Math Question - One of Those Silly Ones From Article

[Nominal Animal]

In general encodings do not enable operations on the encoded information and do not allow the information to be transformed into something different. Notations do enable such operations and transformations.

Agreed.  Encoding maps symbols, but notation maps entire systems: symbols and operators/operations/mappings.

The difference between "notation" and "program code" is interesting, as "notation" tends to be used exclusively when humans do the interpretation (music, math, rally route marking, etc.), and "program code" when a machine does the interpretation.  I can definitely see why both are considered "languages" (by some/many).

The difference between encoding and notation is very vague, of course.  For example, Unicode is an encoding between visual glyphs and their numerical representation in digital systems.  RPN is notation and not encoding because it defines not only the symbols but also the operations the symbols refer to.

I suggest encoding is contained within notation; with whether operations or mappings that affect other symbols are included, and whether it is self-contained or relies on other rules for full utility, being the separator between 'notation' and 'encoding'.

This is, of course, only my current understanding based on the observed effective use (in mostly technical and scientific contexts) and the dictionary definitions.

Why would anyone want to use prefix notation?

Why would anyone want to speak any other language but English?

Your question does not answer my question.

It is a rhetorical question, whose answer also answers your own question:

Because they find it useful.

Just because you find something natural and obvious and comfortable, does not mean it is so for everyone else too.

For example, my native language is Finnish, which has plenty of grammatical cases; is typologically agglutinative; uses suffixal affixation; and just about everything is inflected.  Something like "Can you see my car?" is just two words in Finnish, "Näetkö autoni?", and not because the expression is simpler: "Can you see your car?" is "Näetkö autosi?", and "Can they see my car?" is "Näkevätkö he autoni?".  I'm pretty good (but nowhere near perfect) in Finnish grammar, even with possessive suffix ("autoni" instead of "minun auto" for "my car") which seems very difficult for even those who write Finnish as a job.

Nevertheless, gendered nouns in Swedish and German completely throw me in a loop.  And no, it is not a matter of "not learning enough": I've got years of Swedish studies completed with good grades, but still the only way to get the noun genders right, is for me to memorize every damn noun and their inflection and gender.

To expand a bit: As explained by Łukasiewicz, the principle of his (prefix) notation was to write the functors before the arguments to avoid brackets, and that he had employed his notation in his logical papers since 1929.  The postfix notation, on the other hand, came decades later; in 1941 in Germany, gaining some traction in the 1950s and much more in 1960s.

To explore the reasons why others might find prefix notation easier than postfix notation, we'd need to delve into language, and especially the differences in word ordering.  For example, although many do not realize it at all, English adheres strictly to adjective order, with opinion, size, age, shape, colour, origin, material, and purpose in that order.  Saying 'red large old box' will feel unnatural or as if emphasizing one of the adjectives to native English speakers; they will always prefer to 'large old red box'.

[switchabl]

"BODMAS" is an actual method in mathematics and hence whatever field (subsets) using it including engineering.. its a protocol (or precedence) laid out by experience people to avoid confusion/ambiguity/different semantics outcome.

It's not, it's just one of several common conventions and relying on it is a possible cause of confusion and ambiguity.

It is in fact very common to find e.g. 1/(2x) written as 1/2x in scientific publications and blindly applying "BODMAS" would lead you to the wrong interpretation. And this is not just carelessness, it is actually in the Physical Review style guide (https://cdn.journals.aps.org/files/styleguide-pr.pdf) and it used to be in the AMS one as well (https://web.archive.org/web/19980212112429/http://www.ams.org/authors/guide-reviewers.html).

In my experience, mathematicians in particular are not usually sticklers for these kind of "rules" and use whatever notation and shorthands they find most convenient.

[IanB]

In general encodings do not enable operations on the encoded information and do not allow the information to be transformed into something different. Notations do enable such operations and transformations.

Agreed.  Encoding maps symbols, but notation maps entire systems: symbols and operators/operations/mappings.

I think your definition of encoding is too narrow. It can be that narrow for you if you wish it to be, but I don't see it. Operators, operations and mappings are also represented by symbols.

Why are computer programs called "code" (from encoding) and not called "notes" (from notation)? Computer programs contain operations and operators, or they would do nothing useful.

In language, why is switching fluently between languages called "code switching"? Human languages are the archetype of complex systems of expression.

[IanB]

For example, although many do not realize it at all, English adheres strictly to adjective order, with opinion, size, age, shape, colour, origin, material, and purpose in that order.  Saying 'red large old box' will feel unnatural or as if emphasizing one of the adjectives to native English speakers; they will always prefer to 'large old red box'.

Personally, I would much prefer a nice large old rectangular red Chinese wooden puzzle box 

[tggzzz]

In general encodings do not enable operations on the encoded information and do not allow the information to be transformed into something different. Notations do enable such operations and transformations.

Yeah, I'm just gonna have to say bullshit.

Computer programs are an encoding of information, and tools like compilers readily transform them by translations and optimizations into quite different outputs.

You are an encoding of information, both directly in biochemical ways and also indirectly in emergent phenomena that nobody understands.

Image files and audio files are an encoding of information, and they also are transformed into compressed forms and different formats.

This was made real to me years ago when I compared the source code and the compiled output of a benchmark program I wrote on a DEC Alpha machine. The compiled output was cryptic and bore no resemblance to the program I wrote. It was transformed beyond recognition. Yet it gave the correct results.

Anybody that thinks there is more that a vague relationship between computers and mathematics is living in a state of sin. Ask any mathematician!

[Nominal Animal]

In my experience, mathematicians in particular are not usually sticklers for these kind of "rules" and use whatever notation and shorthands they find most convenient.

Physics too, even more so.  Many a physicist have developed their own notation to express their ideas better; just consider Dirac's bra-ket notation now commonly used in quantum mechanics.  The purpose is to convey the underlying idea/discovery/formula, and if that takes some fiddling with the notation, so be it: just describe the notation first (preferably via an otherwise ambiguous or complicated expression), and get on with it.

Language is hard, because the mapping isn't concept ↔ word, it is something like emotion and experience ↔ term cluster ↔ expression, with all parts in constant flux.  You do not really describe anything exactly; you more like sketch things out with terms and expressions that somehow delineate/circumscribe/limit the concept you're trying to convey, with each one being interpreted or understood slightly differently by each reader/listener.

I think your definition of encoding is too narrow. It can be that narrow for you if you wish it to be, but I don't see it.

Well, start by reviewing the definitions of encoding and notation in some dictionary you consider authoritative/reliable/trustworthy, and go on from there.

My definition may be narrow, but it does conform to how I understand their English dictionary definitions, and matches the real world technical/scientific use I've encountered.  It is not just my "opinion" based on a gut feeling; it is the most useful/effective/reasonable definition I can find.

Why are computer programs called "code" (from encoding) and not called "notes" (from notation)?

Why isn't English written phonetically, like Finnish and Latin are?  For historical reasons.

I suspect the reason for "code"/"coding" can be found by investigating the punched card era, and how the word "programming" was used among the laypeople (including university administration).

For the same reason as we Finns call an electron microscope "elektronimikroskooppi" instead of the much more descriptive and logical "sähkälehitutähystin", I refuse to assume there is actual logic or rational thought behind exactly which words or terms are used for which thing.  (This includes the fact that popularity is no guarantee of quality or fitness for a purpose; only of popularity.)

In language, why is switching fluently between languages called "code switching"?

That is the first time I have ever heard that term used for that.  I also find using "code" for language (except for spy cipher stuff) really uncommon and odd.

Could this – gasp! – be culturally dependent?  Oh noes.

[Mechatrommer]

It's not, it's just one of several common conventions and relying on it is a possible cause of confusion and ambiguity.

we can say that to other conventions too. confusion occurs when sender/receiver use different convention/encoding/decoding/grammar/protocol scheme... so convention A is just as confusing as convention B, or C or whatever... they are all are possible cause of confusion and ambiguity.

It is in fact very common to find e.g. 1/(2x) written as 1/2x in scientific publications and blindly applying "BODMAS" would lead you to the wrong interpretation. And this is not just carelessness, it is actually in the Physical Review style guide (https://cdn.journals.aps.org/files/styleguide-pr.pdf)

i'm not sure at which part you are referring to in the pdf, all i see are formulas following BODMAS convention. anyway how they are going to write if what they meant is (1/2)x without bracket? are ncl and the rest wrong? https://www.ncl.ac.uk/webtemplate/ask-assets/external/maths-resources/numeracy/order-of-operations-bodmas.html

and it used to be in the AMS one as well (https://web.archive.org/web/19980212112429/http://www.ams.org/authors/guide-reviewers.html).

used to be?

In my experience, mathematicians in particular are not usually sticklers for these kind of "rules" and use whatever notation and shorthands they find most convenient.

you havent seen them, or they havent shoot each other's foot yet... when ambiguity occurs, they will try to find a way to correct their laziness.... we wouldnt call us lazy if everybody of us are lazy... cheers.

[Nominal Animal]

It is in fact very common to find e.g. 1/(2x) written as 1/2x in scientific publications and blindly applying "BODMAS" would lead you to the wrong interpretation. And this is not just carelessness, it is actually in the Physical Review style guide (https://cdn.journals.aps.org/files/styleguide-pr.pdf)

i'm not sure at which part you are referring to in the pdf, all i see are formulas following BODMAS convention.

Page 21, 2. Fractions; Apply the following guidelines; (e) When slashing fractions, respect the following conventions. In mathematical formulas this is the accepted order of operations: (1) raising to a power, (2) multiplication, (3) division, (4) addition and subtraction.

It means that in Physical Review style notation, in a subexpression containing a slashed fraction, multiplication has higher precedence than division.

anyway how they are going to write if what they meant is (1/2)x without bracket?

\$\frac{1}{2} x\$ or \$\frac{x}{2}\$, according to the style guide.

are ncl and the rest wrong?

No, they just specify a different set of rules for the notation, i.e. a different notation.

There is no "right" or "wrong" notation; they're just different.

they havent shoot each other's foot yet... when ambiguity occurs, they will try to find a way to correct their laziness.... we wouldnt call us lazy if everybody of us are lazy...

Many, many different notations are used in math and in particular its subfields.  Ambiguity is not "allowed"; they are one thing that reviewers are supposed to note and ask the author to clarify.  In reference books, the exact notation used in the book is almost always described; if not, it is considered a bad choice or poor form, generally speaking.  Often the notation is defined just in passing: "The mathematical expressions use the same notation as Mathematical Methods for Physicists by G. B. Arfken, H. J. Weber, and F. E. Harris."

[TimFox]

Joke: 
A student goes into a mathematics professor.
He says "when I write the equation in this manner, it is ambiguous".
The professor answers, "don't write it that way".

[CatalinaWOW]

It is in fact very common to find e.g. 1/(2x) written as 1/2x in scientific publications and blindly applying "BODMAS" would lead you to the wrong interpretation. And this is not just carelessness, it is actually in the Physical Review style guide (https://cdn.journals.aps.org/files/styleguide-pr.pdf)

i'm not sure at which part you are referring to in the pdf, all i see are formulas following BODMAS convention.

Page 21, 2. Fractions; Apply the following guidelines; (e) When slashing fractions, respect the following conventions. In mathematical formulas this is the accepted order of operations: (1) raising to a power, (2) multiplication, (3) division, (4) addition and subtraction.

It means that in Physical Review style notation, in a subexpression containing a slashed fraction, multiplication has higher precedence than division.

anyway how they are going to write if what they meant is (1/2)x without bracket?

\$\frac{1}{2} x\$ or \$\frac{x}{2}\$, according to the style guide.

are ncl and the rest wrong?

No, they just specify a different set of rules for the notation, i.e. a different notation.

There is no "right" or "wrong" notation; they're just different.

they havent shoot each other's foot yet... when ambiguity occurs, they will try to find a way to correct their laziness.... we wouldnt call us lazy if everybody of us are lazy...

Many, many different notations are used in math and in particular its subfields.  Ambiguity is not "allowed"; they are one thing that reviewers are supposed to note and ask the author to clarify.  In reference books, the exact notation used in the book is almost always described; if not, it is considered a bad choice or poor form, generally speaking.  Often the notation is defined just in passing: "The mathematical expressions use the same notation as Mathematical Methods for Physicists by G. B. Arfken, H. J. Weber, and F. E. Harris."

The hidden subtext here is that BODMAS and similar rules are most useful when an expression is written in a single line of text where all characters are the same size.  As soon as you can write equations using multiple fonts with superscripts, subscripts and other typesetting tools a whole new toolbox of clarification opens up.  And it is a world where BODMAS frequently hurts more than it helps.

What a terrible legacy the punch card and typewriter left for us.

[ledtester]

I can see the reasonableness of allowing 1/2x to stand for 1/(2x). After all, any expression (a/b)c may be written as ac/b. I.e. if you want "one-half x" you can just write x/2.

[coppice]

The hidden subtext here is that BODMAS and similar rules are most useful when an expression is written in a single line of text where all characters are the same size.  As soon as you can write equations using multiple fonts with superscripts, subscripts and other typesetting tools a whole new toolbox of clarification opens up.  And it is a world where BODMAS frequently hurts more than it helps.

Yep. That's it in a nutshell. Who would have thought that an algorithm worked out for trivial cases could be problematic as soon as you add a little complexity? 

[tggzzz]

The hidden subtext here is that BODMAS and similar rules are most useful when an expression is written in a single line of text where all characters are the same size.  As soon as you can write equations using multiple fonts with superscripts, subscripts and other typesetting tools a whole new toolbox of clarification opens up.  And it is a world where BODMAS frequently hurts more than it helps.

What a terrible legacy the punch card and typewriter left for us.

BEDMAS has nothing to do with typeface, human language, machine language, nachine tooling or anything like that. It is a pure mathematics concept concerning mathematical operations.

Other mathematical concepts include planes, lines, points. Or maybe you would like to claim that planes are screwed up because of machining tolerances, and lines because of pen widths?

You are, of course, free to invent your own rules for evaluating expressions - and many have. But the onus is then on you to state and explain (and preferably justify) your rules before using them.

[coppice]

BEDMAS has nothing to do with typeface, human language, machine language, nachine tooling or anything like that. It is a pure mathematics concept concerning mathematical operations.

Er, no. The principles behind BEDMAS are abstract and universal. BEDMAS itself is an in class scheme being taught within a notational framework. Two very different things. Its the split that causes so many stupid arguments about "maths is universal" versus "maths is a social construct". Practical maths is both. The foundation underlying it is universal, and we will see the same things at play with any race we might encounter roaming the universe. Notation is needed to communicate those universal concepts, and that's the socially constructed part. BEDMAS merges the two.

[tggzzz]

BEDMAS has nothing to do with typeface, human language, machine language, nachine tooling or anything like that. It is a pure mathematics concept concerning mathematical operations.

Er, no. The principles behind BEDMAS are abstract and universal. BEDMAS itself is an in class scheme being taught within a notational framework. Two very different things. Its the split that causes so many stupid arguments about "maths is universal" versus "maths is a social construct". Practical maths is both. The foundation underlying it is universal, and we will see the same things at play with any race we might encounter roaming the universe. Notation is needed to communicate those universal concepts, and that's the socially constructed part. BEDMAS merges the two.

I really don't understand the validity of "BEDMAS is a social construct".

But then I don't understand the validity of PoMoLitCrit either, e.g. https://www.info.ucl.ac.be/~pvr/decon.html

[coppice]

I really don't understand the validity of "BEDMAS is a social construct".

The sum of product underpinnings of BEDMAS come from nature. The rest of the package (i.e. the notation on the page) was created by humans, multiple times, in different parts of the world, with a very different look on the page. Whilst we still have many human languages, the whole world has largely focussed around just one of the socially constructed notations for maths in modern times. So, it can be confused with being universal. It isn't.

But then I don't understand the validity of PoMoLitCrit either, e.g. https://www.info.ucl.ac.be/~pvr/decon.html

It has no validity. Its designed to be like the can of worms any science or engineering discipline tries hard to move itself away from.

[Nominal Animal]

What a terrible legacy the punch card and typewriter left for us.

Also consider what effect our writing styles (not just Latin/Western, but also ideographic like Chinese) has had.

Consider what effect it would have on phonetic writing, if we were to just design a single vowel glyph with features matching how the vowel is generated in the human vocal apparatus?  (IPA vowel classification is two-dimensional: front/central/back, and close/close-mid/open-mid/open; so one possibility would be a rectangle with a diagonal stroke, the two intersection points defining the vowel.)  Any generally physically possible vowel could be represented by continuously variant glyph directly.  Length and gliding into another vowel type could be described using a merging glyph, for example sharing one edge of the rectangle, and tone with umlauts (as in Pinyin for Chinese).

Languages – including mathematical notation – are full of history, and have very little to no intentional design.  We could do better, but we choose not to.

We humans really do have a long way ahead, before we can consider ourselves a rational species, when one looks at what we do instead of what we claim we are.

[TimFox]

As I understand it (not being fluent in Latin), the Latin or Roman alphabet is a reasonably efficient encoding of pronunciation in the classic Latin language.
As Romance languages evolved from Latin, the alphabet was used for Spanish, French, Italian, etc. and sort-of worked.
It was then applied to Germanic, Slavic, etc. languages and became a problem requiring lots of diacritics to work at all.
English then mixed all of the above together to result in chaotic orthography.

[IanB]

Consider what effect it would have on phonetic writing, if we were to just design a single vowel glyph with features matching how the vowel is generated in the human vocal apparatus?

But do we want phonetic writing?

Consider, for example, the Norwegian word "ikke". Some people pronounce it with a hard "k" sound, while others pronounce it with a soft "ch" sound (as in German ich). Should the spelling change according to the speaker?

[TimFox]

In the lyrics of Ira Gershwin:  "Let's call the whole thing off."

[Nominal Animal]

As I understand it (not being fluent in Latin), the Latin or Roman alphabet is a reasonably efficient encoding of pronunciation in the classic Latin language.

Finnish, too.  There are a few diphtongs (ae, oe) that meld together, as do ng and nk, but otherwise Finnish is written phonetically using the Latin alphabet.  We do use two additional vowels ä and ö, with ä corresponding to the a in 'back' (whereas a corresponds to the a in 'dark'), ö to the i in 'bird'.  Doubled vowels are long, and doubled consonants refer to a delay/long consonant.

Even so, compare to e.g. Hangul, which is written in syllabic blocks, making correct pronunciation (rhythm-wise) much easier.  Even though I cannot read Korean (or know Hangul alphabet), I do believe it is superior notation compared to the Latin alphabet.

I was mostly thinking about languages like English (especially because of its British/American/Australian variants), French, German, and Swedish.  I know how to pronounce småhagel (Ikea USB charger BigClive did a teardown on a month or two ago) or blåkläder, but it seems to pose difficulties to English speakers.

But do we want phonetic writing?

The use of Pinyin when learning Chinese indicates it might make learning a language easier.  I am not suggesting it would be forced on everybody in all languages (that shit never works), but as another tool: another notation, if you will.  (Except in this case it would be just another encoding of the pronunciation, really.)

It would be useful for foreign names, for example, if it was standardized and widely used.  Consider a name badge, with your name on it using your own orthography, with the standard phonetic form below.

IPA is too esoteric and complicated to be actually useful for this.  I've looked at it a couple of times, but every detail needs to be memorized because there is no logical variation, only separate glyphs for each thing that cannot be predicted or extrapolated from the others.

[TimFox]

Hangul was invented by the scholarly king Sejong the Great in AD 1443, and is an efficient encoding of the Korean language.
I encountered the Romaji spelling rules for using the Latin alphabet for the Japanese language, which originated with a missionary.
A problem for Americans first encountering this very reasonable set of rules is pronouncing vowels;   I learned to pronounce all vowels in that spelling as in Italian.
(Only an idiot would say "as in English", since that is ill-determined.)
There is a wonderful Japanese dairy-based soft drink (tastes something like yogurt) spelled "Calpis", which is often pronounced "Karupisu".
If you ask for it at a vendor, and pronounce the "a" is in "Calvin" and the "i" as in "if", the Japanese vendor will not recognize the word.
Try "a" and "i" as in "adagio", and it works.

[helius]

Languages – including mathematical notation – are full of history, and have very little to no intentional design.  We could do better, but we choose not to.

As an example, the notations for rising and falling powers are ambiguous: some authors use opposite notations for the two concepts.

[IanB]

There is a wonderful Japanese dairy-based soft drink (tastes something like yogurt) spelled "Calpis"

I thought you were going to mention Pocari Sweat, of which I am always pondering, "What is a Pocari?"

Now you also have me wondering,  "What is a Cal?"   

[Edit: add missing grin]

[TimFox]

"Pocari" is a trademark; the manufacturer says "Pocari" was named for its light nuance and bright sound, and it has no particular meaning.
(Like "Kodak".)
"Sweat" refers to the electrolytes lost in perspiration that need to be replenished in bodily fluids.
Another popular "sports drink" in Japan is "NCAA".
Of course, Gatorade is the popular US beverage, where "Gator" refers to the University of Florida football team.
The "cal" in Calpis means "calcium", found in dairy products.