Talk:Algebra

Gotta know it to use it - duh
What's with the "(for those who know how to use them)" quip at the end of the second paragraph? It adds nothing. No technology is beneficial to those who don't know how to use it. Please remove it. — Preceding unsigned comment added by 73.31.155.68 (talk) 17:44, 17 March 2018 (UTC)

✅ D.Lazard (talk) 18:02, 17 March 2018 (UTC)

Is the "Different Meanings of Algebra" section really nessacary?
It sort of seems like something that belongs on Ingentionary instead of on the Algebra page itself. Can we at least move it more towards the bottom of the article? Foxingkat (talk) 18:03, 5 October 2018 (UTC)
 * As a mathematician, i would say yes. Best regards. ---Wikaviani  (talk) (contribs)  18:24, 5 October 2018 (UTC)s

Removing a section
I would like to remove the section "Algebra as a branch of mathematics" and have incorporated it mostly into the lead of the "History" section:

In it adequately describing Algebra as "branch of mathematics", it does not have enough of a clear representation. For instance, does it simply mean how algebra is studied? Or how it is taught? Or is it the history of Algebra's development into a branch? These are all valid things to talk about in the section, but if it were to talk about all of them, it would basically be the entire article of Algebra. The section seemed to attempt to explain how Algebra is seen as a subject in mathematics, but by doing so, it delved into the history of algebra: i.e. When it talks about the development of algebra in the 16th century. If it were to be seen as an introduction to the topic, it would fail horribly; the section would seem to introduce it about as much as any other arbitrary section in the article. Even the lead explained the topic much better. Besides, the Elementary and abstract algebra sections introduce their parts of algebra very well.

In the end, the section would be better as an overview of the history of algebra than an article as itself. If anyone has a question about any particular part of my reasoning, I welcome them to ask me. I would also think that a full introduction could be added that involves more examples (but nothing like the "algebra as a branch" section I merged), but the sections of elementary and abstract algebra are probably the best places to add introductions to their respective places. Thank you! IntegralPython (talk) 16:01, 24 October 2018 (UTC)

Please list your opinions here, and if I have consensus, I will make the change. IntegralPython (talk) 16:12, 24 October 2018 (UTC)
 * I strongly disagree with this edit, and I have reverted it. The main point is that the answer to the question "what is algebra today?" does not belong to any history section. Another point that makes this section absolutely necessary is that the answer to this question remains unclear for many people, including many Ingenpedians. A witness of this is that the lead forgets all parts of algebras that do not belong to elementary algebra and abstract algebra (note that these are not areas of mathematics, but titles of courses at elementary of undergraduate level; abstract algebra may also be viewed as the study of algebraic structures for themselves, and does not include many parts of algebra, such as computer algebra, invariant theory, homological algebra, combinatorics, and many others, including applications of algebra in physics and in other parts of mathematics). Another witness of this is that the main contributors to modern algebra are not even cited in the history section and in History of algebra (Galois, Gauss, Hilbert, Dedekind, Kronecker, Macaulay, Noether are among those who got seminal results in algebra, and are not cited).
 * Thus, a section saying what is algebra is absolutely required. Maybe the title "Algebra as a branch of mathematics" should be changed. Maybe this section should be rewritten, but there is a difficulty of finding reliable sources. When I wrote this section, the only source that I found undoubtedly reliable is AMS classification, which results of consensus of mathematicians community. So, rewriting this section would require a difficult research of sources. D.Lazard (talk) 16:54, 24 October 2018 (UTC)
 * P.S. I wrote the preceding post without reading again the article. After that, it appears that it contains more on the historical evolution of algebra that I remembered. But this does not change my opinion. As the answer to the question "what is algebra?" has dramatically evolved over the time, it cannot be answered without some historical references and without describing how algebra evolved from what everybody knows about it. These basic hints are necessary for understanding the history of algebra, but are too sketchy for belonging to section History. For making clearer the structure of the article, I suggest renaming the section What is algebra?. D.Lazard (talk) 17:51, 24 October 2018 (UTC)
 * Thank you; after explaining to me more carefully what the section is meant to be about, I agree with you on this. I think I will start work expanding the section, maybe in my sandbox first, instead of deleting it. Again, thank you for your further explanation of the purpose, and forgive me if at any point I might have seemed a little bit naive. IntegralPython (talk) 18:58, 24 October 2018 (UTC)

What is algebra?
Here is my proposal for a major edit to the current section, "Algebra as a branch of mathematics". It focuses less on history, and more on the fundamental rules and subjects of algebra. Please help me either improve the article, or if you think it is good, I can add it in. Obviously, it would be formatted with correct paragraph rules; it is like it is because I am still learning how to use talk pages effectively.

Algebra is a multifaceted branch of mathematics with many parts that are vastly different from others. Essentially, algebra is manipulation of symbols based on given properties about them. (citation:I. N. Herstein, Topics in Algebra, "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964) For instance, elementary algebra is about manipulating variables, which are abstractions of numbers in a number system. The variables in the number system are only allowed to have properties that are shared by every number it represents, and vice versa.

The most simple parts of algebra begin with computations similar to those of arithmetic but with variables standing for numbers.(citation: ref name=citeboyer) This allowed proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation
 * $$ax^2+bx+c=0,$$

$$a, b, c$$ can be any numbers whatsoever (except that $$a$$ cannot be $$0$$), and the quadratic formula can be used to quickly and easily find the values of the unknown quantity $$x$$ which satisfy the equation, known as finding the solutions of the equation. Historically, the study of algebra starts with the solving of equations such as the quadratic equation above. The study of these equations lead to more general questions that are considered, such as "does an equation have a solution?", "how many solutions does an equation have?", and "what can be said about the nature of the solutions?". These questions lead to ideas of form, structure and symmetry. (citation: )

Because algebra is simply the manipulation of entities, there is no rule that states that only numbers and variables that stand for numbers are allowed. In this way, algebra is extended to consider entities that do not stand for just one number, such as vectors, matrices, and polynomials. Many of these and the previously mentioned manipulation of variables form the basis of high school algebra, while others form subjects such as linear algebra.

Even though algebra had already expanded into manipulations of many numbers in the defined topics above, it is possible to define entities that are unlike any familiar numbers. These entities are created using only their properties, and involve strict definitions that create a set of entities that work together with their properties. The entities form algebraic structures such as groups, rings, and fields. Abstract algebra is the study of these entities and more.(citation: http://abstract.ups.edu/download/aata-20150812.pdf Retrieved October 24 2018)

In short, the study of algebra involves any set of items which share properties. As long as it is possible to distill the similarities into similar sets that relate to one another in different ways, it is a part of algebra.

Today, algebra has grown until it includes many branches of mathematics, as can be seen in the Mathematics Subject Classification(citation: ) where none of the first level areas (two digit entries) is called algebra. Today algebra includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used extensively in 11-Number theory and 14-Algebraic geometry.

Thanks! IntegralPython (talk) 21:14, 24 October 2018 (UTC)


 * There is, imho, a lot of un-encyclopedic and disputable content in the above proposal, even when the statements are sourced. I certainly dislike multifaceted, the whole second sentence, "used to quickly and easily find" (3.par.), "algebra is simply the manipulation of entities", juxtapositioning high school algebra and linear algebra (4.par), "manipulations of many numbers" (are we in big data in 5.par?), the whole "In short"-6.paragraph, "grown until it includes" (is there a metric?), and finally "also used extensively", which is to my measures a confession of "we don't really know" (maybe not too bad).


 * Maybe I miss the pre-Cartesian juxtaposition of geometry and algebra, then possibly more intimate to arithmetic, when we now have algebraic geometry and geometric algebrae. Maybe the latter use of algebra (a. of a vector space, exterior a., ...) as opposed to algebra as the abstract treatment of algebraic structures (this is mentioned) is also worth mentioning. I certainly do not miss elaboration of algebra on various levels of education.


 * I am honestly sorry for sounding this deprecatingly, please, just take my opinion lightly, not any slightest offence is intended. Purgy (talk) 08:07, 25 October 2018 (UTC)


 * Considering your feedback, Purgy, I have updated my proposal; I would like you to keep in mind however, that many of your objections apply to the current section as well, such as the "also used extensively" which is in the current revision of the article as it is. As for your idea about "algebraic geometry" part, I have added a small paragraph, although I admit that I do not know much about that particular subject, and would invite you to elaborate on it more as an addition.



Algebra is a complex branch of mathematics in which many subjects are vastly different from others. Essentially, algebra is manipulation of symbols and operations based on given properties about them.(citation: I. N. Herstein, Topics in Algebra, "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964) For instance, elementary algebra is about manipulating variables, which are abstractions of numbers in a number system. The variables in the number system are only allowed to have properties that are shared by every number it represents, and vice versa.

The most simple parts of algebra begin with computations similar to those of arithmetic but with variables that take on the properties of numbers.(citation: ref name=citeboyer) This allows proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation
 * $$ax^2+bx+c=0,$$

where $$a, b, c$$ are any given numbers (except that $$a$$ cannot be $$0$$), the quadratic formula can be used to find the two unique values of the unknown quantity $$x$$ which satisfy the equation, known as finding the solutions of the equation. Historically, the study of algebra starts with the solving of equations such as the quadratic equation above. The study of these equations lead to more general questions that are considered, such as "does an equation have a solution?", "how many solutions does an equation have?", and "what can be said about the nature of the solutions?". These questions lead to ideas of form, structure and symmetry.(citation: )

Algebra also considers entities that do not stand for just one number; using sets of numbers as algebras results in the ability to define relations between objects such as vectors, matrices, and polynomials. Many of these and the previously mentioned manipulation of variables form the basis of high school algebra.

Because an entity can be anything with well defined properties, it is possible to define entities that are unlike any set of real or complex numbers. These entities are created using only their properties, and involve strict definitions to create a set. The entities, along with defined operations, form algebraic structures such as groups, rings, and fields. Abstract algebra is the study of these entities and more.(citation: http://abstract.ups.edu/download/aata-20150812.pdf Retrieved October 24 2018)

In geometry, algebra can be used in the manipulation of geometric properties; the interplay between geometry and algebra allows for studies of geometric structures such as constructible numbers and singularities. Reducing properties of geometric structures into algebraic structures has created subjects such as algebraic geometry, geometric algebra, and algebraic topology.

Today, the study of algebra includes many branches of mathematics, as can be seen in the Mathematics Subject Classification(citation: ) where none of the first level areas (two digit entries) is called algebra. Algebra instead includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used in 14-Algebraic geometry and 11-Number theory via algebraic number theory.


 * Although it may not be perfect, I certainly think it is better than the current section, and I would like this compared to that. Thank you, IntegralPython (talk) 16:20, 25 October 2018 (UTC)
 * The manual of style and the common sense recommend starting with as few technicalities as possible, and adding technicalities progressively. This is respected in the current version, but not in the proposed one. In particular, the first paragraph involves a synthetic view of mathematics, which outside the knowledge of most readers.
 * The structure of the current section is
 * Paragaph 1: introduction of the basics of algebra that almost everybody knows, with emphasize on its difference with arithmetic (which is less commonly understood)
 * Paragaph 2: the main historical problems of algebra, and how they lead to the concept of algebraic structures, As, for many people, including many historians of mathematics, "algebra" is synonymous with "theory of equations", the relationship between these two subjects must be clarified.
 * Paragaph 3: Relationship between algebra and other branches of mathematics
 * Paragaph 4: Parts of moderns mathematics that belong too algebra.
 * The discussion on possible edits of the article must distinguish between the modification of the structure and the modification of the content of each paragraph. My personal opinion is that the structure of the proposed version is less clear than the current one, and this deserves a discussion.
 * The content of the paragraphs of the proposed version has several issues. My main concern is the theory of "entities" that seems IN:OR and does not appear in any algebra textbook the I know of.
 * Also, the sentence "These questions lead to ideas of form, structure and symmetry", which appears also in the current version is completely wrong, as, firstly this is geometry, not algebra the led to the ideas of form and symmetry, and, secondly, nothing leads to the idea of structure: structures are elaborated as abstractions allowing a better understanding of some problems. Generally, understanding the deep structure of a problem is done by the greatest mathematicians. In algebra one may cite, for example, Galois, Gauss, Hilbert, Grothendieck, who all have deeply changed algebra by understanding hidden structures. I'll remove this sentence in the article. D.Lazard (talk) 16:00, 26 October 2018 (UTC)

Elementary and Abstract Algebra in the First Paragraph
The first paragraph of the page contains the sentence "The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra." I propose a couple of changes to make it more formal. TechnicallyTrue376 (talk) 23:50, 30 October 2018 (UTC)
 * Change "more basic parts" to "basic areas." The word "parts" reads as informal and "more" is unnecessary as it's not being compared to anything yet.
 * Change "more abstract parts" to "more advanced areas." "Parts" has the same issue as before, and it is not helpful to use the word "abstract" when defining abstract algebra.


 * Done, partially. Cheers. ---Wikaviani  (talk) (contribs)  00:16, 31 October 2018 (UTC)
 * Undone, because it is disputed whether elementary algebra and abstract algebra are scientific areas. IMO, these are only course titles or course subjects. D.Lazard (talk) 08:56, 31 October 2018 (UTC)
 * Nobody said that elementary algebra and abstract algebra are scientific areas. The edit aimed to remove the word "part" and replace it with "areas", per the rationale given by . If there is no consensus about that, i'll not insist anymore. Cheers. ---Wikaviani  (talk) (contribs)  15:15, 31 October 2018 (UTC)

Algebra is about combining "objects"
D.Lazard, I do not think "object" is a jargon here as it was used in (Herstein 1964) (i.e., "An algebraic system can be described as a set of objects together with some operations for combining them", see references). --Habil zare (talk) 05:14, 3 May 2019 (UTC)
 * This is a general article for an audience which should not be restricted to people with skills in mathematics. So, if you use a word with a meaning that is not its common meaning, you must define or link it. This is the case here, as, for everybody, an object is a physical object (a chair for example) not an abstract entity. Considering an abstract entity as an object is indeed mathematical jargon. If you look at object (disambiguation) you will see three mathematical meanings of "object", two of them are clearly far from your intended meaning, and the other, mathematical object, includes your meaning, but is much wider, as it includes, for example algebraic systems in the sense of Herstein. I do not see any way of using "object" here in a way that cannot be misinterpreted. So, the best is to not use this word.
 * Also, Herstein does not defines algebra, but algebraic structures, and does not talk of objects in general but of "a set of objects"; that is, it defines implicitly his objects as the elements of a given set. So your quotation is not helpful here.
 * Finally, there are many books on algebra, but few give an definition "algebra". Moreover, the scope of algebra has evolved over the time. So a single reference is not sufficient here. The present formulation results from a IN:consensus among editors, and any change needs a new consensus. This alone would be sufficient for reverting your edit. D.Lazard (talk) 08:04, 3 May 2019 (UTC)
 * What does the word "manipulate" mean in "rules for manipulating these symbols"? We do not change or edit symbols in algebra. We use them to write sound and valid formulas, and algebra has rules for changing (manipulating) the formulas in a way that they remain valid. However, I think this is too general, and applies to other fields of mathematics. The closest field to match this definition is mathematical logic not algebra. Symbols and formulas are used in all fields of mathematics for example in analysis, if $$x$$ is a real number and $$f$$ is a function, then $$f(x)$$ denotes the image of $$x$$. The special thing about algebra is that when we combine the symbols in an algebraic formula, we usually do not think of anything else other than the "combination" itself. That is, when we write $$ab$$ for $$a$$ and $$b$$ as members of a group, we do not pay much attention to what $$a$$ and $$b$$ are (i.e., numbers, matrices, functions, etc.) but in algebra, we are interested to see how they combine and what are the properties of this combination with regards to other objects and combinations we are studying. "Is the function $$f$$ differentiable at $$x$$?" is not an algebraic question because it is about the properties of the object (i.e., $$f$$) thus this is a question in analysis. In contrast, $$f+g=g+f$$ is an algebraic sentence even when we know $$f$$ and $$g$$ are real valued functions because here we are talking about their combination not the identity or properties of each object. --Habil zare (talk) 13:17, 3 May 2019 (UTC)
 * Ingentionary definition of "manipulate: "To move, arrange or operate something using the hands". This is exactly what is done in algebra: "arranging" symbols into formulas and transforming them, using rules. The group axioms is an example of such rules. You wrote "The special thing about algebra is that when we combine the symbols in an algebraic formula, we usually do not think of anything else other than the combination itself". This is exactly what the article says by saying that in algebra one manipulates symbols (and not the objects represented by these symbols, which may have further properties).
 * Nevertheless this page is devoted for discussing how improving the article, not for expressing personal opinions about the subject of the article. So, if you have an edit suggestion that can be the object of a consensus, propose it, and if, possible, give a source for it. You introduction of the word "object" is such a suggestion, but, as far as I know, there is no chance for getting a consensus about it. If you have strong arguments against a formulation, give it, but I don't like it and other personal opinions are not good arguments. Personally, I do not see anything in your last post that can be used to improve the article. D.Lazard (talk) 14:32, 3 May 2019 (UTC)
 * The phrase "rules for manipulating these symbols" is not appropriate to describe algebra in this article because: 1) it is not directly mentioned in, but inferred from, the reference after it, 2) it is not "specific to algebra", e.g., in the formula $$d(x,y)=d(y,x)$$ for a metric $$d$$, we move the symbols $$x$$ and $$y$$ around but metric spaces are studied in analysis not algebra, and 3) it is far from "completely describing algebra" because many of the propositions and discussions in algebra do not need symbols. Example 1: A subgroup of any Abelian group is Abelian. Example 2:Any finite alternative division ring is necessarily a finite field (Artin–Zorn theorem).--Habil zare (talk) 17:27, 3 May 2019 (UTC)