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I'm a student of math, my problem is that I tend to overformalize and overanalize some things and definitions and so I keep wasting time and effort in pointless things.

For example, one often finds this in calculus books: $\lim_{x\to a}f(x)=l\iff \text{some proposition with epsilons, delta,etc}$. However from my point of view it should be: $\lim_af=l\iff \text{some proposition with epsilons, delta,etc}$, because the $x$ really doesn't add anything, if you put something like $\lim_{x\to a}f(x)=l$ you are really meaning $\lim_a \{f(x)\}_{x\in N_r(a)}$ where $\{f(x)\}_{x\in N_r(a)}$ is a function defined in a neighborhood around $a$ of radius $r$, which radius? anyone that makes sense because you can prove that if two functions are the same around a point then the limit at that point (if it exists) is the same for both functions. Why do we need this $r$?, the problem with the $\lim_{x\to a}f(x)$ notation is that there is really no function being defined, only just an expresion, that is not the problem with the $\lim_af$ notation cause it operates on a function and a point.

AndrewThomas's comment says what happens if we write $\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ just like $\lim_0$ then what variable approaches to $0$?, I've never said that we should write $\lim_0\frac{f(x+h)-f(x)}{h}$, I proposed $\lim_0\{\frac{\{f(x+h)-f(x)}{h}\}_{h\in N_r(0)}$ which clearly says we are defining the function "on h" but that we are also giving it some imaginary domain $N_r(0)$ for "some $r$" (we can actually eliminate the $0$ out of this domain but that's not the point). He is completely right about the part of showing that a function can't have 2 limits at the same point and that before that you can't use the $=$ symbol formally.

Another example would be the integral, for me $\int_a^b f(x)dx=\int_a^b\{f(x)\}_{x\in [a,b]}$ so you actually have to prove that $\int_a^b f=\int_a^bf(x)dx$ by saying that the integral in $[a,b]$ only depends of the value of $f$ at $[a,b]$ instead of just saying that $\int_a^bf=\int_a^bf(x)dx$ really mean the same.

I also dislike the $\partial f/\partial x$ notation which i think is completely useless and informal (but I tried to formalize it without success).

The same goes for Trivial question about a double summation where I actually wanted to "prove" that but is actually obvious.

So I'm just obsessing over this kind of small things that at the end don't really matter. Even if I get over some small thing the next week I will just obsess over another. In classroom no one cares about doing math "properly" and "formal", they instead just want to learn "how" to do things instead of "why" things are this way even if the notation or the arguments they use are completely nonsense. I understand all of the proofs I read (If those proofs are at my level, of course) but these things just don't let me sleep and work peacefully on the problems.

I remember that when I was young, my mathematics was informal, intuitive BUT logical, and I was having fun doing that. I remember, for example, working on problems on polynomials and giving by granted the fundamental Theorem of Algebra by saying "When I grow up I will be able to fully understand this", but now that the time has come, it is not as fun as I thought it would be.

EDIT: I'm doing fine at college, except at physics courses, where I understand nothing about the math/physicis that is being done and I just go along (I really don't care so it's fine). My health is fine but it could be better if I weren't this obsessed.

English is not my first language so apologizes for any grammar errors.

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closed as primarily opinion-based by hardmath, Ayman Hourieh, Thomas Andrews, Pedro Tamaroff, rschwieb Jan 15 at 19:01

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

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Welcome to Math.SE! Your question is really broad, and while I can tell it has a great deal of importance to you, I'm not sure that a thorough answer can be given here. Notation for mathematical concepts often seeks to strike a balance between abbreviation for the sake of quicker reading and exposing all the elements that went into defining that concept. Thus $\lim_{x\to a} f(x)$ includes the $x$ that you think we could do without, but conceals references to epsilons and deltas. The $x$, by the way, is what in logic we call a "bound variable". This level of abbreviation proves to be apt. –  hardmath Jan 15 at 17:58
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I also think that you associate "creating and/or eliminating a notational morass" with formality. –  Arkamis Jan 15 at 18:40
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That might make sense, until you later try to write $\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$. If you write that as $\lim_0$, you don't know which variable is approaching zero. Notation is meant to communicate, not just formalize. –  Thomas Andrews Jan 15 at 19:01
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Did anybody else notice that this poster used a zero in "how t0 stop"? –  rschwieb Jan 15 at 19:08
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A function $f$ is a set of ordered pairs such that if $(a,b)\in f$ and $(a,c)\in f$ then $b=c$. There I'm defining such a set. –  user121566 Jan 15 at 20:25

8 Answers 8

up vote 9 down vote accepted

I agree that the semantics of much mathematical notation is partly non-sensical from a more strict/severe viewpoint, just as ordinary language has such issues.

How to react? On one hand, popular usage is usually adequate to establish the meaning of a notational convention, even if that convention is a little self-contradictory or insensible. So, one looks for the intent, not (perhaps) what is literally said.

At the same time, yes, one should not try to argue to oneself that notations inherited from long ago necessarily make perfect sense, or are optimal. And, at the same time, it is not clear that one should devote too much time to "improving" notation, especially if it is at the expense of intelligibility to others.

Yes, I agree that $\int_a^b f(x)\,dx$ could/should be written $\int_a^b f$. Nothing is lost. This applies as well to integration over a region $\Omega$ in $\mathbb R^n$: instead of $\int_\Omega f(x)\,dx$, just write $\int_\Omega f$.

And, yes, $\lim_{z\to a}f(z)$ can be written $\lim_a f$. (The cases that one wants $\lim_{y\to a}f(x+iy)$ are not numerical limits, and so naturally do require further distinctions.)

But, in the end, these issues are not so much about mathematics itself, but about notation and semantics... which can, indeed, present formidable issues, which become all the more formidable when entangled with mathematical issues.

I really do think the best way to look at the failures or self-inconsistencies or inefficiencies of mathematical notation and language is that it is simply yet-another human language that has evolved, rather than having been carefully designed. Thus, it is unwise to take notation too literally, but, rather, one must know usage.

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+1. Reminds me of Wittgenstein, who (basically) said that no expression has intrinsic meaning, but that every expression can be given meaning by explaining how it is used. –  nomen Jan 15 at 19:36

I have similar problems as you. I hated the symbol $\mathrm d x$ in integrals for a very long time. Then I discovered that it can be interpreted either as a differential form or a hyperreal number, so I wanted to understand these concepts and interpret any $\int f(x) \mathrm d x$ I saw as an integral of a differential form. But since I didn’t quite grasp the concept of a differential form nor the concept of an integral of a differential form (and I still don’t really understand that), I ran into real big problems actually integrating stuff since our Analysis II lectures.

Okay, so first of all I want to say that I haven’t really amended. It probably got even worse since I discovered that there are basic mathematics in which you can work very formally, like basic analysis or linear algebra. (For which I thought that was impossible as well. But since I discovered, it’s possible for them, I try to find formalism in as many things I can and as often as I can.)

On the other hand, I knew these obsessions were holding me back. I once talked to a friend who’s a mathematician I really admire, and he said to me that there’s a level where you move out of the highly polished theories of undergraduate mathematics and starting to do real mathematics. This is the first thing that helped me: Advanced mathematics just can’t be as polished as the well understood theories on which they are built.

I then read Terry Tao’s blog post on There’s more to mathematics than rigour and proofs which also helped me get an understanding of what mathematicians do and why formalism isn’t as popular as I thought it should be. It also helped me feel that simply playing around with complex ideas and only sketching proofs can be fun and rewarding as well.

So, according to this blog post, many mathematicians undergo three certain stages of which you and I are (stuck) in the second one: To us, rigour, proofs and formalism matters. We want to get things right. He then says, that some of the mathematicians move then on to the third phase in which mathematicians actually reason about the material intuitively.

In that context, if you want to get to do real mathematics, I think it’s okay to have a desire for formalism as long as you learn to overcome it. Yet I also think that it might help you understand things more deeply by questioning the way they are presented and looking into steps in proofs that are skipped. You just shouldn’t stop there and get to the real stuff as well.

I myself am trying to think about material more visually to get there. Maybe it’ll help. I often find myself trying to squeeze my imagination into formalism too hard, which probably isn’t a good idea. I’m also trying to use more descriptive language for what I do, explaining ideas by the means of analogies or metaphors.

Sometimes when a step in a proof requires some subtle, little, yet involved argument, I try to make out only the key ideas, express them as good as I can, and leave out the messy details. When I encounter something which I haven’t fully understood yet, I seek the key properties and only work with them telling myself: This is something which probably works that way, let’s just assume it does and see what we get.

The idea behind this is that I explicitly state to myself that I don’t understand the concept or definition as well as I want to understand it, so I that I don’t feel like I’m pretending to understand it. This enables me to actually work with things despite not properly understanding them.

I don’t know whether this helps. My advice to you, however, is to definitely read the above mentioned blog post by Terry Tao. Another advice: There are some textbooks which manage to present good notation/high-standard-formalism with a concise treatment of material. I very much like Commutative Algebra by Atiyah and MacDonald in that regard.

Well, the actual link above didn’t work (probably because it contains a unicode symbol), so I created a redirect via tinyurl. Here is the orginal link:

https://terrytao.wordpress.com/career-advice/there’s-more-to-mathematics-than-rigour-and-proofs/
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Let $f(z)$ be a complex valued function. Let $z = x +iy$.

What would $\lim_a f$ represent. $\lim_{x\to a} f(z)$? $\lim_{x\to \textrm{Re}(a)} f(z)$? $\lim_{y\to a} f(z)$?

Likewise, what does $\int_a^b f$ mean if $f : \mathbb{R}^n \to \mathbb{R}^m$? There's a reason we specify the variable of integration, even if it seems unnecessary in the single-variable case.

Generally speaking, we want our notational standards to work for as many things as possible. We don't want one notation for the single-variable real case, and another for the multi-variable real case, and yet another for the single-variable complex case, and so on.

I suppose what you're trying to do is over-formalize things, but I suspect that what's really happening is that you're not appreciating that the extant notations are precisely as formal as they need to be.

When a theorem says, "there is a neighborhood with some radius $r$," that's not an informal statement just because they don't say what $r$ is. It's a very formal statement. It means, "there definitely exists some radius $r$ that creates a neighborhood that does the thing we need it to do, but we don't care what $r$ is, we just care that it exists."

Lack of precise definitions of the parts does not mean lack of formality. If I point at a Ford Mustang and say "that car is a Ford Mustang," I'm not wrong, even if I don't know who made the sparkplugs.

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I don't really agree with what you're saying here. $\lim_a f$ is perfectly unambiguous notation; its the statement "Let $f(z)$ be a complex valued function" that is incorrect, or at least problematic. –  goblin Jan 15 at 18:02
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So, your proposal is to define ad hoc notation in lieu of typing the two extra symbols $z \to$, the presence of which neither changes the meaning or elides clarity, because it is somehow not formal enough to include them? –  Arkamis Jan 15 at 19:22
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@JLA, I am not misunderstanding what the notation means; see my answer if you need to be convince of this. In any event, progress occurs because someone tried something they have never seen anywhere. Its true that, in the multivariable case, dummy variables are the best solution; and that, in the single-variable case, they're redundant. This is a general principle, and applies in all branches of mathematics. –  goblin Jan 15 at 20:03
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That's a bold statement that is unfortunately false. –  Arkamis Jan 15 at 20:04
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The inclusion of \mapsto doesn't increase formalism, it increases pedantry. If I say "consider the zeros of the function $f(x) = x^2+2x-4$, it's entirely clear precisely what I mean, even if I omit $\mapsto$. Just like it's entirely clear what I mean when I say "my mail was delivered by truk", even though I mis-typed "truck." Symbolic precision in mathematical writing exists only to resolve ambiguity, just like grammatical precision in expository writing exists only to resolve ambiguity. To confuse precise notational formalism with mathematical formalism is a disservice to all involved. –  Arkamis Jan 16 at 0:17

Start to study something very beautiful and very difficult. In the end, the thirst of understanding will exceed obsession for details (only the useless ones).

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Personally, I think "over-analyzing" things is okay (in math, not life!). The people who push mathematics forward are often the one's that ask: "Wait. What do we really mean by this?" For example, I think Cantor falls into this category (what do we really mean by "infinite set") as do Cauchy and Weierstrass (what do we really mean by "infinitesimally close") as do Samuel Eilenberg and Saunders Mac Lane (what do we really mean by "natural")?

So, I think so-called "over-analysis" is less problematic than you seem to believe.

However, if its impacting your grades, then that is another story, and you'll need to work something out.

Anyway, the key is to focus on mathematics that is sufficiently formal for you, while every-so-often attempting to learn some mathematics that you find uncomfortably informal. I think that, if you keep doing this, your perspective broadens, and more and more "apparently informal" mathematics becomes acceptable.

Also, you may find the answers here relevant.


By the way, a small suggestion. When you come across a problematic definition in a book, the first thing to ask is not, "How do I fix this definition?" Its really: "How do I formalize this definition?" Pretend the definition is fine, and see if you can backward engineer a formal theory or notation in which the definition goes through "as is." Much can be learned in this way.

Here's an example of this.

Suppose we have a function $f : \mathbb{R}^2 \rightarrow \mathbb{R}$. Then if $(x,y) \in \mathbb{R}^2$, it follows that $f(x,y)$ is a real number. Its not a function at all, right? So when people say "The function $f(x,y),$" they're simply wrong.

But wait! We should pretend they're right, and try to backward engineer a formal viewpoint such that they really are right.

Okay, well what is $\mathbb{R}^2$? It can be viewed as the set of all ways of assigning a real number to each element of $\{0,1\}$. Well, what is so special about $0$ and $1$? What can't they be $x$ and $y$ (symbols, not real numbers; assume $x \neq y$). This gives us an idea: Write $\mathbb{R}^{\{x,y\}}$ for the set of all ways of assigning a real number to each element of $\{x,y\}.$

So if $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ is a function and $x$ and $y$ are symbols, we can write $f(x,y)$ for the obvious function $\mathbb{R}^{\{x,y\}} \rightarrow \mathbb{R}$. (Try formalizing this).

So now we have a perspective from which talk of "the function $f(x,y)$" is actually perfectly acceptable.

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-1 Just because you found a way to define the point $f(x,y)$ as a function doesn't mean that is what people mean. Getting the point across is more important. –  Michael Greinecker Jan 15 at 19:33
    
@MichaelGreinecker, very few formalization of someone elses work are ever precisely what the original writer meant. Which is good, because otherwise formalization would be incapable of generating any progress. By the way, if you want to do equational logic in finite product categories, then in general you cannot view $f(x,y)$ as "evaluation of $f$ at the point $(x,y)$," because your objects needn't have any points. So you have to interpret $f(x,y)$ as a function $X^{\{x,y\}} \rightarrow X$. Ergo, I think this particular formalization is definitely a step forward. –  goblin Jan 15 at 19:42
    
If someone says $x^2$ is convex, they usually do not mean that a point is convex, which is formally meaningless. And even though you may rightfuly expect students to be more rigourous than, it is not like one can never comprehend something formally unrigorous. Language is for communicating. –  Michael Greinecker Jan 15 at 19:51
    
@MichaelGreinecker, I agree. What makes you think I disagree? –  goblin Jan 15 at 19:55
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@MichaelGreinecker, I am not trying to argue that we should intentionally refuse to understand what other people are saying just because they're speaking informally; what I'm saying is, there is value in formalizing abuses of notation so that, from our point of view at least, they are no longer abuses of notation. –  goblin Jan 15 at 20:13

There is a formalism you might be interested in to resolve your problem with the $dx$ in integration. Namely, from one perspective, you are not really integrating the function $f$ over an interval, you are integrating the differential form $f(x)dx$ over a parameterized interval. Here $f(x)dx$ is a legit mathematical object all on its own. In particular, is a function which takes a point $x$ and a vector $\Delta x$ and returns $f(x)\Delta x$. The advantage of formulating things this way is that integration of this gadget is actually independent of the parameterization of the interval you are integrating over - i.e. the chain rule is folded naturally into the definition of differential form and integration.

Another advantage is that these gadgets generalize: if you ever had trouble understanding line integrals, surface integrals, Green's theorem, etc, you may find that learning about differential forms resolves a lot of your difficulties.

My main point is that sometimes notations seem complicated because they are actually made to deal with somewhat more complex situations than where you are presently learning them. If all anyone ever wanted to do was integrate functions of one variable over intervals, then the $dx$ notation is a little overkill. When you want to integrate something over a $7$ dimensional manifold, it becomes essential.

This also applies to your statement about limits. The notation $\displaystyle\lim_{x \to a} f(x)$ is a little overkill when there is only one variable, but as soon as you have more than one variable it becomes essential. $\displaystyle\lim_{x \to 0} \left(x+y\right)$ is very different from $\displaystyle\lim_{y \to 0} \left( x+y \right)$

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I don't really agree with this answer. If you do analysis on manifolds then you are entitled to think about things this way. But integration of functions on subsets of $\mathbb{R}$ really can be thought of as integration of functions, and it has other generalizations as well. If one integrates with respect to Haar measure on an arbitrary locally compact group, one really is integrating functions, not differential forms, right? Also, you can integrate a function on a subset of $\mathbb{R}^n$ which is much more general than a submanifold. Can you do that for differential forms? –  Pete L. Clark Jan 15 at 18:48
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@PeteL.Clark You are right. I do think that the integration we teach to students in a first introduction to calculus is more from the "forms" point of view, however. After all, we do not integrate over a subset of the reals, but a parameterized interval ($\int_a^b = -\int_b^a$). And the chain rule is basically the only technique of integration. Really I just meant to point out that the $dx$ is really important, not superfluous, and has a good reason to be there. I made a stronger statement than I normally would to be provocative, and hopefully get the OP to look up differential forms! –  Steven Gubkin Jan 15 at 19:52
    
@PeteL.Clark I have changed my answer substantially. Hopefully it is more agreeable. –  Steven Gubkin Jan 15 at 20:09
    
Yes, I find it more agreeable; thanks. Amusingly, today I covered integration by substitution in my caclulus class. I did it first in Newton's notation, then in Leibniz notation, to emphasize how much easier the "$dx$" somehow makes things. Students were nearly silent on the Newtonian way and nearly jumping to answer it the Leibniz way. So yes, the perspective of differential forms and "equivariance under change of variables" is certainly an immensely fruitful one. –  Pete L. Clark Jan 15 at 23:51
    
P.S.: Thanks for reminding me to talk about $\int_b^a = - \int_a^b$! –  Pete L. Clark Jan 15 at 23:53

As Tevye explains in "Fiddler on the Roof" "... why do we do these things? No one knows; it's ... TRADITION!" Mathematical notations are introduced as shorthand. Some have stood the test of time, others have not. Some are good, others less so. It is fine to ponder better notations. It is another thing to convince mathematicians that your notation should be adopted over that which exists. Anyone doubting how significant a task this might be should consider the case of the great French consortium Bourbaki. He tried to set mathematical notation and terminology for a broad swath of modern mathematics. In some cases he was successful (e.g., surjective, injective, etc. for functions, although onto, one-to-one, etc. are still widely used), in others not. I once heard Diedonne, one of the great early members of Bourbaki, declaim that the term Hopf algebra was totally inappropriate because Hopf had almost nothing to do with studying them (or some such reason). I forget now the proposed alternative, which I guess proves my point, because we still call them Hopf algebras.

In the long run, to me, it's all about communication. If we all stick more or less to the same notational conventions, it's a whole lot easier for each of us to understand what the rest of us are saying.

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Short answer: Details likely wouldn't be presented if they weren't important in proving statements given particular axioms.

Longer answer: You've clearly gone through a number of classes in mathematics. Your first experience with mathematics was likely something like $1+1=2.$ One chocolate plus one chocolate gives two chocolates. This is easy enough for most five-year-olds to understand.

I recall some mathematics texts spend hundreds of pages to prove that $1+1=2$. Same statement, but a far greater level of detail. To a five-year-old, these statements are just a bludgeon that they can hit their seven-year-old sister with. But to a professional mathematician, they're essential parts of the proof.

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Why the downvote? –  John Jan 15 at 19:33

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