some 3rd object are the same as each other (transitivity)
        • if body A balances body B, & body B balances body C, then A balances C
      • 2 – by adding objects successively we must be able to make a standard series of one member of which will be the same in respect of the property of any other object we want to measure
        • by placing a body in one pan, & continually adding it to others, collections can be built up which will balance any other body placed in the other pan
      • 3 – equals added to equals produce equal sums
        • If body A balances body B, & body C balances body D, then A & C in the same pan will balance B & D in the other pan
    • These properties make measurement possible & useful
      • It is possible to represent them by numerals to represent numbers & we use these numbers to represent qualities of the bodies above
        • They should be able to be combined for these purposes
  • Laws of Measurement
    • What is the nature of these rules?
      • Laws established by definite experiment
        • Called “rules” because it’s not clear that they’re truly inviolable laws in their application to number
        • But laws in their application to other measurable properties (length, weight)
      • They must be determined by experiment like any other law is proved true
        • Might just appear to be true but we just can’t take it on faith – may only apply in certain conditions
      • Experimental laws – can’t be known apart from definite experiment & observation of external world (NOT SELF-EVIDENT)
    • These are a most important part of experimental science
      • 1st step is to find a way to find a process to measure properties investigated
        • Greeks studied length, weight, volume, area, etc. – probably studied in Egypt & Babylonia
        • Greeks measured force to establish laws of lever & mechanical systems
      • But true method wasn’t established until the 17th century in Galileo’s laws of the pendulum
        • Modern science advanced it by Cavendish & Coulomb, Öerstad & Ampère, Ohm & Kirchhoff
    • Has there ever been a failure to discover the necessary laws?
      • Many properties aren’t measurable in this way – some are more measurable than others
      • Most important to say that measurement depends on experimental laws & facts of the external world
  • Multiplication
    • Weights can be divided into parts & this requires the use of fractions
      • If there’s a collection of bodies 4 number & each one has a weight of 3, we use multiplication of number in quantity & the number in quality (pounds) as 4 x 3 = 12
        • We can sum these up 3 + 3 + 3 + 3 = 12
    • Multiplication represents a definite experimental operation – combination of many collections into 1
      • Division is a direct result of multiplication
        • We even develop new numerals (fractions) to deal with this new operation
    • This couldn’t have been done without experimental inquiry
  • Derived Measurement
    • As said before, measurement is the assignment of numbers to represent properties
      • Need to consider ways that are wholly depended on the fundamental process if the numerals are to represent real properties & actually tell us something significant about the things they are describing
      • For example, Density
        • We all have an idea of what it means but what is truly meant when we said that iron is 8x denser than wood or mercury is 13.5x denser than water?
        • Density is not quite as common sense of an idea as weight
          • But a process was created to give it a fully functional/testable meaning
        • We don’t mean combining 13.5x more water than mercury that it will have the same density as mercury
          • Water will keep its density as it is
          • Density is a specific characteristic of all water together
        • Density’s feature is totally different than the additive feature we discussed before
          • It must be derived in a specific way
            • A common volume is specified between the 2 different bodies & the weight is compared in ratios
              • Same volume of mercury as water will weight 13.5x more than same volume of water
  • Measurement & Order
    • When numerals are characterized, they have a sense of order based on their sequence
      • 2 follows 1 & is before 3
      • 3 follows 2 & is before 4, etc.
    • We use this sort of characteristic order all the time – numbering the pages of a book, houses on a street, etc.
      • Not because we care how many there are in the absolute but in use of finding the page or the house
    • In the liquids density example, we can say mercury is denser than water by seeing that one floats on top of the other
      • If we did that with all liquids, we could make a definitive order of density by putting them in a series of most dense to least dense, or vice versa
    • This use of orders presupposes that this ordering/ranking is always true
      • To be able to say this we need to test if it is true
    • This can be used for many things
      • Hardness is measured by the effect of scraping on object on another & seeing the effect
        • A scale is set up to determine which things are harder than others
    • Some properties can’t be tested this way
      • Colors can be ordered in lightness or purity of a shade of color
        • Not in one single scale order
  • Numerical Laws
    • Arrangement in an order & assignment of numerals in the order of properties are measurement represent something physically different
      • If properties A, B, C, D are arranged in that order, you can’t assign arbitrary numbers that don’t reflect that order
      • The numerals representing a property can be given numerals representing an order
        • These properties can have derivative properties which can be given order as well
    • The invention of this process for properties is not suited for fundamental measurement is a great achievement for scientific investigation
      • Process was not invented by common sense but wasn’t until 18th century when it became widespread
      • Most characteristic feature of density is that it’s the same for all bodies of the same substance
        • Impossible to measure by the fundamental process
          • This assumption had to be determined & investigations were made based on it
  • Importance of Measurement
    • Assignment enables us to distinguish easily & minutely between different but similar properties
    • Terms between laws express relationships themselves based on laws & represent collections of other terms related by laws
      • When we measure a property, by the fundamental process or a derived process, the numeral which we assign to represent it is assigned as the result of experimental laws
        • The assignment implies the laws
        • We should expect to find that other laws could be discovered relating the numerals so assigned to each other or to something else
        • If we assigned numerals arbitrarily, without reference to laws & implying now laws, then we shouldn’t find other laws involving these numerals
    • Distinction between fundamentals & derived measurement
      • It is important because the first one alone makes the second one possible
      • It’s possible for a property to be measurable by both processes which both yield a definite order where one body has more & another has less
      • Properties involved in the numerical law must be such that they are fundamentally measurable – otherwise a law cannot be established.

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