Rueckgrat von Wirklichkeit
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From: Marko Amnell on 21 Jul 2010 18:58 "Marko Amnell" <marko.amnell(a)kolumbus.fi> wrote in message 37d04746-9212-4504-8bb6-4fa1246d510e(a)a30g2000yqn.googlegroups.com... > > On page 604 of Thomas Pynchon's novel _Against the Day_, > the beautiful Yashmeen Halfcourt asks David Hilbert about > the Hilbert-Polya Conjecture. Yashmeen asks Hilbert if the > zeroes of the Riemann zeta-function might be "correlated > with eigenvalues of some Hermitian operator yet to be > determined." Yashmeen Halfcourt also uses the phrase > "Rueckgrat von Wirklichkeit": > > "'There is also this ... spine of reality.' Afterward she would > remember she actually said 'Rueckgrat von Wirklichzeit.' > 'Though the members of a Hermitian may be complex, > the eigenvalues are real. The entries on the main diagonal > are real. The zeta-function zeroes which lie along > Real part = 1/2 are symmetrical about the real axis, and > so ...' She hesitated. She had _seen it_, for the moment, > so clearly." > > What is the origin and significance of the phrase > "Rueckgrat von Wirklichkeit"? Did Pynchon invent it > or is he quoting someone else, possibly Hilbert or Polya? I don't know the answers to the above questions, but a 1982 letter from Polya to Andrew Odlyzko reproduced at http://www.dtc.umn.edu/~odlyzko/polya/index.html seems relevant. Polya writes that: "I spent two years in Goettingen ending around the beginning of 1914. I tried to learn analytic number theory from Landau. He asked me one day: "You know some physics. Do you know a physical reason that the Riemann hypothesis should be true." This would be the case, I answered, if the nontrivial zeros of the Xi-function were so connected with the physical problem that the Riemann hypothesis would be equivalent to the fact that all the eigenvalues of the physical problem are real. I never published this remark, but somehow it became known and it is still remembered." > Pynchon Wiki says that the phrase is "probably a reference > to the main diagonal of a Hermitian matrix, which can > contain only real numbers." What sounds reasonable, > but might Pynchon also be suggesting that the Riemann > Hypothesis is connected to physical reality? As Pynchon > Wiki goes on to note: > > "The most recent refinement of the Hilbert-Polya Conjecture > is that the operator in question arises from the quantization > of a chaotic dynamical system. The zeroes would then > correspond to the possible energy levels of the system. > This would answer Hilbert's prompt to Yasmeen "some > equation" on this page, and would provide the link > between prime numbers and chaos intuited by Gunther > on page 597." > > http://against-the-day.pynchonwiki.com/wiki/index.php?title=ATD_588-614#Page_604 >
From: David Bernier on 21 Jul 2010 20:48 Marko Amnell wrote: > "Marko Amnell"<marko.amnell(a)kolumbus.fi> wrote in message > 37d04746-9212-4504-8bb6-4fa1246d510e(a)a30g2000yqn.googlegroups.com... >> >> On page 604 of Thomas Pynchon's novel _Against the Day_, >> the beautiful Yashmeen Halfcourt asks David Hilbert about >> the Hilbert-Polya Conjecture. Yashmeen asks Hilbert if the >> zeroes of the Riemann zeta-function might be "correlated >> with eigenvalues of some Hermitian operator yet to be >> determined." Yashmeen Halfcourt also uses the phrase >> "Rueckgrat von Wirklichkeit": >> >> "'There is also this ... spine of reality.' Afterward she would >> remember she actually said 'Rueckgrat von Wirklichzeit.' >> 'Though the members of a Hermitian may be complex, >> the eigenvalues are real. The entries on the main diagonal >> are real. The zeta-function zeroes which lie along >> Real part = 1/2 are symmetrical about the real axis, and >> so ...' She hesitated. She had _seen it_, for the moment, >> so clearly." >> >> What is the origin and significance of the phrase >> "Rueckgrat von Wirklichkeit"? Did Pynchon invent it >> or is he quoting someone else, possibly Hilbert or Polya? > > I don't know the answers to the above questions, > but a 1982 letter from Polya to Andrew Odlyzko > reproduced at http://www.dtc.umn.edu/~odlyzko/polya/index.html > seems relevant. Polya writes that: > > "I spent two years in Goettingen ending around the > beginning of 1914. I tried to learn analytic number theory > from Landau. He asked me one day: "You know some physics. > Do you know a physical reason that the Riemann hypothesis > should be true." This would be the case, I answered, if the > nontrivial zeros of the Xi-function were so connected with > the physical problem that the Riemann hypothesis would > be equivalent to the fact that all the eigenvalues of the > physical problem are real. I never published this remark, but > somehow it became known and it is still remembered." In the work of Conrey, Farmer, Keating, Rubinstein and Snaith on moments of L functions and Random Matrix Theory, one simple statement struck me: < http://arxiv.org/abs/math/0206018> page 14, number (5): "Equating the density of the Riemann zeros at height t with the density of the random matrix eigenvalues suggests the familiar equivalence N = log(t/(2pi))." Or: N ~== log(t/(2Pi)) . Here N is the dimension of random unitary matrices, and t>0 designates a region near y = t for the map: p: [0, oo) --> C p(y) := zeta(1/2 + i*y). So if t = 7005, N ~= 7.02 . So for t~= 7005, one would "model" using 7x7 random unitary matrices. Gram points are separated by about 2pi/(log(t/2pi)), so I think I understand the density equality. So using u = t/(2Pi) , the equivalence becomes: N ~== log(u). The "dimension" grows logarithmically with u. If there is a natural Hilbert-Polya hermitian operator, I'm wondering how this growth of "dimension" in random matrix theory ties in, but can't think of anything. David Bernier >> Pynchon Wiki says that the phrase is "probably a reference >> to the main diagonal of a Hermitian matrix, which can >> contain only real numbers." What sounds reasonable, >> but might Pynchon also be suggesting that the Riemann >> Hypothesis is connected to physical reality? As Pynchon >> Wiki goes on to note: >> >> "The most recent refinement of the Hilbert-Polya Conjecture >> is that the operator in question arises from the quantization >> of a chaotic dynamical system. The zeroes would then >> correspond to the possible energy levels of the system. >> This would answer Hilbert's prompt to Yasmeen "some >> equation" on this page, and would provide the link >> between prime numbers and chaos intuited by Gunther >> on page 597." >> >> http://against-the-day.pynchonwiki.com/wiki/index.php?title=ATD_588-614#Page_604 >> > >
From: Marko Amnell on 22 Jul 2010 06:58 On Jul 22, 3:48 am, David Bernier <david...(a)videotron.ca> wrote: > Marko Amnell wrote: > > "Marko Amnell"<marko.amn...(a)kolumbus.fi> wrote in message > > 37d04746-9212-4504-8bb6-4fa1246d5...(a)a30g2000yqn.googlegroups.com... > > >> On page 604 of Thomas Pynchon's novel _Against the Day_, > >> the beautiful Yashmeen Halfcourt asks David Hilbert about > >> the Hilbert-Polya Conjecture. Yashmeen asks Hilbert if the > >> zeroes of the Riemann zeta-function might be "correlated > >> with eigenvalues of some Hermitian operator yet to be > >> determined." Yashmeen Halfcourt also uses the phrase > >> "Rueckgrat von Wirklichkeit": > > >> "'There is also this ... spine of reality.' Afterward she would > >> remember she actually said 'Rueckgrat von Wirklichzeit.' > >> 'Though the members of a Hermitian may be complex, > >> the eigenvalues are real. The entries on the main diagonal > >> are real. The zeta-function zeroes which lie along > >> Real part = 1/2 are symmetrical about the real axis, and > >> so ...' She hesitated. She had _seen it_, for the moment, > >> so clearly." > > >> What is the origin and significance of the phrase > >> "Rueckgrat von Wirklichkeit"? Did Pynchon invent it > >> or is he quoting someone else, possibly Hilbert or Polya? > > > I don't know the answers to the above questions, > > but a 1982 letter from Polya to Andrew Odlyzko > > reproduced athttp://www.dtc.umn.edu/~odlyzko/polya/index.html > > seems relevant. Polya writes that: > > > "I spent two years in Goettingen ending around the > > beginning of 1914. I tried to learn analytic number theory > > from Landau. He asked me one day: "You know some physics. > > Do you know a physical reason that the Riemann hypothesis > > should be true." This would be the case, I answered, if the > > nontrivial zeros of the Xi-function were so connected with > > the physical problem that the Riemann hypothesis would > > be equivalent to the fact that all the eigenvalues of the > > physical problem are real. I never published this remark, but > > somehow it became known and it is still remembered." The Polya letter is nice because it is the only known evidence about the origins of the Hilbert-Polya Conjecture, and it comes from a letter written by Polya in his 90s, when his eyesight is failing, and yet he chooses to reply to Odlyzko in his own handwriting, providing a personal link to pre-First World War Goettingen (also explored by Pynchon in his novel), a bit like the old lady Rose in the film "Titanic". > > In the work of Conrey, Farmer, Keating, Rubinstein and Snaith > on moments of L functions and Random Matrix Theory, > one simple statement struck me: > > <http://arxiv.org/abs/math/0206018> > > page 14, number (5): > > "Equating the density of the Riemann zeros at height t with the > density of the random matrix eigenvalues suggests the familiar > equivalence N = log(t/(2pi))." > > Or: N ~== log(t/(2Pi)) . > > Here N is the dimension of random unitary matrices, and t>0 > designates a region near y = t for the map: > > p: [0, oo) --> C > > p(y) := zeta(1/2 + i*y). > > So if t = 7005, N ~= 7.02 . So for t~= 7005, one would "model" > using 7x7 random unitary matrices. > > Gram points are separated by about 2pi/(log(t/2pi)), > so I think I understand the density equality. > > So using u = t/(2Pi) , the equivalence becomes: > > N ~== log(u). The "dimension" grows logarithmically with u. > > If there is a natural Hilbert-Polya hermitian operator, At least it's an idea, a clue, something to work with, and it has inspired work from the respectable (Alain Connes) to the fringe element -- people like Carlos Castro, whose claim to have proven RH is discussed in this old sci.math thread: http://groups.google.com/group/sci.math/browse_thread/thread/9bec4304c7a1eaff/1fbb87f2a60e7f5f?lnk=gst&q=castro+riemann#1fbb87f2a60e7f5f > I'm wondering how this growth of "dimension" in random > matrix theory ties in, but can't think of anything. > > David Bernier > > > > >> Pynchon Wiki says that the phrase is "probably a reference > >> to the main diagonal of a Hermitian matrix, which can > >> contain only real numbers." What sounds reasonable, > >> but might Pynchon also be suggesting that the Riemann > >> Hypothesis is connected to physical reality? As Pynchon > >> Wiki goes on to note: > > >> "The most recent refinement of the Hilbert-Polya Conjecture > >> is that the operator in question arises from the quantization > >> of a chaotic dynamical system. The zeroes would then > >> correspond to the possible energy levels of the system. > >> This would answer Hilbert's prompt to Yasmeen "some > >> equation" on this page, and would provide the link > >> between prime numbers and chaos intuited by Gunther > >> on page 597."
From: David Bernier on 22 Jul 2010 16:30 Marko Amnell wrote: > On Jul 22, 3:48 am, David Bernier<david...(a)videotron.ca> wrote: >> Marko Amnell wrote: >>> "Marko Amnell"<marko.amn...(a)kolumbus.fi> wrote in message >>> 37d04746-9212-4504-8bb6-4fa1246d5...(a)a30g2000yqn.googlegroups.com... >> >>>> On page 604 of Thomas Pynchon's novel _Against the Day_, >>>> the beautiful Yashmeen Halfcourt asks David Hilbert about >>>> the Hilbert-Polya Conjecture. Yashmeen asks Hilbert if the >>>> zeroes of the Riemann zeta-function might be "correlated >>>> with eigenvalues of some Hermitian operator yet to be >>>> determined." Yashmeen Halfcourt also uses the phrase >>>> "Rueckgrat von Wirklichkeit": >> >>>> "'There is also this ... spine of reality.' Afterward she would >>>> remember she actually said 'Rueckgrat von Wirklichzeit.' >>>> 'Though the members of a Hermitian may be complex, >>>> the eigenvalues are real. The entries on the main diagonal >>>> are real. The zeta-function zeroes which lie along >>>> Real part = 1/2 are symmetrical about the real axis, and >>>> so ...' She hesitated. She had _seen it_, for the moment, >>>> so clearly." >> >>>> What is the origin and significance of the phrase >>>> "Rueckgrat von Wirklichkeit"? Did Pynchon invent it >>>> or is he quoting someone else, possibly Hilbert or Polya? >> >>> I don't know the answers to the above questions, >>> but a 1982 letter from Polya to Andrew Odlyzko >>> reproduced athttp://www.dtc.umn.edu/~odlyzko/polya/index.html >>> seems relevant. Polya writes that: >> >>> "I spent two years in Goettingen ending around the >>> beginning of 1914. I tried to learn analytic number theory >>> from Landau. He asked me one day: "You know some physics. >>> Do you know a physical reason that the Riemann hypothesis >>> should be true." This would be the case, I answered, if the >>> nontrivial zeros of the Xi-function were so connected with >>> the physical problem that the Riemann hypothesis would >>> be equivalent to the fact that all the eigenvalues of the >>> physical problem are real. I never published this remark, but >>> somehow it became known and it is still remembered." > > The Polya letter is nice because it is the only known evidence > about the origins of the Hilbert-Polya Conjecture, and it comes > from a letter written by Polya in his 90s, when his eyesight is > failing, and yet he chooses to reply to Odlyzko in his own > handwriting, providing a personal link to pre-First World War > Goettingen (also explored by Pynchon in his novel), > a bit like the old lady Rose in the film "Titanic". I seems to remember having read, at A. Odlyzko's web-site, his letter to Polya and a scan of Polya's reply. So I guess the period dates from at least 90 years ago (Polya's time with Landau). >> >> In the work of Conrey, Farmer, Keating, Rubinstein and Snaith >> on moments of L functions and Random Matrix Theory, >> one simple statement struck me: >> >> <http://arxiv.org/abs/math/0206018> >> >> page 14, number (5): >> >> "Equating the density of the Riemann zeros at height t with the >> density of the random matrix eigenvalues suggests the familiar >> equivalence N = log(t/(2pi))." >> >> Or: N ~== log(t/(2Pi)) . >> >> Here N is the dimension of random unitary matrices, and t>0 >> designates a region near y = t for the map: >> >> p: [0, oo) --> C >> >> p(y) := zeta(1/2 + i*y). >> >> So if t = 7005, N ~= 7.02 . So for t~= 7005, one would "model" >> using 7x7 random unitary matrices. >> >> Gram points are separated by about 2pi/(log(t/2pi)), >> so I think I understand the density equality. >> >> So using u = t/(2Pi) , the equivalence becomes: >> >> N ~== log(u). The "dimension" grows logarithmically with u. >> >> If there is a natural Hilbert-Polya hermitian operator, > > At least it's an idea, a clue, something to work with, > and it has inspired work from the respectable > (Alain Connes) to the fringe element -- people > like Carlos Castro, whose claim to have proven > RH is discussed in this old sci.math thread: > > http://groups.google.com/group/sci.math/browse_thread/thread/9bec4304c7a1eaff/1fbb87f2a60e7f5f?lnk=gst&q=castro+riemann#1fbb87f2a60e7f5f I didn't intend to suggest it was something like a fringe idea. By the way, Freeman Dyson (the physicist) who worked on random unitary matrices applied to nuclear energy levels and also had a chance encounter with number theorist Hugh Montgomery, wrote an article for the Bulletin of the AMS with the word "Frogs" in the title. He mentions quasi-crystals in his article, but I didn't understand much: "Dyson (2009) suggested trying to prove the Riemann hypothesis by classifying, or at least studying, 1-dimensional quasicrystals." Reference: < http://en.wikipedia.org/wiki/Riemann_hypothesis > --> Quasicrystals in "Attempts to prove RH". David Bernier >> I'm wondering how this growth of "dimension" in random >> matrix theory ties in, but can't think of anything. >> >> David Bernier >> >> >> >>>> Pynchon Wiki says that the phrase is "probably a reference >>>> to the main diagonal of a Hermitian matrix, which can >>>> contain only real numbers." What sounds reasonable, >>>> but might Pynchon also be suggesting that the Riemann >>>> Hypothesis is connected to physical reality? As Pynchon >>>> Wiki goes on to note: >> >>>> "The most recent refinement of the Hilbert-Polya Conjecture >>>> is that the operator in question arises from the quantization >>>> of a chaotic dynamical system. The zeroes would then >>>> correspond to the possible energy levels of the system. >>>> This would answer Hilbert's prompt to Yasmeen "some >>>> equation" on this page, and would provide the link >>>> between prime numbers and chaos intuited by Gunther >>>> on page 597."
From: Marko Amnell on 23 Jul 2010 04:09
"David Bernier" <david250(a)videotron.ca> wrote in message i2ab75030q(a)news6.newsguy.com... > Marko Amnell wrote: >> On Jul 22, 3:48 am, David Bernier<david...(a)videotron.ca> wrote: >>> Marko Amnell wrote: >>>> "Marko Amnell"<marko.amn...(a)kolumbus.fi> wrote in message >>>> 37d04746-9212-4504-8bb6-4fa1246d5...(a)a30g2000yqn.googlegroups.com... >>> >>>>> On page 604 of Thomas Pynchon's novel _Against the Day_, >>>>> the beautiful Yashmeen Halfcourt asks David Hilbert about >>>>> the Hilbert-Polya Conjecture. Yashmeen asks Hilbert if the >>>>> zeroes of the Riemann zeta-function might be "correlated >>>>> with eigenvalues of some Hermitian operator yet to be >>>>> determined." Yashmeen Halfcourt also uses the phrase >>>>> "Rueckgrat von Wirklichkeit": >>> >>>>> "'There is also this ... spine of reality.' Afterward she would >>>>> remember she actually said 'Rueckgrat von Wirklichzeit.' >>>>> 'Though the members of a Hermitian may be complex, >>>>> the eigenvalues are real. The entries on the main diagonal >>>>> are real. The zeta-function zeroes which lie along >>>>> Real part = 1/2 are symmetrical about the real axis, and >>>>> so ...' She hesitated. She had _seen it_, for the moment, >>>>> so clearly." >>> >>>>> What is the origin and significance of the phrase >>>>> "Rueckgrat von Wirklichkeit"? Did Pynchon invent it >>>>> or is he quoting someone else, possibly Hilbert or Polya? >>> >>>> I don't know the answers to the above questions, >>>> but a 1982 letter from Polya to Andrew Odlyzko >>>> reproduced at >>>> http://www.dtc.umn.edu/~odlyzko/polya/index.html >>>> seems relevant. Polya writes that: >>> >>>> "I spent two years in Goettingen ending around the >>>> beginning of 1914. I tried to learn analytic number theory >>>> from Landau. He asked me one day: "You know some physics. >>>> Do you know a physical reason that the Riemann hypothesis >>>> should be true." This would be the case, I answered, if the >>>> nontrivial zeros of the Xi-function were so connected with >>>> the physical problem that the Riemann hypothesis would >>>> be equivalent to the fact that all the eigenvalues of the >>>> physical problem are real. I never published this remark, but >>>> somehow it became known and it is still remembered." >> >> The Polya letter is nice because it is the only known evidence >> about the origins of the Hilbert-Polya Conjecture, and it comes >> from a letter written by Polya in his 90s, when his eyesight is >> failing, and yet he chooses to reply to Odlyzko in his own >> handwriting, providing a personal link to pre-First World War >> Goettingen (also explored by Pynchon in his novel), >> a bit like the old lady Rose in the film "Titanic". > > I seems to remember having read, at A. Odlyzko's web-site, > his letter to Polya and a scan of Polya's reply. So I guess > the period dates from at least 90 years ago (Polya's time with Landau). Yes, that's the link I gave above: http://www.dtc.umn.edu/~odlyzko/polya/index.html >>> In the work of Conrey, Farmer, Keating, Rubinstein and Snaith >>> on moments of L functions and Random Matrix Theory, >>> one simple statement struck me: >>> >>> <http://arxiv.org/abs/math/0206018> >>> >>> page 14, number (5): >>> >>> "Equating the density of the Riemann zeros at height t with the >>> density of the random matrix eigenvalues suggests the familiar >>> equivalence N = log(t/(2pi))." >>> >>> Or: N ~== log(t/(2Pi)) . >>> >>> Here N is the dimension of random unitary matrices, and t>0 >>> designates a region near y = t for the map: >>> >>> p: [0, oo) --> C >>> >>> p(y) := zeta(1/2 + i*y). >>> >>> So if t = 7005, N ~= 7.02 . So for t~= 7005, one would "model" >>> using 7x7 random unitary matrices. >>> >>> Gram points are separated by about 2pi/(log(t/2pi)), >>> so I think I understand the density equality. >>> >>> So using u = t/(2Pi) , the equivalence becomes: >>> >>> N ~== log(u). The "dimension" grows logarithmically with u. >>> >>> If there is a natural Hilbert-Polya hermitian operator, >> >> At least it's an idea, a clue, something to work with, >> and it has inspired work from the respectable >> (Alain Connes) to the fringe element -- people >> like Carlos Castro, whose claim to have proven >> RH is discussed in this old sci.math thread: >> >> http://groups.google.com/group/sci.math/browse_thread/thread/9bec4304c7a1eaff/1fbb87f2a60e7f5f?lnk=gst&q=castro+riemann#1fbb87f2a60e7f5f > > I didn't intend to suggest it was something like a fringe idea. Nor did I. It seems like one of the more promising ideas one might pursue to try to prove the Riemann Hypothesis. It is listed first by Wikipedia (your link below) among the "Attempts to prove the Riemann hypothesis," with the other strategies including the Lee-Yang theorem, Tur�n's result, Noncommutative geometry (Alain Connes), Hilbert spaces of entire functions (Louis de Branges), Quasicrystals and Multiple zeta functions. But it has to be said that despite the hype about some of these approaches, such as Connes's work, none of the strategies has been particularly fruitful. The zero-free region of the Riemann zeta-function is virtually the same now as it was in 1906. So by one objective measure, there has been virtually no progress for over a century. > By the way, Freeman Dyson (the physicist) who worked on > random unitary matrices applied to nuclear energy levels and > also had a chance encounter with number theorist Hugh Montgomery, > wrote an article for the Bulletin of the AMS with the word "Frogs" > in the title. "Birds and Frogs". Peter Woit talks about it here: http://www.math.columbia.edu/~woit/wordpress/?p=1506 > He mentions quasi-crystals in his article, but I didn't > understand much: > > "Dyson (2009) suggested trying to prove the Riemann hypothesis by > classifying, or at least studying, 1-dimensional quasicrystals." > > Reference: > < http://en.wikipedia.org/wiki/Riemann_hypothesis > > --> Quasicrystals in "Attempts to prove RH". > >>> I'm wondering how this growth of "dimension" in random >>> matrix theory ties in, but can't think of anything. >>> >>>>> Pynchon Wiki says that the phrase is "probably a reference >>>>> to the main diagonal of a Hermitian matrix, which can >>>>> contain only real numbers." What sounds reasonable, >>>>> but might Pynchon also be suggesting that the Riemann >>>>> Hypothesis is connected to physical reality? As Pynchon >>>>> Wiki goes on to note: >>> >>>>> "The most recent refinement of the Hilbert-Polya Conjecture >>>>> is that the operator in question arises from the quantization >>>>> of a chaotic dynamical system. The zeroes would then >>>>> correspond to the possible energy levels of the system. >>>>> This would answer Hilbert's prompt to Yasmeen "some >>>>> equation" on this page, and would provide the link >>>>> between prime numbers and chaos intuited by Gunther >>>>> on page 597." |