Thread: "forbidden notations", are they forbidden, and if so, why ?

I am getting repetitious so this will be my final post.

There are no rules about informal or private thinking. Language, however, and all forms of public communication are extra individual; they rely on a shared understanding of what symbols mean.

Mathematicians have developed the least ambiguous system of symbolic representation. Part of mathematical education involves teaching (1) that system and (2) the need for very careful definition and strict logical thinking.

$f(x)$ by itself is an undefined term and is a meaningless concatenation of meaningless shapes without extra individual conventions. When teaching French you don't permit students to say "le dog" as a translation of "the dog," but insist on "le chien."

If I specify $f(x) = \dfrac{1}{x}\ if\ x \ne 0$, then $f(0)$

is an undefined term without any known meaning. It may be 2 or perhaps $sin(x)$. It can be anything. There is nothing to be said about something that is not even defined.

I hate standard analysis. I find it horribly ugly. I may find it personally

convenient to remember the chain rule as $\dfrac{dy}{dx} = \dfrac{dy}{\cancel{du}} * \dfrac{\cancel{du}}{dx}$,

but I would not use it in a math class on standard analysis because those symbols have been defined as operators, not fractions.

Having read a bit of Zoroaster's previous post I can say that mathematical notation in general is not a formal logic system that works purely by symbolic manipulation. It is a proper language that describes concepts and what you write should reflect the proper use of the concepts and objects in mathematics.

When you Wind in the Willows, you do not attempt to view it as a collection of symbols that form words on the page which are themselves part of sentences. That would be a nonsensical way of interpreting the writing. Instead you think of the objects and concepts described by the story, Mole, Rat, Mr Toad, the riverbank and the Wild Wood. You use the words and sentences to build an understanding of the world Grahame is describing. You use them to imbue standard concepts such as "a rat", "a toad" and "a wood" with the particular properties that he specifies so that he can give you an understanding about the more specific objects and concepts of which he write: "Rat", "Mr Toad" and "the Wild Wood".

Of course, when I talk about "Rat" and "Mr Toad" there is an implicit assumption that you understand the objects of which I am speaking. There are objects having formal descriptions within the book Wind in the Willows. In that way we can discuss what Mole might do under given circumstances because we have a shared understanding of what "Mole" is. We can have such a discussion with specifying everything written about "Mole" in the book because we have this shorthand available to us, but the writings in the book are implicit in the conversation.

What you are trying to do is to look at the word "Mole" and imbue it with your own ideas without having read Wind in the Willows. You are making certain assumptions about Mole, but you don't know whether they are accurate and nobody else knows what those assumptions are because you are not drawing on the common resource: Wind in the Willows.

Originally Posted by Zoroaster

But I don't see why f(0) is not representing formally the tree that is written 0/0.

Because there is nothing in mathematics that 0/0 represents and the function $\displaystyle f$ is not defined at zero. It is not correct to imply, as you are, that the function is defined at zero and has the "value" represented by 0/0. There is no $\displaystyle f(0)$, only a hole.

Originally Posted by Zoroaster

and nobody was telling you that you shouldn't use that expression when x was to be replaced by 0 (on the contrary)

They were telling you that by the implied formal definition of the function. You just haven't listened to that part of the definition.

Originally Posted by Zoroaster

If one wanted you to avoid writing f(0), one should have specified already that the expression f(x) = x^2/x is not to be used when x = 0. But that was exactly the question to be answered: for what values of x does this identification make sense ?

And this is exactly the sort of confusion that can be introduced by taking shortcuts with notation.

Originally Posted by Archie

And this perfectly illustrates the point that if you do not properly learn the correct syntax and how to use it, breaking the rules is likely to cause confusion. It is only by understanding that $\displaystyle f(x) = \frac{x^2}{x}$ implicitly includes all the details concerning the domain and range of the function that we are able to properly use the notation.

I understand perfectly well what you are saying, but let us not forget that I'm talking about the exact situation where the domain is not known. I fully agree with you that if one specifies a function like you do, then it is very simple:
f(0) is not an existing number, simply because 0 is not in the domain of f, and hence, there is no couple (0,?) that belongs to f ; as such, f(0) pointing to the "right hand element" of that couple, it is not there, so f(0) is non-existing.

However, if the domain is not known, we have to explore it. And if the only thing that is given is a *formal notation* like x^2/x, then my idea is that we have to treat that formal notation as, exactly that: a formal notation. The exercise, in my mind, is then exactly to find out for what substitutions of x by a number, this formal notation x^2/x, can resolve into a number. If it does, then "by the nature of the question", such a number belongs to the domain.

In other words, by writing f(x) = x^2/x, one didn't give you a function, but just a formal expression, and the question is: "for what numbers in the place of x, can you make a function out of it ?"

Let me illustrate this with another kind of "function", one in the structure "english language". I see a problem as given above, very similar to the following problem: "consider the formal expression f(X) = Xar ; where X is a letter of the alphabet, and f is a function in the Collins dictionary. What is the domain of f ?"

As such, the logical thing to do, in my mind, is to run X throught the alphabet, and to see when the obtained expression is an english word:

f(a) = aar (no)
f(b) = bar (yes)
f(c) = car (yes)
f(d) = dar (no)
f(e) = ear (yes)
f(f) = far (yes)

....

So in the end, it turns out that the domain of f are the letters {b,c,e,f,...}

But we did write f(a), it was correctly substituted, it didn't yield a word in the English dictionary, and this indicated that a was not an element of the domain of f, and hence that f(a) didn't exist, as the image of a function, but of course, it existed as the substitution rule in a formal expression we started out with.

Isn't this how people see things ?

Originally Posted by Archie

Having read a bit of Zoroaster's previous post I can say that mathematical notation in general is not a formal logic system that works purely by symbolic manipulation. It is a proper language that describes concepts and what you write should reflect the proper use of the concepts and objects in mathematics.

I think this is the gist of the troubles I seem to be having with certain aspects. I do see mathematical notation as a succession of formal manipulations, *inspired* by abstract ideas behind it (a Platonic view, let's say). I would think that most of mathematical notation is the application of appropriate transformation rules of formal expressions, depending on the structure one is working in (keeping in mind that these structures are inspired by "existing" abstract ideas we cannot really write down).

So to me, when one writes 2x+x, this is in the first place seen as a succession of 4 symbols (a 2, an x, a +, and an x), but is then syntaxically interpreted as a tree structure (a plus on the root, one sub-tree with an implicit "times", a 2-leaf, and an x leaf, and the other side, simply an x leaf).

Whether one can now transform this tree into another one, which is usually called "calculating" depends on the structure one is in, and in how much the sub-trees satisfy the conditions of the transformation rules. If in the end, you end up with a single atomic identifier, you've "solved' your problem ; if you reduced the complexity of your tree, you've "simplified" your question.

Of course, we keep in mind that when the rules apply, these leaves and sub-trees "represent" Platonic mathematical "objects" in these structures, but these just guide our preferred application of tree transformation rules (which come down to transformations of written symbolic sequences).

Apparently, this is not how most people see "doing math" then.

The symbol sequence can, under appropriate conditions, be transformed into something like 3x, but not necessarily. It depends on the allowed transformation rules in the structure, and the conditions on the sub trees.

As such, x^2/x cannot be simplified into x. But x^2/x can be transformed into x if we are working in a field, and we add the condition (x is not the absorbing element).

So I can always write x^2/x. And I can transform this into "x ; x in field and not absorbing element." If later, x turns out not to be in a field, or to be the absorbing element, then my identity was not valid, and it simply means that this exploratory branch has to be abandoned.

I just found more or less what I was intuitively thinking in the following wiki article:

https://en.wikipedia.org/wiki/Expression_(mathematics)

I'm close in my position to the following section:
"Semantics is the study of meaning. Formal semantics is about attaching meaning to expressions.

In algebra, an expression may be used to designate a value, which might depend on values assigned to variables occurring in the expression. The determination of this value depends on the semantics attached to the symbols of the expression. These semantic rules may declare that certain expressions do not designate any value (for instance when they involve division by 0); such expressions are said to have an undefined value, but they are well-formed expressions nonetheless. In general the meaning of expressions is not limited to designating values; for instance, an expression might designate a condition, or an equation that is to be solved, or it can be viewed as an object in its own right that can be manipulated according to certain rules. "

Originally Posted by Zoroaster

f(a) = aar (no)
f(b) = bar (yes)
f(c) = car (yes)
f(d) = dar (no)
f(e) = ear (yes)
f(f) = far (yes)

Isn't this how people see things ?

It may approximate how some people work the problem, but that doesn't make it formally correct and it doesn't mean that there is no possibility of confusion.

It doesn't matter how much you'd like things to be different, what you are writing is strictly incorrect even though, informally, it may accurately express your ideas. Like "to boldly go" in the 1970s.

We could, accurately, express the ideas as "the denominator of the expression $\displaystyle \frac{x^2}{x}$ is zero when $\displaystyle x=0$ and therefore the expression is undefined at this point. Thus zero is not in the domain of the function $\displaystyle f$ and $\displaystyle f(0)$ does not exist.