https://blog.plover.com/math/se/notation.html
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Mark Dominus Wed, 09 Mar 2022
(Tao Min Xiu )
mjd@pobox.com Bad but interesting mathematical notation idea
[TOP] Zaz Brown showed up on Math SE yesterday with a
proposal to make mathematical notation more uniform.
About me It's been pointed out several times that the
expressions
RSS Atom
$$y^n = x \qquad n = \log_y x \qquad y=\sqrt[n]x $$
12 recent
entries all mean the same thing, and yet look completely
different. This has led to proposals to try to unify
Olaf's new menu the three notations, although none has gone anywhere.
item (For example, this Math SE thread .)
U.S. surnames
with no vowels !!\def\o{\overline}\def\u{\underline}!!
Best
occupational In this new thread, M. Brown has an interesting
name ever? observation: exponentiation also unifies addition and
My horse Pongo multiplication. So write !!\o x!! to mean !!e^x!!,
There is a Unix and !!\u x!! to mean !!\ln x!!, and leave
error device multiplication as it is. Now !!x^y!! can be written
Why no Unix as !!\o{\u x y}!! and !!x+y!! can be written as !!\u
error device? {\bar x \! \bar y}!!.
Bad but
interesting Well, this is a terrible idea, and I'll explain why I
mathematical think so in some detail. But I really hope nobody
notation idea will think I mean this as any sort of criticism of
Quicker and its author. I have a lot of ideas too, and most of
easier ways to them are amazingly bad, way worse than this one.
get more light Having bad ideas doesn't make someone a bad person.
I vent my rage And just because an idea is bad, doesn't mean it
at dumbass Math wasn't worth considering; thinking about ideas is how
SE comments you decide which ones are bad and which aren't.
More about the M. Brown's idea was interesting enough for me to
axioms of think about it and write an article. That's a
infinity and compliment, not a criticism.
the empty set
My mistake I'm deeply interested in notation. I think
about errors in mathematicians don't yet understand the power of
the mathematical notation and what it does. We use it,
presentation of but we don't understand it. I've observed before that
axiomatic set you can solve algebraic equations or calculus
theory problems just by "pushing around the symbols". But
"Shall" and why can you do that? Where is the meaning, and how do
"will" strike the symbols capture the meaning? How does that work?
back from The fact that symbols in general can somehow convey
beyond the meaning is a deep philosophical mystery, not just in
grave mathematics but in all communication, and nobody
understands how it works. Mathematical symbols can be
Archive: even more amazing: they don't just tell you what
other people were thinking, they tell you things
2022: JFMA themselves. You rearrange them in a certain way and
2021: JFMAMJ they smile and whisper secrets: "now you can see this
JASOND function is everywhere zero", "this is evidently
2020: JFMAMJ unbounded" or "the result is undefined when !!\lvert
JASOND x_1\rvert > \frac 23!!". It's almost as if the
2019: JFMAMJ symbols are doing some of the thinking for you.
JASOND
2018: JFMAMJ Anyway this particular idea is not good, but maybe we
JASOND can learn something from its failure modes?
2017: JFMAMJ
JASOND Here's how you would write !!x^2+x!!: $$\u{\o{\o{2\u
2016: JFMAMJ x}}{\o x}}$$
JASOND
2015: JFMAMJ Zaz Brown suggested that this expression might be
JASOND better written as !!x{\u{\o x \o 1}}!!, which is
2014: JFMAMJ analogous to !!x(x+1)!!, but I think that reply
JASOND misses a very important point: you need to be able to
2013: JFMAMJ write both expressions so that you can equate them,
JASOND or transform one into the other. The expression !!x
2012: JFMAMJ (x+1)!! is useful because you can see at a glance
JASOND that it is composite for all integer !!x!! larger
2011: JFMAMJ than 1, and actually twice a composite for
JASOND sufficiently large !!x!!. (This is the kind of thing
2010: JFMAMJ I had in mind when I said the symbols whisper secrets
JASOND to you.) !!x^2+x!! is useful in different ways: you
2009: JFMAMJ can see that it's !!\Theta(x^2)!! and it's !!(x+1)^2
JASOND - (x+1)!! and so on. Both are useful and you need to
2008: JFMAMJ be able to turn one into the other easily. Good
JASOND notation facilitates that sort of conversion.
2007: JFMAMJ
JASOND M. Brown's proposal actually has at least two
2006: JFMAMJ components. One component is its choice of
JASOND multiplication, exponentials and logarithms as the
2005: OND only first-class citizens. The other is the specific
way that was chosen to write these, with the over-
and underbars. This second component is no good at
--------------- all, for purely typographic reasons. These three
Subtopics: expressions look almost identical but have completely
different meanings: $$ \u{\o a\, \o c}\qquad \u{\o {
Mathematics 215 ac}} \qquad \o{\u a\, \u c}.$$
Programming 80
Language 77 In fact, the two on the right were almost
Misc 57 indistinguishable until I told MathJax to put in some
Book 46 extra space. I'm sure you can imagine similar
Tech 40 problems with !!\u{\o{\o{2\u x}}}{\o x}!! turning
Oops 29 into !!\u{\o{\o{2\u x x}}}!! or !!\u{\o{\o{2\u x }
Unix 26 x}}!! or whatever. Think of how easy it is to drop a
Cosmic Call 25 minus sign; this is much worse.
Haskell 22
Physics 21 [ Addendum 20220308: Earlier, I had said that !!x+y!!
Etymology 21 could be written as !!\u{\bar x\bar y}!!. A Gentle
Law 16 Reader pointed out that the bar on the bottom wasn't
Perl 16 connected but should have been, as on the far right
of this screenshot:
[mjd-univer]
Higher-Order Screenshot of blog text "x+y can be written as (xy)
Perl Blosxom (xy)" where in each case both the x and the y have
overbars, and the whole thing has an underbar, except
Comments that on the right the underbar has a tiny break, and
disabled on the left the x and y have been squished together
uncomfortably to eliminate the break in the underbar.
I meant it to be connected and what I wrote asked for
it to be connected, but MathJax, which formats the
math formulas on the blog, didn't connect it. To
remove the gap, I had to explicitly subtract space
between the !!x!! and the !!y!!. ]
But maybe the other component of the proposal has
something to it and we will find out what it is if we
fix the typographic problem with the bars. What's a
good alternative?
Maybe !!\o x = x^\bullet!! and !!\u x = x_\bullet!! ?
On the one hand we get the nice property that !!x^\
bullet_\bullet = x!!. But I think the dots would make
my head swim. Perhaps !!\o x = x\top!! and !!\u x = x
\bot!!? Let's try.
Good notation facilitates transformation of
expressions into equal expressions. The !!\top\bot!!
notation allows us to easily express the simple
identities $$a\top\bot \quad = \quad a\bot\top \quad
= \quad a.$$ That kind of thing is good, although the
dots did it better. But I couldn't find anything else
like it.
Let's see what the distributive law looks like. In
standard notation it is $$a(b+c) = ab + ac.$$ In the
original bar notation it was $$a\u{\o b\o c} = \u{\o
{ab}\, \o{ac}}.$$ This looks uncouth but perhaps
would not be worse once one got used to it.
With the !!\top\bot!! idea we have
$$ a(b\top c\top)\bot = ((ab)\top(ac)\top)\bot. $$
I had been hoping that by making the !!\top!! and !!\
bot!! symbols postfix we'd be able to avoid
parentheses. That didn't happen: without the
parentheses you can't distinguish between !!(ab)\
top!! and !!a(b\top)!!. Postfix notation is famous
for allowing you to omit parentheses, but that's only
if your operators all have fixed arity. Here the
invisible variadic multiplication ruins that. And
making it visible dyadic multiplication is not really
an improvement:
$$ ab\top c\top\cdot\cdot\bot = ab\cdot\top ac\cdot \
top\cdot \bot. $$
You know what I think would happen if we actually
tried to use this idea? Someone would very quickly
invent an abbreviation for !!\u{\o {x_1}\, \o {x_2} \
cdots \o{x_k}}!!, I don't know, something like "!!x_1
+ x_2 + \ldots + x_k!!" maybe. (It looks crazy, I
know, but it might just work.) Because people might
like to discuss the fact that $$ \u{\o 2\, \o 3 } =
5$$ and without an addition sign there seems to be no
way to explain why this should be.
Well, I have been turning away from the real issue
for a while now, but !!a(b\top c\top)\bot = !! !!
((ab)\top(ac)\top)\bot!! forces me to confront it.
The standard expression of the distributive law
equates a computation with two operations and another
with three. The computations expressed by the new
notation involve five and six operations
respectively. Put this way, the distributive law is
no longer simple!
This reminds me of the earlier suggestion that if !!x
^2+x!! is too complicated, one can write !!x(x+1)!!
instead. But expressions don't only express a result,
they express a way of arriving at that result. The
purpose of an equation is to state that two different
computations arrive at the same result. Yes, it's
true that $$a+b = \ln e^ae^b,$$ but the two
computations are not the same! If they were, the
statement would be vacuous. Instead, it says that the
simple computation on the left arrives at the same
result as the complicated one on the right, an
interesting thing to know. "!!2+3=5!!" might imply
that !!e^2\cdot e^3=e^5!! but it doesn't say the same
thing.
Here's my takeaway from consideration of the Zaz
Brown proposal:
It's not sufficient for a system of notation to
have a way of expressing every result; it has to
be able to express every possible computation.
Put that way, other instructive examples come to
mind. Consider Egyptian fractions. It's known that
every rational number between !!0!! and !!1!! can be
written in the form $$\frac1{a_1} + \frac1{a_2} + \
ldots + \frac1{a_n}$$ where !!\{ a_i\}!! is a
strictly increasing sequence of positive integers.
For example $$\frac 7{23} = \frac 14 + \frac1{19} + \
frac1{583} + \frac1{1019084}$$ or with a bit more
ingenuity, $$\frac7{23} = \frac16 + \frac1{12} + \
frac1{23} + \frac1{138} + \frac1{276},$$ longer but
less messy. The ancient Egyptians did in fact write
numbers this way, and when they wanted to calculate
!!2\cdot\frac17!!, they had to look it up in a table,
because writing !!\frac27!! was not an expressible
computation, it had to be expressed in terms of
reciprocals and sums, so !!2\cdot\frac 17 = \frac14 +
\frac1{28}!!. They could write all the numbers, but
they couldn't write all the ways of making the
numbers.
(Neither can we. We can write the real root of !!x^
3-2!! as !!\sqrt[3]2!!, but there is no effective
notation for the real root of !!x^5+x-1!!. The best
we can do is something like "!!0.75488\ldots!!",
which is even less effective than how the Egyptians
had to write !!\frac27!! as !!\frac14+\frac1{28}!!.)
Anyway I think my conclusion from all this is that a
practical mathematical notation really must have a
symbol for addition, which is not at all surprising.
But it was fun and interesting to see what happened
without it. It didn't work well, but maybe the next
idea will be better.
Thanks again, Zaz Brown.
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