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The author essentially says that the quintic has no closed form solution which is true regardless of the exp-minus-log function. The purpose of this blog post i

The author essentially says that the quintic has no closed form solution which is true regardless of the exp-minus-log function. The purpose of this blog post is lost on me. Can anyone please explain this further? It seems like he’s moving the goalposts.

DevelopingElk 6mo ago
His claim is that we exp-minus-log cannot compute the root of an arbitrary quintic. If you consider the root of an arbitrary quintic "elementary" the exp-minus-log can't represent all elementary functions. I think it really comes down to what set of functions you are calling "elementary".
- throwanem 6mo ago
  The author discusses this in his third paragraph, and states explicitly in his fourth that he considers the result faulty for its unrealistically narrow definition of elementarity. (I'm not a mathematician, so don't expect me to have an opinion as far as that goes. But the author also writes well in English, and that language we do share.)
  - bawolff 6mo ago
    Well the author saysin that paragraph: > In layman’s terms, I do not consider the “Exp-Minus-Log” function to be the continuous analog of the Boolean NAND gate or the universal quantum CCNOT/CSWAP gates. But is there actually a combination of NANDs that find the roots of an arbitrary quintic? I always thought the answer was no but admittedly this is above my math level.
    - reikonomusha 6mo ago
      Combinations of the NAND gate can express any Boolean function. The Toffoli (CCNOT) or Fredkin (CSWAP) can express any reversible Boolean function, which is important in quantum computing where all gates must be unitary (and therefore reversible). The posited analog is that EML would be the "universal operator" for continuous functions.
    - meindnoch 6mo ago
      Solving polynomials over finite fields is trivial. Just try all combinations.
      - eru 6mo ago
        You probably want a fast algorithm. Compare https://arxiv.org/abs/1108.1791 https://arxiv.org/abs/1108.1791 and why computational complexity is often more interesting that computability.
      - bawolff 6mo ago
        Sure, i guess i should have said something like with a polynomial circuit size or something. However by the same token couldn't you use the same brute force approach with exp minus log? What im really asking, are NAND gates really different here?
        zeroonetwothree 6mo ago
        How can you brute force real numbers?
        bawolff 6mo ago
        I meant for finite fields like the person i was responding to said.
    - zeroonetwothree 6mo ago
      Yes, NAND gates can implement root finding algorithms for arbitrary polynomials. For example a variant of Newton’s method can be used (there are also better algorithms for circuits specifically). This can be done in polynomial time as well. This is fairly obvious if you think about that your computer can do the same thing and it’s just a fancy circuit.
      - floxy 6mo ago
        Surely you can use EML to do root finding approximations also.
petters 6mo ago
Yes, that blog post could have been much shorter….
AlotOfReading 6mo ago
The argument is that a universal basis would be capable of solving arbitrary polynomial roots. The rest is an argument that the group constructed by eml is solveable, and hence not all the standard elementary functions. It wouldn't be a math discussion without people using at least two wildly different definitions.
- deleted 6mo ago
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markgall 6mo ago
Can anyone provide a link that "Some are going as far as to suggest that the entire foundations of computer engineering and machine learning should be re-built as a result of this", or anything similarly grandiose? I am a professional mathematician, though nowhere near this kind of thing. The result seems amusing enough, but it doesn't really strike me as something that would be surprising. I confess that this thread is the first I've heard of it...
- saithound 6mo ago
  It's a fun, but unsurprising undergrad-level result. It got picked up and overhyped on HN [1] and /r/math [2] earlier this week. Some of my favorites: DoctorOetker: "I'm still reading this, but if this checks out, this is one of the most significant discoveries in years." cryptonektor: "Given this amazing work, an efficient EML operator HW implementation could revolutionize a bunch of things." zephen: "This is about continuous math, not ones and zeroes. Assuming peer review proves it out, this is outstanding." [1] https://news.ycombinator.com/item?id=47746610 https://news.ycombinator.com/item?id=47746610 [2] https://www.reddit.com/r/math/comments/1sk63n5/all_elementary_functions_from_a_single_binary/ https://www.reddit.com/r/math/comments/1sk63n5/all_elementar...
  - renewiltord 6mo ago
    This result itself is being described in those terms[1]: > If this is true, then this blog post debunking EML is going to up-end all of mathematics for the next century. This is very concerning for mathematics in general. 1: https://news.ycombinator.com/item?id=47775105 https://news.ycombinator.com/item?id=47775105
    - seanhunter 6mo ago
      Why on earth would it upend all of mathematics? Secondly, even if it did that, why would that be concerning for mathematics?
      - renewiltord 6mo ago
reikonomusha 6mo ago
"The quintic has no closed form solution" is a theorem that is more precisely stated (in the usual capstone Galois proof) as follows: The quintic has no closed form solution in terms of arbitrary compositions of rational numbers, arithmetic, and Nth roots. We can absolutely express closed form solutions to the quintic if we broaden our repertoire of functions, such as with the Bring radical. The post's argument is different than the usual Galois theory result about the unsolvability of the quintic, in that it shows a property that must be true about all EML(x,y)-derived functions, and a hypothetical quintic-solver-function does not have that property, so no function we add to our repertoire via EML will solve it (or any other function, elementary or not, that lacks this property).
- lotaezenwa 6mo ago
  Cool explanation, thanks!
- cyberax 6mo ago
  Bring radicals are just cheating. You can't solve an equation? Why not just introduce a function that is equal to the solution of the equation! Problem solved.
  - reikonomusha 6mo ago
    This fundamental "cheat" gave rise to some of the most important pure and applied mathematics known. Can't solve the differential equation x^2 - a = 0? Why not just introduce a function sqrt(a) as its solution! Problem solved. Can't solve the differential equation y'' = -y? Why not just introduce a function sin(x) as its solution! Problem solved. A lot of 19th century mathematics was essentially this: discover which equations had solutions in terms of things we already knew about, and if they didn't and it seemed important or interesting enough, make a new name. This is the whole field of so-called "special functions". It's where we also get the elliptic functions, Bessel functions, etc. The definition of "elementary function" comes exactly from this line in inquiry: define a set of functions we think are nice and algebraically tractable, and answer what we can express with them. The biggest classical question was: Do integrals of elementary functions give us elementary functions? The answer is "no" and Liouville gave us a result which tells us what the answer does look like when the result is elementary. Risch gave us an algorithm to compute the answer, when it exists in elementary form.

DevelopingElk 6mo ago

His claim is that we exp-minus-log cannot compute the root of an arbitrary quintic. If you consider the root of an arbitrary quintic "elementary" the exp-minus-log can't represent all elementary functions. I think it really comes down to what set of functions you are calling "elementary".

throwanem 6mo ago
The author discusses this in his third paragraph, and states explicitly in his fourth that he considers the result faulty for its unrealistically narrow definition of elementarity. (I'm not a mathematician, so don't expect me to have an opinion as far as that goes. But the author also writes well in English, and that language we do share.)
- bawolff 6mo ago
  Well the author saysin that paragraph: > In layman’s terms, I do not consider the “Exp-Minus-Log” function to be the continuous analog of the Boolean NAND gate or the universal quantum CCNOT/CSWAP gates. But is there actually a combination of NANDs that find the roots of an arbitrary quintic? I always thought the answer was no but admittedly this is above my math level.
  - reikonomusha 6mo ago
    Combinations of the NAND gate can express any Boolean function. The Toffoli (CCNOT) or Fredkin (CSWAP) can express any reversible Boolean function, which is important in quantum computing where all gates must be unitary (and therefore reversible). The posited analog is that EML would be the "universal operator" for continuous functions.
  - meindnoch 6mo ago
    Solving polynomials over finite fields is trivial. Just try all combinations.
    - eru 6mo ago
      You probably want a fast algorithm. Compare https://arxiv.org/abs/1108.1791 https://arxiv.org/abs/1108.1791 and why computational complexity is often more interesting that computability.
    - bawolff 6mo ago
      Sure, i guess i should have said something like with a polynomial circuit size or something. However by the same token couldn't you use the same brute force approach with exp minus log? What im really asking, are NAND gates really different here?
      - zeroonetwothree 6mo ago
        How can you brute force real numbers?
        bawolff 6mo ago
        I meant for finite fields like the person i was responding to said.
  - zeroonetwothree 6mo ago
    Yes, NAND gates can implement root finding algorithms for arbitrary polynomials. For example a variant of Newton’s method can be used (there are also better algorithms for circuits specifically). This can be done in polynomial time as well. This is fairly obvious if you think about that your computer can do the same thing and it’s just a fancy circuit.
    - floxy 6mo ago
      Surely you can use EML to do root finding approximations also.

petters 6mo ago

Yes, that blog post could have been much shorter….

AlotOfReading 6mo ago

The argument is that a universal basis would be capable of solving arbitrary polynomial roots. The rest is an argument that the group constructed by eml is solveable, and hence not all the standard elementary functions. It wouldn't be a math discussion without people using at least two wildly different definitions.

deleted 6mo ago
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deleted 6mo ago

[deleted]

markgall 6mo ago

Can anyone provide a link that "Some are going as far as to suggest that the entire foundations of computer engineering and machine learning should be re-built as a result of this", or anything similarly grandiose? I am a professional mathematician, though nowhere near this kind of thing. The result seems amusing enough, but it doesn't really strike me as something that would be surprising. I confess that this thread is the first I've heard of it...

saithound 6mo ago
It's a fun, but unsurprising undergrad-level result. It got picked up and overhyped on HN [1] and /r/math [2] earlier this week. Some of my favorites: DoctorOetker: "I'm still reading this, but if this checks out, this is one of the most significant discoveries in years." cryptonektor: "Given this amazing work, an efficient EML operator HW implementation could revolutionize a bunch of things." zephen: "This is about continuous math, not ones and zeroes. Assuming peer review proves it out, this is outstanding." [1] https://news.ycombinator.com/item?id=47746610 https://news.ycombinator.com/item?id=47746610 [2] https://www.reddit.com/r/math/comments/1sk63n5/all_elementary_functions_from_a_single_binary/ https://www.reddit.com/r/math/comments/1sk63n5/all_elementar...
- renewiltord 6mo ago
  This result itself is being described in those terms[1]: > If this is true, then this blog post debunking EML is going to up-end all of mathematics for the next century. This is very concerning for mathematics in general. 1: https://news.ycombinator.com/item?id=47775105 https://news.ycombinator.com/item?id=47775105
  - seanhunter 6mo ago
    Why on earth would it upend all of mathematics? Secondly, even if it did that, why would that be concerning for mathematics?
    - renewiltord 6mo ago

reikonomusha 6mo ago

"The quintic has no closed form solution" is a theorem that is more precisely stated (in the usual capstone Galois proof) as follows: The quintic has no closed form solution in terms of arbitrary compositions of rational numbers, arithmetic, and Nth roots. We can absolutely express closed form solutions to the quintic if we broaden our repertoire of functions, such as with the Bring radical. The post's argument is different than the usual Galois theory result about the unsolvability of the quintic, in that it shows a property that must be true about all EML(x,y)-derived functions, and a hypothetical quintic-solver-function does not have that property, so no function we add to our repertoire via EML will solve it (or any other function, elementary or not, that lacks this property).

lotaezenwa 6mo ago
Cool explanation, thanks!
cyberax 6mo ago
Bring radicals are just cheating. You can't solve an equation? Why not just introduce a function that is equal to the solution of the equation! Problem solved.
- reikonomusha 6mo ago
  This fundamental "cheat" gave rise to some of the most important pure and applied mathematics known. Can't solve the differential equation x^2 - a = 0? Why not just introduce a function sqrt(a) as its solution! Problem solved. Can't solve the differential equation y'' = -y? Why not just introduce a function sin(x) as its solution! Problem solved. A lot of 19th century mathematics was essentially this: discover which equations had solutions in terms of things we already knew about, and if they didn't and it seemed important or interesting enough, make a new name. This is the whole field of so-called "special functions". It's where we also get the elliptic functions, Bessel functions, etc. The definition of "elementary function" comes exactly from this line in inquiry: define a set of functions we think are nice and algebraically tractable, and answer what we can express with them. The biggest classical question was: Do integrals of elementary functions give us elementary functions? The answer is "no" and Liouville gave us a result which tells us what the answer does look like when the result is elementary. Risch gave us an algorithm to compute the answer, when it exists in elementary form.