“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

- the word “number” can have 2 different meanings
- 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”

- Numerals are distinguishable objects we build our standard series of but adding them to previous members
- 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)

- the numbers can be used to compare the overall equality, superiority or inferiority of groups being counted
- 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

- 1 – 2 objects which are the same in respect to that property as some 3rd object are the same as each other (transitivity)
- 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

- It is possible to represent them by numerals to represent numbers & we use these numbers to represent qualities of the bodies above

- If a property is measurable it must be that
- 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)

- Laws established by definite experiment
- 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

- 1st step is to find a way to find a process to measure properties investigated
- 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

- What is the nature of these rules?
- 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

- 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
- 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

- Division is a direct result of multiplication
- This couldn’t have been done without experimental inquiry

- Weights can be divided into parts & this requires the use of fractions
- 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

- A common volume is specified between the 2 different bodies & the weight is compared in ratios

- It must be derived in a specific way

- As said before, measurement is the assignment of numbers to represent properties
- 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

- Hardness is measured by the effect of scraping on object on another & seeing the effect
- 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

- Colors can be ordered in lightness or purity of a shade of color

- When numerals are characterized, they have a sense of order based on their sequence
- 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

- Impossible to measure by the fundamental process

- Arrangement in an order & assignment of numerals in the order of properties are measurement represent something physically different
- 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

- 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
- 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.