Why aren't mixed ratios allowed in Euclid's Elements ?

[prosteve037]

Why aren't mixed ratios allowed in Euclid's "Elements"?

Why did Euclid forbid the comparing of different kinds of magnitudes? And was it the same for numbers? Or were ratios specifically meant for magnitudes (lines, planes, solids)

What are the consequences of comparing 2 different kinds of magnitudes? I mean, we eventually figured out that the concept of velocity and rate change could be useful but why did it take so long to realize this?

 

[HallsofIvy]

They simply don't make sense. What sense does it make to say that "x meters" is larger or smaller than "y square meters"? Or that "x seconds" is larger or smaller than "y kilograms"?

 

[SteveL27]

They simply don't make sense. What sense does it make to say that "x meters" is larger or smaller than "y square meters"? Or that "x seconds" is larger or smaller than "y kilograms"?

I'd rather have a million dollars than 3 apples.

But then you could argue, well, dollars and apples are not the same but yet they can be converted to each other via the market price, which is well known.

Then I would argue that everything can be converted to everything else via market value. So Euclid was wrong. What do you think about that?

I can't think of any two (physical) things that can't be compared to each other. Would you rather have six cars or three cans of tuna fish? Easy comparison, right?

But even mathematically you can make the same argument. In the theory of order, we can put a partial order, a total order, a well order, etc. on any set. There's no requirement that the elements of the underlying set have to be "like each other."

So I'd say that Euclid is wrong in every day life and also mathematically.

 

[AlephZero]

I'd rather have a million dollars than 3 apples.

Not if they were Zimbabwean dollars, which after several devalations ended up worth $10^{-25}$ of their original value.

Maybe Euclid was right to leave "market forces" out of pure math

http://en.wikipedia.org/wiki/Zimbabwean_dollar

 

[Mandlebra]

Why did Euclid forbid the comparing of different kinds of magnitudes? And was it the same for numbers? Or were ratios specifically meant for magnitudes (lines, planes, solids)

What are the consequences of comparing 2 different kinds of magnitudes? I mean, we eventually figured out that the concept of velocity and rate change could be useful but why did it take so long to realize this?

The greeks viewed everything very geometrically. What does it mean for area to be "more" than a line? A volume and an area? A volume and a line?

 

[prosteve037]

The greeks viewed everything very geometrically. What does it mean for area to be "more" than a line? A volume and an area? A volume and a line?

They simply don't make sense. What sense does it make to say that "x meters" is larger or smaller than "y square meters"? Or that "x seconds" is larger or smaller than "y kilograms"?

Ah okay, I see. It seems like the Ancient Greeks made it very clear in keeping magnitudes separate from numbers; they kept physical quantities separate from numbers and instead represented them with magnitudes (figures). Why did they do this though? What makes numbers so special that they felt the need to segregate them from representing other things besides quantities?

 

[prosteve037]

... Unless I'm mistaken and they did in fact represent physical quantities with numbers?

But if this was the case, why then did it take so long for mixed ratios like velocity and/or rate change to be accepted as viable comparisons? This is why I assumed that physical quantities were identified with different "types" (and thus as magnitudes instead of numbers).

 

[mathwonk]

what are you talking about? Have you read Euclid? Can you give me a citation of what you claim?

 

[prosteve037]

what are you talking about? Have you read Euclid? Can you give me a citation of what you claim?

Unfortunately I have a very fragmented understanding of the very little I've read of the Elements. So basically, no I have not Forgive me!

I guess what I'm really asking is how concepts that involve mixed ratios (like velocity and density) came to be when their Euclidean roots forbade such notions. I always assumed (which I know is very nonsensical) that Classical Mechanics used Euclidean Geometry as its mathematical foundation to describing mathematical relations; so I guess I just associated this to how Euclid wrote the Elements.

Secondly, and if this assumption is correct, I was curious as to how this may have affected the discoveries of physical relations. If the physicists of the past had compared unlike quantities before/instead of like quantities, would it have affected their discoveries? I'm sure the implications for this scenario would be few and far between, having little effect on what we know today, but I was just in the mood for some opinions and discussion