Why there is no sign of logic symbols in mathematical texts?

Question

Either in undergraduate or graduate textbooks on Mathematics (Real/Complex Analysis, General Topology, Differential Geometry, ...), I never saw symbols $\Rightarrow$, $\iff$, $\forall$, $\exists$, etc. Instead, I just see their "read as" or explanation such as "if ... then ...", "if and only if", "for all", "there exists", etc. Before I started to study mathematics, I thought I would see a lot of logical symbols, but I didn't see any (even once!) even in the proofs. Why is that so? If they are for mathematical use and they take less space in texts, why are there none of them in mathematical textbooks?

Answer 1

Certain symbols are best used only in certain cases.

The most common place where such symbols are used, at least when not talking about the topic of logic, is likely on a blackboard. That's because they are used in addition with the spoken word. Then the symbols are an abbreviated language.

For example, the "therefore" symbol does not have any formal usage in mathematical logic, and I've hardly ever seen it in print, but it is great for a blackboard argument because the professor accompanies it with he spoken word, "therefore."

Logic symbols for print exist because sometimes we want to reason about logic itself.

Answer 2

Many mathematicians, and I want to be in that number, believe that

Let us fix any $\epsilon>0$. It follows from the assumptions that there exists a positive number $\delta$ with the property that $1/x<\epsilon$ whenever $x>\delta$

is more elegant than

$(\forall \epsilon>0)(\exists \delta>0)(\forall x)(x > \delta \Rightarrow 1/x<\epsilon)$.

Book authors often want to write good books, carefully written, elegant and pleasant to read. Book authors often think of themselves as artists, or professional writers: if, as others said in their answers, good English grants both style and scientific quality to a book, why not use it?

Answer 3

The famous topologist James Munkres requires students in his MIT courses follow these guidelines on good mathematical style. Rule (7) reads

Don't use logical symbols at all. The symbols $\exists, \ni, \forall, \exists !, \vee, \wedge $ as well as the abbreviations s.t., w.r.t., are to be avoided in mathematical writing. In papers in logic, these symbols constitute part of the subject matter and are completely appropriate. In informal mathematical discourse, on blackboard or paper, they are often used as "parts of speech", in a sort of mathematical shorthand. However, they are not allowed by editors in formal mathematical writing.

Just as you wouldn't submit a history paper that is written partly in secretarial shorthand, don't submit a math paper written partly in mathematical shorthand!

Rule (8) adds

One exception is the use of the symbols $\Rightarrow$ (implies) and $\Leftarrow$ (is implied by) and $\Leftrightarrow$ (is equivalent to). One of course does not use these symbols as word-substitutes, any more than one uses $<$ or $+$ or $\cap$ as word-substitutes [e.g., "Consider the set of all numbers $< 1$" or "Consider the $\cap$ of the sets $A$ and $B$"].

But usage is allowed in phrases such as: "We show that (a) $\Rightarrow$ (b) $\Rightarrow$ (c)," or "To show (a) and (b) are equivalent, it suffices to show that (a) $\Rightarrow$ (b) and (b) $\Rightarrow$ (a)".

There is a reason why editors (at least those who are also mathematicians) enforce rule (7) strictly. Most mathematical readers find sentences in which this rule is violated quite unreadable, just as they find secretarial shorthand unreadable. They translate the sentence into the English language (or German, or French, or ...) mentally, before attempting to understand it. ...

Yes!

Answer 4

Proofs in textbooks are supposed to be readable for humans. Using logical symbols can make proofs significantly shorter but also significantly more confusing. The proofs written in natural language with actual words and sentences are much faster to read. Moreover, with words you can explain ideas, not just state truths. Mathematics is not only about finding all the true statements. Textbooks and shorter mathematical texts tell a story, and stories are hard to tell in symbols only.

A proof in words can also be difficult to understand if it never explains why things are done the way they are. If you only use symbols, you don't even have the chance to explain. A symbolic proof explained in words might work, but I find non-symbolic logic often easiest to follow in a proof.

Make a test if you will: Take a proof that is approximately one page long. Convert it to logical symbols so that no English word remains and give it to a friend to read. Can they understand what is going on?

Some things are better written in symbols, some in words, and the position of the borderline is a matter of taste. I believe most people would agree that writing mathematics with symbols only or without symbols of any kind is not a good idea.

Answer 5

I have to disagree with many of the answers here. There is a theme here that "humans are not computers, so it's easier for them to read words than symbols". But why stop at logical symbols? There are many symbols we do use in math in lieu of words. In earlier times, an equation like $x^3+3x=2$ would be written as "the thing cubed plus three times the thing equals two". Perhaps we should all go back to that?

Of course not. Mathematical symbols are here to make our lives easier, not harder - they do so by expressing ideas in a concise and easy-to-work-with form.

The question was also asked if we would like to read a proof that has no words and only symbols. But that is a strawman since nobody was suggesting this - instead, we wish to use a mixture of words and symbols according to what makes sense in the context, and that logical symbols will not be singled out for avoidance.

I can only speculate that the real reason behind this is that logic is, on one hand, ubiquitous in math, and on the other hand, formal mathematical logic was developed much later than algebra. So by the time mathematicians realized they actually can use symbols for logic, it has been widespread to use words for logical connectives, and symbols for the domain-specific subject matter.

And of course, traditions are hard to change - once everyone is used to one way, it's harder for them to parse statements written another way, and they find the traditional way prudent, so they continue the tradition in their own writing, enforce it in their journals, etc., and perpetuate the tradition. But it is important to understand that there is no real reason to eschew logical symbols - it's just a continuation of a tradition that resulted from a historical accident.

PS. $\forall$ and $\exists$ are rare in non-logic texts, but I do see $\implies$ and the like occasionally.

Answer 6

(This is a non-standard answer.)

Suppose we wish to prove a theorem of the form:

$$\varphi_0,\ldots,\varphi_{n-1} \vdash \psi$$

(See here for notation.)

There's basically two ways to go about this:

• In the deductive paradigm, we start by writing down the premises (the $\varphi$'s), and then proceed to use inference methods to write down further formulae. We're done only if we can write $\psi$ on our page without breaking the rules.

• In the algebraic paradigm, inference methods are instead used to manipulate the whole judgement $\varphi_0,\ldots,\varphi_{n-1} \vdash \psi.$ We're done if we can manipulate it so that it says something trivial, like $\alpha \vdash \alpha$, without breaking the rules.

For instance, to prove a trigonometric identity, a high school teacher will often begin by writing down the identity to be proved, say $\tan^2 \theta + 1 = \sec^2 \theta$, and then begin manipulating it. The proof is complete only once the identity to be proved has been transformed to the point where it says something that is already know to be true, like $\cos^2 \theta + \sin^2 \theta = 1$. This is the algebraic paradigm in action.

However, the deductive paradigm is currently dominant; pick up any serious mathematics text, and everything you see will be deductive. This is because the deductive paradigm allows us to write less. Therefore, if we're creating a "static" proof, like a blackboard proof, or a paragraph in a mathematics article (or book), we're essentially forced to use the deductive paradigm. Otherwise, our 1-paragraph proof might end up spanning 5 pages!

So to answer your question: if a book is using the deductive paradigm (and they all are), then there isn't much benefit in using logical notation like $\wedge,\rightarrow$ etc. since we're not going to be manipulating any formulae or judgements directly. This, in my opinion, is the true reason that you rarely see such symbols, although few (if any) authors would explain their choice in this way.

Its worth noting that, if we're interested in slideshow proofs that are presented on a computer, then suddenly the algebraic paradigm suddenly becomes a lot more user-friendly. In this case, the use of logical symbols like $\wedge,\rightarrow$ becomes indispensable, because only by using such symbols do we get rigorous and terse mathematical formulae that are easy to manipulate algebraically.

Answer 7

As a programmer, I find the purpose of these symbols the same as for programming languages - to keep the structure of the code readable and maintainable. As you say, these symbols make the texts significantly shorter, and, in my opinion, I would find short readable proof more interesting than long readable English-words-only proof, where I would sooner or later get lost in the text.

Another problem is different interpretation. English "or" or other connectors sometimes don't exactly follow their mathematical definition, and that can be a problem, especially when translating to other languages. To a mathematician, a simple "∨" should always be clear and free of any different interpretations. I find this more useful than trying to specify which of the 20 meanings of "or" you mean. Symbols also make the text much more readable for non-English mathematicians.

However, I don't want to encourage you to change all connectors to symbols. Only do so when the statement itself has mathematical meaning, like being part of a proof describing properties of various mathematical objects or similar. When you can write the whole statement in mathematical symbols, do it.

Answer 8

Note: OPs question addresses one of many important aspects which an author of a mathematical text(book) has to consider and it should be seen in a wider context around the question:

What is the essence of writing a mathematical book?

I highly appreciate the following $50$ year old answer:

P. R. Halmos: The basic problem in writing mathematics is the same as in writing biology, writing a novel, or writing directions for assembling a harpsichord: the problem is to communicate an idea. To do so, and to do it clearly, you must have something to say, and you must have someone to say it to, you must organize what you want to say, and you must arrange it in the order you want it said in, you must write it, rewrite it, and re-rewrite it several times, and you must be willing to think hard about and work hard on mechanical details such as diction, notation, and punctuation. That' s all there is to it.

So, the main theme is to communicate an idea.

In order to do so we have besides other things take care about diction, notation, and punctuation. But, there may also be subtle (and less subtle) differences to whom this idea should be communicated.

P. R. Halmos: The second principle of good writing is to write for someone. When you decide to write something, ask yourself who it is that you want to reach. Are you writing a diary note to be read by yourself only, a letter to a friend, a research announcement for specialists, or a textbook for undergraduates? The problems are much the same in any case; what varies is the amount of motivation you need to put in, the extent of informality you may allow yourself, the fussiness of the detail that is necessary, and the number of times things have to be repeated. All writing is influenced by the audience, but, given the audience, an author's problem is to communicate with it as best he can.

So, the contents and the way to present contents will usually differ depending on the audience. We should keep in mind, that the amount of motiviation and the extent of informality are key ingredients of good mathematical books, although these aspects are typically beyond the symbolism we use in mathematics.

But, what about the usage of logical symbols?

P. R. Halmos: Here is a sample :

"Prove that any complex number is the product of a non-negative number and a number of modulus $1$."

...

One way to recast the sample sentence of the preceding paragraph is to establish the convention that all "individual variables" range over the set of complex numbers and then write something like $$\forall z\exists p\exists u [(p=|p|) \wedge (|u|=1) \wedge (z=pu)]. $$

I recommend against it. The symbolism of formal logic is indispensable in the discussion of the logic of mathematics, but used as a means of transmitting ideas from one mortal to another it becomes a cumbersome code. The author had to code his thoughts in it (I deny that anybody thinks in terms of $\exists, \forall, \wedge$, and the like), and the reader has to decode what the author wrote ; both steps are a waste of time and an obstruction to understanding. Symbolic presentation, in the sense of either the modern logician or the classical epsilontist, is something that machines can write and few but machines can read.

Note: This article written by one of the great provides many more guidelines. It has the somewhat cheeky title How to write Mathematics and guarantees highly instructive amusement.

I like to finish my answer with his guideline $15:$ Resist symbols

P.R. Halmos: Everything said about words applies, mutatis mutandis, to the even smaller units of mathematical writing, the mathematical symbols. The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it. A good attitude to the preparation of written mathematical exposition is to pretend that it is spoken. Pretend that you are explaining the subject to a friend on a long walk in the woods, with no paper available; fall back on symbolism only when it is really necessary.

Answer 9

It depends on the texts you are reading. There are still places where the logic symbols are used frequently; of course in mathematical logic (for an MSE example see here); some professors still encourage the use of logic symbols to shorten a proof, see the discussion here.

Answer 10

All my math related university textbooks made large use of logic symbols.

The worst ones were actually almost only logic symbols, which is not bad per se, but as someone noted symbols only state things, a little writing in a less formal language (note: implying math symbols are a language) can help understand.

Conclusion: best is to state with symbols and then help understand by explaining stuff with words.

@ThomasAndrews There is a huge problem with this statement you made:

"The most formal proofs are all symbols. We choose to not use them because humans are not computers".

Math symbols are a LANGUAGE, specifically one where every "word" (symbol) has a very precise meaning (not like other human languages, where you can have a word have different meanings in the same context - e.g. inclusive and exclusive "or").

Some symbols can have different meanings in different contextes, but that is fine as the context makes the meaning well defined (it happens in other human languages too).

A symbol express a concept, sometimes a concept is expressed in more than one word in a language and a proper 1 (word) to 1 (word) mapping does not exist (= is read "is equal to" in english, but it is only 1 word in italian). Perfectly normal if you ever studied a foreign language, so nobody should have an issue with this.

The issue with symbols is not that they are not human readable, the issue is that they are a language most of us never became proficient (enough) in. The very first thing you need to learn when studying new languages is to stop thinking in the old one: it is not a matter of being computers, it is a matter of not being bad at other languages (math or any other "human" languages).

Once again, the issue with the symbol language (maybe because I am not good enough myself) it does not help to visualize the concept so that you can really understand what it says (and I mean more than understanding the causality chain if you want). Things that help visualization are e.g. examples (pun not intended). For visualization purposes, a "normal" language is more efficient, since you do not need "precise meaning" of each word, but rather what I would define as "richness" in expressiveness - something that, thanks to being less precise, can express a concept in a less verbose manner. In this way you get the idea "fast" and can clarify it with the help of the symbolic statement (so english first, and symbols later is my preferred approach).

Answer 11

I find that symbols are only really helpful to the extent that they don't actually change the sentence. Writing "Let epsilon be..." and "Let $\varepsilon$ be..." are identical, because the collection of strokes translates to the same sentence in my head, which is also how I would naturally describe the idea to myself and others. The second form may be slightly better because the different font and unusual letter of $\varepsilon$ serve to draw visual attention to the central mathematical actor at hand, rather than blending in with the supporting dialogue. The form of conciseness that still clearly matches the reader's internal monologue is undoubtedly helpful.

The issue, I think, is when you need to "translate" logical symbols, rather than merely read them. If I say "all roots of unity have magnitude one", that's a natural sentence. "All roots of unity $z$ have $|z|=1$" is more or less the same, since the order of concepts is still the order of the natural phrasing; translating the symbols into English as I go left to right still yields a natural sentence in my head. If I go all the way and say: $$\forall z\in \mathbb{C}\ \forall n\in\mathbb{N} \ \ z^n=1\implies zz^*=1$$ this looks much less like what I wrote originally, though it's still clearly identical. In visual order it reads more like "For all complex numbers $z$ and for all natural numbers $n$, $z$ raised to the nth power being equal to one implies that the complex number times its conjugate is one". That's clearly a very muddled way of putting things. Anyone's mental reading of this sentence packages the phrasing into another form as they scan it, so you may as well just write that in the first place. Turning the symbols into reasonable sentence involves mentally rearranging them, which is extra work for both the reader and writer. In this case it's not much work, since there aren't really nested quantifiers and the individual predicates are simple, but it also doesn't get you any extra clarity either. The only thing that can happen is that you lose a reader.

As an aside, note that the phrase "roots of unity" got turned into its defining expression. In general, mathematical English can bundle detailed definitions into simple phrases that help compartmentalize both the writing and the reader's thinking appropriately. Mathematical logic is a more minimalist language, so such terms often have to be explicitly expanded everywhere they appear if you refuse to write the English phrase. That would easily turn into a nightmare if you actually stuck to it for a whole text.

Answer 12

Since "no sign of logic symbols in mathematics texts" is exaggerated, we assume this phrase is replaced by "so little use of logic symbols in most mathematics texts". The answer to the question is: (a) dogmatism, (b) various myths, (c) the state of development of symbolic logic. Let's expand.

(a) Dogmatism: the failure to recognize that different people benefit from different means of expression. For one's own use, one may choose what suits best. When writing for others, looking for a balance that suits the audience helps understanding. This obviously depends on (c). Prohibiting logic symbols (John Munkres) is dogmatic and stifles progress regarding (c).

(b) Myth 0: logic symbols are for logic texts only. If logic texts fail to produce notations that are useful for the remainder of mathematics, their usefulness is limited. Fortunately, the situation is not that bleak. Myth 1: symbolism is for computers. In fact, programming languages use just ASCII. Symbolism was developed by and for humans long before computing machines existed.

(c) The state of development of symbolic logic. In algebra, development of symbolism started with Diophantus nearly 2000 years ago and reached its current form about 300 years ago. Using symbols as mere secretarial shorthand is called syncopation, and has no advantages for reasoning. However, mature symbolism also has well-designed rules such that the shape of the expressions provides guidance in reasoning. In analysis/calculus, for instance, the shape of expressions often helps deciding the next step in the symbolic calculation of an integral. Books such Gries and Schneider's (see Google) show how symbolic logic can provide similar advantages for routine reasoning in all branches of mathematics. Of course, this is a development of recent decades and is constantly improving. As already said, forcing (or even encouraging) students to avoid logic symbols is anathema to learning.