Attempting to create a generalized graph of mathematics

[ALine]

I want this thread to be an attempt to create a generalized overall mapping of mathematics as we know it at this current time.

Its basis is that all models developed by a human is itself a model, therefore I do not want to delve to deeply into the philosophy of the subject matter, whereas I would like for this to be an overall model of existing mathematics from beginner to expert levels all explained and viewed as a general "mapping." Showing the interconnections between all of the different mathematical concepts formed which have been agreed upon. Showing all the twists and turns, all the connections, all of the branches that lead to no where. 

This is a big project, I am not blind to that. However I believe that providing an overall visual mapping of mathematics as a whole will allow for a sort of "key" for future persons, a minute or a millennia away, to quickly and easily view the processes that exist so that they themselves can go back and see the holes in their knowledge. While also quite possibly coming up with new ideas and adding to the overall mapping of the analogous "cinematics math's universe."

The goal of this project is for anyone, anywhere, at anytime to be able to learn all of mathematics in a minute. To be able to master it and contribute to it in a matter of moments.

I would also like for this to grow into something more, one in which instead of conceptualizing the concept of mathematics and guessing what exists based on a developed mental mapping, there is one that actually exists. Where you can add your own name to. Like a gigantic tree of knowledge or something.

Thank you for your time

[Ghideon]

4 minutes ago, ALine said:

I want this thread to be an attempt to create a generalized overall mapping of mathematics as we know it at this current time.

Not sure I grasp the idea completely but maybe the "the map of mathematics" could act as a starting point or act as a rough guide? It is a one page drawing showing how many concepts such as pure mathematics, applied mathematics, number systems, topology and many other fits together:

https://www.flickr.com/photos/95869671@N08/32264483720

 

[ALine]

3 minutes ago, Ghideon said:

Not sure I grasp the idea completely but maybe the "the map of mathematics" could act as a starting point or act as a rough guide? It is a one page drawing showing how many concepts such as pure mathematics, applied mathematics, number systems, topology and many other fits together:

Apologies for not being specific enough. 

Analogously, you know how there exists a blueprint and design for a car, and all you have to do is follow it in order to build that said car? And in the design process there are prototypes which have been tested and failed to pass the test? 

That idea, but with all of mathematics that is currently known.

Essentially a mapped out manual for all of mathematics. 

This however is an analogy only to transfer a idea abstractly. I can try and convey the idea for concretely, however it would take a little bit.

 

Just now, ALine said:

maybe the "the map of mathematics" could act as a starting point or act as a rough guide? It is a one page drawing showing how many concepts such as pure mathematics, applied mathematics, number systems, topology and many other fits together:

Ye, I have already taken a look at it and it shows how all of the fields are related together in categorically. I was thinking something more like an overall "process." Like writing an sharing a program, but instead of programs its theorems and proofs.

 

[studiot]

4 hours ago, ALine said:

I want this thread to be an attempt to create a generalized overall mapping of mathematics as we know it at this current time.

Its basis is that all models developed by a human is itself a model, therefore I do not want to delve to deeply into the philosophy of the subject matter, whereas I would like for this to be an overall model of existing mathematics from beginner to expert levels all explained and viewed as a general "mapping." Showing the interconnections between all of the different mathematical concepts formed which have been agreed upon. Showing all the twists and turns, all the connections, all of the branches that lead to no where. 

This is a big project, I am not blind to that. However I believe that providing an overall visual mapping of mathematics as a whole will allow for a sort of "key" for future persons, a minute or a millennia away, to quickly and easily view the processes that exist so that they themselves can go back and see the holes in their knowledge. While also quite possibly coming up with new ideas and adding to the overall mapping of the analogous "cinematics math's universe."

The goal of this project is for anyone, anywhere, at anytime to be able to learn all of mathematics in a minute. To be able to master it and contribute to it in a matter of moments.

I would also like for this to grow into something more, one in which instead of conceptualizing the concept of mathematics and guessing what exists based on a developed mental mapping, there is one that actually exists. Where you can add your own name to. Like a gigantic tree of knowledge or something.

Thank you for your time

I think I get the general idea.

+1 for ambition, I will remember to log on again in 3020 to find out how you got on.

 

Meanwhile here are a few thoughts.

If you have access to a library look at the contents pages of compendia of mathematics books.

I have a couple by that name, one by Manzel (2 vols) and one by Meyler and Sutton.

There is also the Princeton Companion to Mathematics

and the Cambridge Encyclopedia of Mathematics.

The contents pages should give you some subject headings  to think about.

Also some authors publish dependencies or dependecy diagrams such as "chapters 4 -7 should be read before chapter 11". though this is more common in Engineering than Mathematics.

However such information from either subject could be very useful, as it shows what depnds on what.

 

As regards the subject areas themselves I suggest you don't use the tree analogy.

This reuqires that the 'branches' are separate areas or subjects.

In truth there is considerable overlap and I don't thionk there is a single 'branch' that could stand alone by itself.
There is considerable overlap,
For instance although Geometry does not require measurement, you could not do Geometry without numbers even for shape and form. How else could your distinguish triangles, squares, pentagons hexagons etc?

So I suggest you go for a tiled presentation, perhaps a bit more formal than in Ghedieons diagram (+1 for finding that).
With suitable overlap or overlay you can sowh the interactions.

 

Go well in your endeavour.

 

 

Edited July 3, 2020 by studiot

[studiot]

Again some history (of mathematics) books publish timelines,

These can also be a valuable source of information and inspiration.

[Strange]

8 hours ago, ALine said:

I want this thread to be an attempt to create a generalized overall mapping of mathematics as we know it at this current time.

Looks like someone beat you to it: https://www.quantamagazine.org/the-map-of-mathematics-20200213/

[ALine]

4 hours ago, studiot said:

Again some history (of mathematics) books publish timelines,

These can also be a valuable source of information and inspiration.

thank you for the resource!

6 hours ago, studiot said:

So I suggest you go for a tiled presentation, perhaps a bit more formal than in Ghedieons diagram (+1 for finding that).
With suitable overlap or overlay you can sowh the interactions

tried to find it and could not.

2 hours ago, Strange said:

Looks like someone beat you to it: https://www.quantamagazine.org/the-map-of-mathematics-20200213/

(image source: https://ytmnd-fads.fandom.com/wiki/Noooooo!)

But I was kind of thinking more of a "theory and definition relationship mapping" kind of thing. Where It shows the interconnected nature of different existing definitions and theory's.

That is very impressive though, need to take  a closer look at it.

[ALine]

Starting off I was thinking about having this mapping be in a sort of having the propositions and then showing the definitions used to build up to those propositions like so...

 

Where all of the different definitions, axioms, and propositions are all connected in a progressive like fashion. Visualizing the definitions and proofs together.

This however will most likely change due to inaccuracies. 

 

maybe something more like this

[studiot]

1 hour ago, ALine said:

Starting off I was thinking about having this mapping be in a sort of having the propositions and then showing the definitions used to build up to those propositions like so...

 

Where all of the different definitions, axioms, and propositions are all connected in a progressive like fashion. Visualizing the definitions and proofs together.

This however will most likely change due to inaccuracies. 

 

maybe something more like this

 

I don't know if you have heard of David Hilbert but he tried to do just this.

The axiomatisation of Mathematics.

You should read about his fate.

https://en.wikipedia.org/wiki/David_Hilbert

[ALine]

21 hours ago, studiot said:

I don't know if you have heard of David Hilbert but he tried to do just this.

The axiomatisation of Mathematics.

You should read about his fate.

S1: Ye, my professor talked about him during lecture last semester. Also heard that he went insane trying it. Do not know if it is true or not, could not find any reference to hospitalization in the wiki. Only said that the axiomatization of Mathematics was being continuously revised by him to the 7th edition until his death I read. 

What I am wanting to do is use the existing axioms and theorems out there and show the connections between them through their uses. From that I am wanting to show the arising fields of study which emerge from those said connections. Like an every building tower using existing theorems as the building blocks and proofs as the glue analogously.

Ex: algebra is a set of processes which come with certain defined rule or processes. Because of this the field of algebra may not be that large. Just a repeating of those said rules or processes in order to solve given problems. I do not wish to capture a theorems application however. I wish to capture its formation only so that connective holes can be seen by those who do not really have a relatively large understanding of mathematics "like myself" in there own knowledge of mathematics as a whole.

p.s. I think I used that example totally wrong.

p.s.s this explanation feels under developed so I will be working on it more and more as this thread progresses.

[studiot]

8 minutes ago, ALine said:

S1: Ye, my professor talked about him during lecture last semester. Also heard that he went insane trying it. Do not know if it is true or not, could not find any reference to hospitalization in the wiki. Only said that the axiomatization of Mathematics was being continuously revised by him to the 7th edition until his death I read. 

What I am wanting to do is use the existing axioms and theorems out there and show the connections between them through their uses. From that I am wanting to show the arising fields of study which emerge from those said connections. Like an every building tower using existing theorems as the building blocks and proofs as the glue analogously.

Ex: algebra is a set of processes which come with certain defined rule or processes. Because of this the field of algebra may not be that large. Just a repeating of those said rules or processes in order to solve given problems. I do not wish to capture a theorems application however. I wish to capture its formation only so that connective holes can be seen by those who do not really have a relatively large understanding of mathematics "like myself" in there own knowledge of mathematics as a whole.

p.s. I think I used that example totally wrong.

p.s.s this explanation feels under developed so I will be working on it more and more as this thread progresses.

Yeah it is certainly a big project, with a capital B that will take a lot of work.

I am genuinely worried about this idea that one thing sits on another.

This is true for some things, but for most subject areas you need to know a bit of other areas to do anything.

When you have developed the new subject a bit you often then find you need to/can  develop the 'supporting' subject a bit further and in turn can progress the top one.
That is how we learn mathematics (and other technical subjects).

What did you think of my tiling v tree comment ?

I note Ghideon's picture does some of that.

[ALine]

3 minutes ago, studiot said:

Yeah it is certainly a big project, with a capital B that will take a lot of work.

I am genuinely worried about this idea that one thing sits on another.

yeah same, I am actually pretty apprehensive about it to be honest, was thinking this would be like a start of something that would take a long time to develop over my career as a future mathematician. As I learn about the different fields of study in mathematics I would add to it and what not. Like a pet project to share with the community that the community can take part in.  

 

6 minutes ago, studiot said:

This is true for some things, but for most subject areas you need to know a bit of other areas to do anything.

When you have developed the new subject a bit you often then find you need to/can  develop the 'supporting' subject a bit further and in turn can progress the top one.
That is how we learn mathematics (and other technical subjects).

This is something that speaks to me on an analogous "spiritual level" so to speak.  That supporting subject thing always gets to me. Like when you learn one thing then realize that there is something always underneath it so you have to keep trying to find the bottom of it, but always appear to be exactly where you last left off.

 

12 minutes ago, studiot said:

What did you think of my tiling v tree comment ?

I note Ghideon's picture does some of that.

Sorry about not responding to this comment sooner.

I was not really sure what you mean by tiling, would it be possible to provide some sort of graphic or further explanation?

Also was never able to find the Ghideon picture so I guess no plus one

I will try again in a second though. 

[Ghideon]

1 hour ago, ALine said:

Also was never able to find the Ghideon picture

 

If the link I provided does not work (it works for me) Google for 'The Map of Mathematics' Poster by Dominic Walkman.

I'm not sure about posting the actual picture here on the forum due to copyrights.

The link again: https://www.flickr.com/photos/95869671@N08/32264483720

 

 

[ALine]

17 minutes ago, Ghideon said:

 

If the link I provided does not work (it works for me) Google for 'The Map of Mathematics' Poster by Dominic Walkman.

I'm not sure about posting the actual picture here on the forum due to copyrights.

The link again: https://www.flickr.com/photos/95869671@N08/32264483720

Oh, oh god. I am stupid. I thought you meant that there was an image CALLED Ghideon. I was googling there for about 30 minutes. My bad Ghideon.

[ALine]

On 7/5/2020 at 12:03 PM, studiot said:

What did you think of my tiling v tree comment ?

Apologies for the delay, I think that it is a good idea. It would reduce the amount of complexity and work required to make this thing.

Thinking about the project now I am probably going to start off by creating a layman's mapping based on what I currently know

then expand it with the help of others as I mature my knowledge in mathematics.

 

[studiot]

45 minutes ago, ALine said:

Apologies for the delay, I think that it is a good idea. It would reduce the amount of complexity and work required to make this thing.

Thinking about the project now I am probably going to start off by creating a layman's mapping based on what I currently know

then expand it with the help of others as I mature my knowledge in mathematics.

 

As I think your project is very worthwhile I have been giving some serious consideration to explain my tiling comment. +1

I have been rather busy today but I will post soon on that, using these pillars of Maths Arithmetic, Algebra, Geometry and set Theory in relation to trigonometry and symmetry.

Meanwhile perhaps you would like to think about this category  ?

Mathematical notation and symbols. This is often the Cinderalla category, but it pervades all of Mathematics.

[ALine]

3 minutes ago, studiot said:

As I think your project is very worthwhile I have been giving some serious consideration to explain my tiling comment. +1

I have been rather busy today but I will post soon on that, using these pillars of Maths Arithmetic, Algebra, Geometry and set Theory in relation to trigonometry and symmetry.

Meanwhile perhaps you would like to think about this category  ?

Mathematical notation and symbols. This is often the Cinderalla category, but it pervades all of Mathematics.

I will do some research and think it over, thank you for taking the time to assist.

[ALine]

I have thought about it for a little bit, mathematical notation and symbols, and have come to the conclusion that they seem as if they extend very much into multiple different fields of study. Encompassing even outside of mathematics such as in physics and the other sciences. 

I remember reading that mathematics is a science in and of itself, however I would need to preform more research on that one. 

Symbols, in my opinion, are conceptual representations of some physical or virtual object which expresses some given action or process maybe. In terms of mathematics, because it is a pattern defining field of study it would make sense that it would utilize patterns more often then not.

So maybe mathematical notation is itself pattern connections to form more pattern connections to develop entirely new "universes" from simple rules and behaviors.

However very much unsure as this is just my thoughts without much research.

 

[ALine]

looking back at the image ghideon linked I am thinking that this should be a tiled mapping which does not incorporate the progression of time. Also I am thinking that categories are still present, however not in as a distinct form. I feel as if listing different areas of mathematics takes away from the blended nature of it, showing the intrinsic beauty which can be found from the flowing of concepts together. The tiling suggestion is still a good method of interrelating mathematical concepts together, however I would also like to see where one thing connects to another instead of leaving just a void for the viewers mind to wonder what could be in that space.

I am also thinking about looking back at previous research papers in the fields of mathematics as far back as I can find in order to see if I can find the different connections that I mentioned seeing that these are the origins of the concepts. 

[studiot]

I intend to post a diagram (it's too scruffy at the moment) later tonight.

 

Whilst you are still online look at these links

https://en.wikipedia.org/wiki/Mathematics

https://en.wikipedia.org/wiki/Glossary_of_mathematics

 

[ALine]

2 hours ago, studiot said:

I intend to post a diagram (it's too scruffy at the moment) later tonight.

 

Whilst you are still online look at these links

https://en.wikipedia.org/wiki/Mathematics

https://en.wikipedia.org/wiki/Glossary_of_mathematics

ok cool! 

have taken a taken a look at learned some cool stuff, like did not know about the Pythagorean theorem being so widely used and it being one of the only ones to be so widely used. It gives me the impression that math is more of an artform with accidental applicable properties.

 

 

[studiot]

OK so here is the beginnings of my suggestion for one small corner of your map.

What is Mathematics about ?  What does it do?

 

Well Mathematics is about mathematical objects and what we can do with them according to rules of logic.

So what are mathematical objects ?

Well they are objects like points and lines and sets and functions and shapes and angles and numbers and.........

So what can mathematics do with them?

Well it can describe

Properties of objects such a symmetry,

Arrangements  such as an array (matrix object) , arranging points to make a square and so on.

Relationshipes between two or more objects such as 10 > 3

Combine objects to generate another (different ) object such as 9x = iy) making a complex number

Transform one object into another for some purpose eg taking logarithms

 

Some of these operations are important when used to make the branch of maths called Geometry;
Others support number theory and yet others support Algebra.

 

So where does Trigonometry fit in?

Well sinx is defined as opposite/hypotenuse and is a number so number theory comes in.
 

Yet you can perform arithmetic operations such that sin²x + co² x =1 so arithmetic is onvolved

and again

[math]\sin x = x - \frac{{{x^3}}}{{3!}} + \frac{{{x^5}}}{{5!}}...[/math]

So summation of algebraic series brings in algebra

 

How are we doing ?

 

[ALine]

wow, this is awesome! 

from my understanding it looks like a solid grounding for a project like this.

 

I will need to read over your post in more detail to gain deeper understanding thought. Some stuff to unpack

 

[ALine]

Edit: "though" instead of "thought"

 

[ALine]

Ok, I think I have gained a full grasp on your tile method, you are not saying that mathematical objects are not the foundations of mathematics, whereas it is only a block used where logic acts as the connectors metaphorically speaking. 

And you pose a good question, where does symmetry play in terms of mathematics? And what is symmetry from a mathematicians perspective?

Now all that's left to do is to show that all fields of mathematics naturally arise from this understanding. Somehow making a conical mapping visualizing the interlacing's.