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topological group (obsolete)

Type of Math Object: 
Definition
Major Section: 
Reference

Mathematics Subject Classification

22A05 no label found

Comments

I'm not submitting this as a correction to the entry
http://planetmath.org/encyclopedia/TopologicalGroup.html

because the opinion below, I expect, will be controversial,
and the above entry is not even a particularly egregious example
of what I'm going to criticize.
But I have always wanted to bring the topic up for discussion.

======================================
The use of pretentious formal notation
======================================

Why is it necessary to write

``A topological group is a triple $(G,\cdot,\mathcal{T})$ ...''

instead of

``A topological group is a group equipped with a topology
under which the group multiplication and the group inverse are continuous mappings.''

or if you think that is not precise enough:

``A topological group is a group, equipped with a topology $\mathcal{T}$ under which the group
multiplication $(x,y) \mapsto x \cdot y$
is continuous in the product topology $\mathcal{T} \times \mathcal{T}$,
and the inverse $x \mapsto x^{-1}$ is continuous
in the topology $\mathcal{T}$.''

My point is, why is it necessary to introduce the notion
of triples? Is the syntax, that you put $G$ and $\cdot$ before $\mathcal{T}$, really relevant to the concept of topological groups?
(a tuple is order-sensitive)

Are you using extra mathematical symbols to make yourself look
sophisticated, like this entry?

http://planetmath.org/?op=getobj&from=objects&id=7374

which seems to have been written by mathematician-wannabe
computer scientists

Let me emphasize how ridiculuous this is.
This is how computer-science-formalese would define the English word ``library'':

``A library is a 2-element tuple (library, books),
where library is an object representing the building
holding the books and books is a set of objects representing books. ''

Of course, it would be ridiculuous to suggest that we should always prefer informal notation to formal notation. But in this case,
I think a good rule of thumb would be:

``Do you need to manipulate the objects _as tuples_?''

For example, we can reasonably (and frequently we do)
talk about equality of Cartesian coordinates:

``two point on the plane $(x_1,y_1)$ and $(x_2,y_2)$ coincide
if and if $(x_1,y_1) = (x_2, y_2)$
(meaning that $x_1 = x_2$ and $y_1 = y_2$)''

On the other hand, do you ever need to use an equation like the following

$(G, \cdot, \mathcal{T}) = (H, \cdot', \mathcal{S})$

If not, then I'd suggest that we chuck the tuple notation.
It's irrelevant.

// Steve

Addendum:

Another good rule of thumb to when to use a tuple or not:

Do you need to project the object, or
index the tuple with a numerical index?

For example:

$\pi_k(x)$ or $x_k$ extracts the $k$th coordinate of the tuple $x$

One may also ask if we need to work on the tuple as a whole

e.g. $x \cdot y$ takes the dot product of the row vectors $x$ and $y$
$\lVert x \rVert$ takes the norm of $x$

and so forth. If so, then obviously, use the tuple.

The use of the tuples in formalese fails these tests.
Would you write: topological group $H = (G, \cdot, \mathcal{T})$,
$\mathcal{T} = H_3$ !?

I'm a fan of having both formalizations and informalizations.

They're both useful.

apk

I think you should read the "spirit of ubuntu" posting I just wrote up
(http://planetmath.org/?op=getmsg&id=10087). Why are you using the
word "pretentious" in this discussion? A triple is not pretentious.
Perhaps some justification for the notation should be given (e.g. why,
precisely, is G listed first in the triple, why are triples used in
the first place).

In this case, I would simply ask the author to explain this choice of
notation, rather than asserting that this choice is pretentious and
making an unfounded allegation ("Are you using extra mathematical
symbols to make yourself look sophisticated, like this entry?").

You can approach the matter more respectfully by posing the essentially
technical question: why are you using triples?

the tuple notation is quite standard in mathematics. i find
it quite easy to take in at a glance what is being defined.

i don't think an author needs to explain why he uses
common notation. instead, perhaps there should be a discussion in
the mathematical community of why this notation is used.

personally, i think a good entry has both a formal and
informal definition, as aaron said.

-kyle

>
> Why is it necessary to write
>
> ``A topological group is a triple $(G,\cdot,\mathcal{T})$
> ...''
>

I think this should be read as follows: A topological group
is a new mathematical object depending on three objects;
a group G with product $\cdot$, and a topology $T$ on $G$ such
that ...

In other words, the triple "notation" is just a way
of emphasizing what the new defined object depends on. I
don't like it very much, although I have probably used it myself
sometimes. For example, of:

(X,d) metric space.
Let $(X,d)$ be a metric space.
Let $X$ be a metric space with metric $d$.

the first one is convenient on a blackboard, but
latter is probably the best writing.

> ``A topological group is a group equipped with a topology
> under which the group multiplication and the group inverse
> are continuous mappings.''
>

To this I would add a note about the product topology.
Multiplication should be continuous as a map $G\times G\to G$
under the product topology for $G\times G$. On the other
hand, if the definition would be followed by some examples,
and properties this could be mentioned there.

> Of course, it would be ridiculuous to suggest that we should
> always prefer informal notation to formal notation. But in
> this case,
> I think a good rule of thumb would be:
>
> ``Do you need to manipulate the objects _as tuples_?''
>

This is a good point. A common mistake is to introduce
a lot of notation (intended for the proof) in the theorem
formulation. THis just clutters up the formulation
and makes it difficult to grasp.

stevecheng writes:

> I'm not submitting this as a correction to the entry
> http://planetmath.org/encyclopedia/TopologicalGroup.html
> because the opinion below, I expect, will be controversial,
> and the above entry is not even a particularly egregious
> example of what I'm going to criticize.

So why didn't you post it in a forum instead of attaching it to this entry? If you were hoping that the author would see it here, you should note that he hasn't even logged in for over a year. If you really want him to know what you think of his triples, you could try e-mailing him.

You really couldn't have picked a worse entry to attach your post to: I adopted this entry yesterday, and I intend to delete it, replacing it with a new one. I am doing this (rather than just modifying the existing entry) in order to make it clear that I am the sole author (since there is nothing in the previous author's version that I want to keep). This way the entry can easily be relicensed when PlanetMath moves away from the flawed FDL, as I hope it will.

I've already written the new entry:

http://planetmath.org/encyclopedia/TopologicalGroup2.html

When I delete the old entry this whole thread will presumably disappear. I will leave it a few hours before doing this. Maybe Aaron can save the thread by moving it to a suitable forum.

Alright, I admit I wrote that piece on impulse, after seeing
yet another ``is a triple...'' definition and being
annoyed at it. I really should have refined my opinion
--- although most replies have not actually addressed the points
of my argument other than essentially ``I agree'', ``I disagree''.
I apologize for the offence that I caused.

Now I want to state my refined opinion:

After thinking about it a bit, I'm don't have anything against
using a notation of the form $(X, \Sigma, \mu)$ (to take
a common example) to inform the reader of what the letters mean.
For example,

``Let $(X, \Sigma, \mu)$ be a measure space. Then ...''

What I have a problem with is with definitions like:

``A measure space _is a triple_ $(X, \mu, \Sigma)$ where''

for the phrase _*is* a triple_ seems to emphasize the syntax
rather then the concept being defined,
and all the important concept is relegated to a sentence
subclause. The canned phrase _is a tuple_ is essentially
a cop-out when the author has nothing meaningful to say
about the tuple $(X, \mu, \Sigma)$ itself,
but only wants to talk about its constituent parts $X$,
$\Sigma$, and $\mu$, and that the parts are to be used together.

I don't think in the majority of cases this is
what the author wants; he only writes that because he is often
exposed to this kind of writing and think of no reason not to imitate it, and/or he is not thinking carefully about what he is writing.

My accusation that it was pretentious was really a rhetorical question, although I regret I directed it too specifically to the Toplogical groups entry that I made an example of.
It is a rhetorical question that I sometimes ask myself when I proofread my writing --- is it too pretentious written?
That question is also a jab at those instances
(not the Topological groups entry)
where the excessive formality seems to be deliberate
rather than the author not realizing that the writing can be improved.

As someone has pointed out, the triple _notation_
is fairly standard. I agree. But is the
triple-based _definition_ standard?
I haven't looked so thoroughly to be sure one way or another.
In any case, my opinion is that it is a bad style of writing.
It is not wrong _mathematically_, of course.
On the other hand, there are lots of mathematics books out there,
written by experts whose rigour and reputation I don't haven't the foggiest doubt of, but whose style of exposition is terrible, and difficult
for a reader who is not already familiar with the work to follow.
Even something as basic as typesetting the work
with margins separating section and theorem numbers from the main text, and spacing between different sections of the text,
--- all would make the text easier to read ---
is not done, even though this is the responsibility of the publisher side rather than the author.

I think it is important, for the criticism, to make the distinction between the triples *notation* and *definition*,
and this is not a mere concession.
Notation is a means of communicating, as concise and precise
as possible. It should also be as unified as possible
(to reasonable limits). The $(X,\Sigma,\mu)$ notation
serves this purpose. But it could have well have been written
$\lbrace X, \Sigma, \mu \rbrace$, which would mean essentially the same thing, although it is not usually written this way.
And as such, it is important not to codify a choice
of notation in the definition using phrases like ``is a triple...''.

As another example,
the derivative (function) of $f$ is not the symbol the $f'$, nor
the symbol $\frac{df}{dx}$, it is a function, denoted by $f'$, such that
$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$.

In the case of the measure space example, I would consider writing:

``A measure space $(X, \Sigma, \mu)$ is a set X,
together with a sigma-algebra $\Sigma$ on $X$, and a measure $\mu\colon \Sigma \to \mathbb{R}$.''

Note that I say that a measure space *is* a set X.
This corresponds with the English usage, and informal usage;
after all, the word ``space'' is in the term ``measure space'',
and it is the noun that is being modified by the term ``measure''.
It is incongruent if you think the word *is* means *equal*,
(though compare that with the statement ``$G$ is a group'').

Another wording that avoids the word *is* is the following:

``A measure space $(X, \Sigma, \mu)$ consists of a set X,
a sigma-algebra $\Sigma$ on it, and a measure $\mu$
on that sigma-algebra.''

Let me make a final analogy: that of the $\forall$ notation.
Everybody knows about it; there isn't even any misunderstanding
if an author uses it in a mathematical exposition (not about formal logic). But is it a good idea to use it as a substitute
for the English phrase ``for all'' everywhere?
Most people would say no. It adds no meaning, not all
that much more concise. It's just not good stylistically.

To summarize, my objection to the ``is a triple''-type definition
falls along the same lines: it is not good style.

// Steve

> After thinking about it a bit, I'm don't have anything
> against
> using a notation of the form $(X, \Sigma, \mu)$ (to take
> a common example) to inform the reader of what the letters
> mean.
> For example,
>
> ``Let $(X, \Sigma, \mu)$ be a measure space. Then ...''
>

Agreed. But, on PM, a "measure space" is denoted by
$ (E, \mathcal{B}, \mu)$.
See

http://planetmath.org/encyclopedia/CountablyAdditive2.html

So, on PM one needs to be careful what notation one uses.

>
> ``A measure space _is a triple_ $(X, \mu, \Sigma)$ where''
>
..
..

> ``A measure space $(X, \Sigma, \mu)$ is a set X,
> together with a sigma-algebra $\Sigma$ on $X$, and a measure
> $\mu\colon \Sigma \to \mathbb{R}$.''
>
> Note that I say that a measure space *is* a set X.

I would agree that the first is better.

> Let me make a final analogy: that of the $\forall$ notation.
> Everybody knows about it; there isn't even any
> misunderstanding
> if an author uses it in a mathematical exposition (not about
> formal logic). But is it a good idea to use it as a
> substitute
> for the English phrase ``for all'' everywhere?
> Most people would say no. It adds no meaning, not all
> that much more concise. It's just not good stylistically.

Many times you can just delete it. For example, compare
$$
x\in \mathcal{F}(y), \quad \forall y\in C.
$$
and
$$
x\in \mathcal{F}(y), \quad y\in C.
$$

>
> To summarize, my objection to the ``is a triple''-type
> definition
> falls along the same lines: it is not good style.

Rudin (Real and complex analysis) also discusses the
"triple"-notation. As an example he describes the
real numbers as the quadruple $(R,\cdot,+,<)$.

> Rudin (Real and complex analysis) also discusses the
> "triple"-notation. As an example he describes the
> real numbers as the quadruple $(\mathbb{R},â‹…,+,<)$.

Good that you mention this example.
I would argue that it is formal dressing that doesn't add anything
to a more descriptive, syntax-independent, definition.

First, the notation is actually not so precise as
it might first seem.
After all, it doesn't say anything about the symbols
$1, 0, /, \cdot^{-1}, >, \neq$, etc.
even though, say $\cdot^{-1}$ could conceivably
be taken as meaning subtraction. And would you remember
that $\cdot$ goes before $+$, and what would you think
of this statement:

``The real numbers are the quadruple $(\mathbb{R}, +, \cdot, <)$.

I have switched the symbols $+$ and $\cdot$! Would that be a mere typo, or does it indicate something significant?
(i.e. is the ordering of those symbols
and that they have been put in a tuple, really
relevant to the definition of real numbers?)

And you couldn't quite say ``5 belongs to the real numbers'', can you?
Because real numbers aren't the set $\mathbb{R}$ according to this definition; rather it is this funny tuple.

// Steve

> Another wording that avoids the word *is* is the following:

>``A measure space $(X, \Sigma, \mu)$ consists of a set X,
>a sigma-algebra $\Sigma$ on it, and a measure $\mu$
>on that sigma-algebra.''

I'm surprised you find this palatable. Doesn't this implicitly define a measure space to be a set consisting of three elements? I see little implicit difference between this definition and the following:

``A measure space is a set ${X, \Sigma, \mu}$ consisting of a set X,
a sigma-algebra $\Sigma$ on it, and a measure $\mu$
on that sigma-algebra.''

The one (admittedly non-trivial) distinction is that in the first case there is an implicit ordering in the notation, whereas in the latter, as long as you know which object is which, you could write it in any order. If that's your main point, then I would certainly concede that an unfortunate byproduct of the tuple notation is that it implies an ordering of the required data. On the other hand, mathematical writing is filled with ad hoc conventions for organizing data (consider left- and right-actions, and composition of permutations), none of which are typically regarded as bad style.

In general, I strongly prefer the tuple definition. I think it's important to remember exactly what pieces of data you get when you specify an object, and then to think of an object just as that collection of data, at least until the concept is internalized. One has no hope, for example, of understanding a scheme, if one simply thinks of it as a set or a topological space (since then the spec of every field is the same scheme!). I think thinking of a scheme (and many other objects as well) as a collection of information organized in a tuple is a good step toward an actual understanding of the concept.

Cam

I'm glad you brought up the following point, giving me another opportunity to sharpen my argument.

> >``A measure space $(X, \Sigma, \mu)$ consists of a set X,
> >a sigma-algebra $\Sigma$ on it, and a measure $\mu$
> >on that sigma-algebra.''
>
> I'm surprised you find this palatable. Doesn't this
> implicitly define a measure space to be a set consisting of
> three elements?

The difference is in style. (I explain that below.)

> I see little implicit difference between
> this definition and the following:
>
> ``A measure space is a set ${X, \Sigma, \mu}$ consisting of
> a set X,
> a sigma-algebra $\Sigma$ on it, and a measure $\mu$
> on that sigma-algebra.''

The difference is that the first quote de-emphasizes
the set or tuple used to hold the three objects under discussion,
because it is mere syntax. When you read the quote, the verb ``consists'' alerts you to the fact that some objects will follow,
and those objects are important because they are
(grammatical) objects. Whereas

``blah is a tuple, where (....)''

says that blah *is* a tuple, and all the important information
is buried inside a subsidiary "where" subclause.

Surely you would agree that
there is a difference between a container,
and the objects held inside that container. We want
to focus the attention on the objects held on the container
(in this case, the set $X$, the sigma-algebra $\Sigma$,
and the measure $\mu$), not on the set
$\lbrace X, \Sigma, \mu\rbrace$
or the tuple $(X, \Sigma, \mu)$,
which in most situations is just syntax.

Let me give another example. Consider the definition:

``A real number is an equivalence class of Cauchy sequences of rational numbers, where (...definition of the equivalence...)''

This is alright as a definition if it is to be used in an exposition
that constructs the real numbers out of Cauchy sequences of rational numbers. But would this be a good definition for a general encyclopedia entry? I would think not.
The fact that when you ``look inside'' a real number
you see Cauchy sequences of rational numbers is irrelevant
to what the real numbers *are*. I could have used some other construction where real numbers are decimal sequences. The problem
with the ``*is* a tuple'' definitions is similar: the definition focuses undue attention on the syntax.

// Steve

> The fact that when you ``look inside'' a real number you see Cauchy
> sequences of rational numbers is irrelevant to what the real numbers
> *are*.

Alright then, what is a real number, really?

The only way I know to answer this question is to first pick some
axiomatic framework and work within it. In one framework, real
numbers are equivalence classes of Cauchy sequences. In another
framework, they are Dedekind cuts. In a third, they are objects which
satisfy certain properties.

Of course, one should complememnt this with a description of how to
relate these different axiomatic descriptions. Which sequence
corresponds to which set of rational numbers? How does one assign a
Cauchy sequence or a Dedekind cut to an object satisfying the axioms?

In a way, this is much like motion in Physics. To describe the motion
of an object, we need to first pick a frame of reference, then say
what is going on in that reference frame and be careful to not
indiscrimainately mix up things from different reference frames
without first transforming them into a common reference frame.

Just as it is meaningless to ask whether an object is at rest except
witin the context of a particular reference frame, so I would say that
this business of looking inside real numbers is similarly meaningless
except in the context of a specific axiomatic framework. In the first
system, you see sequences; in the second you sets of rational numbers;
in the third, you don't see anything becasue looking inside is
undefined.

I would say that some of the isssues raised here aren't matters of
style, they're matters of logical precision and rigor.

> On the other hand, mathematical writing is filled with ad hoc
> conventions for organizing data (consider left- and right-actions,
> and composition of permutations), none of which are typically
> regarded as bad style.

While these choices are arbitrary (because everything is covariant
under permutation of terms in the tuplet) it is necessarry to make
some arbitrary choice of convention and stick with it consistently.
For instance, a ring is defined by a triplet (R, +, x), where "+" and
"x" are binary operations on the set R. While it doesn't matter
whether I choose to write "(R, +, x)" or "(R, x, +)", I need to make a
choice and stick with it consistently. If I switch "+" with "x" in
the disstrtibutive law for instance, I get a false statement.

> In general, I strongly prefer the tuple definition. I think it's
> important to remember exactly what pieces of data you get when you
> specify an object, and then to think of an object just as that
> collection of data, at least until the concept is internalized.

More importantly, this is a matter of proper axiomatic presentation.
In order to describe an axiomatic system, one needs to know the
primitive terms and the axioms. The items in these triplets are the
primitive terms. Unless one specifies the primitive terms, one has
not completely specified one's system.

I think that this entry is well-written and rigorous and do not
understand why it should be deleted. As I see it, there is no reason
for not having both entries --- if someone wants a more intuitive
definition with more words than symbols, one can look at one entry,
while if one wants a precise, formal definition of the same, one has
the other entry. Both sentries serve useful purposes and I hope Yark
will relent and let both entries stay. If there is an issue of two
entries, then I would say that the proper solution is to add some sort
of metadata which would let people filter based on their preferences.
In addition to such distinctions as formal vs. expository, one could
also hav such things as level of difficulty (beginner, intermediate,
advanced), etc.

rspuzio writes:

> Both sentries serve useful purposes and I hope
> Yark will relent and let both entries stay.

What do you think is useful about the old entry? If I include the triple notation in the new entry, would that make the old one redundant in your eyes?

In general I think that duplicate entries on PlanetMath are a good thing (though obviously it's better to write an entry about something new rather than something that's already covered). But I had no intention of duplicating this entry, and I doubt anyone would even have noticed that I had replaced the old entry with a new one if I hadn't had to keep the old one around for while to prevent this thread from disappearing. (Is it really impossible for an admin to move the thread to a forum?)

I am not going to keep this entry, but I could transfer it to you if you really think it's worth keeping.

> I am not going to keep this entry, but I could transfer it to you if
> you really think it's worth keeping.

I'll take it.

Yet another problem surrounding this benighted entry --- Yark offered
me ownership, of the object, I clicked "accept", the computer said I
was given ownership, but did not actually transfer ownership to me.
This is the second time I havetried to accept the transfer of
ownership --- the first time was a few adays ago; I thought all went
wel but got another e-mail posting askiing me to accept ownership of
the object.

The reason I am posting this here is because of the bug Yark noticed
with the bug reporting system.

I assumed something had gone wrong with the first transfer request, as I hadn't got any notice back about acceptance or rejection of the transfer, and the entry was still owned by me. So I sent a second request today, which is why you got two. Both transfer requests appeared to go normally from this end, but I didn't receive either of the acceptance notices that I should have got.

Even though this discussion is months old, let me add my input to it, because quite a few of my own entries use the "pretentious" triple notation, even though this entry in particular is not mine.

It is completely clear from the discussion that you (stevecheng, the originator of this complaint) have never studied anything that I would consider to be extremely advanced mathematics. In saying this, I do not mean to denigrate your knowledge or to discourage you in any way. Indeed, I strongly encourage you to continue learning more mathematics, because it is the best way for you to outgrow your aversion to triples definitions. I guarantee you that, as you advance, there will come a point in your studies when you will find yourself *unable* to learn new definitions unless you use triples.

The reason is that, as definitions increase in complexity, it becomes impossible for the human mind to categorize all the elements of the definition properly unless the individual components of the definition are fully laid out, in a precise fashion, using "pretentious" formal notation. If you are still at the level of topological groups or measure spaces, then I can see how you might think triples are unnecessary. However, once you get to definitions such as group schemes or algebraic stacks, most mathematicians (with _rare_ exceptions) need to have a definition that specifies every component of the object, with formal notation, in order to even have any hope of learning what the object means.

If it should turn out that you are one of the exceedingly rare individuals who can learn group schemes or algebraic stacks without using tuple notation, then, well, people like that are such a small minority that it would be counterproductive to target PlanetMath towards those people at the expense of the majority.

The reason I use tuple notation even in comparatively simple definitions such as measure space is because I believe it is good practice to get readers into the habit of accepting formal notation in definitions, in order to prepare them for later on in their studies when such techniques become indispensible.

It might be said that pretentious, formal mathematics requires pretentious formal notation!

But joking aside, I too believe that the initial seeming artificiality of much of mathematical notation is greatly compensated by the benefits of it.

However, it is certainly possible to overdo it. I personally do detest a mode of mathematical writing that insists on using strings of symbols where a few phrases would do much better. They might be even more evocative of the spirit underlying the concept(s) being presented.

Overly formalistic writing reminds me of the statement made by philosopher / epigrammatist La Rochefoucauld (?) that "language was given to men that they might conceal their thoughts"! (I'm almost certainly not quoting very faithfully).

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