System to represent objects in Mathematics

[pairofstrings]

Hi.

Usually, Computer Programmers use Flow Charts, Algorithms, or UML diagrams to build a great software or system. In the same manner, in Mathematics, what do Mathematicians use to build a great system that they want to build.

Category Theory is at the highest level of abstraction; then comes Set Theory which can be used to talk about the system in little more detail?

Could you please demonstrate Category Theory and Set Theory with an example?

My attempt to know the System in Math that let's me communicate with other Mathematicians.

Example: There is a book. I can study the information that it has. To describe what the book is about I will use Category Theory and to see what words are on what pages I would use Set Theory?

In short, I want to know the System in Mathematics that let's me communicate my ideas with other mathematicians.

Thanks.

 

[jedishrfu]

There is only the common notation and conventions agreed upon by mathematicians worldwide who want to communicate and collaborate.

There is no math system in use akin to what programmers might use to design and document their programming systems. Even these programmer tools are seldom used except when deconstructing and reverse engineering a software system.

@fresh_42 is better able to comment on this topic.

 

[Mark44]

Example: There is a book. I can study the information that it has. To describe what the book is about I will use Category Theory and to see what words are on what pages I would use Set Theory?

I don't think this is a good metaphor to describe the differences between these two areas of mathematics..

Have you done any basic research? For category theory, you might start with this wiki page: https://en.wikipedia.org/wiki/Category_theory.

 

[fresh_42]

Category Theory is at the highest level of abstraction; ...

According to which order?

... then comes Set Theory which can be used to talk about the system in little more detail?

There is no category theory before set theory. Math is somehow the formalism of our human characteristic of pattern recognition. So basic arithmetic and the concepts of sets is basically the foundation everything else is built upon, category theory included. There is no certain system of categorization or language to communicate in. Each area has its own code. Of course they all usually use first order logic, which in the end is the language you might be looking for.

Could you please demonstrate Category Theory and Set Theory with an example?

$\mathcal{F}\, : \,(\text{ vector spaces },\text{ linear functions }) \longrightarrow (\text{ sets },\text{ mappings })$ is the forget functor in category theory and the Boolean algebra $(\mathcal{P}(\mathbb{R}),\cap ,\cup ,\{\,\,\}^C)$ is an example of set theoretic considerations.

The main difficulty with your question is, that it's not clear what you mean as mathematics doesn't work this way. Last time mathematicians have tried resulted in a mathematical revolution, cp. Hilbert's program, Russel, Gödel.

As soon as you will have properly and well defined what you're talking about, it can be called mathematics. The difference to the real world is, that in life nobody makes the effort to do so, and furthermore attaches undefined and implicit meanings to words and concepts.

 

[pairofstrings]

Assume that there is a Rocket. The Rocket lifts off from initial point 'C' and then it moves from point 'A' to point 'B' and then it returns from point 'B' to point 'A'. Now the Rocket is back at the point 'C' - returns to the same point from where it was first launched.

How do I say about the quantity, structure, space, change of the Rocket in Mathematics at the highest level of abstraction so that I can give a rough idea about the Rocket for the same to my Math friends?

 

[jbriggs444]

Assume that there is a Rocket. The Rocket lifts off from initial point 'C' and then it moves from point 'A' to point 'B' and then it returns from point 'B' to point 'A'. Now the Rocket is back at the point 'C' - returns to the same point from where it was first launched.
View attachment 225784
How do I say about the quantity, structure, space, change of the Rocket in Mathematics at the highest level of abstraction so that I can give a rough idea about the Rocket for the same to my Math friends?

It's a fools errand.

There is no fixed and objective mapping between mathematical statements and physical reality. Typically, one picks a correspondence between some mathematical model and some aspects of the real world. Like the correspondence between the number of sheep in a pasture and the natural numbers. One then proceeds to reason about sheep by reasoning about natural numbers.

What behavior in your rocket scenario do you wish to reason about? One possibility is that you want to know after 1,000,000 rocket rides starting at point A, will the rocket end up at A, B or C. There is a useful mapping for that.

 

[Stephen Tashi]

Hi.

Usually, Computer Programmers use Flow Charts, Algorithms, or UML diagrams to build a great software or system. In the same manner, in Mathematics, what do Mathematicians use to build a great system that they want to build.

In short, I want to know the System in Mathematics that let's me communicate my ideas with other mathematicians.

Your examples from computer programming are precisely defined methods of communication. There is no generally accepted formal system for constructing and communicating mathematics. Mathematics is communicated using a "natural language" (such as English) together with certain conventions of notation and conventions about terminology. Each special branch of mathematics has its own conventions of notation and terminology.

Common to all mathematics is the use of logic and the understanding of how the "game" of mathematics is played. Logic involves things like understanding how to reason with quantifiers "for each" and "there exists", how to do proof by induction etc. Understanding the "game" of mathematics involves things like understanding that definitions must be precise and that definitions are theoretically arbitrary, not something that is proven.

Some mathematicians are interested in developing concise sets of axioms that are sufficient to develop all of mathematics and this work may be conducted with formal symbolic languages. Some mathematicians are interested in formulating mathematical ideas so that computers can prove theorems. However, the majority of mathematics is not expressed in a formal language or a computer language.

 

[pairofstrings]

Okay, but what I am saying is: assume that if I have a Physics experiment that I want to perform then I take a sheet of paper and write down on it the following things:
1. Aim
2. Hypothesis
3. Apparatus
4. Method
5. Observation
6. Conclusion

In the same manner, if I want to build a System by using Mathematics and I need Algebra, Trigonometry, Vector Space then Category Theory or Set Theory could be a system that would let me convey the requisites like Algebra, Trigonometry, Vector Space etc, on a piece of paper, just like what I write on a paper when I perform Physics experiment?

Thanks!

 

[fresh_42]

Okay, but what I am saying is: assume that if I have a Physics experiment that I want to perform then I take a sheet of paper and write down on it the following things:
1. Aim
2. Hypothesis
3. Apparatus
4. Method
5. Observation
6. Conclusion

Sounds good. However, usually there is also a part, often in the introduction, where the following can be found within the world of existing results in this area. For short: What's known and what's new?

In the same manner, if I want to build a System by using Mathematics and I need Algebra, Trigonometry, Vector Space then Category Theory or Set Theory could be a system that would let me convey the requisites like Algebra, Trigonometry, Vector Space etc, on a piece of paper, just like what I write on a paper when I perform Physics experiment?

What do you mean? You can assume that your readers will already know, what a ring is, what the cosine means, what linearity is, and even what a contravariant functor and certainly power sets are. So there is no need to repeat those. On the other hand, if you want to write down a result from scratch, you'll have to start by defining your language and symbols, in which case I would first read a book on the theory of formal languages.

And you can always simply go to arxiv.org, chose an arbitrary mathematical article and analyse how it is written. (I would only recommend to select a couple of them in order to reduce the risk of selecting a strangely written one.)

 

[FactChecker]

I would assume that Category theory is good for generalizing abstract concepts, but not a very good (efficient?/effective?) way to define specific concrete mathematical concepts. In other words, going from the specific to define the abstract is easier than going from the abstract to define the specific. By its very definition, the abstract approach has intentionally stripped away and avoided specifics.

 

[fresh_42]

I would assume that Category theory is good for generalizing abstract concepts, but not a very good (efficient?/effective?) way to define specific concrete mathematical concepts. In other words, going from the specific to define the abstract is easier than going from the abstract to define the specific. By its very definition, the abstract approach has intentionally stripped away and avoided specifics.

I agree and it reflects how the common people among us understand complex subjects; an example in mind. I just want to add a statement which I incidentally came upon just yesterday. It is from van der Waerden's obituary for Emmy Noether (1935):

Franz Lemmermeyer und Peter Roquette: Helmut Hasse und Emmy Noether - Universitätsverlag Göttingen 2006

"The maxim from which Emmy Noether has always been guided can be formulated as follows: All relations between numbers, functions, and operations only become transparent, generalizable, and really fruitful when they are detached from their particular objects and reduced to general conceptual contexts. For her, this guiding principle was not the result of her experience of the scope of scientific methods, but an a priori principle of her thinking. She could not accept and process any theorem, proof in her mind until it was abstractly conceived and thereby made transparent to her spirit eye."

So for the gifted, abstraction seem to be very necessary at times.

 

[Stephen Tashi]

Okay, but what I am saying is: assume that if I have a Physics experiment that I want to perform then I take a sheet of paper and write down on it the following things:

In the same manner, if I want to build a System by using Mathematics and I need Algebra, Trigonometry, Vector Space then Category Theory or Set Theory could be a system that would let me convey the requisites like Algebra, Trigonometry, Vector Space etc, on a piece of paper, just like what I write on a paper when I perform Physics experiment?

I don't know what you mean by "build a System".

There is such a thing as "experimental mathematics", but for the most part, describing experiments and doing mathematics are very different processes.

When we describe an experiment, we don't present a summary or diagram of all the knowledge necessary to understand the description. For example, we usually take it for granted that the reader is famililar with the system of physical units used and the mathematical notation used to express the physical laws mentioned in the hypothesis.

1. Aim
2. Hypothesis
3. Apparatus
4. Method
5. Observation
6. Conclusion

Are you seeking an outline for writing a mathematical paper?

 

[jedishrfu]

There is one comparable system of notation that was developed for computers, the APL language which allowed one to describe mathematical formulas in a computable form. I think Iverson (inventor of APL) tried to systematize math with a common notation suitable for computation too.

http://cl-informatik.uibk.ac.at/teaching/ws12/bob/reports/FA.pdf

http://www.math.bas.bg/bantchev/place/apl.html

At one time, IBM's opcodes for System 370 were defined in terms of APL symbols.

Lastly, with respect to physics and math:

https://www.amazon.com/dp/0691154562/?tag=pfamazon01-20

The author shows how many math concepts can be derived from physical experiments.

 

[pairofstrings]

Are you seeking an outline for writing a mathematical paper?

No.

What I am trying to know is that if I need a Circle then I could do "something" to get a Circle.
Do we have a name for this "something"? Yes, that something could be Arithmetic, Algebra, Space, et cetera.
So, if I want to tell others to "build" a Circle then I can guide them by saying: "Use Arithmetic, Algebra, Space.. to build your Circle, please".
Why I am doing this? I'm doing this because I don't want my Circle to be dependent on Arithmetic, Algebra, Space, et cetera.
The question is what else would I do besides Arithmetic, Algebra, Space et cetera to build a Circle.
So, if I abstract out Arithmetic, Algebra, Space, et cetera then I get more opportunities to "build" a Circle.
Arithmetic, Algebra, Space et cetera are theories.
Do we have something in Mathematics that gives me a flexibility in choosing a theory to build a Circle?
Category Theory? Set Theory? Can the elements of a Set be equations?
Can Category Theory and Set Theory tell me how I can swap these theories?

 

[fresh_42]

What I am trying to know is that if I need a Circle then I could do "something" to get a Circle. Do we have a name for this "something"?

Yes, set theory and arithmetic: $\mathcal{C}=\{\,(x,y)\in \mathbb{R}^2\,|\,x^2+y^2-1=0\,\}$. Everything beyond this point, depends on what you want to have the circle for: differential geometry, algebraic geometry, geometry, topology, group theory, analysis, number theory and whatever else.

 

[FactChecker]

I'm afraid that the very definition of a circle must be given in a way that only makes sense in a particular context. I don't think you can define a circle without using Arithmetic, Algebra, Space, et cetera. So trying to "abstract that away" will probably fail.

 

[pairofstrings]

I'm afraid that the very definition of a circle must be given in a way that only makes sense in a particular context. I don't think you can define a circle without using Arithmetic, Algebra, Space, et cetera. So trying to "abstract that away" will probably fail.

I am not asking anyone to define a circle without using Arithmetic, Algebra, Space, et cetera.
I am just asking how to write "abstract that away" on a piece of paper in Mathematics. Category Theory? Set Theory?

 

[fresh_42]

What does "abstract that away" mean? It would be clearer if you said it in Samoyedic and we used google translate. I have not the least idea what you are talking about. There is no abstraction beyond set theory and arithmetic. You could invent one, but what for?

 

[FactChecker]

I am not asking anyone to define a circle without using Arithmetic, Algebra, Space, et cetera.
I am just asking how to write "abstract that away" on a piece of paper in Mathematics. Category Theory? Set Theory?

There certainly are things more abstract than algebra, etc., but the question is whether they are concrete enough to define what a "circle" is. I think that most people would not be satisfied with anything which can not specify curvature, distance from a center point, etc. In topology, you would be able to define a simple closed curve, but that is not enough to be called a circle. In general, you can't be very abstract and still retain specific geometric and algebraic properties.

 

[pairofstrings]

What does "abstract that away" mean?

Assume that I have ten distinct theories for each mathematical operation: Addition, Subtraction, Multiplication, Division.
How do I write on a piece of paper that I have opted for ninth theory to perform addition? That's it. This is what I want to know: I want to present my thoughts in standard ways for communication. So, how do I say that I have chosen ninth theory to perform addition in a standard way?

Little more explanation to my question:

If you look at the code below, you will notice that IWater is of two types: normal water and soda water. In the same manner, how do I say to someone that I have such flexibility of choosing theories on a piece of paper?

using System;

namespace ConsoleApp1
{
    public interface IWater
    {
        string ThisWater();
    }

    public class Water
    {
        IWater _obj;
        public Water(IWater _obj)
        {

            this._obj = _obj;
        }
        public void MyWater()
        {
            string myString = _obj.ThisWater();
            Console.WriteLine(myString);
        }
    }

    public class NormalWater : IWater
    {
        public string ThisWater()
        {
            return "Normal Water";
        }
    }

    public class SodaWater : IWater
    {
        public string ThisWater()
        {
            return "Soda Water";
        }
    }
    class Program
    {
        static void Main(string[] args)
        {
            Water nwater = new Water(new NormalWater());
            nwater.MyWater();
            Water swater = new Water(new SodaWater());
            swater.MyWater();
            Console.ReadLine();
        }
    }
}

 

[fresh_42]

These operations are binary operations on certain types of objects aka sets with certain properties aka arithmetic rules. Rules and sets can vary a lot, so the binary operation is all what they have in common: $\circ \, : \,\mathcal{U} \times \mathcal{V} \longrightarrow \mathcal{W}$. Normally $(*)\quad\mathcal{U}=\mathcal{V}=\mathcal{W}$ but not necessarily, as e.g. ring modules or inner products. Of course, it makes not a lot of sense only to view such an operation without any specifications of the sets or the rules. Even if we chose categorial terms, we normally need to specify which category. If we only define categories which allow a binary operation of its objects, it will be necessary to at least define something about the allowed objects as e.g. $(*)$. A binary operation between any of its objects is so arbitrary that few can be said in such a generality. It will be a collection of categories, rather than one on its own. But in principle, you can do this. However, it will be challenging even to define morphisms properly.

 

[Mark44]

Assume that I have ten distinct theories for each mathematical operation: Addition, Subtraction, Multiplication, Division.
How do I write on a piece of paper that I have opted for ninth theory to perform addition? That's it. This is what I want to understand.

Let's look at addition only.
We have several different rules (these are NOT theories) for addition, including

1. Addition of positive numbers
2. Addition of signed (positive or negative) numbers
3. Addition of complex numbers
4. Addition of vectors
5. Addition of matrices
6. Addition of functions
7. ... and possibly others that don't spring to mind immediately.

You don't just arbitrarily pick one of these as the "best" -- you use the one that works for the set of things you're considering. So the context in which you're working dictates which form of "addition" you should use.

If you look at the code below, you will notice that IWater is of two types: normal water and soda water. In the same manner, how do I say to someone that I have such flexibility of choosing theories on a piece of paper?

Going back to my example of addition (only), you don't have flexibility in choosing which operation to use. Whatever set of things you're working with will dictate which operation you have to use. If you need to write something on a piece of paper, it would be a big if ... else if ... else if ... decision structure, where the type of things you're adding determines which rule for addition you will use.

My question for you is, why are you doing this? What's your purpose?

 

[fresh_42]

... and the addition doesn't even have to be an addition in the classical sense. It could as well be the union on sets or the OR in Boolean algebras, or simply the only operation which is just written with a $+$.

 

[pairofstrings]

My question for you is, why are you doing this? What's your purpose?

I'm curious and I want a PhD. in Mathematics.

 

[fresh_42]

I'm curious and I want a PhD. in Mathematics.

Good luck!

By the way, many objects which define a category on their own can be realized as homomorphic images of tensor spaces. Here you have a natural construction $\mathcal{U} \otimes \mathcal{V}$ from which you can build a quotient for the defining rules.

 

[pairofstrings]

Thanks.

 

[Mark44]

It could as well be the union on sets or the OR in Boolean algebras

Yep, I thought of those as well, but my list seemed long enough already to make my point.

 

[fresh_42]

Yep, I thought of those as well, but my list seemed long enough already to make my point.

And the list of multiplications isn't any shorter.

 

[FactChecker]

If you have never had abstract algebra, you should look at that. It studies addition and multiplication in very abstract terms. That is the only way to realize what is essential for those operations to make sense and what is not essential.

 

[Mark44]

I'm curious and I want a PhD. in Mathematics.

It's good to have goals, but in order to attain them, you need to have a realistic plan of action and follow through on it. Based on some of your recent threads, you are not very far along in your mathematics studies.

As examples, in this thread you asked about the difference between a function and a relation -- https://www.physicsforums.com/threads/functions-and-relations.937968/
In this thread, you asked about the roots of a quadratic equation -- https://www.physicsforums.com/threads/what-are-the-applications-of-roots-of-a-polynomial.919059/
In this thread, you weren't clear on the difference between measures of length vs. area -- https://www.physicsforums.com/threa...way-into-formula-of-area-of-rectangle.778700/
Questions like these suggest that you aren't very far along in mathematical knowledge, so getting a PhD will be a long road.

 

[FactChecker]

Based on some of your recent threads, you are not very far along in your mathematics studies.

Fortunately, @pairofstrings has given us his level of education (working on undergrad) in his profile. I really appreciate that. So many do not give any information. I would say that it is time for him to start including as many math classes as possible before he graduates.

 

[Mark44]

I would say that it is time for him to start including as many math classes as possible before he graduates.

Strongly agree.

 

[jonk75]

$\mathcal{C}=\{\,(x,y)\in \mathbb{R}\,|\,x^2+y^2-1=0\,\}$

In the above, it should be $(x,y)\in \mathbb{R}^2$.

 

[pairofstrings]

Okay. I want to be able to write equations and I am working towards it.

Thanks everyone.

 

[fresh_42]

In the above, it should be $(x,y)\in \mathbb{R}^2$.

Indeed. I corrected it, thank you.