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Re: [ANE-2] Whose tomb ?

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  • OAK
    Wonderful statistical analysis -- thanks a lot. And an additional factor that nobody seems take in consideration is that we are probably not looking at a
    Message 1 of 21 , Mar 1, 2007
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      Wonderful statistical analysis -- thanks a lot.

      And an additional factor that nobody seems take in consideration is that we are probably not looking at a single generation of deaths which all of the statistical analysis seems to be based upon. It is far more likely we are looking at several generations. Obviously this consideration would lower the probability factors drastically. In addition since we have no dates of the death order who knows whether Joseph of Talpiot may not have died in 90 and Mary of Talpiot in 25 and Jose the imagined son of Jesus in 10 etc. Not having such data, the span of time and the death order, makes statistical analysis at best complete conjecture it would seem, which, I guess is fitting with the rest of the extravagant unscholarly claims.

      Everett Oakley
      Houston, Texas

      [Non-text portions of this message have been removed]
    • Peter T. Daniels
      You must be referring to a different Jesus ... -- Peter T. Daniels grammatim@verizon.net ... From: OAK To: ANE-2@yahoogroups.com Sent:
      Message 2 of 21 , Mar 2, 2007
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        You must be referring to a different Jesus ...

        --
        Peter T. Daniels grammatim@...



        ----- Original Message ----
        From: OAK <eoakley3@...>
        To: ANE-2@yahoogroups.com
        Sent: Thursday, March 1, 2007 6:55:38 PM
        Subject: Re: [ANE-2] Whose tomb ?

        Wonderful statistical analysis -- thanks a lot.

        And an additional factor that nobody seems take in consideration is that we are probably not looking at a single generation of deaths which all of the statistical analysis seems to be based upon. It is far more likely we are looking at several generations. Obviously this consideration would lower the probability factors drastically. In addition since we have no dates of the death order who knows whether Joseph of Talpiot may not have died in 90 and Mary of Talpiot in 25 and Jose the imagined son of Jesus in 10 etc.

        <...>
      • Peter Nathan
        For the record, Kloner s report from Atiqot notes that remains of at least 35 people were recovered from the tomb, 17 in the 10 ossuaries and 18 others.
        Message 3 of 21 , Mar 2, 2007
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          For the record, Kloner's report from 'Atiqot notes that remains of at
          least 35 people were recovered from the tomb, 17 in the 10 ossuaries and
          18 others. Bearing in mind that the ossuaries had been tampered with,
          the numbers originally in ossuaries would have been higher. So we are
          clearly dealing with a multi-generational family tomb. This would have
          been a small tomb in relation to some of the other discoveries which
          have had up to 80 people interred.

          The 'Caiaphas tomb" also in Talpiot area had six ossuaries with human
          remains, the rest of the ossuaries had been emptied by grave robbers or
          others. Some of those ossuaries had up to six sets of remains. The
          same is true of the Mr Scopus tomb that Sussman reported in 'Atiqot 1992.

          --
          Best wishes,

          Peter Nathan
          ___________
          Peter Nathan
          476 S. Marengo
          Pasadena
          CA. 91101
          Tel. Office: 626-535-0444/204
          Tel. Home: 626-357-9995
          Facsimile: 626-535-0455
          Cellular: 626-353-7309

          Call me on SKYPE: free internet telephony
          http://www.skype.com (username: peternathan)
        • Ariel L. Szczupak
          At 01:01 PM 3/1/2007, Ariel L. Szczupak wrote: [...] ... The Discovery Channel PDF ...
          Message 4 of 21 , Mar 2, 2007
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            At 01:01 PM 3/1/2007, Ariel L. Szczupak wrote:
            [...]
            >That's a "duh?" result.

            The Discovery Channel PDF ...

            http://dsc.discovery.com/convergence/tomb/explore/media/tomb_evidence.pdf

            ... has [p.13] basically the same text as on the web page but has
            also some computations. I don't understand the last two, but I'll use
            the first three.

            I'll add to it another bit of info from an article in today's Yediot
            Aharonot printed supplement which says that in those calculations the
            population of Jerusalem was assumed to be of 100,000. [Again I think
            the calculations should be made using the country's population, not
            just Jerusalem's, but for the sake of this demonstration I'll use it]

            When they (whomever wrote what's in the PDF) say that the frequency
            of "Jesus son of Joseph" is 1/190, that means that 1 out of 190
            persons will have that name. That means that in a population of
            100,000 persons there'll be 526 persons with this name.

            How do I know it's "persons" and not "males"? Because in the 2nd &
            3rd calculations the frequencies are multiplied as-is without
            separating male-names from-female names or halving the frequencies.
            [If the frequency of "Jesus son of Joseph" is 1 of 190 male-names,
            then the frequency is 1/380 for the general male/female population]

            The 3rd calculation claims that the probability of the four names
            appearing together, as calculated from the frequencies of individual
            names, is 1/2,400,00. That means that if you check the names in
            4-person groups, only one of every 2,400,000 will have those exact four names.

            Impressive, right?

            Now let's see how many 4-person combinations, order not important and
            without repeats (can't have one person dying twice), we can have
            given a population of 100,000?

            The formula (which you'll find in any basic text on statistics) is
            n!/(r!(n-r)!)

            [n is the number of items to choose from, r is the number of items
            that are chosen together each time, ! is factorial (n! = 1*2*3*...*n)

            In this case we have 100,000!/(4!(100,000-4)!), or (courtesy of a
            spreadsheet) 4,166,416,671,249,980,000.

            Every 2,400,000 of these 4,166,416,671,249,980,000 4-person groups
            will have these four names. How many is that?

            1,736,006,946,354.16

            I.e. given the probability they calculated for those four names,
            checking all the possible combinations of 4-person groups (no order,
            no repetitions) in a population of 100,000 will result in some 1.7
            trillion and change such possible groups having those four names.

            Duh? DUH?

            One of the reasons for such a ridiculous result is the 1/4 frequency
            for "Maria". It means that out of the 100,000 people, 25,000 were
            called Maria. Or that every second woman was called Maria. And since
            they call the wind Maria, that's a little bit too much wind.

            So I doubt we'll be able to figure out this "600 to 1" claim without
            a serious publication.



            Ariel.

            [100% bona fide dilettante ... delecto ergo sum!]

            ---
            Ariel L. Szczupak
            AMIS-JLM (Ricercar Ltd.)
            POB 4707, Jerusalem, Israel 91401
            Phone: +972-2-5619660 Fax: +972-2-5634203
            ane.als@...
          • Joe Zias
            The math/probability is a real snow job, e.g for starters pop. of Jrsm ca 30 AD was about 50,000 at the most and a good percentage of those were not even
            Message 5 of 21 , Mar 2, 2007
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              The math/probability is a real snow job, e.g for starters pop. of Jrsm ca 30 AD was about 50,000 at the most and a good percentage of those were not even Jewish. They up the ante to 100,000 as if everyone was Jewish and rig the math game as they have all along. don't buy into it. They have rigged the 'game' from the beginning, their geneaolgy chart is a good example of ignoring the fact that the tomb is an extended family running up to 100 yrs from 30 BC to 70 AD.

              Joe Zias

              "Ariel L. Szczupak" <ane.als@...> wrote: At 01:01 PM 3/1/2007, Ariel L. Szczupak wrote:
              [...]
              >That's a "duh?" result.

              The Discovery Channel PDF ...

              http://dsc.discovery.com/convergence/tomb/explore/media/tomb_evidence.pdf

              ... has [p.13] basically the same text as on the web page but has
              also some computations. I don't understand the last two, but I'll use
              the first three.

              I'll add to it another bit of info from an article in today's Yediot
              Aharonot printed supplement which says that in those calculations the
              population of Jerusalem was assumed to be of 100,000. [Again I think
              the calculations should be made using the country's population, not
              just Jerusalem's, but for the sake of this demonstration I'll use it]

              When they (whomever wrote what's in the PDF) say that the frequency
              of "Jesus son of Joseph" is 1/190, that means that 1 out of 190
              persons will have that name. That means that in a population of
              100,000 persons there'll be 526 persons with this name.

              How do I know it's "persons" and not "males"? Because in the 2nd &
              3rd calculations the frequencies are multiplied as-is without
              separating male-names from-female names or halving the frequencies.
              [If the frequency of "Jesus son of Joseph" is 1 of 190 male-names,
              then the frequency is 1/380 for the general male/female population]

              The 3rd calculation claims that the probability of the four names
              appearing together, as calculated from the frequencies of individual
              names, is 1/2,400,00. That means that if you check the names in
              4-person groups, only one of every 2,400,000 will have those exact four names.

              Impressive, right?

              Now let's see how many 4-person combinations, order not important and
              without repeats (can't have one person dying twice), we can have
              given a population of 100,000?

              The formula (which you'll find in any basic text on statistics) is
              n!/(r!(n-r)!)

              [n is the number of items to choose from, r is the number of items
              that are chosen together each time, ! is factorial (n! = 1*2*3*...*n)

              In this case we have 100,000!/(4!(100,000-4)!), or (courtesy of a
              spreadsheet) 4,166,416,671,249,980,000.

              Every 2,400,000 of these 4,166,416,671,249,980,000 4-person groups
              will have these four names. How many is that?

              1,736,006,946,354.16

              I.e. given the probability they calculated for those four names,
              checking all the possible combinations of 4-person groups (no order,
              no repetitions) in a population of 100,000 will result in some 1.7
              trillion and change such possible groups having those four names.

              Duh? DUH?

              One of the reasons for such a ridiculous result is the 1/4 frequency
              for "Maria". It means that out of the 100,000 people, 25,000 were
              called Maria. Or that every second woman was called Maria. And since
              they call the wind Maria, that's a little bit too much wind.

              So I doubt we'll be able to figure out this "600 to 1" claim without
              a serious publication.

              Ariel.

              [100% bona fide dilettante ... delecto ergo sum!]

              ---
              Ariel L. Szczupak
              AMIS-JLM (Ricercar Ltd.)
              POB 4707, Jerusalem, Israel 91401
              Phone: +972-2-5619660 Fax: +972-2-5634203
              ane.als@...






              [Non-text portions of this message have been removed]
            • nancy in chicago
              ... From: Doug Weller much missing The film producers have tested the DNA of one of the Mary s and discovered it is not maternally
              Message 6 of 21 , Mar 2, 2007
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                ---------- Original Message-----------
                From: Doug Weller <dweller@...>

                much missing

                The film producers have tested the DNA of one of the Mary's and
                discovered it is not maternally related to Jesus' DNA.

                ------- End of Original Message-------

                Umm... what Jesus do they have DNA from?  Could they tell which one of the bones was actually Jesus in the ossuary? 

                Otherwise, all that they are saying is that Mary married into the family -- which I would think more likely than having a grown daughter in the ossuary.

                Personally, having the DNA of THE Jesus of Nazareth is probably as likely as having the DNA of THE Harry Potter (and the latter is impossible).

                Nancy Jones
                Interested Amateur

                "Outside of a dog, a book is man's best friend; inside of a dog, it's toodark
                to read."  Groucho Marx




                [Non-text portions of this message have been removed]
              • Doug Weller
                Have people been following this blog? http://benwitherington.blogspot.com/ and
                Message 7 of 21 , Mar 2, 2007
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                  Have people been following this blog?
                  http://benwitherington.blogspot.com/
                  and
                  http://benwitherington.blogspot.com/2007/02/jesus-tomb-titanic-talpiot-tomb-theory.html

                  Doug Weller
                  --
                  Doug Weller Moderator, sci.archaeology.moderated
                  Director and Moderator The Hall of Ma'at http://www.thehallofmaat.com
                  Doug and Helen's Dogs: http://www.dougandhelen.com
                  Doug's Archaeology Site: http://www.ramtops.co.uk
                • Jim West
                  If they have they should know a couple of things about Ben. First, he is among the very few who supported and still supports the authenticity of the James
                  Message 8 of 21 , Mar 2, 2007
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                    If they have they should know a couple of things about Ben. First, he
                    is among the very few who supported and still supports the authenticity
                    of the James Ossuary inscription. Second, the very persons involved in
                    the Talpiot Tomb program were involved with him when he peddled the
                    James Ossuary.

                    In other words- he's all for the James Ossuary (with its questionable
                    inscription) and he's all against the latest "big news". He is, in
                    other words, hardly a disinterested bystander.


                    Doug Weller wrote:
                    > Have people been following this blog?
                    > http://benwitherington.blogspot.com/
                    > and
                    > http://benwitherington.blogspot.com/2007/02/jesus-tomb-titanic-talpiot-tomb-theory.html
                    >
                    > Doug Weller

                    --
                    Jim West, ThD

                    http://drjewest.googlepages.com/ -- Biblical Studies Resources
                    http://drjimwest.wordpress.com -- Weblog
                  • Ariel L. Szczupak
                    ... For a population of 25,000 the numbers are 6,780,056,542.97 Jesus groups out of 16,272,135,703,118,800 possible 4-person combinations. Still ridiculous.
                    Message 9 of 21 , Mar 2, 2007
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                      At 07:33 PM 3/2/2007, Joe Zias wrote:

                      >The math/probability is a real snow job, e.g for starters pop. of
                      >Jrsm ca 30 AD was about 50,000 at the most and a good percentage of
                      >those were not even Jewish.

                      For a population of 25,000 the numbers are 6,780,056,542.97 "Jesus
                      groups" out of 16,272,135,703,118,800 possible 4-person combinations.

                      Still ridiculous.

                      >They up the ante to 100,000 as if everyone was Jewish and rig the
                      >math game as they have all along. don't buy into it. They have
                      >rigged the 'game' from the beginning, their geneaolgy chart is a
                      >good example of ignoring the fact that the tomb is an extended
                      >family running up to 100 yrs from 30 BC to 70 AD.

                      Of course I don't buy into it. The official web site linking the
                      pirates' skull-and-bones flag to the tomb was a "subtle" hint as to
                      how serious this was.

                      But I focus on the statistics for a reason.

                      In statistics the issues are much simpler than in the interpretation
                      of archeological evidence or of ancient texts. That's why it was so
                      simple, once I had some calculations to work with, to reduce their
                      claims to absurdity.

                      But the real reason is that we, humans, have a blind spot when it
                      comes to assessing odds.

                      Our brains can do amazing things. Practically everyone can crumple a
                      page of paper into a very irregular shape and throw it into a waste
                      basket 3 feet away - a trajectory that is expressed in complex
                      aerodynamic equations. But when it comes to odds, our brains don't
                      have that ability. If we had there would be no casinos. I know,
                      through learning & training, to asses odds. At the same time when I
                      talked last week to a high school classmate and found out that the
                      names of two of his daughters are the same as my wife's and my
                      youngest daughter's I had this "what an amazing coincidence" internal
                      sensation. Even though I fully realized that there's nothing
                      surprising since these are very common names, another part of me was
                      saying "wow!"

                      Most people will be able to follow the pro/con archeological and
                      historical argumentation, if they wish to, but the "600 to 1" claim
                      takes advantage of a cognitive blind spot.

                      Is that wrong? Depends what you do with it. Magicians take advantage
                      of our blind spots all the time, and so do con artists. Questions is
                      where are Jacobivici et al on this scale.

                      When Uri Geler claims supernatural powers instead of trickery and
                      slight of hand, he is being dishonest, but he still presents his act
                      as entertainment. Dan Brown wrote a novel, not a scientific paper.
                      Swift claimed in "Gulliver's Travel" that the story is true, but he
                      didn't present it to his generation's Discovery Channel, the
                      Geographical Society, but published it as a novel. In all such cases
                      people know from the context they are about to be entertained, not
                      about to learn truths about reality.

                      Jacobivici, on the other hand, presents his "masterpiece" in another
                      context, a "this is real" context. For me the money he makes is not
                      honest wages for honest entertainment but the dishonest proceeds of a
                      con job. And it's not a con job that takes advantage of someone's
                      moral defects (greed etc), but one that takes advantage of this
                      statistical blind spot, like conning a blind man into buying a
                      "beautiful" picture. A dirty con.

                      And when I say "Jecobivici et al" I include everyone making money out
                      of this pseudo-scientific circus. That includes every reporter that
                      makes in the back of his mind a "it's a good story so I will not
                      check if it's just supermarket tabloid material" decision. And
                      specifically it includes the Discovery Channel. If they air it they
                      are, to me, con artists.

                      May they all be blessed in getting the same respect they give to others.



                      Ariel.

                      [100% bona fide dilettante ... delecto ergo sum!]

                      ---
                      Ariel L. Szczupak
                      AMIS-JLM (Ricercar Ltd.)
                      POB 4707, Jerusalem, Israel 91401
                      Phone: +972-2-5619660 Fax: +972-2-5634203
                      ane.als@...
                    • Schmuel
                      Hi Folks, I want to thank Doug for this analysis he gave the forum a few days ago. If there is any follow-up, or if it has been posted on a blog, please let us
                      Message 10 of 21 , Mar 7, 2007
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                        Hi Folks,

                        I want to thank Doug for this analysis he gave the forum a few days ago.
                        If there is any follow-up, or if it has been posted on a blog, please let us know.
                        The only other interesting discussion of the statistics that I have seen
                        was by Joe D'Mello and was on the NTGateway blog of Mark Goodacre.
                        While they are in some senses complementary I believe that Doug has done
                        the best overall job of really trying to wrestle directly with the statistical issues,
                        writing with a real savvy and flair and comprehension. D'Mello emphasized more
                        the *interpretation* of the existing numbers given by the film makers rather than trying
                        to find a proper and accurate *formulation*. And the latter is ultimately far more important.

                        The fly in the ointment of much of the discussion (and was never even remotely
                        discussed in all the brouhaha) is the difficulty of what is sometimes called "post facto
                        probability" calculations. This pops up in all sorts of venues and defining the proper
                        input parameters is very dicey since there already is a bullesye target (what actually
                        happened) staring you in the face. So it is very difficult to go back to *before* what
                        happened.. happened.. and develop the proper formulations and questions that give
                        a sound resultant calculation number.

                        To give an example of a post facto probability (one that was never really
                        done, afaik) in the public arena that actually made a lot of sense and could
                        have well-defined input .. when Hillary made her $100g on futures trading
                        one might be able to go back and say .. given a-b-c .. what is the likelihood
                        that she could gain that amount by *luck* as opposed to a little "help" from
                        the person entering the profits and loses (ie. perhaps doing a little winnowing
                        out of losses to help her accrue gains.)

                        However the variables in the ossuary situation are a lot more difficult. I will
                        give an example of one. Jose is an extraordinary name in one sense, in that
                        it did not show up on ossuaries much. On the other hand it is all over the
                        Talmud and other ancient Hebraic realms and was apparently simply much like
                        "Joe" is today to Joseph, a shorthand, a nickname. John Gill discusses this as well.
                        And most graves today do not have a Joe, they have Joseph. So if you hit a grave site
                        today with "Joe" how easy would it be for you to figure out what Joseph or Joe was
                        involved ?.. (answer, very difficult .. especially as most people with one name are
                        sometimes called the other).

                        Difficulties like this are all over the ossuary calculations.

                        Enough for now. Thanks, Doug.

                        Shalom,
                        Steven Avery
                        Queens, NY
                        http://groups.yahoo.com/group/Messianic_Apologetic


                        Doug Weller -
                        >I just found this statistical analysis - like the other one, it assumes the names are correct.
                        >I have computed that the probability of the tomb in the Talpiot
                        >district of Jerusalem being the family tomb of the Jesus of Nazareth
                        >is at least 12 to 1 *against*. Apparently the makers of the movie
                        >calculated the probability that more than one family living in ancient
                        >Jerusalem would produce a cluster of names like the ones discovered in
                        >the tomb in the Talpiot district of Jerusalem, and found that this
                        >probability is very small and that therefore this must be the tomb of
                        >Jesus of Nazareth's family. But I think they asked the wrong question.
                        >The right question is: How many families living in ancient Jerusalem
                        >would produce a cluster of names in a tomb that would appear to be as
                        >similar to the names in Jesus' family as the cluster of names actually
                        >found? And the answer is that more than 12 families would have
                        >produced such remarkable cluster of names in a tomb.
                        >
                        >Here is how I computed this number. According to the gospels Jesus'
                        >family consisted of Joseph and Mary, Jesus, and four male brothers of
                        >Jesus named James, Joseph, Simon, and Judas (besides unnamed female
                        >siblings). We also know the approximate frequency of names in ancient
                        >Palestine. According to
                        >http://benwitherington.blogspot.com/2007/02/jesus-tomb-titanic-talpiot-tomb-theory.html
                        >
                        >these are: 9.2%, 8.3%, 6.2% and 3.8% for Simon, Joseph, Judas and
                        >Jesus respectively for male names, and Mary's name frequency is a
                        >whooping 21.3% for female names. The tomb discovered in Talpiot
                        >contained 10 ossuaries, of which 6 carried inscriptions. The relevant
                        >inscriptions here are "Jesus son of Joseph", "Mary", "Mary", and
                        >"Joseph". These inscriptions were in different languages and used
                        >different forms for these names, but that's about it. (see:
                        >http://www.sciencedaily.com/releases/2007/02/070228135009.htm ) I read
                        >somewhere that first century Jerusalem had about 50,000 inhabitants.
                        >As the period in question spanned various generations I used a
                        >population of 10,000 families.
                        >
                        >I wrote a computer program simulation that actually randomly produced
                        >10,000 families of 10 members each (keeping the right name
                        >frequencies) and then proceeded to compute the following average
                        >numbers:
                        >
                        >111 families would have a Jesus son of Joseph. I understand that in
                        >fact other ossuaries have been found with the inscription "Jesus son
                        >of Joseph".
                        >
                        >75 families would moreover have at least one Mary.
                        >
                        >71 families would moreover have one more name that belongs to Jesus'
                        >family, be it Joseph, Simon or Judas, or maybe a second Mary
                        >(supposedly Mary Magdalene).
                        >
                        >43 families would have two more such names. One of such clusters might
                        >be: [Jesus son of Joseph, Mary, Mary, Joseph], as is the one
                        >discovered in the Talpiot tomb.
                        >
                        >And 16 families would have three more such names. Here is in detail
                        >the very first family case my simulation produced: An unnamed (i.e.
                        >with no relevant names) couple have three children: a daughter Mary, a
                        >son Joseph, and an unnamed second daughter. Their daughter Mary
                        >marries Simon and produces an unnamed daughter. Their son Joseph
                        >marries an unnamed wife and produces two children, Jesus and Mary.
                        >Voila: A family cluster of 10 whose tomb might have had inscriptions
                        >"Jesus son of Joseph", "Mary", "Joseph", "Mary", and "Simon" - all
                        >names related to the circle of Jesus of Nazareth, but this is not
                        >Jesus Christ's family. Nevertheless this cluster would appear to be
                        >even more statistically conspicuous than the one discovered in the
                        >Talpiot tomb.
                        >
                        >The film producers have tested the DNA of one of the Mary's and
                        >discovered it is not maternally related to Jesus' DNA. I compute that
                        >adding this condition we still get 12 families. Here is the very first
                        >such case my simulation produced: A Jesus (whose father was named
                        >Joseph) marries a Mary and has 4 children: Simon, Jesus, and two more unnamed ones. Their son Simon marries a second Mary but have no children that would be buried in the family tomb. One unnamed daughter marries Joseph and has a daughter Mary. That's the second Mary who is also not maternally related to "Jesus son of Joseph". - So, any of these 12 families might have produced tomb even more conspicuous than the one found, but at most one of these families could be Jesus'. Hence the chance of the Talpiot tomb being Jesus is less than 1/12.
                        >
                        >Further: Taking into account that Jesus' family was not from
                        >Jerusalem, that his family was too poor to afford a family tomb, that
                        >if Jesus' bones were put in an ossuary one would expect the ossuary
                        >itself or the inscription on it to be more special in some way, and
                        >that if Jesus' body was buried in a tomb to decompose and then put in
                        >an ossuary then probably somebody would have found out back then when
                        >so much was made of Jesus' bodily ascension to heaven - taking all
                        >that into account the probability of the Talpiot tomb being of Jesus
                        >is much less than 12 to 1 against. Finally, if the movie producers
                        >really believed that this was Jesus' tomb one would expect that they
                        >would have asked neutral professional archeologists to evaluate their
                        >evidence or argumentation - which they haven't done.
                        >
                        >There is some more arguments, such as the "James son of Joseph brother of Jesus" ossuary (see:
                        >http://en.wikipedia.org/wiki/James_Ossuary )
                        >having come from this same tomb, but the archaeologist who first
                        >studied the tomb flatly denies it (see:
                        >http://www.jpost.com/servlet/Satellite?c=JPArticle&cid=1171894527185&pagename=JPost%2FJPArticle%2FShowFull )
                        >and the archeological report on the Talpiot tomb counts six ossuaries
                        >with inscriptions, all accounted for (go to
                        >http://dsc.discovery.com/convergence/tomb/explore/explore.html
                        >and
                        >click on "Enter the Tomb", then on "Download Documents" and then on
                        >"Download PDF").
                        >
                        >I did the above computations in a hurry and it's possible that I have
                        >committed some mistake. If you send me an email to
                        >dianelos@... I will gladly send you a copy of the program I
                        >wrote, so that you can check it yourself. It's written in Pascal, and
                        >it's a simple 150 lines program that any programmer can read.
                        >
                        >Doug Weller
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