Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Re: Proving a definition of multiplication (wrong) by induction
Replies: 0

 Jonathan J. Crabtree Posts: 355 From: Melbourne Australia Registered: 12/19/10
Re: Proving a definition of multiplication (wrong) by induction
Posted: Apr 17, 2013 10:51 PM

FYI the post below post was a continuation from...

Jonathan Crabtree

> Recursive definitions often have the word 'times' in
> them. Being recursive, such definitions can be proven
> by the principle (axiom) of mathematical induction.
>
> FALSE HYPOTHESIS
> 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.
>
> ie ab = a added to itself b times
>
> This can be restated recursively as ab = itself + a(b
> times)
>
> When assessing the truth of a statement, the first
> step is to remove all pronouns.
>
> In this case the word 'itself' is a third person
> singular reflexive pronoun.
>
> So we first restate the definition ab = a added to
> itself b times.
>
> ab = a added to itself b times
>
> becomes...
>
> ab = a added to a b times.
>
> The theorem is thus written algebraically...
>
> ab = a + ab
>
> ...which straight away makes no sense!
>
> We have an extra 'a' in the definition!
>
> PROOF ACCEPTED RECURSIVE DEFINITION OF MULTIPLICATION
> IS FALSE BY MATHEMATICAL INDUCTION
> STEP 1) From the definition ab = a added to itself b
> times, the theorem is the proposition P(n): ab = a
> +ab
>
> STEP 2) Show that P(base case) is true.
>
> Let n = 1. Therefore for the proposition P(1) we let
> b = 1
>
> a(1) = a + a(1)
> a does NOT equal a + a
> a does NOT equal 2a
>
> The base case fails and we CANNOT prove ab = a added
> to itself by mathematical induction!
>
> Therefore the commonly accepted and quoted dictionary
> definition of multiplication is FALSE.
>
> *
> http://www.collinsdictionary.com/dictionary/english/multiplication
>
> TRUE HYPOTHESIS
> Now let us prove another recursive repeated addition
> definition of multiplication known as an arithmetic
> progression true.
>
> {a, a + a, a + a + a, a + a + a + a, ...}
>
> Here our first term is 'a' and our common difference
> is also 'a'.
>
> Our nth term in this arithmetic progression is
> defined algebraically as
>
> an (Note: both the 'n' in 'an' and the '1' in 'a1'
> are subscript)
>
> an = a1 + (n-1)a
>
> PROOF OF 'ARITHMETIC PROGRESSION' RECURSIVE
> DEFINITION = TRUE BY MATHEMATICAL INDUCTION
> STEP 1) From the definition an = a1 + (n-1)a the
> theorem is the proposition P(n): an = a1 + (n-1)a
> STEP 2) Show that P(base case) is true.
>
> Let n = 1. Therefore for the proposition P(1) we get
> a(1) = a1 + (1-1)a
> a1 = a1 + (0)a
> a1 = a1, which is TRUE.
>
> STEP 3) Inductive Step
> Assume for some integer k, P(k) = a1 + (k-1)a1 (again
> note the '1' in 'a1' is subscript)
>
> STEP 4) Show P(k+1) = a1 + (k+1-1)a1
>
> STEP 5) Proof of 3) Inductive Step
> a1(k+1) = a1 + (k+1-1)a1
> a1(k+1) = a1 + (k)a1
> a1(k+1) = a1(1+k)
> This by commutative law of addition
> a1(k+1) = a1(k+1), which is TRUE.
>
> STEP 6) Therefore P(k+1) is TRUE when P(k) is true,
> and therefore P(n) is TRUE for all natural numbers
> and an = a1 + (n-1)a for all natural numbers 'n'.
>
> CONTRAST THE DEFINITIONS
>
> ab = a added to itself b times
> an = a added to itself n-1 times
>
> Let b=n
>
> ab = a added to itself b-1 times is TRUE
> ab = a added to itself b times is FALSE
>
> CONCLUSION
> The definition of multiplication attributed to Euclid
> since 1570 has been FALSE and the contradiction is
> both revealed and proven.
>
> http://jonathancrabtree.com/euclid/1570_First_English_Euclidean_Definition_Of_Multiplication_by_Henry_Billingsley_Definition_16.jpg
>
> ^ Quote: 'Take the terminology. Does 'seven times
> three' mean 'seven added to itself three times'? Or
> 'three added to itself seven times'?
>
> Five added to one three times is 16.
> Seven added to itself three times is 28
> Three added to itself seven times is 24
>
> Seven multiplied by three, according to the true
> original Greek of Euclid, is simply...
>
> ...seven placed three times!
>
> ab = a placed b times, or in more modern 16th century
> English...
>
> ab = a taken b times
>
> The 'addition bit' was an optional extra to be done
> AFTER the multiplication. Euclid was multiplying line
> segments, not numbers.
>
> Multiplication NEVER WAS defined as repeated binary
>
> Euclid was observing magnitudes in action not
> counting multitudes. Euclid was undertaking unary
> actions and observing the result.
>
> Quantity is a function of number and size.
> Multiplication varies number while scaling varies
> size.
>
> So basically, our arithmetical pedagogy has been
> fundamentally flawed for 443 years.
>
> In conclusion, this note can be viewed as a recall
> notice.
>
> ab never was a added to 'itself' b times.
>
> Teachers and parents would be well advised to
> demonstrate multiplication as the joining of equal
> line segments starting from zero on the number line.
>
> So...
>
> an introductory explanation could become
>
> ab = a added to zero b times.
>
> This can then evolve into rationals and integers as
> required.
>
> ^ Source: Steven Strogatz, Professor of Applied
> Mathematics
> http://www.scientificamerican.com/article.cfm?id=commuting-strogatz-excerpt