You can have a lambda calculus, the calculus of the real numbers or a logical calculus but on the other hand you could also have an algebra of sets, a Lie algebra, or a linear algebra.  Is there any convention which dictates whether something is an algebra or a calculus?  From my understanding, a calculus seems to give you information about objects you might encounter within a space, while an algebra gives you information about the interactions and structure within the space as a whole. Is this correct?
123250
I sometimes see phrases like 'the relational algebra' or 'the lambda calculus'. What is the difference between an algebra and a calculus?
49364
Why is it that math so focuses on the subclass of relations known as functions? I.e. why is it so useful for us in nearly all branches of mathematics to focus on relations which are left-total and left-unique? Left- (or even right-) totality seem to be intuitive, since if an element doesn't appear in the domain, we might throw it out. But why left-uniqueness?  I'm looking for something like a "moral explanation" of why they would be the most useful subclass of relations.  My apologies if this is a previous question; I looked and didn't find much.
62605
I know they are important in abstract algebra, but why do people study them? Why are they so important to study? Do they make certain things easier to understand?
122027
"Ring" and "semiring" are concepts defined both in algebra and set theory.  In Algebra  A  is a set R equipped with two binary operations + and · called addition and multiplication, that Addition (+) is abelian, Multiplication (⋅) is associative, Multiplication distributes over addition, and Multiplicative identity (1) exists.  A ,  is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.   In Set theory  A  is a family of sets closed under unions and set-theoretic differences.  That is, it obeys the two properties $$A \setminus B \in \mathcal{R} $$ $$A \cup B \in \mathcal{R}$$ This implies that it is also closed under intersections, $$A \cap B \in \mathcal{R}$$  A  is a non-empty collection S of sets such that   $\emptyset \in S$ If $E \in S$ and $F \in S$ then $E \cap F \in S$. If $E \in S$ and $F \in S$ then there exists a finite number of mutually disjoint sets $C_i \in S$ for $i=1,\ldots,n$ such that $E    \setminus F = \bigcup_{i=1}^n C_i$.   I wonder, are such definitions somehow linked or equivalent? or they are just coincidence that both fields of mathematics used the same terms?
51139
I am interested in understanding how mathematics is divided into many categories, such as what categories are particular cases of what, what categories do not or have little overlap with what. This is meaningful to me, because it helps me get a big picture and not mess up  many categories.   Quoted from :     think if "Algebra", "Geometry",   "Analysis", "topology, "Number   Theory", etc. as 'first-level   subjects'; then you have algebraic   number theory, algebraic topology,   analytic geometry, etc., as   'second-level subjects.' Now we have   algebraic arithmetic geometry, a   'third level subject'   I was wondering what criterion is used to divide mathematics into the first level subjects?  My understanding for these subjects are:  The objects studied in algebra are sets with operators with some properties, and mapping between such sets. So algebra is dealing with general and abstract objects.  Geometry is, quoted from Wikipedia:      a branch of mathematics   concerned with questions of shape,   size, relative position of figures,   and the properties of space.   To make these questions meaningful, is the space a general inner product space? Or must it be a particular one, an Euclidean space $\mathbb{E}^n$? In either case, geometry is dealing with some kind of topological vector space, which seems to be more concrete than algebra.  The subjects studied in analysis are derivatives and integrals of some mapping between some sets( others that I miss?). To make derivative concept meaningful, the domain and codomain of the mapping must be Banach spaces (?); to make integral concept meaningful, the domain and codomain of the mapping must be measure space and Banach space respectively(?).   Topology is about neighborhood of each element in a set, defined as a class of subsets that are closed under arbitrary union and finite intersection. This is also quite general and abstract.  Number theory is about properties of natural, integer, rational, real, complex, algebraic numbers, that are represented in terms of four specific operators $+, -, \times, \div$. This is quite concrete.  In summary, the first-level subjects algebra, geometry, analysis, topology and number theory seem not stand at the same level of abstraction or concreteness. Is there a criterion or reason for dividing mathematical topics into these first-level subjects? There are also other categories of mathematics, such as set theory, category theory, logic and measure theory, which especially the first three seem quite general and each does not very much overlap with other categories of mathematics, including algebra, geometry, analysis, topology and number theory. So what kind of criterion is used to form these other categories? Are there other criteria for forming mathematics categories?   Thanks and regards!
61538
From a physicist's perspective there are several situations in which somehow arbitrary choices of mathematical structures can be made. One can describe a system from different perspectives, etc. without changing anything in the physical properties. (see gauge invariance, picture changing etc.) In the situation when these choices impose a mathematical structure one has limitations in the subsequent choices (are these limitations physical???). However, there are situations when this prescription does not hold. The best example is in this case the Black-Hole complementarity principle (if it is well defined in the first place). So, the question about how mathematical structures "interact" in a logical way is not a bad one. Again: what is the interaction between, say, locality and Hausdorff-ity in the case of a black hole? Or what is the true relation between metrizability and hausdorff-ity of a space in the context of a black hole? The question about how mathematical structures assigned (innocently) at some point interact in different situations is in my opinion a quite relevant one...
120688
I'm thinking of $f$ as a function either from the reals to the reals or from the naturals to the naturals.   Edit: Okay, what if $f$ is from the naturals to the naturals?
54183
When I make a claim in a proof that a mathematical entity exists, is this no more than saying that the theory I'm working within is consistent, and that all the steps upto that point in the proof are allowed moves in the theory?   By mathematical entity I mean a number, say 3; or an algebraic system such as the group of rotations of the square or the ring of natural numbers; or the category of varieties (a variety is an algebraic system with a fixed number of operations). It could also be a circle, or the category of all manifolds. It could be set theory axiomatized in ZFC.  Not only do these entities relate to each other (for example, after 2 is 3; the group of rotations of a square is related to that of a cube) but also they have structure (3=2+1 in the ring of natural numbers; or the circle, a 1-dimensional manifold is made up of 0-dimensional points)    Existence is a loaded word, we experience in our everyday lives with the existence of chairs and tables or the colour red. It seems very odd that the same word is used for entities that are unavailable directly to the senses, but at the same time it is 'oddly' appropriate.   Taking the platonic perspective - that these entities exist in their own world - and that we access this world through an effort of will. Then supposing that our own physical universe does have a complete mathematical theory. Then the entities in this theory, as well as the theory itself appear to exist in this other separate world, and our own physical world (with us in it as we are physical beings) appears to be incarnated in someway in this other world. This seems odd. Perhaps this suggests that the universe cannot have an underlying mathematical theory?  On the other hand, taking an entirely formal point of view seems (to me) deeply unsatisying. Its an interesting/entertaining point of view, granted.   For one thing, a formalisation appears to be only a perspective or an approximation. For example I have a circle in mind, I could choose to formalise this as a figure of plane geometry as Euclid originally did or a 1-dimensional smooth manifold. It is approximate, as formalisations have changed over time, and are likely to keep on doing so.
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I posted this question nearly 10 days ago, but am still really not satisfied with the answers I got, I have no prior education in abstract algebra, group theory, or other abstractions, and most of the answers I got involved defining sets, groups and other things I am unfamiliar with, my question was,  How does one know the notion of real numbers is compatible with the axioms defined for complex numbers, ie how does one know that by defining an operator '$i$' with the property that $i^2=-1$, we will not in some way contradict some statement that is an outcome of the real numbers.  For example if I defined an operator x with the property that $x^{2n}=-1$, and $x^{2n+1}=1$, for all integers n, this operator is not consistent when used compatibly with properties of the real numbers, since I would have $x^2=-1$, $x^3=1$,thus $x^5=-1$, but I defined $x^5$ to be equal to 1.  How do I know I wont encounter such a contradiction based upon the axioms of the complex numbers.  I reiterate here again, Please don't use terminology, or abstract math, that hasn't been taught in say a thorough collage level calculus 1 course.  Here's a link to the same question I asked 8 or so days ago:
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I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra).  Mathematics is a system of axioms which you choose yourself for a set of undefined entities, such that those entities satisfy certain basic rules you laid down in the first place on your own.  Now using these laid-down rules and a set of other rules for a subject called logic which was established similarly, you define certain quantities and name them using the undefined entities and then go on to prove certain statements called theorems.  Now what is a proof exactly? Suppose in an exam, I am asked to prove Pythagoras' theorem. Then I prove it using only one certain system of axioms and logic. It isn't proved in all the axiom-systems in which it could possibly hold true, and what stops me from making another set of axioms that have Pythagoras' theorem as an axiom, and then just state in my system/exam "this is an axiom, hence can't be proven".  EDIT : How is the term "wrong" defined in mathematics then ? You can say that proving fermat's last theorem using the number-theory axioms was a difficult task but then it can be taken as an axiom in another set of axioms .  Is mathematics as rigorous and as thought-through as it is believed and expected to be ? It seems to me that there many loopholes in problems as well as the subject in-itself, but there is a false backbone of rigour that seems true until you start questioning the very fundamentals .
56427
If a function is a combination of other functions whose derivatives are known via composition, addition, etc., the derivative can be calculated using the chain rule and the like. But even the product of integrals can't be expressed in general in terms of the integral of the products, and forget about composition! Why is this?
19500
What is the difference between "algebra" and "field"? In term of definition in Abstract algebra. (In probability theory, sigma-algebra is a synonym of sigma-field, does this imply  algebra is the same as field?)
67867
I understand that  of a function, but while I can easily see how to apply this convention to work with functions instead of numbers in some circumstances, e.g.,  $$\sin(x)\text{ becomes } \sin\text{,}$$ $$\cos(x)\text{ becomes } \cos\text{, and}$$ $$g(x)\text{ becomes } g\text{,}$$ it isn't clear to me how to to express the functions corresponding to the numbers $$x^{2}\text{, or}$$ $$2\cdot x\text{.}$$  Short of defining new symbols, like $S$ or $I$ (which I've seen for the function corresponding to $x \mapsto x$), can one say things like $$\left(\cdot\right)^{2}\text{ for }x^{2}\text{, or}$$ $$2\cdot \left(\cdot\right)\text{ or }2\left(\cdot\right)\text{ or }2\cdot\text{ for }2\cdot x\text{?}$$  What is the general idiom for expressing such functions concisely?
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Do we use abstract algebra in real life?   Is there any application of abstract algebra in real life?
38213
I have the following question:    $\textbf{Question}:$ Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two algebras. Is $\mathcal{F}_1 \cap \mathcal{F}_2$ a $\sigma$-algebra?  I believe the answer is no. I know the definition of an algebra and the definition of $\sigma$-algebra.    I'm assuming I need to show that the intersection of the finite unions of each algebra is finite, but then I get stuck trying to work out how to show it is not a $\sigma$-algebra.  All help is appreciated.  Many thanks, John
124023
(c) Let $\mathcal{L}^*\subseteq\mathcal{L}^{\text{FOPC}}$ be a first order language of predicate calculus consisting of three unary function symbols $f,g,h$.  (i) Write down an $\mathcal{L}^*$-sentence $\phi$ such if $\mathcal{A}$ is an $\mathcal{L}^*$-structure with $\mathcal{A}\vDash\phi$ then the restriction of $h_\mathcal{A}$ to the range of $f_\mathcal{A}$ gives a one-to-one function onto the range of $g_\mathcal{A}$.  (ii) Write down an $\mathcal{L}^*$-sentence $\psi$ such that if $\mathcal{A}$ is an $\mathcal{L}^*$-structure with $\mathcal{A}\vDash\psi$ and the domain $A$ of $\mathcal{A}$ is finite then the cardinality $|A|$ is a square.    Part (i) is easy enough, but I'm having trouble with part (ii). I assume that you can use part (i). My initial idea was to use $h$ to ensure $|Range(f_A)| =|Range(g_A)|$ but I'm not sure where to go from there. Thanks.
100895
Which of the two books is suited for a student looking to learn how to write proofs? I have a working knowledge of calculus and linear algebra but I'm not good at writing proofs. My intention is to learn proofs in general, not necessarily for the two.       The reason I ask is that the latter is suggested on a highly voted question here but the former has a more apt name. The reviews aren't helping. Please don't suggest books other than these.