Is Multiplication Really Just Repeated Addition?

[disregardthat]

Yes, division can be a wholly constructive operation (namely, repeated subtraction) but only because a further "natural" step has been taken in breaking the symmetry of the number line by choosing a base 10 numbering system.

You should explain what you mean by "breaking the symmetry" of the number line. Real numbers are not defined as "what makes up the number line" if that is what you are driving at.

So it seems to me that an extra geometrical argument has been introduced at this point. Whereas the numberline is a linear additive concept, we are now laying over the top of it a geometric expansion which gives us "counting in orders of magnitude and decimal scale".

The number line is a "linear additive concept"? What does that mean? The definition of real numbers is not referring to the number line. The number line is used as an analogy or intuition of real numbers, as it e.g. captures the geometric interpretation of the intermediate value theorem.

Now of course I am sure people will say they see no issue here because all the points on the numberline exist. So 1.3, or pi, are just as natural as entities as 1,2,3.

Why would there be an issue? You should explain that first.

But that is what I am musing about. Some extra constraint appears needed to break the naive symmetry of the numberline. The challenge was to connect something that is essentially discrete (a string of points) with what also had to be essentially continuous (a line) and breaking the scale of counting in this way, using a base as a further constraint, seems like the way it has been done.

What symmetries are you talking about, and how are they broken? A base representation is not a constraint (or what do you mean by that?), it is just what it is called: a representation.

The numberline is founded on the notion of "one-ness". And that is a symmetric or single-scale concept. But as soon as you introduce an asymmetry, a symmetry-breaking constraint - such as any base system starting even from base 2 - then there is something new. A connection is forged between the original point-like discreteness and the continuity implied by a numberline. Scale is broken geometrically over all scales. Allowing then measurement down to the "finest grain".

You should describe what you mean mathematically. "One-ness" is meaningless as a mathematical "foundational notion" as you put it, unless you describe mathematically what you mean by it.

 

[Hurkyl]

Try being helpful. And if you don't wish to be, simply don't respond.

It really is good advice. It's way too easy to be a never-ending fount of nonsense if you never even try to connect your thoughts back to mathematics.

It would also make my job easier and more fun -- it would mean fewer threads I have to step in and moderate, and more threads that I might enjoy helping someone learn something or formulate their own ideas.

(For the sake of clarity -- yes, this thread is nearing the point where I would lock it. On another day I might have locked it already)

 

[apeiron]

...more threads that I might enjoy helping someone learn something or formulate their own ideas.

You say that, but you aren't giving me the slightest help in connecting my prose to your maths.

From the start, I simply pointed out the bit of the OP that intrigued me and asked about how that would be handled. I was expecting someone to say, that is just the xyz conjecture, or whatever, a very standard issue you can go google to discover more.

The only response slightly like that was Studiot's, and it took several goes and a few insults to discover something that indeed gave me a better insight.

If I could package up my own thoughts in terms that you would consider properly mathematical, of course I would. But I still feel the gist was clear enough.

And when I did make further effort to explain myself more clearly that your response - "I can't guess what you mean from this description." - just appears childish and mean spirited.

It's your choice to lock the thread. Just don't insult me anymore by pretending that you have made an effort, whereas I have not.

 

[brydustin]

Of course there will always be much said on something so ridiculous as a post like this... everyone wants to give their two cents and anyone can. With that said...
Multiplication IS repeated addition for children because they are children.

 

[Hurkyl]

Multiplication IS repeated addition for children because they are children.

I don't think you give children enough credit -- they can form abstract concepts ideas too. In fact, I understand they're generally better at it than grown-ups.

Of course, if you drill a child into mentally substituting "repeated addition" whenever he sees a multiplication symbol, that would make it very difficult for him to mentally form the notion of multiplication being an operation in its own right.

 

[coolul007]

The correct answer is: multiplication defines area. It can be an area Pi x e, or .4 x .005

 

[TylerH]

The correct answer is: multiplication defines area. It can be an area Pi x e, or .4 x .005

How do you know area (of a rectangle in R2) doesn't define mulitplication?

 

[Robert1986]

The correct answer is: multiplication defines area. It can be an area Pi x e, or .4 x .005

OK, multiplication defines area. Big whoop! Read the title of the thread; read the posts in the thread. We're not discussing what multiplication defines; we're discussing what defines multiplication. If you mean that multiplication is defined as area, then you are very wrong. At best, your "definition" works when you multiply two real numbers. When you do three, you are (in your terms) talking about volume. Then, what about when you multiply complex numbers? Or, what if you are not multiplying numbers at all? What if you are dealing with matricies? Or composition of functions?

 

[Stephen Tashi]

Try being helpful. And if you don't wish to be, simply don't respond.

If you think you can become the thread controller, you're dreaming.

 

[coolul007]

OK, multiplication defines area. Big whoop! Read the title of the thread; read the posts in the thread. We're not discussing what multiplication defines; we're discussing what defines multiplication. If you mean that multiplication is defined as area, then you are very wrong. At best, your "definition" works when you multiply two real numbers. When you do three, you are (in your terms) talking about volume. Then, what about when you multiply complex numbers? Or, what if you are not multiplying numbers at all? What if you are dealing with matricies? Or composition of functions?

Well maybe, I should have said area/volume/? defines multiplication. Functions define a result, geometrically, whether in hyperspace or Euclidean planes. Matrices define multidimensional space. All are consistent with the definition.

 

[Hurkyl]

How about just saying "multiplication and area are related by the fact that the product of the lengths of two sides of a rectangle is equal to the area of the rectangle", rather than dogmatically trying to assert one defines the other?

 

[coolul007]

How about just saying "multiplication and area are related by the fact that the product of the lengths of two sides of a rectangle is equal to the area of the rectangle", rather than dogmatically trying to assert one defines the other?

I wasn't being dogmatic, I was trying t answer the question/discussion with a consistent definition that would work for the original poster. Repeated addition is taught as it is a convenient way to describe to a 7 year old what is taking place. We don't always teach things in the correct way, so that we don't cause inconsistencies down the road. I am a fan of teaching primes and prime factorization at an early age. That will ease most elementary problems that children struggle with, I.E. fractions/rationals, however, that won't happen as our elementary educational system is not enlightened enough, but I digress.

 

[TylerH]

Does the origin of a function have any bearing on what we say its definition is? If it does, I think we could say that multiplication is defined as repeated addition in N, with extensions for everything else. I doubt the cavedudes were thinking pi * e when they invented/discovered multiplication.

 

[Stephen Tashi]

Neo-Platonisitic discussions certainly become contentious! When threads are posted along the lines of "Is .9999 =1?", "Are infinitismials nonzero?", "Is sqrt(-1) an actual number?" they do well as light mathematical chit-chat. Everyone gets to express their own personal intuitions. I think it's interesting to put the real mathematical definitions of things aside and hear about how people imagine things.

However, there are always some participants that take the topic seriously and insist that they are the ones who see the THE TRUTH. Formal mathematical definitions may be dull, but after reading attempts at discussing math as serious personal philosophy, I begin to appreciate them.

 

[Hurkyl]

I think it's interesting to put the real mathematical definitions of things aside and hear about how people imagine things.

I actually view this as a sort of trick question. I've always viewed one of the greatest strengths of mathematics is the ability to seamlessly flow back and forth between 'pictures'.

If someone answers the question "how do you imagine multiplication" with something like "I imagine it as talking about area!", that's a bad thing -- they've limited themselves to one particular 'picture'.

This person will probably have success at applying multiplication to area problems, and have some success applying area to multiplication problems.

However, this person will also face unnecessary difficulty in applying multiplication to problems that aren't related to area, or converting multiplication problems into other sorts of problems.

 

[coolul007]

I actually view this as a sort of trick question. I've always viewed one of the greatest strengths of mathematics is the ability to seamlessly flow back and forth between 'pictures'.

If someone answers the question "how do you imagine multiplication" with something like "I imagine it as talking about area!", that's a bad thing -- they've limited themselves to one particular 'picture'.

This person will probably have success at applying multiplication to area problems, and have some success applying area to multiplication problems.

However, this person will also face unnecessary difficulty in applying multiplication to problems that aren't related to area, or converting multiplication problems into other sorts of problems.

What I find interesting, many of you have tried to turn this into a personal issue instead of a mathematical one. It is not about people it is whether the 'definition" fits the problem. If it does use it, if it doesn't, find a better one.

 

[Stephen Tashi]

I actually view this as a sort of trick question. I've always viewed one of the greatest strengths of mathematics is the ability to seamlessly flow back and forth between 'pictures'.

If someone answers the question "how do you imagine multiplication" with something like "I imagine it as talking about area!", that's a bad thing -- they've limited themselves to one particular 'picture'.

I agree that having a variety of pictures is a good thing. I disagree that there are any serious consequences from the way people answer questions like this. For one thing, the way that people imagine things probably isn't as consistent as the way they answer questions about their imagination. I can't imagine this type of thread damaging the career of any budding mathematicians. If they have the talent to do math, they will enjoy these discussions in the way that people enjoy discussions about "What's your favorite beer?" or "Who's your favorite author?".

 

[hillzagold]

Won't someone think of the children?

Ahem. The author in the column from the initial post had to make 5 separate entries explaining to emailers who disagreed with him. I myself initially disagreed with him and believed that multiplication was repeated addition and exponentiation was repeated multiplication. When I had to do 2x3.3, I broke it up into 2 + 2 + 2 + .3x2, and went my whole life. But reading Devlin's column made my realize that I was defining multiplication as repeated addition, and was defining this instance of repeated addition using multiplication! I was pretty confused until Devlin explained a key difference being multiplied values can have different units, such as kilowatts and hours, and I could finally accept multiplication as it's own distinct operation.

As one poster previously mentioned, seeing multiplication and exponentiation as repeated addition made understanding i²=-1 completely impossible for me. It's presumably still impossible for at least 95% of America. But some posters argue that it's necessary to that children can master multiplication at all. May I propose we try and make a pros and cons list?Pros:
This is the status quo, at it is what almost the entire country, including many of the teachers, honestly believes.
It's apparently easier to learn and understand, to those simple 1st grader minds. (?)
It can be retaught properly later, maybe in middle school or high school or even college. (?)

Cons:
It is mathematically flawed, according to any mathematician you can find who speaks your language.
It makes more advanced concepts, from fractions up to complex numbers and beyond, more confusing.(?)
It actually can't be retaught properly later, with a dependable success rate. (?)

Would anyone like to contribute to this list in any way?

 

[coolul007]

I think of the children, being taught one thing and then having to "unlearn" it later in favor of a better concept later, is the confusing part. And what if the better concept never comes. The state of lower education is not in favor of the rigor of mathematics, but some "touchy feely" form to make the children have a false sense of confidence. Mathematics is a system that does not have a lot of "stand alone" concepts. (I will get a large reaction for that last statement) I tutor high school students, the part most of them are missing is the foundation for understanding the concepts being taught to them. The modern trend is also to rename a lot of concepts, some good some bad, fractions, rational numbers, etc. Vocabulary and English should be taught in that class not mathematics.

 

[Studiot]

Won't someone think of the children?

Motivation is all.

Of course this applies to all ages, not only children.

It is easy to demotivate.
Just teach a list of arbitrary definitions and statements, really designed to keep pupils quiet and sitting up straight.

No go down the pub and ask a darts player who has just scored 347 and who flunked school, what he need to finish and I guarantee he will know more quickly than any 5 maths professors put together.

 

[coolul007]

The dole is always a counterexample to staying in school...

 

[Studiot]

The dole is always a counterexample to staying in school...

That's pretty disrespectful of the young.

The average youth unemployment rate (through no fault of their own) currently stands at 25% and reaches over 50% in some areas.

Those who do get jobs do so, not on their academic record but by knowing the rich and powerful, if recent news articles are to be believed.

go well

 

[dalcde]

I think that multiplication is the unique operation which satisfies the distributive property, and, when applied to integers, is equivalent to repeated addition.

 

[agentredlum]

try to multiply e by pi. e from exponential 1 and pi = 3.14...you'd see it's not a repeated addition at all.

I think it is...add e to itself 3 times, then add some more 'stuff'