Monday, October 26, 2015

Semantics of Elementary Arithmetic Expressions

Musing on the Semantics of Arithmetic Expressions

What is meant by the arithmetic expressions that we read and write? I certainly do not intend to present anything new, just report some observations made while working out what arithmetic can mean, with an eye toward the possibilities when numbers denote more than just themselves. I want to look at elementary mathematics as an active endeavor where you start by writing an expression for something you want in terms of what you know, and whose evaluation plays as a movie in your mind. 
For addition and multiplication what actions flow out from the form of their expression? In the infix notation used at this level, expressions are of the form: "(operand operator operand)"; the parentheses are just my notation to improve blog readability. It has long been noted that the semantics are ambiguous: do we read (2 + 3) as instruction to calculate, or is it just another way to write (5).  If the meaning is calculation, we use a process that takes two numbers and produces a third. If the expression is to just to be rewritten, then 
we look in a big table we carry in our heads. This is what most of us do for small numbers
These are real questions when you are trying to instruct a computer; they are also real questions whenever their answer is not clear from the context. It was clear in elementary education before we tested students on material long after it was taught.  For small numbers elementary math instruction first teaches a calculation process and later assumes a student has the big look-up table.

For our purposes, the arithmetic expression shall mean the operator denotes calculation with operands: the expression (2 + 3) means "add 3 to 2 " and produce the sum. How do we know it does not mean "add 2 to 3"? In a sense we do not care, the sum produced by any process that we know about produces the same result, and in fact we could have chosen the reverse order, but the two processes are different: they treat their two operands differently, and we must choose one. This is reflected in our usual interpretation of the expression for addition as "add N2(1) to N1(2)" and not "produce the sum of the two operands"; or for multiplication: "multiply N1 by N2" and not "produce the product of the two operands".
The classical words for the two operands in both addition and multiplication are different from each other and are suggestive of the calculation process:

addition: Augend + Addend = Sum
subtraction: Minuend - Subtrahend = Difference

multiplication: Multiplicand x Multiplier = Product
Division: Dividend / Divisor = Quotient

...contain a wealth of Latin. If you happen to know Latin, you will understand these meanings more deeply. For example "minuend" comes from a form meaning "about to be lessoned" and "subtrahend" comes from a form meaning "about to be taken away". In general, "-nd" will carry the meaning "about to be --ed".

Interesting! What appear to be exotic labels suggests a practical way to look at the calculation process. I choose to read "Augend" as the operand about to be augmented, with "Addend" about to be the one doing the augmenting. The "Multiplicand" is the operand about to be multiplied, and "Multiplier" is the one that will do the multiplying

This is consistent with the operational meaning: choose one operand to be acted on and the other to act on it. For addition one operand plays the passive role, waiting to be augmented by the second operand. In multiplication, again one operand plays the passive role, waiting to be multiplied ( stretched either lengthwise or crosswise in an amount determined) by the second operand. The classical nomenclature suggests actions and actors.

Any suggestions for different words for the operands? Surely you can improve these: ("abt" is "about to be", "at" is "about to")

Addition:             (atbAdded-to + atAdd) 
Multiplication:    (atbMult-ed + atMultiply)


Addition:            (atbAdd-edTo + atbAdd-ingTo)
Multiplication:   (atbMultipli-ed     x  atbMultipli-er) .

These position-specific "words"  for "operand" are all adjectives, each one now modifies an unvoiced "operand"; e.g.       "[the] atbAdd-edTo [operand]" in
"(the atbAdd-edTo 2 + the atbAdd-ingTo 3)" for the expression    (2 + 3).

A more practical notation might forgo the "about to be":

Addition:            (To 2 + Add 3)
Multiplication:   (Mult 2 x By 3)

and this suggests the more elegant

Addition:            (To 2 , Add 3) means "add 3 to 2", or               "with 3 augment 2".
Multiplication:    (Mult 2 , By 3) means "multiply 2 by 3",              or "with 3 multiply 2".

None of these feel quite right; but the real "problem" is now apparent. "add" and "multiply" are not symmetrical. "Add" refers to what is is going to be added, not to what is about to be added to; "multiply" refers to what is about to be multiplied, not to what it is going to be multiplied by:
  • (2 + 3) means add 3 to 2, 
  • (2 x 3) means multiply 2  by 3.
In short, one adds "to", while one multiplies "by".

Things might be less confusing if we could use symmetric terminology for (operand , operator, operand). For addition: 
  • the name of the operator "+" could be "augmentation":
  • the left operand is the "augment"-this-operand,
  • the right operator is the do-it-"with"-this operand; ie,
Addition expression: 
(augment-this-operand  +  by-this-operand)

Multiplication expression: 
(multiply-this-operand  x  by-this-operand).

Do you think the order of the operands should be reversed? In the order shown above the operands on the right are those most likely considered prior to those on the right.

For both addition and multiplication there is generally the connotation that after the calculation the left operand has been changed, and the right one did the changing; however, neither the sum nor the product depend on the operand order: addition and multiplication commute. To reduce confusion in subsequent work, however, I will try to keep the changed operand on the left.

There is a major difference between addition and multiplication: In multiplication we must first consider the changing operand because Multiplication distributes the change it specifies over the entire changed operand while, for addition, the augmentation can be effected by one "attachment".  In a sense multiplication is melding; and addition, an add-on. At this juncture all this is premature; perhaps you can see it. It will come up in later posts, and I may go back and change the first post where I think some of it appears more naturally. 

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