Date: Apr 17, 2013 10:51 PM Author: Jonathan Crabtree Subject: Re: Proving a definition of multiplication (wrong) by induction FYI the post below post was a continuation from...

http://mathforum.org/kb/thread.jspa?forumID=206&threadID=2431535&messageID=8183575#8183575

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

> addition.

>

> 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