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[Antone]

ORIGINATOR'S NOTE: This thread can be seen as a continuation of my thread:
Is 0 an exception to the recipr
If it were possible to post two separate posts, I'd have posted this in that thread. And I would encourage those who respond here to look at that thread as well.
However, I think this topic can also stand alone, as it's own topic, even if you haven't read that thread.

The Nature of Multiplication:
[Addition] and [subtraction] are functions where you manipulate LIKE things to get a LIKE [sum] or [difference].
For instance you might add [apples] to [apples] to get the [number of apples].

By contrast, [multiplication] and [division] are functions where you manipulate DIFFERENT things to get a DIFFERENT [product] or [quotient].
For instance you might multiply [rows] by [columns] to get the [number of chairs]. [Rows] are different from [columns], and both are distinctly different from [chairs]. Thus, there are three different and distinct units involved in a multiplication problem.

This is rather obvious when it comes to division, but I think it's worth explicitly pointing out that multiplication has the same basic structure, with respect to the three different units.

Multiplying [10 rows] by [10 rows] does not give you a meaningful number, because there's no way to know how many 'chairs' are in the rows.

The [row term] needs a reciprocal aspect,

I define a reciprocal aspect as: something that is both alike and different. For example, in our current discussion [rows] are reciprocal to [columns] because they are alike in that they both contain chairs; but they are also different in that they are laid out perpendicular to one another.

Because the terms are reciprocal to one another, we can use them to create a grid-like structure. All multiplication problems necessarily involve such reciprocal aspects. For example, [MPH] is how fast you're going. But it is determined by multiplying [distance] by [time]. Distance and time are alike, because they define the nature of space--but they are clearly different as well. And neither resembles speed, (i.e. MPH).

If you keep this [reciprocal structure] in mind during physics class, it can often give you additional insight into which things are related and how--just observe which things are multiplied together. They will usually be reciprocal in nature, with respect to the product they produce.

Dividing by 0:
The above fact plays into my discussion (on the other thread) of Dividing by zero. Consider:

Multiplication
[1 row] x [2 columns] = [2 chairs].
Division
[2 chair] ÷ [1 row] = [2 columns]... 2÷1=2
[2 chair] ÷ [2 columns] = [1 row]... 2÷2=1

Now, lets consider multiplying and dividing by [0].

Multiplication
[2 row] x [0 columns] = [0 chairs].
Division
[0 chair] ÷ [1 row] = [0 columns]... 0÷1=0
[0 chair] ÷ [2 columns] = [0 row]... 0÷2=0

At first glance, this might seem to imply that you can't [divide by 0], but I believe a philosophical argument can be made that you can. You just have to convert what it is you are talking about from the real world to the conceptual world. How does this happen?

First,notice that in the multiplication problem

[2 row] x [0 columns] = [0 chairs]... 2x0=0

The chairs in the [2 rows] are necessarily imaginary. There aren't any [columns], so there can't be any [actual chairs], but the unit of measure still has to be chairs--else the rows and columns wouldn't share a like aspect, and the multiplication problem wouldn't represent the necessary grid structure.

This obviously implies that the chairs in the [2 rows] are imaginary chairs. Thus, if you think of the product
in terms of [imaginary chairs] instead of [actual chairs] then the answer to

[2 row] ÷ [0 columns] = [n chairs]... 2÷0=n

can be any number at all. Why?
Because the [number of imaginary chairs] in the [2 rows] has not been specified by a [non 0 number of columns].

But a single concept can represent any number of actual objects. For example, the concept [chair]is exemplified by any actual chair. This was true when the first chair was built--at which point there was only [1 actual chair] and it will be true no matter how many chairs are eventually built before time runs out. That single concept, stands for any and all of those chairs, past present and future.

Thus, if we let [n] equal any number, we can rewrite the multiplication problem as:

[2 row] x [0 columns] = [n imaginary chairs]... 2x0=n

And the division becomes

[n imaginary chairs]÷[0 columns]=[2 rows]... n÷0=2

And, since it wouldn't matter how many rows were involved, the multiplication answer would still be [n imaginary chairs], the [2 rows] answer is arbitrary. So we can replace it as,

[n imaginary chairs]÷[0 columns]=[n rows]... n÷0=x

Where [x] is any specific number.
Thus, if we replace [n] with a specific number, we can get (for example)

[1 imaginary chairs]÷[0 columns]=[1 rows]... 1÷0=1

What the row contains is an unspecified number of imaginary chairs. But no matter how many instances of an 'imaginary chairs' the row contains, they are the same single concept: [an imaginary chair]. Thus, [1=n]. Which is why you can "prove" using 0 in the denominator that 1=2 (or any other number you chose).

This is all a bit confusing, which is why it is best to leave [0] out of the denominator position when performing math. But (at least at a philosophical level) I don't think it is entirely accurate to say that [0 in the denominator] is undefined. It's just that, unlike other numerical equations, it can be given several possible interpretations.

[Mishra.av]

Why should the addition and subtraction involve similar types e.g. 3 apples+4 apples =7 apples.
What about 2 cats + 3 dogs = 5 animals
or
5men+3women+5children=15 persons

[Belinda]

Or four rows of five each multiplied by five rows of three each?

[Antone]

What about 2 cats + 3 dogs = 5 animals
or
5 men + 5 women + 5 children = 15 people

When you add [2 cats] + [3 dogs] = [5 animals], what you are really doing is adding
[2 animals--who happen to be cats] plus
[3 animals--who happen to be dogs].

What you are adding are animals all along. We can do this on the fly, (without explicitly stating the obvious facts) because we know that both cats and dogs are (by definition) animals. And so when we make the transition from [cats + dogs] to [animals] it can be done at an intuitive level. But we are still counting like things: animals.

Again, what are [2 cats] + [3 nails]... We can always find something that any two groups share--perhaps [5 objects]. But again, if we do, we are counting objects... not cats or nails. And so again, what we are counting are necessarily similar.

If you still insist that you are right, then fine. Lets put all of our money in a pot. And I'll count out who gets what... and to show my enormous generosity I'll give you things at a ratio of [100 to 1]. i.e. I'll give you 100 [pennies] for ever 1 [$100 bill] I take. That's obviously fair, since it doesn't matter what we add together, right?

Of course, each cat is unique, as well. So in a sense, we are not adding 'like' things when we add cats either. They are only 'alike' conceptually--just as the [cats] and [dogs] are alike conceptually--in that they are [animals]. So it's all relative--a matter of degrees. And in the strictest possible sense, it is impossible to add anything--because there cannot be anything that is exactly identical to itself. Thus, there is only ever one of any given thing.

[Evilwill32]

Hmmm...this thread makes me uncomfortable. Physics student eh? I have not studied physics at a high level, but have a feeling you are attempting to generalise a concept we called 'dimensional consistency' to everyday objects such as apples. Also, it seems you are attempting to extend the concept of mathematical operations such as addition and multiplication, specifically which are performed on numbers, to concepts which are not included in this domain. Maybe it would be clearer if I elucidated this mroe clearly:-

When I think of multiplying a row of 10 chairs by a column of 10 chairs to get a 10*10 grid of chairs (i.e. 100) I go through 2 steps:-

1. (Mathematical): I perform the multiplication 10*10, which importantly acts over numbers (i.e. an arithmetic operation)

2. (Conceptual): I assign a meaning to this number 100 using my grasp of concepts of chairs, row and column

The thing I object to in your analysis is that you conflate 1. and 2. and attempt to perform them simultaneously. When we add 1 apple to 1 apple, we are perfomring mathematical operations with arithmetic and then add a visualisation of this in the real world (ie. physically having 2 apples). Hence the visualisation is only an aid for what's going on with the mathematics and is not the mathematics itself. For a better explanation, I suggest you have a look at "Goedel, Escher Bach" by Douglas Hofstadter, which specifically looks at the concepts of things like addition and multiplication from the viewpoint of formal systems, and how formal systems such as mathematics generate meaning through isomorphisms or similarities with the real world. It is specifically this real world - formal systems (mathematics) relationship that he examines in certain parts of the book, although I do believe the book is a must-read, despite its length.

wow. three mathematics posts. what's gotten into me.

[Antone]

Hmmm...this thread makes me uncomfortable. Physics student eh?

Most of your post makes me think that it is in reference to my posts--but this comment doesn't fit at all. I am NOT what I would call a physics or math student--except in the sense that these subjects interest me and I have done some reading. But virtually everything I know beyond basic algebra I learned by reading, studying and (most importantly) THINKING on my own. I have no high-level formal training in these areas.

I ... have a feeling you are attempting to generalize a concept we called 'dimensional consistency' to everyday objects such as apples.

First, thanks for using this term... I haven't heard it before, and will try to look it up to see what the "professionals" have to say on the subject. What it means seems pretty obvious, but I've learned that the obvious meaning is often not the meaning that is actually given to technical terms.

SECONDLY, (without actually knowing what the term means) I would guess that I agree with your assessment. And yes, I am saying that the principle (or a similar one) naturally applies to things in the non-mathematical realm.

When I think of multiplying a row of 10 chairs by a column of 10 chairs to get a 10*10 grid of chairs (i.e. 100)

Notice that you have set up the very same scenario that I suggested. You are multiplying two reciprocally opposite types of things.
1) rows of chairs, and
2) columns of chairs.

Like all reciprocal things, these are alike in that they both contain chairs. But they are opposites in that [rows] and [columns] are oriented at a right angle to one another. Thus, your example illustrates my very point.

If we remove the chairs from the context of rows and columns, and tried to multiply just [10 by 10] it would be a meaningless exercise of nonsense.

steps:-
1. (Mathematical): I perform the multiplication 10*10, which importantly acts over numbers (i.e. an arithmetic operation)
2. (Conceptual): I assign a meaning to this number 100 using my grasp of concepts of chairs, rows and columns.

I agree.

If I were teaching a child how to multiply, these are the steps I would teach them. But that doesn't mean that this is what we are actually doing. In fact, when we multiply [10 chairs by 10 chairs] what we are really doing is multiplying [10 rows by 10 columns]... One way to verify this is to show that it doesn't matter what we put into those rows and columns... for instance, it could have been [10 band members by 10 band members] giving us a band with 100 members. We're multiplying reciprocal things, but we call them by the aspect that they share... [10 chairs] or [10 band members] because this keeps what the final unit of the answer will be firmly in our mind. And, since the [rows and columns] aspect never changes, its easy to let it slip into the background and be overshadowed by the unit that will be reflected in the answer.

But here is my long version of the WHOLE process of finding the number of chairs in a 10 by 10 array.

1) We must ensure that no [row] or [column] is longer or shorter than the other. (If it is we can still do the math, but we must attach an addition or a subtraction element to our final equation.)
2) we observe the number of chairs in a column, [10]. This represents the number of rows in the array. One chair in the [column] for each [row in the array]. So we know we'll be multiply [10 rows by x columns]
3) we observe the number of chairs in a row, [10]. This represents the number of columns in the array. One chair in the [row] for for each [column in the array]. So we know we'll be multiply [10 columns by x rows].
4) Combining (2) and (3)... we multiply [10 columns by 10 rows] and use our memorized knowledge of the multiplications table to deduce that the answer is [100 x-units]
5) Because we know that both the columns and the rows contained chairs, we know that the [x-unit] is [chairs]. Thus, our answer is [100 chairs].

Most of this process we do at an intuitive level, without thinking or even realizing that we're doing it. We simply say, "Ten by ten chairs." and (as you suggested) we recall from the multiplications table that 10 by 10 is 100. So that's 100 chairs.

But the rows and columns aspects are still critically important--even if we are not consciously aware that we are relying on them. This is easy to understand: just try to multiply [10 chairs] that are not in an array by [10 chairs that are also not in an array...

What is the answer?

There is none. The question is not even mathematically meaningful.

We can attack this from another perspective, to show that units always matter. Suppose, for example, that I have four bills, {$1, $2, $5, $10}. I can put the first bill down and say "That's one" then put the second down and say, "That's two." I am right if I am referring to the number of [bills], but I am wrong if I am referring to the number of [dollars].

Again, when multiplying bills we must have reciprocal aspects. {$2, $2, $2, $2} satisfies this requirement for the [dollar unit] because have four [different bills] which are all the same [dollar unit]. Again, the unit that is the same is the unit that carries over to the answer, so when we multiply [$2 x 4] we get [$8].

By contrast, {$1, $2, $5, $10} satisfies the reciprocal requirement for the [bill unit] because we have four [different dollar units] which are all the same in that they are [all bill-units]. Again, the unit that is the same is the unit that carries over to the answer, so when we multiply [4 x 1 bill unit] we get [4 bills].

Again, all of this we do at an intuitive level. We don't have to think about what it is we are doing. AND (as is often the case) thinking too much about something that we do intuitively can interfere with our ability to do it. If you've learned to type (not hunt and peck) try to think about each letter that you type, instead of just thinking words. When you think each letter it slows you down and can force you to make more mistakes. This is because conscious thought interferes with the intuitive (or SUBconscious) process of typing.

I believe that I have exposed the intuitive process of multiplying to conscious thought--and because it is the first time you've thought about it, it makes you feel uncomfortable--for much the same reason that consciously thinking about each typed letter makes a typist uncomfortable and inefficient.

When we add 1 apple to 1 apple, we are performing mathematical operations with arithmetic and then add a visualization of this in the real world

I disagree, I believe that what you are relegating to a visualization aid is really the most crucial and fundamental part of the whole math process.

Yes, we can perform abstract math... [10 * 10 = 100], and we don't need to attach this equation to [chairs] or [band members]. But even though the math is abstract, it is still intuitively understood that (whatever we are adding) we are adding similar units. That's why [1] is referred to as the identity element; it defines what the 'unit' for all the other numbers are.

When we are doing abstract math, this is an arbitrary unit. It doesn't matter what that unit is--as long as it is the same. This is what allows us to insert any unit into a math equation. In essence, we are saying
[10x * 10x = 100x]. We can replace [x] with any unit we want. And because this is intuitively obvious, we again do not stop to think about it--we just do it.

But [2] is only [2] if it is a [second of what 1 was]. If [1] is an atom, we can't bring along a solar system and refer to their union as [2]--unless we define their collective units extremely vaguely, such as [2 things], etc.

I suggest you have a look at "Goedel, Escher Bach" by Douglas Hofstadter, which specifically looks at the concepts of things like addition and multiplication from the viewpoint of formal systems, and how formal systems such as mathematics generate meaning through isomorphisms or similarities with the real world.

I agree that Hofstadter's book is a good one... but nothing he says disagrees with what I've said. He doesn't necessarily spell it out like I do--but he doesn't contradict my premise either... so I'm not sure what your point is in bringing up his work.

The point of a formal system is to remove any interpretation from the system. All that can be employed are typographical elements, the rules for combining these elements, the rules for manipulating these elements, and the 'rules' or axioms that tell us which combination of elements we can start with.

Such a formal system has no connection at all with the real physical world--severing that connection is the whole point of creating the formal system in the first place. Nor do we use a formal system when we perform a math problem--even if we're doing abstract math.

In the formal system itself, the units are built into the system. For example, [1] may be defined as ['] and [2] may be defined as ['']. [2] would NOT be defined in such a
case as ['#], because [#] is a new and different unit, and so it would render the meaning of [2] inconsistent and meaningless. also, there would be no way of predicting what [3] should be. Should it be ['##] or ['#'] or any number of other variants. There's no possible way to predict. But when [1] is ['] and [2] is ['']... then it becomes fairly obvious that [3] must be ['''], and so forth.

Thus, the unit for each number is [']--exactly what [1] is--again, that's why they call [1] the identity element. If we know what [1] is, we know what the unit of all the other numbers must be.

[Belinda]

You can do multiplication sums on identical things, not on things that are dissimilar, as I expect Antone would agree.

A small child takes it for granted that when the teacher or parent says 'these are all apples' that they are all the same sort of things, and differ from other things. The small child therefore begins to findout what attributes the so-called apples have that make them all the same things. The attributes that the child discovers that apply to apples, say hardness, edibility, sweetness, having a core etc, differentiate the apples from lemons.

These attributes are judged by human cultures of beliefs to be significant attributes. For instance some exotic culture might group lemons and apples together and be able to do addition and multiplication with what our culture calls different fruits. The attributes of any set of objects are arbitrarily defined by the culture and either don't inhere in the objects, or if they do inhere in the objects the attributes are selected for their significance for the humans who do the selecting.

[Antone]

You can do multiplication sums on identical things, not on things that are dissimilar, as I expect Antone would agree.

To do any math function (addition, multiplication, subtraction, division) the things being manipulated must be both the same and different. For example, apples are conceptually the same because we have (by cultural convention) defined them to be the same. But they are different in that they are not identical in size, shape, color, flavor, etc.

For addition, this is all that is required. But Multiplication must always creates an array--and every array has at least 2 dimensions. So in addition to the reciprocal nature of the objects themselves (that was required in addition) the math process itself must have a reciprocal characteristic that must be satisfied. The array must contain conceptually similar objects, but it must also have columns and rows, etc.

In physics, we find the same thing, but the reciprocal nature of the things being multiplied is often more vague. For instance, we can state how fast we're going in MPH. We get this number by multiplying [miles x hours] Miles are a measure of distance (or space); hours are a measure of time. Space and time are clearly reciprocal--as Einstein demonstrated. More specifically, in this example, they are both used to define the travel from [point A] to [point B]. One measures the movement between these points with respect to space and the other measures it with respect to time.

Now, which unit of measure do we keep? Both aspects are part of the space-time continuum. So we must keep both units of measure, which is why both (SPACE and TIME) units show up in the answer MILESperHOUR.

This is one of the more obvious examples, but I would suggest that any physics formula (that contains x) is based on a similar reciprocal relationship. But keep in mind that any given thing has many characteristics--and to meaningfully multiply any 2 things we only need to find two of those characteristics: [one which they share in common] and [one which they hold in opposition].

I challenge anyone to offer up such a physics equation that doesn't utilize two reciprocal aspects.

In fact, I think that physicists (subconsciously?) use my RULES to create their formulas in the first place. They observe the relationships that nature has, and then they write them down. Which explains why some people are so good at finding such relationships--and others are not. Those who intuitively understand the [relationships] and the [rules] are the best at finding and expressing such relationships in mathematical formula.

some ... culture might group lemons and apples together and be able to do addition and multiplication with what our culture calls different fruits.

True. Real life examples include: Eskimos, because they live in a land of snow and ice (and the quality of the snow is important to them)--have something like 42 names for different types of snow. And Some primitive cultures only distinguish between three distinct colors--grouping all other together under the same names.

The attributes of any set of objects are arbitrarily defined by the culture and either don't inhere in the objects, or if they do inhere in the objects the attributes are selected for their significance for the humans who do the selecting.

To a certain extent, I agree. But things do have certain characteristics independent of our conceptualization of those characteristics--otherwise physicists couldn't come up with their formulas. Which appear to be universal across cultural boundaries.

[Belinda]

Antone wrote:

For example, apples are conceptually the same because we have (by cultural convention) defined them to be the same. But they are different in that they are not identical in size, shape, color, flavor, etc.

For the purpose of computations we arbitrally abstractan attribute or set of attributes. Therefore the size, shape, colour, flavour of the apples are irrelevant attributes of the set unless we decide that any of these attributes be relevant.If e.g. shape of the apples were to be relevant we would compute only with apples that fitted specified shape parameters.
Shape of apples is probably relevant for growers who supply supermarkets that want to attract choosy cutomers, but shape of apples in a sample that the primary school teacher presents as a set to the children is probably not an attribute that is of interest to the children whose concept of apples for the purpose of computing them is less choosy than that of supermarkets.

I can picture some exotic culture in which any fruit that is green is deemed neither apple not pear but a sacred object and tabu to eat.In some other exotic culture, for another instance,leafy vegetables that have not been washed are deemed to be animal flesh. What these and similar imaginary and real examples show is that categories are not defined by any essential attributes but are concepts pertaining to social groups and are mediated through the social group's language.

For example, apples are conceptually the same because we have (by cultural convention) defined them to be the same.

But the same mathematical methods can be used by all social groups who are inducted into this useful tool of abstraction.' Abstract ' implies not culture-bound.

[Evilwill32]

theres one thing i want to ask before i go any further with this and it would help to be honest. you seem to mention the word 'reciprocal nature' a lot and say that they are your ideas. to what extent are your conjectures influenced by the work of Dewey B. Larson?

[Antone]

theres one thing i want to ask before i go any further with this and it would help to be honest. you seem to mention the word 'reciprocal nature' a lot and say that they are your ideas. to what extent are your conjectures influenced by the work of Dewey B. Larson?

I have read only a little of Larson's work--and I find it interesting. But what little I am familiar with (of his ideas)does not seem to be all that similar to my ideas. They are roughly compatible, I would say--much as multiplication is compatible with division, but they don't have a whole lot to do with each other beyond that, other than being parts of math of course.

In any case, I developed my ideas well before reading Larson's work--and (as I recall) it was (in large part) the "reciprocal" term that was largely responsible for attracting my attention to his work. I was hoping his work would have more in common with my own than it did.