GS Chandy
Posts:
7,942
From:
Hyderabad, Mumbai/Bangalore, India
Registered:
9/29/05


Re: Proving a definition of multiplication (wrong) by induction
Posted:
Feb 5, 2013 12:07 AM


Responding to Jonathan Crabtree's post dt. Feb 5, 2013 7:51 AM (pasted in full below my signature for ready reference): > > If you can't accept the definition of multiplication > 'ab = a added to itself b times' is false, might you > at least all agree this definition of multiplication > is not well defined? > I agree with you that there are several unresolved issues with the way things in 'elementary' math are defined/ discussed  for example the instance of multiplication that you have been discussing does seem to be suffering from defective definition. (If I recall rightly from my work many years ago in graduate school [mathematics], there are also quite a number of such issues in 'advanced' [i.e. non'elementary'] math).
I do NOT agree that you have shown a practical means to resolve such issues. Any appropriate resolution of such issues would, I believe, have to work from a 'meta'level in order to help us actually resolve them all (or at least most of them) in real life on the ground, in practice. Else, we are likely to end up foncusing children and students about math; remember that most students graduate from school fearing/ loathing math.
THAT is the issue actually to be resolved!
For practical means to resolve such 'meta'issues, check out the tools described at the attachments to my message at http://mathforum.org/kb/thread.jspa?threadID=2419536
(To be entirely honest about the matter, I observe that a couple of Mathteach participants [one of whom is now unfortunately absent from our discussions] had raised entirely spurious/phony objections to the tools described. I had several times demonstrated that their objections were entirely spurious).
GSC Jonathan Crabtree posted Feb 5, 2013 7:51 AM > > > > > On Feb 4, 2013, at 5:51 PM, Dave L. Renfro > > wrote: > > > > > In a formal treatment one would clear up any > > possible ambiguity > > > that arises in the use of a term such as this. > To > > me, it's a > > > simple matter of writing > > > > > > a + a + a + ... + a > > > Indexing also shows up in board games, like do you > start counting on the square you are on, or the next > one. I remember my son's confusion (when he was 5). > You actually NEED this ambiguity early on in order > to > distinguish differences in context. It is a critical > and essential part of higher level reasoning that > should be embraced. > Bob Hansen > > ========== > > Thank you Dave and Bob. > > I suggest many parents would prefer the definitions > taught to their children, and especially algorithmic > definitions at that, be correct. > > Surely a definition is not a 'word problem' given by > teachers designed to be ambiguous in order to force > e parents and students to work out what it means? > > After saying ab = a added to itself b times, > mathematicians often write something like... > > ab = a + a + a + ... + a > .......\......................../ > ...............b times > > and this is where an ambiguity exists. > > Are there b additions or b addends? There can't be > BOTH b additions AND b addends as in this case, > addition is a binary operation acting on two > arguments. > > The binary + sign results in n1 additions for n > addends in multiplication. > > In multiplication ab, the multiplicand a appears as > an addend b times. In exponentiation a^b the base a > appears as a factor b times. > > A twostep pedagogical approach for the conjunct > operation of multiplication may be... > > STEP 1: MULTIPLICATION OF A, B TIMES > ab = a a a a ... a > > STEP 2: QUANTIFICATION OF AB > > ...........b addends of a > ......./...........................\ > ab = a + a + a + a +... a > .......\.........................../ > ..........(b  1) additions > > Similarly in exponentiation, there are n1 > multiplications for n factors. > > .............b factors of a > ........./.........................\ > a^b = a x a x a x a x... a > .........\........................./ > .........(b  1) multiplications > > Some mathematicians say 2^3 is two multiplied by > itself three times while others say 2^3 is two > multiplied by itself two times. It can't be both. Is > there a risk this situation might make mathematicians > look stupid? > > The above diagrams are crudely drawn, yet you can > imagine the horizontal braces { } above and below the > depictions of ab and a^b. > > You can then move onto linear growth from zero and > exponential growth from 1 in subsequent lessons. > > ab = 0 + a + a + a + a + ... + a for b a's > > and > > a^b = 1 x a x a x a x a x ... x a for b a's > > Only when you model linear and exponential growth > from their additive and multiplicative identities do > you have the same number of binary operations as you > have arguments. In both cases the a still appears, as > an addend in multiplication or a factor in > exponentiation, b times. > > With the use of the identities, you reveal the > essence of multiplication (when defined via repeated > addition) and exponentiation (when defined via > repeated multiplication) especially with respect to > time. > > In short, mathematicians must get definitions right > so the pedagogy can follow. Using illdefined > definitions such as ab = a added to itself b times > just confuses children and their teachers. > > All the above supports the case for math professors > (who should both know better and lead by example) > being sent to the naughty corner for repeating, > without thinking, ab equals 'a added to itself b > times'. > > Mathematicians often hold a traditional view that > definitions are to be accepted without proof on the > assumption somebody would have checked > > Yet for practical arithmetic it is reasonable to > reject a definition that leads to contradictions. > > A single counter example proves a theorem false. It > follows from an axiom of real arithmetic that 1 x 1 = > 1. > > >From ab = a added to itself b times, 1 x 1 = 1 + 1 > i.e. 1 added to itself 1 time, but 1 + 1 = 2 so 1 x 1 > = 2 which is false, so the theorem ab = a added to > itself b times is false. > > I believe the logic of all the above is correct. I'm > sure people will let me know if I am mistaken. > > If you can't accept the definition of multiplication > 'ab = a added to itself b times' is false, might you > at least all agree this definition of multiplication > is not well defined? > > Thank you > Jonathan Crabtree
Message was edited by: GS Chandy

