Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



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...
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 + (n1)a > > PROOF OF 'ARITHMETIC PROGRESSION' RECURSIVE > DEFINITION = TRUE BY MATHEMATICAL INDUCTION > STEP 1) From the definition an = a1 + (n1)a the > theorem is the proposition P(n): an = a1 + (n1)a > STEP 2) Show that P(base case) is true. > > Let n = 1. Therefore for the proposition P(1) we get > a(1) = a1 + (11)a > a1 = a1 + (0)a > a1 = a1, which is TRUE. > > STEP 3) Inductive Step > Assume for some integer k, P(k) = a1 + (k1)a1 (again > note the '1' in 'a1' is subscript) > > STEP 4) Show P(k+1) = a1 + (k+11)a1 > > STEP 5) Proof of 3) Inductive Step > a1(k+1) = a1 + (k+11)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 + (n1)a for all natural numbers 'n'. > > CONTRAST THE DEFINITIONS > > ab = a added to itself b times > an = a added to itself n1 times > > Let b=n > > ab = a added to itself b1 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=commutingstrogatzexcerpt



