Date: Jan 29, 2013 5:38 PM
Author: Jonathan Crabtree
Subject: Re: Proving a definition of multiplication (wrong) by induction
Jonathan Crabtree wrote (in part):
> > Multiplication* an arithmetical operation, defined
> initially in terms of repeated addition, usually written a × b, a.b, or ab, by which the product of two quantities is calculated: to multiply a by positive integral b is to add a to itself b times.
i.e. ab = a added to itself b times
This definition fails proof by induction.
Dave said in part
> I don't follow your argument. Assuming that something
> "fails proof by induction" , it does not follow that the result is not true.
Thank you for thinking about this.
Assume ab = a added to itself b times is a proposition.
As you know, proof by mathematical induction has only two steps requiring the proposition P(n) is true for every positive integer.
BASE CASE: Show the proposition P(1) is true.
INDUCTIVE STEP: Assume P(k) is true for any positive integer k, and show that under that assumption, P(k + 1) is true.
After STEP 1 AND STEP 2 we conclude (by the principle of mathematical induction) that for every positive integer n, P(n) is true.
BUT! The propositions P(n) and P(k) CANNOT be proven false by the inductive step because the BASE CASE shows that the proposition P(1) is FALSE.
The INDUCTIVE STEP cannot be taken on the positive integers.
The first step is to rephrase the proposition by replacing the third person singular reflexive pronoun 'itself' with a.
Then restated more clearly;
ab = a added to a b times
P(n) = a added to itself b times
BASE CASE: Show that the proposition P(1) is true.
Substitute b with the multiplicative identity 1.
Therefore ab => a(1) = a
a = a added to a 1 time (the proposition)
a does NOT equal a + a
The BASE CASE is false.
The proposition ab = a added to itself b times is FALSE.
It fails the BASE CASE and therefore ab = a added to itself b times CANNOT be a definition of multiplication.
To be rigorous, we should define addition as well as multiplication. So rather than provide a formal proof of addition which is more than 1000 lines, I will remind others of the 'essence' of addition.
Addition is the combining TWO numbers into ONE number. i.e. addition is binary
In '+ 1' the placing of the + 1 on your screen is a unary additive operation. The addition sign operates on only one argument.
Now assume I have blanked your screen and then this appears...
'1 + 1'
In the above the addition sign is a binary additive operation on two arguments (addends or more strictly, an augend and an addend).
+ 1 IS NOT 1 added TO itself as it is singular and unary in nature
1 + 1 IS a singular binary addition on two arguments
1 + 1 IS 1 added to 1 one time, nothing more, nothing less.
Mathematicians have always counted the addends in the expanded definition of ab as repeated addition. That has prevented people checking the definition as well as the mantra 'definitions are not provable'.
In ab the addend appears b times and is added b-1 times.
The same error occurs with exponentiation when non-mathematicians say 2^3 is two multiplied by 'itself' three times.
a^b is a multiplied by itself b-1 times. (Citing both Euler and de Morgan)
2^3 is the factor two appearing in a multiplication three times. 2^3 is two multiplied by itself twice, or 3-1 times.
2^3 = 2 x 2 x 2 ie three factors of 2 connected vie two binary operations of multiplication.
2 x 3 = 2 + 2 + 2 ie three addends of 2 connected via two binary operations of addition.
ab = a added to itself b-1 times.
Yes this could be be written more formally, yet it won't make the conclusion any different.
The definition of multiplication 'ab = ab added to itself b times' people have been repeating for 442 years is FALSE.
If any of the above requires refinement, please point out how and where.
Thank you again.