math, are those conditions in the definition of a field, natural?

So, in algebra, you have a field. It's a set and 2 functions f and g, of the form f(a,b) and g(a,b), and X nesting properties of f and g (called commutativity, associativity, distributive, invertible, etc).

So, what happens if we have more than 2 functions, 3 parameters each? So, such study is called universal algebra. (i haven't studied, but i wonder what happens there, in general. The mix of nesting of function, i gather, would create more complex concept similar to associativity and distributive, involving 3 functions, but we don't have a name.)

But, WHAT is the fundamental nature, that real number (a field) is this specific X nesting properties? What is it, that real numbers, which we consider as naturally occurring or developed, form this “field” with such specific nesting properties?

am thinking, there must be some logical answer.

to describe my questions further, for example, real number developed because, first we have counting, 1, 2, 3, then naturally we developed 0, then we have rational, which is ratio, e.g. 1/2 as cutting a pie, then we have negative numbers (from, say, I OWE YOU). And from rational we discovered irrational, as in pythagorean. So there, we have real numbers. And, addition came from simple counting. Multiplication can be considered as a short for repeated additions.

so, am guessing, addition, and multiplication (repeated addition) necessitates the commutativity, associativity, distributive, properties?

now, having written this out, it seems obvious and is the answer to my own question.

... because, by looking at the definition of field, i've always thought, it's somehow arbitrary and complicated. I'd be interested, in a systematic approach of studying structures, e.g. a set, with n operation of m-arity, starting with n=1 and m=1. Then, we develop, all possible ways of nesting n such functions of m arity, so that associativity, commutativity (order of arg), distributivity, are just 3 of the possible properties.

but i gather that, universal algebra may began like this, but actually has become a bit something else.

similar situation is group theory

Group, is much simpler than field. But if you look at definition of group, you see that, it seems also arbitrary and complex.

But then, if you look at symmetry, such as symmetry of polyhedra, you see that all the requirement of group is necessary, and no more, no less.

but the question remain, why is the group definition seems arbitrary?

i mean, is there some point of view, so the associativity (a • b) • c = a • (b • c) requirement, dissolved as if it is natural?

again, i really like to see, a combinatorial exploration of all such possible condition of n functions of m arity. (as in universal algebra)

perhaps after seeing that, then one can judge, comparatively, whether the associativity condition is natural.

now, to be sure, function with 1 or 2 arity is actually perhaps the most natural. And 1 or 2 functions is also pretty bare. As opposed to, a structure with function that has 3 arity, or more than 2 functions.

If you just have 1 function, or just function wit 1 arity, than it may become too simple to have interesting things going on.

but again, would like to see a systematic combinatorial list of the conditions that may arise of n functions and m arity.

So, in algebra, you have a field. It's a set and 2 functions f and g, of the form f(a,b) and g(a,b), and X nesting properties of f and g (called commutativity, associativity, distributive, invertible, etc).

So, what happens if we have more than 2 functions, 3 parameters each? So, such study is called universal algebra. (i haven't studied, but i wonder what happens there, in general. The mix of nesting of function, i gather, would create more complex concept similar to associativity and distributive, involving 3 functions, but we don't have a name.)

But, WHAT is the fundamental nature, that real number (a field) is this specific X nesting properties? What is it, that real numbers, which we consider as naturally occurring or developed, form this “field” with such specific nesting properties?

am thinking, there must be some logical answer.

to describe my questions further, for example, real number developed because, first we have counting, 1, 2, 3, then naturally we developed 0, then we have rational, which is ratio, e.g. 1/2 as cutting a pie, then we have negative numbers (from, say, I OWE YOU). And from rational we discovered irrational, as in pythagorean. So there, we have real numbers. And, addition came from simple counting. Multiplication can be considered as a short for repeated additions.

so, am guessing, addition, and multiplication (repeated addition) necessitates the commutativity, associativity, distributive, properties?

now, having written this out, it seems obvious and is the answer to my own question.

... because, by looking at the definition of field, i've always thought, it's somehow arbitrary and complicated. I'd be interested, in a systematic approach of studying structures, e.g. a set, with n operation of m-arity, starting with n=1 and m=1. Then, we develop, all possible ways of nesting n such functions of m arity, so that associativity, commutativity (order of arg), distributivity, are just 3 of the possible properties.

but i gather that, universal algebra may began like this, but actually has become a bit something else.

similar situation is group theory

Group, is much simpler than field. But if you look at definition of group, you see that, it seems also arbitrary and complex.

But then, if you look at symmetry, such as symmetry of polyhedra, you see that all the requirement of group is necessary, and no more, no less.

but the question remain, why is the group definition seems arbitrary?

i mean, is there some point of view, so the associativity (a • b) • c = a • (b • c) requirement, dissolved as if it is natural?

again, i really like to see, a combinatorial exploration of all such possible condition of n functions of m arity. (as in universal algebra)

perhaps after seeing that, then one can judge, comparatively, whether the associativity condition is natural.

now, to be sure, function with 1 or 2 arity is actually perhaps the most natural. And 1 or 2 functions is also pretty bare. As opposed to, a structure with function that has 3 arity, or more than 2 functions.

If you just have 1 function, or just function wit 1 arity, than it may become too simple to have interesting things going on.

but again, would like to see a systematic combinatorial list of the conditions that may arise of n functions and m arity.

- I've never seen a systematic analysis of all possible equations that can hold on n functions of arity m. What universal algebra actually has done is:

1) prove lots of great theorems that unify and generalize the results we know for familiar algebraic structures like groups, rings etc.

2) classify, to some extent, the possible equations that can be added to the usual equations governing familiar structures like groups or rings.

For example, a**semigroup**is a set with an associative binary operation. A**band**is a semigroup where element obeys xx = x. People have classified all extra equations that one can add here - that is, people have classified all "varieties of bands". You can see some of them here:

https://en.wikipedia.org/wiki/Band_(mathematics)#Lattice_of_varieties5d - +John Baez that's the best answer!

yesterday, after reading your reply, i went and read semigroup on wikipedia. With your mention, reading it makes a lot sense (because before, i was randomly reading them, eg magna, monoid, loop,). There's tons of info just on that semigroup page! and then i went on to read band. What a name! love these names, group field ring band.

And i also learned 2 new definitions of idempotent! Before, i only knew f(f(x)) == f(x). Now i know it also has definition for f of arity 2, and idempotent operator eg f(x,x)==x for all x in domain.

lattice i've tried to scan on wikipedia a couple of times in past year but i never understood. I imagine it's abstraction of integer points on the plane. Or, was it field + order? or ring + order?

the subject of universal algebra and model theory interest me greatly. afaik, model theory is algebraic structure with formal logic approach. Do you know good free books on these you can recommend? cuz wandering thru wikipedia can be quite bewildering. (or, does category theory kinda took them over now? I still don't know anything technical about category theory.)

PS was gonna read more before i reply, but thought i better do now else it gets old.4d - +John Baez i had been thinking, i was wondering if you'd give me sort of a mentorship?

about maybe 10 min a week answering my question, with a focus on a topic. Am interested in logic, model theory, type theory, with ultimate goal of understanding homotopy type theory, or something similar where i can write computer code to do math proof related thing. (understand the subject matter is my first interest. i.e. math foundation via computer based (or computable) proof system)

so am thinking, with your guide, we pick a topic or a book of your choice, i go thru, and maybe ask you question once in a while.

am not sure this would work out, maybe once i have a goal i wander off. But anyway, i thought i ask you. Let me know what you think.

or, last year you answered my question about a pdf your student wrote on category theory. Maybe i can help proof read it? is there something i can do that can be useful to you and while i learn the subject?4d - +Xah Lee - this sounds like an interesting idea. I have a suggestion. Can we do this publicly, either here or on one of my blogs? I think this would be fun. You'd ask questions, and I would answer them, and other people would learn along with you, and ask other questions, and other people would answer them too...

We should pick a topic. I don't like programming much so I'm not good at it. But a lot of category theorists like it, so the topic could include programming, and other category theorists would help out.

You're interested in homotopy type theory. I'm not very interested in that, though it's interesting. But my student Christian Williams and I are working for a company called Pyrofex, run by my former student Mike Stay and another guy, helping them write papers on category theory and computer science. Christian asked Mike:*P.S. Almost forgot - how can I begin playing with process calculi in Scala? Definitely want to start working with them by hand. Do you have any example programs you could show me?*

Mike answered:*Not really, sorry. I'd suggest using K framework instead. There are**lots of small calculi in the tutorials that illustrate K's facilities for modeling programming language features, and I can give you an**almost-complete implementation of rho calculus.*

Later Christian told me:*I'm starting to try out the K Framework, to play with calculi.*

I'm interested in the category theory underlying the pi calculus and rho calculus, which are attempts to formalize the foundations of distributed computing. There's a lot I don't understand about that, which I want to figure out.

I personally have no interest in implementing these calculi on the K framework, and I don't even know what the K framework is! But you might enjoy doing that. Maybe you could help Christian with that.4d - +John Baez great! and public is good!

so, do i just post to google plus and tag you? or, let me know the appropriate forum to use. (the n-category cafe??)

yes, let's pick a topic.

to outline my interest in logic, basically we can do an intro course on any of 1st order logic, model theory, universal algebra, or type theory, sequent calculus. My eventual goal is to understand godel's incompleteness theorems or Tarski's undefinability theorem, and understand Curry–Howard correspondence. Am particularly interested in anything with formal language perspective, i.e. the approach are amendable to be run on computer eventually. (not in any hurry to actually write programs though.)

Give me a list of topics, Wikipedia article, or perhaps a list of homework for a week. Or something like that. (one of your course syllabus?) I'll do them 1 hour a day.

i haven't really heard of process calculi, k-frame work, rho calculus. (i think in your “recent” posts process calculi is mentioned)

ok, so they are about distributed computing.

Why the interest in distributed computing? because they are very much needed today?

But i'd be interested to pick up category theory. (category theory, seems to me, “abstraction raised to the power of 4” ^_^ with few utility. e.g. not really the foundation of computer based math, nor directly practical in everyday math. Am i totally wrong about that?)

scala i heard of a lot, though not thrilled to code it, anything from Java, since it runs on jvm.

i thought these lines of work would be done in haskell, ocaml, coq, etc. Which i'm somewhat familiar except coq, and very much interested to dive in.

(differential geometry, and complex analysis, as intro course , would be great too, but these two i know what sequence to follow.)

i saw Mike's work, that nested game of life on YouTube. One of the greatest spine chilling thing i've seen. (i spent years playing game of life in 1990s) Later learned he's the author via your posts.3d - +Xah Lee - okay, let me think a little about what to talk about. It should probably be something involving category theory and computation - for example, the Curry-Howard correspondence, the lambda calculus, and stuff like that.

"category theory, seems to me, “abstraction raised to the power of 4” ^_^ with few utility. e.g. not really the foundation of computer based math, nor directly practical in everyday math. Am i totally wrong about that?)"

I don't know what "everyday math" is. I don't need category theory to balance my checkbook. But I need it to do modern mathematics: it clarifies and connects everything, and eliminates unnecessary grunge.3d

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