The article mentions reports to this effect dating back to at least Plutarch. Modern scholarship, however, found no scientific evidence for it. (The article mentions 1892 French excavations, 1904 A. P. Oppe, 1948 Oxford Classical Dictionary, 1950 Pierre Amandry).
Your 1980 English teacher might possibly even have read E. R. Dodds' The Greeks and the Irrational (1951) which, in addition to dismissing the vapor account as myth, gives a good statement of why it is irrelevant to trying to understand such phenomena:
As to the famous "vapours" to which the Pythia's inspiration was once confidently ascribed, they are a Hellenistic invention, as Wilamowitz was, I think, the first to point out [65]. Plutarch, who knew the facts, saw the difficulty of the vapour theory, and seems finally to have rejected it altogether; but like the Stoic philosophers, nineteenth-century scholars seized with relief on a nice solid materialist explanation. French excavations showed that there are to-day no vapours, and no chasm from which vapours could once have come [66]. Explanations of this type are really quite needless; if one or two living scholars still cling to them [67], it is only because they ignore the evidence of anthropology and abnormal psychology.
E. R. Dodds The Greeks and the Irrational, III: The Blessings of Madness, pp.73-4
The evidence supporting the "myth" is (relatively) new. Quite fascinating how geologist and author de Boer discovered the fissure in 1981 but, having read Plutarch, assumed it was already known and only in 1995 learned that it was not known to modern science while discussing it with archaeologist John R. Hale under the influence of some wine (which is when they resolved to team up and do a thourough investigation).
As an admirer of Dodds' scholarship, I also can't resist noting that of the 311 pages of the book, 129 comprise the 1099 annotations (three of which appeared in the citation above). Not quite hyperlinks, but enough in quantity and quality for me to judge him the Knuth of his field.
I was also thinking about the feedback effects of measurement, but from a different angle. Also, my physics & chemistry are admittedly weak:
Does anybody know if liquid Gallium, or any other suitable replacement, conducts electricity? Then one could measure the current induced by an electric field, or vice versa, to infer the temperature. I think these might provide less 'feedback' than an electron microscope, if the field and current measurement were also nano-localized.
Or maybe if is there a liquid dielectric of suitable scale, the temperature could somehow be inferred from this (using a wall of several adjacent nanotubes perhaps to create a varying area)?
Just some speculative questions... Of course, it'd be cool someday to see a nanocomputer.
It doesn't sound to me like the poster wants optimized code to do one thing thousands of times. It does sound like they want a framework to try out different problems and learn mathematical algorithms and programming. Many good suggestions have been mentioned. One other, similar to Matlab or Maple, is MuPad (http://www.sciface.com), which has free trial versions and supports many OS's.
If you say that you want the sum of all numbers between 12 and 12^12^12 with no 7's in them, any semi-properly trained (or is it brainwashed, I'm not sure) math person like me will tell you first that they are interpreting the question thus:
Find the sum of all positive integers between 12 and 12^(12^12)inclusive which have no 7 digits in their decimal representation.
Next, they will ask you whether you want leading digits, i.e. like what a truncated floating point representation would give you, or whether you want an exact integer solution. Given that the sum would have on the order of 18*10^12 digits, they would probably assume you want a floating point representation of it.
Then they would tell you it is approximately ten to the above number, and pat themselves on the back, confident that one really doesn't need to know anything more (why the hell would one?).
All of this can be done on paper in less than 5 minutes, with the use of a calculator with base ten logarithms.
[Hint: if N=the upper bound and k=log(N) to the base ten, and ignoring the "trivial" lower bound, the fraction of numbers between 1 and N with no 7's is going to be about (0.9)^k, while the average value is going to be about (N+1)/2. This gives a sum of.5*90^k where k = 12^12 * log(12).]
The above reasoning is good enough to nail down the order of magnitude of the number; in fact, you can get a little more precise (see below).
I agree it would probably be extremely hard if not impossible to write a general engine for use with arbitrary math problems (having say a well-defined real or complex number as an answer). For one thing, to decide even whether to use asymptotics and give a real-valued numerical estimate, to solve exactly as an arbitrary-precision integer, or a "closed form" expression (using combinatorics or number theory or Fourier analysis and accepted formulas like Bessel functions or Stirling numbers as appropriate). Nevertheless, within a reasonable range of computability, programs like Mathematica already do just that. They probably don't solve this problem, because they are engineered to solve using known algorithms for commonly needed types of solutions.
(Now I am going to just dream:)
To really solve such problems efficiently, one would need a database or research query mechanism to classify problems by known methods of solution in specific problem domains. That would really be cool if such an infrastructure existed that a program could tell you what methods are known that even apply to your problem, beyond a mere predefined set of problem ranges. Does anyone know if such 'expert systems' exist or are conceivable in the foreseeable future?
P.S. A slightly titillating but related problem is the sum of all k-digit positive integers containing at least one 7, allowing leading 0's for simplicity. For example, if this sum is T_k, then the sum S above is approximately
S = N*(N+1)/2 - 11*12/2 - T_k
where the approximation stems from the fact that k is not integer. One can give a ballpark estimate for T_k of about.5*90^k using the above heuristic, but a precise formula can also be given (either using the inclusion-exclusion principle and binomial coeffiecient formulas, or some other combinatorics argument). T_k is a given ratio of the sum S_k of all k-digit integers 00...0 to 99...9:
What all this shows is that the dominant term of T_k is 14/81 * (90)^k, which vanishes compared to S_k and S, so that not only the magnitude or logarithm, but the actual mantissa of the number in question (S), can in fact be computed fairly accurately if we ignore the fact that k is not an integer.
The point? A question: does anyone forsee the day that a computer will be able identify, on the fly, what sort of reasoning is possible and appropriate to general mathematical questions??? My quess is that we are far too idealistic about questions like this, which involve AI, because we are far too naive (and unconscious) about the sophistication of our own thinking. However, it is very conceivable that an expert system can account for known methods of reasoning and be able to discriminate as to when they are appropriate for given questions.
Lastly, lest the original poster may think we have forgotten him or her, I'll have to agree with others that many languages are useful, but in math, often a pencil and paper (or a desktop calculator) -- and some mathematical reasoning -- is all you're going to need to solve a problem or to entertain yourself. It does help to have a little background in computability and numerical analysis, but playing around with ANY computing language will also give you background that would help in learning these subjects. Personally I have used C/C++, mathematica, maple, fortran, assembly, Commodore basic, quickbasic, a TI-82, an old HP calculator, or even javascript or the windows NT command shell(!) to explore, compute, demonstrate or entertain myself with math questions (depending on what was at hand).
Ordinarily, searching for '6408 manual site:ibm.com' on Google would be your best bet, since generally the manufacturers should have the best documentation. In this case, however, the closest available manual available on IBM's site seems to be the user's manual for the 6400 (ftp://ftp.software.ibm.com/printers/manuals/6400/ s5445641.pdf, from the manuals page, http://www.printers.ibm.com/R5PSC.NSF/Web/6400m, which also has a link to the setup manual). This is probably because the 6408 is not manufactured by IBM (http://www.printers.ibm.com/R5PSC.NSF/Web/linemat rixattach):
The 6408 printer model A00 features PC Parallel, Dataproducts Parallel, and RS-232C Serial connections. The 6408 CTA model features ASCII Serial/Parallel system attachments. A model conversion is available from Model CT0 to Model CTA via MES. There are external I-Data LAN attachments available for 6408 model CTA printers. This product is manufactured by i-data of Denmark and distributed by IBM. This device allows the IBM 6408/CTA printers to attach to a 4 or 16 MBps Token-ring or Ethernet LAN for use with PSF/2 or PSF/6000, PSF/MVS and PSF/400. The Token Ring hardware and software is ordered as RPQ #8B4313. The RPQ for Ethernet hardware and software is RPQ #8B4314. Please contact your marketing team for assistance in configuring and ordering any model conversions or RPQ's. The recommended printer drivers for the 6408 model CTA or model A00 are: IBM Proprinter IIIXL, Epson FX1050, or generic text driver for non-graphical printing.
What IBM does offer specifically for the 6408 is at:
http://www.printers.ibm.com/R5FAQ.NSF/6400?OpenV ie w&Start=1&Count=200&Expand=2.5#2.5
They also state:
Question:
What publications are available for Code V printing on an IBM 6408 Model CTA printer?
Answer:
The publication, Model CTA Code V - Printronix Emulation User's Manual (Form Number G246-0086), is available.
To order, contact your local IBM printing systems representative or an authorized IBM reseller. Or contact
IBM Publications at 1-800-879-2755 (in U.S. and Canada only).
This begins to seem a little bit more difficult than meets the eye. So, why is it so hard to find?
Slashdot Mirror
Your 1980 English teacher might possibly even have read E. R. Dodds' The Greeks and the Irrational (1951) which, in addition to dismissing the vapor account as myth, gives a good statement of why it is irrelevant to trying to understand such phenomena:
The evidence supporting the "myth" is (relatively) new. Quite fascinating how geologist and author de Boer discovered the fissure in 1981 but, having read Plutarch, assumed it was already known and only in 1995 learned that it was not known to modern science while discussing it with archaeologist John R. Hale under the influence of some wine (which is when they resolved to team up and do a thourough investigation).
As an admirer of Dodds' scholarship, I also can't resist noting that of the 311 pages of the book, 129 comprise the 1099 annotations (three of which appeared in the citation above). Not quite hyperlinks, but enough in quantity and quality for me to judge him the Knuth of his field.
I was also thinking about the feedback effects of measurement, but from a different angle. Also, my physics & chemistry are admittedly weak:
Does anybody know if liquid Gallium, or any other suitable replacement, conducts electricity? Then one could measure the current induced by an electric field, or vice versa, to infer the temperature. I think these might provide less 'feedback' than an electron microscope, if the field and current measurement were also nano-localized.
Or maybe if is there a liquid dielectric of suitable scale, the temperature could somehow be inferred from this (using a wall of several adjacent nanotubes perhaps to create a varying area)?
Just some speculative questions... Of course, it'd be cool someday to see a nanocomputer.
It doesn't sound to me like the poster wants optimized code to do one thing thousands of times. It does sound like they want a framework to try out different problems and learn mathematical algorithms and programming. Many good suggestions have been mentioned. One other, similar to Matlab or Maple, is MuPad (http://www.sciface.com), which has free trial versions and supports many OS's.
If you say that you want the sum of all numbers between 12 and 12^12^12 with no 7's in them, any semi-properly trained (or is it brainwashed, I'm not sure) math person like me will tell you first that they are interpreting the question thus:
Find the sum of all positive integers between 12 and 12^(12^12) inclusive which have no 7 digits in their decimal representation .
Next, they will ask you whether you want leading digits, i.e. like what a truncated floating point representation would give you, or whether you want an exact integer solution. Given that the sum would have on the order of 18*10^12 digits, they would probably assume you want a floating point representation of it.
Then they would tell you it is approximately ten to the above number, and pat themselves on the back, confident that one really doesn't need to know anything more (why the hell would one?).
All of this can be done on paper in less than 5 minutes, with the use of a calculator with base ten logarithms.
[Hint: if N=the upper bound and k=log(N) to the base ten, and ignoring the "trivial" lower bound, the fraction of numbers between 1 and N with no 7's is going to be about (0.9)^k, while the average value is going to be about (N+1)/2. This gives a sum of .5*90^k where k = 12^12 * log(12).]
The above reasoning is good enough to nail down the order of magnitude of the number; in fact, you can get a little more precise (see below).
I agree it would probably be extremely hard if not impossible to write a general engine for use with arbitrary math problems (having say a well-defined real or complex number as an answer). For one thing, to decide even whether to use asymptotics and give a real-valued numerical estimate, to solve exactly as an arbitrary-precision integer, or a "closed form" expression (using combinatorics or number theory or Fourier analysis and accepted formulas like Bessel functions or Stirling numbers as appropriate). Nevertheless, within a reasonable range of computability, programs like Mathematica already do just that. They probably don't solve this problem, because they are engineered to solve using known algorithms for commonly needed types of solutions.
(Now I am going to just dream:) To really solve such problems efficiently, one would need a database or research query mechanism to classify problems by known methods of solution in specific problem domains. That would really be cool if such an infrastructure existed that a program could tell you what methods are known that even apply to your problem, beyond a mere predefined set of problem ranges. Does anyone know if such 'expert systems' exist or are conceivable in the foreseeable future?
P.S. A slightly titillating but related problem is the sum of all k-digit positive integers containing at least one 7, allowing leading 0's for simplicity. For example, if this sum is T_k, then the sum S above is approximately
where the approximation stems from the fact that k is not integer. One can give a ballpark estimate for T_k of aboutThe point? A question: does anyone forsee the day that a computer will be able identify, on the fly, what sort of reasoning is possible and appropriate to general mathematical questions??? My quess is that we are far too idealistic about questions like this, which involve AI, because we are far too naive (and unconscious) about the sophistication of our own thinking. However, it is very conceivable that an expert system can account for known methods of reasoning and be able to discriminate as to when they are appropriate for given questions.
Lastly, lest the original poster may think we have forgotten him or her, I'll have to agree with others that many languages are useful, but in math, often a pencil and paper (or a desktop calculator) -- and some mathematical reasoning -- is all you're going to need to solve a problem or to entertain yourself. It does help to have a little background in computability and numerical analysis, but playing around with ANY computing language will also give you background that would help in learning these subjects. Personally I have used C/C++, mathematica, maple, fortran, assembly, Commodore basic, quickbasic, a TI-82, an old HP calculator, or even javascript or the windows NT command shell(!) to explore, compute, demonstrate or entertain myself with math questions (depending on what was at hand).
Ordinarily, searching for '6408 manual site:ibm.com' on Google would be your best bet, since generally the manufacturers should have the best documentation. In this case, however, the closest available manual available on IBM's site seems to be the user's manual for the 6400 (ftp://ftp.software.ibm.com/printers/manuals/6400/ s5445641.pdf, from the manuals page, http://www.printers.ibm.com/R5PSC.NSF/Web/6400m, which also has a link to the setup manual). This is probably because the 6408 is not manufactured by IBM (http://www.printers.ibm.com/R5PSC.NSF/Web/linemat rixattach):
V ie w&Start=1&Count=200&Expand=2.5#2.5
The 6408 printer model A00 features PC Parallel, Dataproducts Parallel, and RS-232C Serial connections. The 6408 CTA model features ASCII Serial/Parallel system attachments. A model conversion is available from Model CT0 to Model CTA via MES. There are external I-Data LAN attachments available for 6408 model CTA printers. This product is manufactured by i-data of Denmark and distributed by IBM. This device allows the IBM 6408/CTA printers to attach to a 4 or 16 MBps Token-ring or Ethernet LAN for use with PSF/2 or PSF/6000, PSF/MVS and PSF/400. The Token Ring hardware and software is ordered as RPQ #8B4313. The RPQ for Ethernet hardware and software is RPQ #8B4314. Please contact your marketing team for assistance in configuring and ordering any model conversions or RPQ's. The recommended printer drivers for the 6408 model CTA or model A00 are: IBM Proprinter IIIXL, Epson FX1050, or generic text driver for non-graphical printing.
What IBM does offer specifically for the 6408 is at:
http://www.printers.ibm.com/R5FAQ.NSF/6400?Open
They also state:
Question:
What publications are available for Code V printing on an IBM 6408 Model CTA printer?
Answer:
The publication, Model CTA Code V - Printronix Emulation User's Manual (Form Number G246-0086), is available.
To order, contact your local IBM printing systems representative or an authorized IBM reseller. Or contact
IBM Publications at 1-800-879-2755 (in U.S. and Canada only).
This begins to seem a little bit more difficult than meets the eye. So, why is it so hard to find?