17 ms·

> My concern is that the word “elementary” in the title carries a much broader meaning in standard mathematical usage, and in this meaning, the paper’s title do

by rnhmjoj 5mo ago

> My concern is that the word “elementary” in the title carries a much broader meaning in standard mathematical usage, and in this meaning, the paper’s title does not hold. > Elementary functions typically include arbitrary polynomial roots, and EML terms cannot express them. If you take a real analysis class, the elementary functions will be defined exactly as the author of the EML paper does. I've actually just learnt that some consider roots of arbitrary polynomials being part of the elementary functions before, but I'm a physicist and only ever took some undergraduate mathematics classes. Nonetheless, calling these elementary feels a bit of stretch considering that the word literally means basic stuff, something that a beginner will learn first.

reikonomusha 5mo ago
The definition of "elementary function" typically includes functions which solve polynomials, like the Bring radical. The definition was developed and is most fitting in algebraic contexts where algebraic structure is meaningful, like Liouvillian structure theorems, algorithmic integration, and computer algebra. See e.g. - Page 2 and the following example of https://billcookmath.com/courses/math4010-spring2016/math4010-spring2016-differential_algebra.pdf https://billcookmath.com/courses/math4010-spring2016/math401... (2016) - Ritt's Integration in Finite Terms: Liouville's Theory of Elementary Methods (1948) It's not frequent that analysis books will define the class of elementary functions rigorously, but instead refer to examples of them informally.
- thaumasiotes 5mo ago
  > See e.g. Page 2 and the following example of https://billcookmath.com/courses/math4010-spring2016/math4010-spring2016-differential_algebra.pdf https://billcookmath.com/courses/math4010-spring2016/math401... (2016) There appears to be a typo in that example; I assume "Essentially elementary functions are the functions that can be built from ℂ and f(x) = x" should say something more like "the functions that can be built from ℂ and f(x) = y".
  - reikonomusha 5mo ago
    Not a typo! Think of f(x) = x as a seed function that can be used to build other functions. It's one way to avoid talking about "variables" as a "data type" and just keep everything about functions. We can make a function like x + x*exp(log(x)) by "formally" writing f + f*(exp∘log) where + and * are understood to produce new functions. Sort of Haskell-y.
- Joker_vD 5mo ago
  > The definition of "elementary function" typically includes functions which solve polynomials, like the Bring radical. What. Does that "typical definition" of elementary function includes elliptic functions as well, by any chance?
  - reikonomusha 5mo ago
    Not that I've seen.
burnished 5mo ago
All I know is that when a class starts with 'elementary' or 'fundamentals of' you had best buckle up.
- TeMPOraL 5mo ago
  Introduction to ...
  - Nevermark 5mo ago
    That's code for 101.
    - TeMPOraL 5mo ago
      No. It's code for the thickest, densest book on the subject that you're ever gonna not read, as it actually assumes you're experienced in the subject and goes into everything except intro level topics. See e.g. Petzold, et al.
      - mr_mitm 5mo ago
        I'm getting flashbacks to Spivak, who wrote a 2000 page "introduction" to differential geometry.
        reikonomusha 5mo ago
        To be fair to Spivak, he did say it was comprehensive introduction. :)
- quchen 5mo ago
  Algebraic too. There's also the opposite in physics though, "modern" means from the 60s with square roots drawn in manually.
chii 5mo ago
jargon are words being used that don't carry the typical laymen definition, but a specific one from the domain of said jargon. If a written piece is intended for an audience who knows the jargon, then it's fine to use jargon - in fact it's appropriate and succinct. If it was intended for the laymen, then jargon is inappropriate. But it seems you're lamenting that this jargon is wrong and that it shouldn't be jargon!?
mcmoor 5mo ago
I don't know if I read this right, but I thought it's proven that "elementary functions" can't solve 5th degree or higher polynomial, so I'm confused how it's interpreted if elementary functions also include arbitrary polynomial roots. Or is it different elementary functions?
- eru 5mo ago
  The term 'elementary function' doesn't really have a single universally agreed on strict definition. Definitions are either a bit fuzzy, or not universally agreed on. Though interestingly https://en.wikipedia.org/wiki/Elementary_function https://en.wikipedia.org/wiki/Elementary_function says "More generally, in modern mathematics, elementary functions comprise the set of [...]". Though at least Wikipedia thinks that 'modern mathematics' has a consensus; of course, there's no guarantee that whoever you are talking to uses the 'modern mathematics' definition that Wikipedia brings up.
- adrian_b 5mo ago
  That theorem is not formulated about "elementary functions". It says that polynomial equations of the 5th degrees or higher cannot, in general, be solved using "radicals". While something like "polynomials" or "radicals" has a clear meaning, which are the "elementary functions" is a matter of convention. The usual convention is to include all algebraic functions and a few selected transcendental functions. In "all algebraic functions", are included the rational functions, the radicals and the functions that compute solutions of arbitrary polynomial equations. Some conventions used for "elementary functions" describe the expressions that you can use to write such "elementary functions", in which case not all algebraic functions are included, but only those written by combining rational functions with radicals. For an algebraic function that computes a solution of a general polynomial equation, which cannot be expressed with radicals, you cannot write an explicit formula, but you can write the function only implicitly, by writing the corresponding polynomial equation. So the difference between the 2 kinds of conventions about which are "the elementary functions" is usually based on whether only explicitly-written functions are considered, or also implicit functions.
  - 5mo ago
SideQuark 5mo ago
> If you take a real analysis class, the elementary functions will be defined exactly as the author of the EML paper does. I just looked through many of the best known real analysis texts, and not a single one defines them this way. This list included the texts by Royden, Terence Tao, Rudin, Spivak, Bartle & Sherbert, Pugh, and a few others.... Can you cite a single text book that has this definition you claim is in every real analysis course? I find all evidence points to the opposite.
- rnhmjoj 5mo ago
  I guess you're right, I was probably mislead this whole time. I went through my old analysis class book [1] and there doesn't seem to be an explicit definition of elementary functions. The best I can find is this paragraph (I translate from italian): > The elementary functions of analysis, that is powers, roots, exponentials, logarithms and their inverses, functions obtained from the former by arithmetic operations or composition, admit the limit f(p) for x → p, for any p in their set of definition. The study of such functions, which is not limited to the sole real functions of real variable, is carried out naturally in the setting of metric spaces. That said, I'm relatively sure that a definition was given in class and it didn't include arbitrary roots: despite being notoriously difficult, the exam didn't require students to draw the graph of any elementary function including implicitly-defined algebraic roots. I picked up another one of the old recommended books [2] and it seems to be similarly vague; while the book currently taught in my university [3], gives this definition: > The following functions (from ℂ to ℂ) are called the elementary functions of the Analysis: > 1) Rational functions (integral or fractional) > 2) Algebraic functions (explicit or implicit) > 3) The exponential function > 4) The logarithm function > 5) All those functions that can be obtained by combining a finite number of times the functions of kind 1)...4). So, roots of arbitrary polynomials implicitly defined are indeed considered elementary. I never knew this. [1]: https://search.worldcat.org/title/1261811544 https://search.worldcat.org/title/1261811544 [2]: https://search.worldcat.org/title/801297519 https://search.worldcat.org/title/801297519 [3]: https://search.worldcat.org/title/935666878 https://search.worldcat.org/title/935666878
  - rnhmjoj 5mo ago
fnordpiglet 5mo ago
In math elementary usually means fundamental or foundational not elementary school. The root word is element and the relationship to “simple subject” is tangential and more related to its teaching the elemental topics for a lifetime education than definitionally cross discipline.

by rnhmjoj 5mo ago

reikonomusha 5mo ago

The definition of "elementary function" typically includes functions which solve polynomials, like the Bring radical. The definition was developed and is most fitting in algebraic contexts where algebraic structure is meaningful, like Liouvillian structure theorems, algorithmic integration, and computer algebra. See e.g. - Page 2 and the following example of https://billcookmath.com/courses/math4010-spring2016/math4010-spring2016-differential_algebra.pdf https://billcookmath.com/courses/math4010-spring2016/math401... (2016) - Ritt's Integration in Finite Terms: Liouville's Theory of Elementary Methods (1948) It's not frequent that analysis books will define the class of elementary functions rigorously, but instead refer to examples of them informally.

thaumasiotes 5mo ago
> See e.g. Page 2 and the following example of https://billcookmath.com/courses/math4010-spring2016/math4010-spring2016-differential_algebra.pdf https://billcookmath.com/courses/math4010-spring2016/math401... (2016) There appears to be a typo in that example; I assume "Essentially elementary functions are the functions that can be built from ℂ and f(x) = x" should say something more like "the functions that can be built from ℂ and f(x) = y".
- reikonomusha 5mo ago
  Not a typo! Think of f(x) = x as a seed function that can be used to build other functions. It's one way to avoid talking about "variables" as a "data type" and just keep everything about functions. We can make a function like x + x*exp(log(x)) by "formally" writing f + f*(exp∘log) where + and * are understood to produce new functions. Sort of Haskell-y.
Joker_vD 5mo ago
> The definition of "elementary function" typically includes functions which solve polynomials, like the Bring radical. What. Does that "typical definition" of elementary function includes elliptic functions as well, by any chance?
- reikonomusha 5mo ago
  Not that I've seen.

burnished 5mo ago

All I know is that when a class starts with 'elementary' or 'fundamentals of' you had best buckle up.

TeMPOraL 5mo ago
Introduction to ...
- Nevermark 5mo ago
  That's code for 101.
  - TeMPOraL 5mo ago
    No. It's code for the thickest, densest book on the subject that you're ever gonna not read, as it actually assumes you're experienced in the subject and goes into everything except intro level topics. See e.g. Petzold, et al.
    - mr_mitm 5mo ago
      I'm getting flashbacks to Spivak, who wrote a 2000 page "introduction" to differential geometry.
      - reikonomusha 5mo ago
        To be fair to Spivak, he did say it was comprehensive introduction. :)
quchen 5mo ago
Algebraic too. There's also the opposite in physics though, "modern" means from the 60s with square roots drawn in manually.

chii 5mo ago

jargon are words being used that don't carry the typical laymen definition, but a specific one from the domain of said jargon. If a written piece is intended for an audience who knows the jargon, then it's fine to use jargon - in fact it's appropriate and succinct. If it was intended for the laymen, then jargon is inappropriate. But it seems you're lamenting that this jargon is wrong and that it shouldn't be jargon!?

mcmoor 5mo ago

I don't know if I read this right, but I thought it's proven that "elementary functions" can't solve 5th degree or higher polynomial, so I'm confused how it's interpreted if elementary functions also include arbitrary polynomial roots. Or is it different elementary functions?

eru 5mo ago
The term 'elementary function' doesn't really have a single universally agreed on strict definition. Definitions are either a bit fuzzy, or not universally agreed on. Though interestingly https://en.wikipedia.org/wiki/Elementary_function https://en.wikipedia.org/wiki/Elementary_function says "More generally, in modern mathematics, elementary functions comprise the set of [...]". Though at least Wikipedia thinks that 'modern mathematics' has a consensus; of course, there's no guarantee that whoever you are talking to uses the 'modern mathematics' definition that Wikipedia brings up.
adrian_b 5mo ago
That theorem is not formulated about "elementary functions". It says that polynomial equations of the 5th degrees or higher cannot, in general, be solved using "radicals". While something like "polynomials" or "radicals" has a clear meaning, which are the "elementary functions" is a matter of convention. The usual convention is to include all algebraic functions and a few selected transcendental functions. In "all algebraic functions", are included the rational functions, the radicals and the functions that compute solutions of arbitrary polynomial equations. Some conventions used for "elementary functions" describe the expressions that you can use to write such "elementary functions", in which case not all algebraic functions are included, but only those written by combining rational functions with radicals. For an algebraic function that computes a solution of a general polynomial equation, which cannot be expressed with radicals, you cannot write an explicit formula, but you can write the function only implicitly, by writing the corresponding polynomial equation. So the difference between the 2 kinds of conventions about which are "the elementary functions" is usually based on whether only explicitly-written functions are considered, or also implicit functions.
- 5mo ago

SideQuark 5mo ago

> If you take a real analysis class, the elementary functions will be defined exactly as the author of the EML paper does. I just looked through many of the best known real analysis texts, and not a single one defines them this way. This list included the texts by Royden, Terence Tao, Rudin, Spivak, Bartle & Sherbert, Pugh, and a few others.... Can you cite a single text book that has this definition you claim is in every real analysis course? I find all evidence points to the opposite.

rnhmjoj 5mo ago
I guess you're right, I was probably mislead this whole time. I went through my old analysis class book [1] and there doesn't seem to be an explicit definition of elementary functions. The best I can find is this paragraph (I translate from italian): > The elementary functions of analysis, that is powers, roots, exponentials, logarithms and their inverses, functions obtained from the former by arithmetic operations or composition, admit the limit f(p) for x → p, for any p in their set of definition. The study of such functions, which is not limited to the sole real functions of real variable, is carried out naturally in the setting of metric spaces. That said, I'm relatively sure that a definition was given in class and it didn't include arbitrary roots: despite being notoriously difficult, the exam didn't require students to draw the graph of any elementary function including implicitly-defined algebraic roots. I picked up another one of the old recommended books [2] and it seems to be similarly vague; while the book currently taught in my university [3], gives this definition: > The following functions (from ℂ to ℂ) are called the elementary functions of the Analysis: > 1) Rational functions (integral or fractional) > 2) Algebraic functions (explicit or implicit) > 3) The exponential function > 4) The logarithm function > 5) All those functions that can be obtained by combining a finite number of times the functions of kind 1)...4). So, roots of arbitrary polynomials implicitly defined are indeed considered elementary. I never knew this. [1]: https://search.worldcat.org/title/1261811544 https://search.worldcat.org/title/1261811544 [2]: https://search.worldcat.org/title/801297519 https://search.worldcat.org/title/801297519 [3]: https://search.worldcat.org/title/935666878 https://search.worldcat.org/title/935666878
- rnhmjoj 5mo ago

fnordpiglet 5mo ago

In math elementary usually means fundamental or foundational not elementary school. The root word is element and the relationship to “simple subject” is tangential and more related to its teaching the elemental topics for a lifetime education than definitionally cross discipline.