Mathematics and Geometry

[geordief]

I understand all Geometry can be represented in a mathematical  way. 

 

But the two are clearly different.Is Geometry a (sub?) branch of  Mathematics or could it be the other way around?

 

Could  all Mathematics even be represented geometrically perhaps?

 

Can ideas muscle in on the act if the "Materialists"* are allowed full license?

 

 What I  am trying to  ask  in first place is "How,in essence  is Geometry different from Maths ?"

 

 

*if that is the right term for those who claim that all thoughts and ,by extension ideas can be reduced to their physical  interactions...

[beecee]

7 hours ago, geordief said:

I understand all Geometry can be represented in a mathematical  way. 

 

But the two are clearly different.Is Geometry a (sub?) branch of  Mathematics or could it be the other way around?

 

Could  all Mathematics even be represented geometrically perhaps?

 

Can ideas muscle in on the act if the "Materialists"* are allowed full license?

 

 What I  am trying to  ask  in first place is "How,in essence  is Geometry different from Maths ?"

 

 

*if that is the right term for those who claim that all thoughts and ,by extension ideas can be reduced to their physical  interactions...

Geometry of course is a subset of mathematics in general and concerns itself with shapes and dimensions such as triangles, squares, circles etc. Funny my mathematical ability at school was average, but in geometry I was pretty good and among the tops in my class. Not sure what that actually says about me. 

Also worth noting that the late Professor Stephen Hawking, mainly communicated [at least in the early days of the onset of his terrible disease] via figures, shapes and geometry to express his ideas.

[Strange]

12 minutes ago, beecee said:

Geometry of course is a subset of mathematics in general and concerns itself with shapes and dimensions such as triangles, squares, circles etc.

I was going to say this, but then started thinking about the fact that so much mathematics uses concepts from geometry. For example, Fermat's Last Theorem was proved using elliptic functions. They also form the basis of modern cryptography. As the name suggests, they originally came from geometry. Whether they still count as geometry or not isn't something I could comment on.

Also, complex numbers, a fairly abstract extension of the reals in one sense, can also be represented as points on a plane. 

I wonder if all of mathematics could be considered to be geometry. (But that might depend on exactly how one defines "geometry".)

If so, what is the connection to symmetry, which is also fundamental to many branches of mathematics and physics.

 

[studiot]

Hello, Geordie, haven't you asked this question before, or perhaps it wasn't you?

 

The answer depends in part on whether you are talking to a Pure Mathematician or an Applied Mathematician.

A Pure Mathematician might say that Geometry is the branch of Algebra that deals with congruence.
Since (Pure) Mathematicians set out in the early 20th century to recast the whole of geometry in algebraic form.

These same Pure Mathematicians might also say that Topology is the algebra of similarity
And that wider, since similarity includes congruence as a special case, geometry is a branch of the more general topology, where a beer glass and a football are considered to have the same shape.

This arises because you can only define congruence (and congruent shapes) when you have a definition of distance.

For instance corresponding sides of congruent triangles have the same length.

Thus is this Geometry.

But if your triangles only have the same angles, but different coresponding lengths of sides they are similar but not congruent.

 

 

Completely on the other hand an Applied Mathematician would say that yes, Geometry is the Mathematics of Shape and Form.
Indeed I have several important textbooks with that or similar names.

So a Pure mathematician might say to you

The equation of a general inverted Catenary is

[math]y = \frac{\alpha }{2}\left[ {2 - \left( {{e^{ - \frac{x}{\beta }}} + {e^{\frac{x}{\beta }}}} \right)} \right][/math]

Where alpha and beta are coefficients.

But an Applied Mathematician might say,

"That's no good" or "That's no fun"

"But if you put apha = 1.030 and beta = 1.322 you wil have the equation of the St Louis Arch, which ahs featured on television recently.

[geordief]

45 minutes ago, studiot said:

Hello, Geordie, haven't you asked this question before, or perhaps it wasn't you?

Not this one (having just gone through the list of topics I have started). 

It seems like it could be a very fundamental question bearing in mind how some people

seem to be speculating that the entire universe could be a holographic projection of an arrangement of 0s and 1s on the surface of a sphere (unless I have wildly misunderstood that).

 

Anyway ,is there a freestanding entity called "mathematics" and a corresponding freestanding entity such as "geometry"  or are the two joined at the hip  or hips,"bleeding into each other"?

 

The thought crossed my mind that mathematics could be like the source code to geometry  similar to how the page on this screen is  made visible by the html code  "behind" the screen.

2 hours ago, beecee said:

Geometry of course is a subset of mathematics in general and concerns itself with shapes and dimensions such as triangles, squares, circles etc. Funny my mathematical ability at school was average, but in geometry I was pretty good and among the tops in my class. Not sure what that actually says about me. 

Also worth noting that the late Professor Stephen Hawking, mainly communicated [at least in the early days of the onset of his terrible disease] via figures, shapes and geometry to express his ideas.

It would be interesting to have a few examples of that. He expressed mathematical ideas in that way or his ideas more generally?

 

2 hours ago, Strange said:

I was going to say this, but then started thinking about the fact that so much mathematics uses concepts from geometry. For example, Fermat's Last Theorem was proved using elliptic functions. They also form the basis of modern cryptography. As the name suggests, they originally came from geometry. Whether they still count as geometry or not isn't something I could comment on.

Also, complex numbers, a fairly abstract extension of the reals in one sense, can also be represented as points on a plane. 

I wonder if all of mathematics could be considered to be geometry. (But that might depend on exactly how one defines "geometry".)

If so, what is the connection to symmetry, which is also fundamental to many branches of mathematics and physics.

 

Symmetry is closely connected to reflections in a mirror isn't it?

 

Can Geometry be called mathematics using light  as the tool of communication? Is maths  a eunuch in that regard? 

Edited January 6, 2019 by geordief

[studiot]

22 minutes ago, geordief said:

Not this one (having just gone through the list of topics I have started). 

It seems like it could be a very fundamental question bearing in mind how some people

seem to be speculating that the entire universe could be a holographic projection of an arrangement of 0s and 1s on the surface of a sphere (unless I have wildly misunderstood that).

 

Anyway ,is there a freestanding entity called "mathematics" and a corresponding freestanding entity such as "geometry"  or are the two joined at the hip  or hips,"bleeding into each other"?

 

The thought crossed my mind that mathematics could be like the source code to geometry  similar to how the page on this screen is  made visible by the html code  "behind" the screen.

It would be interesting to have a few examples of that. He expressed mathematical ideas in that way or his ideas more generally?

 

Symmetry is closely connected to reflections in a mirror isn't it?

 

Can Geometry be called mathematics using light  as the tool of communication? Is maths  a eunuch in that regard? 

 

Perhaps only a poet can answer you so here is Samuel Taylor Coleridge on Mathematics

 

 

Quote

STC

Dear Brother, I have often been surprized, that Mathematics, the quintessence of Truth, should have found admirers so few and so languid.--Frequent consideration and minute scrutiny have at length unravelled the cause--viz.--that though Reason is feasted, Imagination is starved; whilst Reason is luxuriating in it's proper Paradise, Imagination is wearily travelling on a dreary desart. To assist Reason by the stimulus of Imagination is the design of the following production. In the execution of it much may be objectionable. The verse (particularly in the introduction of the Ode) may be accused of unwarrantable liberties; but they are liberties equally homogeneal with the exactness of Mathematical disquisition, and the boldness of Pindaric daring. I have three strong champions to defend me against the attacks of Criticism: the Novelty, the Difficulty, and the Utility of the Work. I may justly plume myself, that I first have drawn the Nymph Mathesis from the visionary caves of Abstracted Idea, and caused her to unite with Harmony. The first-born of this Union I now present to you: with interested motives indeed--as I expect to receive in return the more valuable offspring of your Muse-- Thine ever, S. T. C.

(Christ's Hospital) March 31, 1791.

 

[beecee]

23 minutes ago, geordief said:

It would be interesting to have a few examples of that. He expressed mathematical ideas in that way or his ideas more generally?

I will see what I can find, but I did pick that up in his book "A Brief History of Time" I'm fairly sure. Light cones illustrations was one way, but I'm again not too sure he was the first to use them. Bit busy right now but will get back with what I can on that.

[beecee]

1 hour ago, beecee said:

I will see what I can find, but I did pick that up in his book "A Brief History of Time" I'm fairly sure. Light cones illustrations was one way, but I'm again not too sure he was the first to use them. Bit busy right now but will get back with what I can on that.

OK, Am unable to find what I am looking for, but from memory, before his disease  had progressed too far, his speech was very slurred and only able to be understood by his wife Jane. He subsequently communicated a lot with geometrical figures, sketches, light cones etc. There was also a video out called a BHoT  it may have been in that......

 

 

 

[Strange]

15 hours ago, geordief said:

seem to be speculating that the entire universe could be a holographic projection of an arrangement of 0s and 1s on the surface of a sphere (unless I have wildly misunderstood that).

That is a very common misrepresentation of the holographic principle (more extreme versions imply that the 3D world doesn't even exist). 

What the holographic principle says is that some information in a 3D volume of space can be represented on a 2D boundary (with no loss of information). The original example was the observation that the entropy of a black hole is proportional to its area, not its volume.

15 hours ago, geordief said:

Symmetry is closely connected to reflections in a mirror isn't it?

That is one of the simple examples (eg. left vs right socks gloves). Another is rotation. There are many others; there is a whole branch of mathematics devoted to it (I really struggled even with the most basic ones when I studied physical chemistry). It appears to be fundamental to quantum theory, where many of the properties are defined in terms of symmetry groups.

15 hours ago, geordief said:

Can Geometry be called mathematics using light  as the tool of communication?

You seem to be thinking of geometry as the stuff we see and played around with on paper (lines, circles, triangles, etc) when we were at school. But I think of those as just a visual representation of some geometry - much of it is hard to represent that way. For example, non-Euclidean geometries in N-dimensional spaces (especially when N is greater than 2 or 3).

[geordief]

@Strange

Yes I see my idea of "geometry" was very limited.

As for the holographic principle,is  it just "some information" and not "all or any"?

And is it just 2d》3d or (as I had supposed ) 2d+1》3d+1  ?

 

Regarding the entropy of a BH  would this be one of the few object's where entropy can apparently  be measured in a reliable way?(from my limited acquaintance with entropy it seems like a roller coaster of a phenomenon ,decreasing and increasing alternately)