[HN Gopher] Julia Robinson helped define the limits of mathemati...
___________________________________________________________________
Julia Robinson helped define the limits of mathematical knowledge
(2019)
Author : rfreytag
Score : 53 points
Date : 2023-01-13 12:53 UTC (1 days ago)
(HTM) web link (www.sciencenews.org)
(TXT) w3m dump (www.sciencenews.org)
| Frummy wrote:
| Interesting to learn she was the little sister of Constance Reid,
| who wrote a fantastic biography of Hilbert which is on my desk
| right now.
| jimhefferon wrote:
| I was sorry to read that recently Martin Davis passed. Along with
| JR, another giant.
| hackandthink wrote:
| A nice article with a nice equation:
|
| 42 = -80538738812075974^3 + 80435758145817515^3 +
| 12602123297335631^3
|
| Douglas Adams would rejoice. (checked it in bc)
|
| Some Background from Wikipedia:
|
| https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich's...
|
| "Hilbert's tenth problem asks for a general algorithm deciding
| the solvability of Diophantine equations. The conjunction of
| Matiyasevich's result with the fact that most recursively
| enumerable languages are not decidable implies that a solution to
| Hilbert's tenth problem is impossible."
| aleks224 wrote:
| The undecidability property proven here doesn't imply that
| there exists at least one Diophantine equation for which we'll
| never know if it's solvable or not, does it?
| hackandthink wrote:
| My understanding:
|
| There's no algorithm to decide. But for any equation we can
| be lucky to find a solution or a proof that there's no
| solution.
|
| But this doesn't prove that there is an equation for which
| we'll never know if it's solvable or not.
| moefh wrote:
| From my understanding, while that's is technically true,
| given a consistent axiomatic system (like ZFC[1], the
| foundation of mathematics we use) there exists a
| diophantine equation that can't be proven to have no
| solutions in that system (even though it has no solutions).
| This mathoverflow answer[2] gives the equation and a link
| to the paper that shows how to calculate the constants (the
| numbers are huge!).
|
| What that means in practice is that although what you wrote
| is true, for some diophantine equations we'd have to come
| up with new axioms to be able to write a proof of the
| inexistence of its solutions. But then, how can we be sure
| that the the new axioms are consistent?
|
| [1] I'm assuming ZFC is consistent; if it's not then it can
| prove anything, including the existence of solutions for
| any equations at all
|
| [2] https://mathoverflow.net/a/81986
| hackandthink wrote:
| Thanks.
|
| I'm somewhat lost, but it seems to work Godel like.
|
| The statement is true (equation has no solution) but we
| can't prove it.
| layer8 wrote:
| See https://news.ycombinator.com/item?id=34384838, I think
| it disproves your last sentence -- at least when assuming
| that all solutions and all proofs of non-existence of a
| solution are expressible in a shared formal language.
| ykonstant wrote:
| In [0], Carl and Moroz give an explicit polynomial in
| 3639528+1 variables such that: a well-formed formula is a
| theorem in the first order predicate calculus if and only if
| the polynomial parametrized by the Diophantine coding of the
| formula (a single natural number) has a solution in
| N^{3639528}.
|
| From this, they get an explicit Diophantine equation such
| that: the Godel-Bernays set theory is consistent if and only
| if that Diophantine equation has no solutions (and thus the
| same is true for ZFC, since NBG is a conservative extension
| of ZFC).
|
| [0]
| https://link.springer.com/article/10.1007/s10958-014-1830-2
| najdan33 wrote:
| Does there exist a set of yes/no problems such that:
|
| - there's no general algorithm that can solve an arbitrary
| problem from the set (the whole thing is undecidable)
|
| - each problem in isolation _can_ be solved. there's no
| single problem that's impossible to solve
| layer8 wrote:
| I don't think so, at least if you assume that each
| concrete solution can be expressed in finite length in a
| formal language with a finite alphabet, and can be
| mechanically checked (which is generally the case for
| mathematical proofs). Because then you could just
| enumerate all strings of that language until you find one
| that describes the solution to the given problem, which
| by your second item would be guaranteed to exist, and
| thus the procedure be guaranteed to terminate,
| contradicting your first item.
| robinhouston wrote:
| If anyone's interested in more context on that particular
| equation, here's an article I wrote for The Aperiodical at the
| time: https://aperiodical.com/2019/09/42-is-the-answer-to-the-
| ques...
| moloch-hai wrote:
| I would have liked for the story to have noted, at least in
| passing, that 33+43+53 = 63.
|
| (I found this in the brilliantly insane "The celestial
| inspirations for Giza, Stonehenge and Washington D.C." by Robin
| Spivey:
|
| https://www.researchgate.net/profile/Robin-Spivey-2/publicat...
|
| Along with that, the Great Pyramid's latitude matches the speed
| of light in m/s divided by 10000. Suffice to say that numerical
| coincidences are more the norm than the exception.)
___________________________________________________________________
(page generated 2023-01-14 23:01 UTC)