When is it Appropriate to Apply Mathematical Operations to Objects?

[sigurdW]

When we join sticks together beside each other we get a good picture of addition.

Let "l" symbolize a stick then l+l=ll illustrates 1+1=2

But if we join sticks together to form one long stick, the joining has not the qualities of addition.

Are there guidelines when we apply mathematical operations to objects?

I thought this was a philosophical question but the thread got closed.

If the question on how we can apply numbers to reality in a safe way is not philosophical

nor mathematical (I predict this thread gets closed as well) then what kind of question is it?

 

[HallsofIvy]

When we join sticks together beside each other we get a good picture of addition.

Let "l" symbolize a stick then l+l=ll illustrates 1+1=2

But if we join sticks together to form one long stick, the joining has not the qualities of addition.

I don't understand what you mean by this. What "qualities of addition" does this not have? I don't see why joining many sticks should be any different from joining two.

Are there guidelines when we apply mathematical operations to objects?

Well, you don't, strictly speaking, "apply mathematicsl operations to objects". We can, and I believe this is what you mean, represent what we are doing to objects as mathematical operations as long as we know what we are doing has all the "qualities" of the mathematical operation.

I thought this was a philosophical question but the thread got closed.

If the question on how we can apply numbers to reality in a safe way is not philosophical

nor mathematical (I predict this thread gets closed as well) then what kind of question is it?

 

[Mark44]

When we join sticks together beside each other we get a good picture of addition.

Let "l" symbolize a stick then l+l=ll illustrates 1+1=2

But if we join sticks together to form one long stick, the joining has not the qualities of addition.

Sure it does. When you join one stick to another, the length of the new stick is equal to the sum of the lengths of the sticks that were used to make it.

In your first example, you are counting sticks. In your second example, you are adding lengths.

 

[Mentallic]

I've once had someone arguing that maths is flawed because if you add 1 raindrop to another raindrop, then you still have 1 raindrop, thus 1+1=1.

 

[sigurdW]

I don't understand what you mean by this. What "qualities of addition" does this not have? I don't see why joining many sticks should be any different from joining two.

I am not sure answering your question won't get the thread closed...But ok: If you join the sticks together into ONE stick then some silly fool might say: Oh look! One stick added to another here results in one stick, twice as long naturally, but still ONE stick so if we don't care about length then 1+1=1, 1+n=1 and generally ax+bx=1 for joining sticks this way...
We can't allow such nonsense, can we? He is misapplying maths. What should we tell him?

Suddenly the rain drop argument drops by...Math isn't flawed in itself, but mustnt some care be taken when you apply mathematics? And I sort of asks: What care? (Join two rabbits and they might not form a constant sum.) Can we for (imagined or not) objects be sure in advance that joining them will conform to addition? Or must we always test?

The first time I thought about the game of "interpreting" the equation: x+x=nx for n not being zero or two, was while reading on transfinite numbers. If the rule of addition for them are their union then one infinite set + another infinite set (of same cardinality) would be an example of n=1...so to make things consistent we define things to be not so. A question here is if we perhaps are over reacting?
A possible example? :Objects classified into four cathegories depending on their having outside and inside or not...Maybe their characteristics give different values for n? Remember this is not claimed to be so! Its mostly a joke:

1 Real objects having both (n=2)
2 Elementary objects lacking insides(n=2) elementary particles
3 Inclusive objects lacking outside(n=1)? the everything, the universe
4 Imaginary objects lacking both (n=0) the nothings

 

[arildno]

Im not sure answering your question won't get the thread closed...But ok: If you join the sticks together into ONE stick then some silly fool might say: Oh look! One stick added to another here results in one stick, twice as long naturally, but still ONE stick so if we don't care about length then 1+1=1, 1+n=1 and generally ax+bx=1 for joining sticks this way...
We can't allow such nonsense, can we? He is misapplying maths. What should we tell him?

Suddenly the rain drop argument drops by...Math isn't flawed in itself, but mustnt some care be taken when you apply mathematics? And I sort of asks: What care? (Join two rabbits and they might not form a constant sum.) Can we for all imagined objects be sure in advance that joining them will conform to addition? Or must we always test?

The care follows when you get observant about the (essentially arbitrary) laws defining some mathematical version of "addition".

For those real world cases where those laws seem to apply as well, then that particular addition operation can be expected to work.

You can, of course, make other types of mathematial "addition" than the usual one.

 

[Number Nine]

What should we tell him?

That addition is clearly the wrong model for what he wants to do. Choose something else.

 

[sigurdW]

Thank you guys, I like your answers.

 

[Curious3141]

I've once had someone arguing that maths is flawed because if you add 1 raindrop to another raindrop, then you still have 1 raindrop, thus 1+1=1.

That's actually not flawed. It's just Boolean Math. 1 raindrop AND 1 raindrop = 1 raindrop!

 

[Mentallic]

That's actually not flawed. It's just Boolean Math. 1 raindrop AND 1 raindrop = 1 raindrop!

Haha I did explain to him that there exist other systems (or you can at least create one) that deals with such problems, but it was fruitless.

It just depends on your audience. For example, I've never really felt compelled to say "oh sure, you can divide by zero... in the projective real line".

 

[chiro]

One easy interpretation is to think of different sticks as orthogonal variables and the legth of the stick as the value of a particular variable. So two different sticks are x=1,y=1 and adding one stick to another is setting x = 2.

 

[DaveC426913]

I've once had someone arguing that maths is flawed because if you add 1 raindrop to another raindrop, then you still have 1 raindrop, thus 1+1=1.

Amusing.

But ask him to rigorously define "raindrop". Make him provide properties such as volume. Then he'll see.