from geometry to units

[michel123456]

This is a question:

Is it possible, starting from a geometric description that uses only proportionality, to construct a system that ends up with some kind of unit, like kilogram, meter, second or ampere?

Intuitively, I would say no.

[Fuzzwood]

Then again, the kilogram was defined by taking a "cube" of water with edges of 10 cm.

[md65536]

This is a question:

Is it possible, starting from a geometric description that uses only proportionality, to construct a system that ends up with some kind of unit, like kilogram, meter, second or ampere?

Intuitively, I would say no.

SI defines some dimensionless quantities like "Plane angle" with a unit of radians. http://en.wikipedia.org/wiki/Physical_quantities

Radians are defined geometrically.

 

So yes, it's possible to end up with some kind of unit, but I don't think it's possible to end up with the units that have dimension, like the ones you mention.

I suppose that deriving physical units from pure numbers might perhaps result in a Theory of Everything???

 

 

 

See also http://en.wikipedia.org/wiki/Dimensionless

 

 

[granpa]

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

 

my favorite is stoney units

 

http://en.wikipedia.org/w/index.php?title=Stoney_units&oldid=364213115

[michel123456]

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

 

my favorite is stoney units

 

http://en.wikipedia.org/w/index.php?title=Stoney_units&oldid=364213115

 

Stoney units, very interesting.

 

From what I see:

from geometry you can derive ratios. Like Pi ([math]\pi[/math]).

Radians, as proposed by Md, are ratios.

 

I suppose from geometry one can also derives integer numbers, like 1 (one rotation?).

Then, applying ratios, one can derive (i suppose) bit by bit all that we actually know about numbers.

In such a way that one can establish an entire mathematical theory.

 

The point is, is it possible from all that, at some point, to get some physical unit, a dimension ?

 

I was wondering:

we know from the physical world, from experience, observation, measurement, that there is a limit to the ratio distance/time, which is the Speed Of light

So, in a geometry that includes time together with space, we know that there is an upper value to some ratio. We know what we must end up with. That is manageable. We can find from pure geometry a maximum ratio, that is a ratio that goes up till a value, then goes down and up again but never transgress a maximum. Like an angle for example, be in radians or any other system.

 

More:

 

we know also that SOL is not only an upper limit to a ratio, it is also a constant. Again, we know what we must end up with.

That is manageable too. We can find from pure geometry a constant ratio. [math]\pi[/math] is the first one that comes to mind, there are others.

 

Now, we need to get from geometry a ratio that is both a maximum and a constant. That is not trivial, [math]\pi[/math]for example is not a maximum value.

 

--------------------------------

And after doing that, the next step will be to consider SOL as a pure dimensionless ratio.

Edited January 22, 2012 by michel123456

[granpa]

math is 1-dimensional

 

math boils down to the next function

next(1) = 2

next(2) = 3

 

geometry is multi-dimensional math.

 

geometry has an extra added axiom defining the length of the hypotenuse

 

in euclidian geometry the length of the hypotenuse is defined as

c^2 = a^2 + b^2

 

in other geometries it can be defined differently.

 

the space we live in is euclidean.

but mathematically speaking there is nothing special about it.

its just one of many possible geometries.

 

pi would have different values in different spaces.

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

Edited January 22, 2012 by granpa

[md65536]

I suppose from geometry one can also derives integer numbers, like 1 (one rotation?).

Then, applying ratios, one can derive (i suppose) bit by bit all that we actually know about numbers.

In such a way that one can establish an entire mathematical theory.

Hm, I think there's something wrong there but I don't know what it is.

It might be that -- I think -- geometry comes from some axioms that are themselves based on the physical world. It might have something to do with naive set theory and/or some proof that mathematics cannot be derived purely logically. At its base there must be some assumptions. Or, perhaps on the other end of the spectrum, is Godel's incompleteness theorem, which I think shows that mathematics (or any other axiomatic system) can't completely consistently describe itself... or something.

 

I have only a vague grasp of this. I'd need to research this further but I probably won't.

 

But I think... neither can you start with something derived, nor can you derive everything you know.

 

Now, we need to get from geometry a ratio that is both a maximum and a constant. That is not trivial, [math]\pi[/math]for example is not a maximum value.

Pi is the maxium normalized angle between two vectors (in 2d and 3d Euclidean space and maybe other spaces).

Tau (2pi) is maybe the maximum normalized value for radians (in Euclidean space I guess).

1 is the maximum value for sin(theta) which in turn is the ratio of y to length of a unit vector rotated around the origin, ie 1 is the maximum y value of a unit vector.

 

 

[DrRocket]

Stoney units, very interesting.

 

From what I see:

from geometry you can derive ratios. Like Pi ([math]\pi[/math]).

Radians, as proposed by Md, are ratios.

 

I suppose from geometry one can also derives integer numbers, like 1 (one rotation?).

Then, applying ratios, one can derive (i suppose) bit by bit all that we actually know about numbers.

In such a way that one can establish an entire mathematical theory.

 

 

You have it backwards.

 

Starting from the natural numbers, in the form of the Peano Axioms, or more rigorously the Zermelo Fraenkel axioms you can construct (not derive but construct) the integers, rational numbers and complex numbers. From that you can construct Euclidean geometry.

 

Now throw in the axiom of choice and you can construct essentially all of the remainder of mathematics.

 

The important thing is that the only assumptions are really the existence the natural numbers and the axiom of choice. Everything else is a logical consequence.

 

If you would like to see the construction through the complex numbers you can find it done in Rudin's Principles of Mathematical Analysis or Landau's Foundations of Analysis. It is not difficult. I have taught it to sophomores and juniors. It is, however, a bit tedious.

[michel123456]

You have it backwards.

 

Starting from the natural numbers, in the form of the Peano Axioms, or more rigorously the Zermelo Fraenkel axioms you can construct (not derive but construct) the integers, rational numbers and complex numbers. From that you can construct Euclidean geometry.

 

Now throw in the axiom of choice and you can construct essentially all of the remainder of mathematics.

 

The important thing is that the only assumptions are really the existence the natural numbers and the axiom of choice. Everything else is a logical consequence.

 

If you would like to see the construction through the complex numbers you can find it done in Rudin's Principles of Mathematical Analysis or Landau's Foundations of Analysis. It is not difficult. I have taught it to sophomores and juniors. It is, however, a bit tedious.

 

I may be wrong, but I thought it was possible to have geometry without numbers.

[md65536]

I may be wrong, but I thought it was possible to have geometry without numbers.

How would you describe the relative size of something without ratios?

How would you express a ratio without numbers? (Pure analogy might do, I suppose.)

 

How would you express area or volume without numbers?

 

Perhaps you could describe some aspects of shapes without numbers.

How would you describe a triangle (don't refer to numbers!)?

 

 

"A number is a mathematical object used to count and measure." [http://en.wikipedia.org/wiki/Number]

The word geometry comes from the words "Earth" and "measure".

Even if you used only analogy -- for example your only measures were "Is it bigger than a breadbox? Is it bigger than an elephant?" -- the set of analogous objects might count as a system of numbers???

 

 

[TonyMcC]

I may be wrong, but I thought it was possible to have geometry without numbers.

I tend to agree. IMO any geometric construction is true for any size the construction is scaled against. For example given a particularly shaped right angle triangle with side ratios 3,4,5 will tell you that 3^2+4^2=5^2. But unless you are given the scale it might well stand for 30^2+40^2=50^2.

I deduce that if you start with a geometric construction, you can only get ratios from it. Not absolute values (again IMO).

Edited January 23, 2012 by TonyMcC

[michel123456]

How would you describe the relative size of something without ratios?

How would you express a ratio without numbers? (Pure analogy might do, I suppose.)

 

How would you express area or volume without numbers?

 

Perhaps you could describe some aspects of shapes without numbers.

How would you describe a triangle (don't refer to numbers!)?

 

 

"A number is a mathematical object used to count and measure." [http://en.wikipedia.org/wiki/Number]

The word geometry comes from the words "Earth" and "measure".

Even if you used only analogy -- for example your only measures were "Is it bigger than a breadbox? Is it bigger than an elephant?" -- the set of analogous objects might count as a system of numbers???

You can do geometry without measuring.

drawing a circle.

drawing a second same circle with center anywhere on the perimeter of the first circle: creating a vesica piscis.

drawing a line joining the 2 centers.

drawing a line joining the intersections of the 2 perimeters: that is dividing the radius in 2.

continuing the radius to the other side of the circle: creating a diameter.

from one intersection point of the diameter with the circle, drawing a random line that intersects the circle anywhere. From the new intersection point, draw a line that joins to the other intersection of the diameter with the circle. Creating a right rectangle.

For example.

[md65536]

I deduce that if you start with a geometric construction, you can only get ratios from it.

What is a ratio?

 

 

 

 

You can do geometry without measuring.

drawing a circle.

drawing a second same circle with center anywhere on the perimeter of the first circle: creating a vesica piscis.

 

How do you draw a circle? I know how to, using a given fixed length, but isn't that a measure or can count as a unit?

 

Or if you're given a circle to begin with, how do you determine its center? Again I know how, but I don't think I could describe it without words like "of the same length", and wouldn't that be a measure?

 

Perhaps congruence is enough to do all this, and wouldn't count as using numbers???

 

 

[michel123456]

What is a ratio?

A mathematical unitless concept of proportion.

 

 

 

 

 

How do you draw a circle? I know how to, using a given fixed length, but isn't that a measure or can count as a unit?

Anything can count as unit, it is just a matter of convention.

 

Or if you're given a circle to begin with, how do you determine its center? Again I know how, but I don't think I could describe it without words like "of the same length", and wouldn't that be a measure?

The center can be anywhere, position doesn't matter. And a compass, basically, has no mensuration. To draw 2 same circles the only thing you have to do is keep the compass open the same way.

 

Perhaps congruence is enough to do all this, and wouldn't count as using numbers???

I don't understand what you mean.

 

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edit: I don't want to push this too far. Maybe you're right, maybe numbers came first.

Edited January 23, 2012 by michel123456

[md65536]

I don't understand what you mean.

 

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edit: I don't want to push this too far. Maybe you're right, maybe numbers came first.

Yes, but it's interesting to figure out if it could be done any other way.

 

 

What I mean with congruence is something like the first paragraph from http://www.highhopes.com/numberandgeometry.html (I don't know the reliability of this source.)

"Counting was not always important. In fact, it was unnecessary when keeping track of such items as sheep. All that was needed was a one-to-one correspondence. If one had fifteen sheep and kept fifteen pebbles in a bag, it was a simple procedure to remove one pebble for each sheep to see if they were all present. An extra pebble meant a lost sheep. One doesn’t have to know how many sheep or pebbles one has – only to check the one-to-one correspondence between them. Using fifteen knots on a string or fifteen knicks on a stick could also do this. (One-to-one correspondence is a precursor of counting yet pervades much of mathematics.)"

 

Similarly, it might not be necessary to express a length as a number if all you need to know is congruence. I can construct a circle by using congruent pairs of points, like the ends of a stick I'm given. I don't need numbers to do this, unless the stick is by definition representative of a number (its length, or the number r, or the number 1 in "stick units").