What does "formal" mean? I know the definition of formal power series, power series and polynomials. But what does the adjective "formal" mean? In google English dictionary, does it mean "9. Of or relating to linguistic or logical form as opposed to function or meaning" or maybe another one in the link? 
Or does "formal" have some mathematical meaning which is other than usual dictionary meaning?
 A: When I was learning about logic as an undergraduate, I recall being told that the word "formal", with respect to "formal languages" meant that the "form" of expressions written in that language had primacy.
In other words, rules for manipulating expressions in a formal language could be given in terms of the form of the expression only, without needing to know to what values the variables in the expression were bound.
So a formal language permits us to use relatively simple pattern-matching algorithms to decide which transformations of an expression are valid at any given time.
In this context, formality is linked to the simplicity of the rules that define the set of valid transformations of an expression.
A: As an example,  formal power series is analyzed without regard to convergence.  Really, what is of interest is the sequence of coefficients.  
A: I see formal used in at least two senses in mathematics.


*

*Rigorous, i.e. "here is a formal proof" as opposed to "here is an informal demonstration."

*"Formal manipulation," that is, manipulating expressions according to certain rules without caring about convergence, etc.


Confusingly they can mean opposite things in certain contexts, although "formal manipulations" can be made rigorous in many cases.
A: The word "formal" in "formal power series" is indicating that you are considering all objects that are algebraically "like a power series". This is opposed to its use in analysis where you spend a lot of time figuring out for which $x$ the series converges. 
Basic analysis goes like this: 
"$\displaystyle\sum_{n=1}^{\infty} x^n$ is a series which converges for $|x|<1$ and therefore the function $f(x) = \displaystyle\sum_{n=1}^{\infty} x^n$ has the domain $|x| < 1$".
You then proceed to use the function and talk about derivatives and integrals on the restricted domain. If the series has very few points of convergence such as $\displaystyle\sum_{n=1}^{\infty} n!x^n$ which converges only for $x=0$, then casting it as the function $g(x) = \displaystyle\sum_{n=1}^{\infty} n!x^n$ can only have domain $x=0$ and its value is $g(0)=0$. Pretty boring function when it comes to derivatives and integrals!
When you study formal power series, you ignore the consideration of convergence and use the series as it is presented as an algebraic entity, so even though $g$ only converges at $x=0$, you ignore that and focus on other properties of the series.
Another common use of the word "formal" is with a "formal system" which is basically a big rulebook for an artificial language comprised of an alphabet (a list of symbols), a grammar (a way of arranging those symbols), and axioms (initial lists of symbols to start from). The word "formal" here is needed because it is very prim and proper and only allows manipulations according to the grammar and axioms; you can't combine symbols in any way like you can in English (for example this ee cummings poem is an "acceptable" combination of the symbols of English, but is also seemingly "wrong" according to our standard grammar).
A: And don't forget the notion of formal space arising in rational homotopy theory.
A: Formal proof systems
One context in which the word "formal" comes in, is that of formal proof systems.
A formal proof system is a way to write theorems and their proof in the computer, such that after it have been done the proof can be automatically verified by a computer.
In such systems, theorems are just sequences of characters (strings), and starting from your axiom strings, you use a few well defined rules to transform those strings mechanically, and obtain new true strings (thus making a proof).
The huge advantage of such systems is that since each proof step is so simple and mechanical, computers can verify proofs, which can be an extremely difficult and error prone task for humans to do!
There are also cases where the proof itself requires tedious verification of thousands of cases, and would be too time consuming for any human. One notable example of this is the four color theorem.
The downside of such systems is that they are much harder to write your proofs in, because in order to communicate with the computer you have to write everything in a very precise way.
I do believe however, that if such systems are done well enough with good tooling and standard libraries, writing a proof should be no harder than writing a computer program, and the benefits would largely outweight the greater difficulty of writing the proof.
For a concrete well presented example, a have a look at Metamath's awesome proof that 2 + 2 = 4 http://us.metamath.org/mpeuni/mmset.html#trivia
Metamath is an older proof system, and there are likely better choices today as I have mentioned at: What is the current state of formalized mathematics? but their web presentation is very nice!
Such proofs require of course defining everything in terms of things that the proof system understands. In the case of Metamath, Zermelo–Fraenkel-like set theory is used to my understanding.
A TL;DR version of the classic set theory approach would be:


*

*we can use sets, forall, exists and modus ponens

*the naturals can be defined in terms of sets like this: https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers

*the rationals can be defined easily as an ordered pair of integers. Ordered pairs can be defined in terms of sets easily with Kiratowski's definition, see also: Please Explain Kuratowski Definition of Ordered Pairs

*the reals can be defined in terms of sets with Dedekind cuts, see also: True Definition of the Real Numbers

*functions are just a set of pairs: Is $f(x) = (x + 1)/(x +2)$ a function?

*once we have reals and functions, note how the epsilon delta definition of limits only uses concepts that we have previously defined: functions, reals, forall and exists! Once you see this, it is easy to believe, that, at least, we can formalize real analysis with this simple system 


The fact that mathematics can be fully formalized is surprising, and was arguably only fully realized at the end of the 19th century, in particular through the seminal Principia Mathematica, and materialized with the invention of computers.
This is in my opinion the property of mathematics that best defines it, and that which clearly separates its preciseness from other arts such as poetry.
Once we have maths formally modelled, one of the coolest results is Gödel's incompleteness theorems, which states that for any reasonable proof system, there are necessarily theorems that cannot be proven neither true nor false starting from any given set of axioms: those theorems are independent from those axioms. Therefore, there are three possible outcomes for any hypothesis: true, false or independent!
Some famous theorems have even been proven to be independent of some famous axioms. One of the most notable is that the Continuum Hypothesis is independent from ZFC! Such independence proofs rely on modelling the proof system inside another proof system, and forcing is one of the main techniques used for this.
A: Formal Logic as a basis for propositional logic and boolean algebra
"Formal" power series seems very niche.
To me, "formal" is related to "formal logic", which is the most foundational mathematic and philosophical basis for essentially all propositional logic and "formal" proofs in algebra and number theory. Others have already described what "formal" means in "formal" proofs: that they need to be rigorous and adhere to systematized rules that build from each other into broader and broader hierarchies.
It starts from philosophical ideas about reality, existence, and truth, and metabolizes into a "formal" systemized expression in logic. These expressions  are tantamount to equivalence relations and various identities that we use in algebra and boolean algebra, that are, in turn, often required to prove things at a more abstract level with confidence that the results are cohesive to everything established at lower levels of the ladder.
These can further be expressed in the form of code to create formal equivalence checking and things of that nature that use compute resources to prove things instead of relying on the poor accuracy of the hand-written detail.
That's what I think when it comes to "formal" in math.
Formal proof-based math to solve Induction and Count
Building off of propositional logic, strong induction is an important technique in proving that a sequence of cases is true, and has strong ties to trees and applications to recursion in functional programming. It can also be used to assert that a function in code should behave in a certain way across a sequence of cases.
Counting in both ways is a formal proof-based methodology that's perhaps the most important technique in combinatorics. It involves proving that two combinational expressions are equivalent by demonstrating that they count the size of the same set. Proving that a function is bijective is a form of double counting, and this tool can also be used to formally prove modular arithmetic theorems like Little Fermat.
Biases and logical fallacies
It's an underappreciated topic, but defining and identifying fallacies and bias is crucial, and this is contingent on formal propositional logic that we've already mentioned.
This became a topic of great note first in classical periods, particularly in Greece in the 5th through 3rd centuries B.C.E. Renewed interest peaked during the European Enlightenment era and Scientific Revolution. It waned and hasn't picked up again in some time. After all, there's social media instead.
The Scientific Method
The entire thing is a formal mathematical exercise. You establish a hypothesis, test it fairly while maintaining a control group as a set of items with which to compare against the test group, and you draw a conclusion based on the output. It must be rigorous enough to be reproducible. To get published, the paper must be screened through a formal review by expert peers in the field of relevance. Others will attempt to recreate your test.
Also, a lot of formal descriptions of imperative math, including algebra, calculus, statistics, and probability theory is communicated in the process.
While it's used for very interesting but funny and niche parts of science, like fish mating habits, it's also used to exhibit a pharmaceutical drug's safety, and other scary important things, as well as findings in emerging fields like robotics, ML, AI, autonomous systems, networked sensor arrays, and gene editing, which are all fascinating, but also equally as scary if they go wrong, justifying significant formal scientific techniques based on formal use of math.
Ethical publication of statistical studies through formal testing
Statistical studies inform us about a whole host of things, but the numbers are meaningless unless they've been vetted through formal tests and logically work free from bias and fallacies (see earlier section). The problem with biases and fallacies when it comes to statistical studies is that bias is hard to recognize in oneself. Which is why these also need to be vetted by peers.
The tests used to analyze the data which has been acquired – such as p-tests, t-test, and chi-sq tests – all have a basis in probability and are, like almost all mathematics, described in excruciating detail.
If they don't undergo the requisite formal math progressions, then the data can be malformed, or the interpretation of the data can be completely wrong.
Formal methods in Safety-Critical design and security
When money, equipment, machinery, property, and lives are on risk, it's critical to go through formal procedures in order to reduce the likelihood of system faults or compromise. Which is why standards (such as IEEE, ISO, and MISRA) exist, are produced and followed by related industries. FTAs, HAZOP examinations, sequence diagrams, use case scenarios, and test plans are all drawn out. Source code and design documents are peer-reviewed. Unit and system test results are deeply scrutinized. And test fleets are burned in stressed to the brink of failure and perfected to 6-sigma of reliability.
Everyone from Otis (a designer of elevators) to Lockheed Martin (specializing in aircraft) commit to some of these industry-standard ways of development because a formally-designed and formally-tested product is far more likely to function as intended compared to one manufactured using shoddy practices.
You might ask what this has to do with math. It has EVERYTHING to do with math! It's all probability theory: what's an acceptable probability that a car fails to decelerate? what's an acceptable probability that a spacecraft's stabilization systems fail? and an acceptable probability of someone being able to access a bank account or commit identity theft? All of these things are defined by formal procedures, calculated with formal rigorous math, and mitigated with techniques that have a basis in probability theory and number theory, such as elections in containment regions, physical redundancy, and combinational intractability. We can't forget the applied side of things, as it saves lives and improves product lifetime of usefulness.
