Author Topic: geometry - long post  (Read 3257 times)

Offline SGOS

Re: geometry - long post
« Reply #15 on: August 23, 2013, 01:21:40 PM »
Quote from: "Plu"
  This is really hard to explain, especially in English, but I'll try anyway...  
Hmmm, for some reason I thought you were British.  Where are you from?

Quote from: "Plu"
I think you are approaching it from the wrong side.
Yeah, that same thought is occurring to me.  I think the discussion about units of measure being arbitrary helps clear up some of my confusion.  I may have to just think about this some more.  I still feel confused, but about what, I'm not entirely sure.  Does that sound weird?

Quote from: "Plu"
The meter is an arbitrary distance. (You probably know that Napoleon first introduced it because he was sick of all the different distance units.)
 Is he like the father of the metric system?

Quote from: "Plu"
However, the concept of distance is not arbitrary. We understand that there is such a thing as distance, and we needed to measure it. So we decided to pick our numbers scale, pick a random distance, and then said "1 distance unit".

Then, when we came across other kinds of abstract concepts, we picked a unit for them as well. We took a random point in time and said "1 time unit".  
 This may be important.  I'm not sure.  Question:  Does gravity have a unit of it's own, or is that something we just calculate from other arbitrary units?

Offline josephpalazzo

Re: geometry - long post
« Reply #16 on: August 23, 2013, 01:47:57 PM »
There's no "inconsistencies" between the different geometries. They can be classified by a single factor of curvature with k = 0 for flat space, +1 for spherical, and -1 for hyperbolic. It turns out that empirically, our 3-D world is flat, while 4-D spacetime (in GR) is hyperbolically curved.

Re: geometry - long post
« Reply #17 on: August 23, 2013, 01:48:22 PM »
Quote from: "Johan"
Science doesn't tell us what's true. Science can only tell us what isn't true and we are then left to assume what is true after we have eliminated all other possibilities as false. And of course there are always other possibilities so the loop is always open.

I'm not sure where you're coming from here.  I think perhaps you mean scientific generalizations cannot be proven true, but certainly there is a role of confirmation in science that enables us to call observations of some phenomenon true.

Some microorganism infect cells; helium is a superfluid at or below 2.17 K; crows are in the phylum chordata.  Without resorting to brain in a vat arguments, what other possibilities do we need to eliminate in order for these statement to be true?
Gentlemen, you can't fight in here! This is the War Room! -- President Merkin Muffley

My mom was a religious fundamentalist. Plus, she didn't have a mouth. It's an unusual combination. -- Bender Bending Rodriguez

Offline josephpalazzo

Re: geometry - long post
« Reply #18 on: August 23, 2013, 02:39:11 PM »
Quote from: "SGOS"
This may be important.  I'm not sure.  Question:  Does gravity have a unit of it's own, or is that something we just calculate from other arbitrary units?

Historically, our units have been quite arbitrary, however we have found that nature has three fundamental units: c, the speed of light,  ? , Planck reduced constant and G, Newton's gravitational constant. Out of these three, we can build any units whether it's for time, space, or mass.

Offline entropy

Re: geometry - long post
« Reply #19 on: August 23, 2013, 05:20:32 PM »
Quote from: "josephpalazzo"
There's no "inconsistencies" between the different geometries. They can be classified by a single factor of curvature with k = 0 for flat space, +1 for spherical, and -1 for hyperbolic. It turns out that empirically, our 3-D world is flat, while 4-D spacetime (in GR) is hyperbolically curved.

I was referring to consistency in axioms. Non-Euclidean geometries substitute different axioms for the parallel postulate in Euclidean geometry.

I will admit that somewhere I got the impression that there are subsystems of each type of geometry (i.e., geometries with more dimensions) that contained axioms that were inconsistent with other subsystems of that type of geometry, but with some quick research I see that is likely not to be true.

Offline aitm

Re: geometry - long post
« Reply #20 on: August 23, 2013, 07:23:43 PM »
another thread without pictures? This sucks..













 :D/
A humans desire to live is exceeded only by their willingness to die for another. Even god cannot equal this magnificent sacrifice. No god has the right to judge them.-first tenant of the Panotheust

Offline josephpalazzo

Re: geometry - long post
« Reply #21 on: August 23, 2013, 07:42:55 PM »
Quote from: "aitm"
another thread without pictures? This sucks..













 :D/

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Offline Solitary

Re: geometry - long post
« Reply #22 on: August 23, 2013, 07:57:08 PM »




Does this help? Solitary
There is nothing more frightful than ignorance in action.

Offline josephpalazzo

Re: geometry - long post
« Reply #23 on: August 24, 2013, 10:23:40 AM »
The more you know... whatever...  :-D

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Offline entropy

Re: geometry - long post
« Reply #24 on: August 24, 2013, 01:28:18 PM »
It has been many years since I took geometry in college, so I had remembered incorrectly - I had thought that the axioms of Euclidean geometry need to be modified as you go through different numbers of dimensions (and the same thing with the other basic types of geometry). I see now that the basic axioms of Euclidean geometry (or non-Euclidian geometries) don't change as the number of dimensions increases.

It's the axioms which are the core of what I'm talking about, not the conclusions that are drawn about geometric shapes based on those axioms. Even though I was off about the effect of more dimensions, it doesn't change the underlying point I was making - that mathematicians don't consider one geometry to be more true or false than another geometry just because it is axiomatically different from another. And there are axiomatic differences between the three different kinds of geometry. The differences are:

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Quote
1.) Euclidean: Given a line L and a point P not on L, there is exactly one line passing through P, parallel to L.
2.) Hyperbolic: Given a line L and a point P not on L, there are at least two lines passing through P, parallel to L.
3.) Elliptic: Given a line L and a point P not on L, there are no lines passing through P, parallel to L.

It is the difference in those axioms in each of the three basic types of geometry that differentiate the three geometries. But mathematicians don't consider one type of geometry "true" and the other two "false". They are just different geometric systems. That's what I meant when I said the mathematicians are "agnostic" about one valid axiomatic system relative to another valid axiomatic system.

Offline josephpalazzo

Re: geometry - long post
« Reply #25 on: August 24, 2013, 02:25:21 PM »
Quote from: "entropy"
It has been many years since I took geometry in college, so I had remembered incorrectly - I had thought that the axioms of Euclidean geometry need to be modified as you go through different numbers of dimensions (and the same thing with the other basic types of geometry). I see now that the basic axioms of Euclidean geometry (or non-Euclidian geometries) don't change as the number of dimensions increases.

It's the axioms which are the core of what I'm talking about, not the conclusions that are drawn about geometric shapes based on those axioms. Even though I was off about the effect of more dimensions, it doesn't change the underlying point I was making - that mathematicians don't consider one geometry to be more true or false than another geometry just because it is axiomatically different from another. And there are axiomatic differences between the three different kinds of geometry. The differences are:

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Quote
1.) Euclidean: Given a line L and a point P not on L, there is exactly one line passing through P, parallel to L.
2.) Hyperbolic: Given a line L and a point P not on L, there are at least two lines passing through P, parallel to L.
3.) Elliptic: Given a line L and a point P not on L, there are no lines passing through P, parallel to L.

It is the difference in those axioms in each of the three basic types of geometry that differentiate the three geometries. But mathematicians don't consider one type of geometry "true" and the other two "false". They are just different geometric systems. That's what I meant when I said the mathematicians are "agnostic" about one valid axiomatic system relative to another valid axiomatic system.

Offline entropy

Re: geometry - long post
« Reply #26 on: August 25, 2013, 09:22:13 AM »
Quote from: "josephpalazzo"

To add to that, mathematicians work with the axioms and see where that can lead. However, geometry of the real world is determined by matter. This is the fundamental gist of General Relativity.

This is why I chose geometry to talk about when trying to make the case that mathematics is not a science; rather mathematics and science are fairly distinct intellectual endeavors even though mathematics is key to the functioning of much of science. Science is about ferreting out the rules of the patterns of physical events in nature. Mathematics is about assuming rules (axioms) and, as you say, "see where that can lead" - and where it leads to is patterns of relationships of mathematical entities. They are both fundamentally about patterns, but science is about finding the patterns in nature and mathematics is about assuming axioms and seeing what patterns develop. It turns out that some of the mathematic patterns work to predict the patterns of physical events in nature, like the General Relativity equation that you note, but that does not imply that mathematics is a science. If anything, it appears to imply that science is a branch of mathematics - except for that pesky "empirical" thing about science. :)

 

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