The Faith Equation: Part Two of the Review

Oct 22 2007 Published by under Bad Probability

The bulk of this part of the review is looking at the total train-wreck that is chapter 4, which contains Bittinger's version of dreadful probabilistic arguments for
why Christianity must be true. But before I do that, I need to take care of one loose
end from part 1. I should have included chapter three in part one of the review, since it's really just a continuation of the paradox rubbish, but I didn't.

The basic idea behind chapter three is that Jesus is the most fundamental
resolution of paradox. All of the most important of the (Bittinger) paradoxes that we
encounter in our lives are ultimately resolved by Jesus. Oh, joy! It's really just more of the same: he keeps playing with his own personal definition of paradox, and using it to create fake problems that he can solve with cheap arguments and fake math. The only thing really worth pointing out specifically
from this chapter is another thoroughly pathetic example of how Bittinger appropriates mathematical
notation to lend a sense of credibility to a piss-poor argument:

lim L(t) = God, where L(t) = life's spiritual experiences

What purpose does this "equation" serve? Does it clarify anything? No. Does it explain anything? No. Is there anything that's meaningfully represented by a function? No. Is there anything, of
any sort, to be gained by inserting a make-believe function?

Yes. It looks good. Instead of just being more of Bittingers self-obsessed babbling,
it looks like he's using his skills as a mathematician to support his arguments, even though
he isn't. It's pure nonsense, wrapped up in mathematical notation, just to make it look as if there was something serious there, rather than the total bullshit.

The other notable thing about this chapter is that Bittinger uses his notion of
"paradox" to define sin. Sin, according to Bittinger, is self-centered behavior, rather than
God-centered behavior. All sin, according to him, ultimately comes down to acting in selfish, self-centered ways. The reason that this is interesting is that I'm not sure if I've ever read
such a totally self-centered, self-obsessed book as this one. This book is a monstrosity of
self-indulgence. As a Jew who's spent time dealing with too many fundamentalist Christians, I've got to say that there's something very fundamentalist christian about the idea of writing an
incredibly self-centered text about how self-centeredness is the root of all evil; I've never seen
any group of people so likely to engage in this kind of blatant hypocrisy about self-centeredness. Other people are always wrong when they're doing something self-centered - but when
the Real True Christian does it, they're not being self-centered, they're just doing what
God wants.


Then we move on to chapter four, and things get fun, in a pathetic sort of way. How many times have I
mocked bad probability arguments? Well, this takes things to a whole new level of badness. Bittinger wants
to use a pretty common argument: that Christianity somehow satisfies a set of prophecies, and that the likelihood of those prophecies being satisfied by a random event is vanishingly small. Like I said, a remarkably common argument.

But Bittinger's approach to it is the worst I've ever seen. He picks nine prophecies
that he claims were satisfied, and slaps together some of the strangest probability numbers
for them that I've ever seen. It's like a compendium of all of the classical mistakes of
probabilistic arguments, all wrapped up in one neat little pile. Back when GM/BM was on blogger, I
proposed my own taxonomy of the fundamental errors of statistical and/or probabilistic arguments. The basic errors are:

  1. Big Numbers: This consists of using our difficulty in really comprehending how huge numbers work to say that beyond a certain probability, things become impossible. So in using this argument, you
    use other tricks to create an incredibly huge number that is allegedly the odds against
    something happening - and then say "See, it's too improbable!".
  2. Perspective errors: using a priori estimates of something to predict the likelihood of
    a specific event that actually occurred. Shuffle a deck of cards, and then ask
    what was the likelihood of getting this specific order?
  3. False independence errors: computing probabilities separately, and then combining them without
    considering how they interact. Given absolutely no information, what's the probability that my birthday is in July? 31/365. Then, separately, what's the probability that my birthday is in summer? 1/4. So the probability of my being born during the summer in July is 1/4×31/365.
  4. Fake numbers: generally part of a big numbers argument. You want to inflate the numbers,
    so you include as many factors as you can. But some of them are hard (or even impossible)
    to figure out. So you just pull numbers out of the air, and throw them into the equation.
  5. Misshapen search space: Make some event look unlikely by pulling a switch; instead of
    computing its probability in the real setting in which the event would occur, create a
    different setting where it's more unlikely, and then compute the probability there.

Bittinger arguably makes every single one of those. In particular, all of his
prophecy arguments are plagued by perspective errors. Every one of them is built by
assuming that Jesus was the messiah, and then going back and computing probabilities of events
a priori. So they all require that you first accept that Jesus was the Messiah, and that the prophecies were about him, as priors - but then calculates the probabilities without that
prior. Classic perspective error. But he also makes all of the others (except, arguably,
the search space, but I think he even does that one!)

I'll show you two examples of how he calculated probabilities for prophecies, and where each category of error occurs in them: the supposed prophesy about the betrayal of Jesus, and the prophecy of
the virgin birth.

For the betrayal, he pulls out an early Jewish text that makes reference to the throwing of 30 pieces of silver; and claims that the betrayal of Jesus in exchange for 30 pieces of silver is a fulfillment of
that prophecy. Here's how he puts together a probability for it:

  1. Time: The prophet lived 500 years before Jesus. The probability of the predicted event occuring
    after 500 years is, apparently, 1 in 1000. (Fake numbers, perspective.)
  2. Currency: There were 7 ways of paying for things: silver, gold, bronze, land, crops, animals, or labor.
    Therefore the odds of correctly predicting silver were 1 in 7. (Again, fake numbers - why
    specifically choose seven for this? Is including "labor" appropriate? Are all seven equally
    probably? It's just a way of adding a factor of seven for no particularly good reason.)
  3. Delivery: The money was thrown down at the betrayer; there are two ways of presenting money: placing it, or throwing it. So the odds of predicting "throwing" are 1 in 2. (Once again - fake numbers. Strangely, this seems like he's missing lots of opportunities. Why are the only options "place" or "throw"? You could hand it, place it, drop it, throw it, etc.)
  4. Location: the money was thrown down either inside or outside the temple - so again, 1 in 2. (Fake numbers/Misshapen space, but in a strange way. Why only those two choices? There are thousands of places
    that the money could have been delivered. But why restrict it this way?)
  5. Quantity: his estimates that the payment could be between 1 and 1000 pieces of silver, and
    end up deciding, without explanation, that the correct balance sets the probability of 30
    pieces of silver as 1 in 100. (Fake numbers, and false independence. The probability of the payoff being "30 pieces of silver" is not independent of the probability of the payoff being silver coins.)
  6. ... and so on

Numbers randomly drawn out of a hat, slapped together arbitrarily, assuming the desired conclusion,
ignoring issues of independence, in order to reach something that supports the desired conclusion.
Can anyone, Christian or not, actually look at that train wreck of a probability estimate, and claim that it should be taken seriously? Could anyone actually find that convincing?

Well, if you think so, I've got news for you: it gets worse, so even if you're willing to accept
the dreck above as some kind of valid mathematical argument, you're going to be surprised by what comes next.

What's the probability of Jesus being the result of a virgin birth? How does he compute a number for that? His first estimate is 1 in 84,850,000. Why? Because that's an estimate of the number of women of
childbearing age in the world at the time. And since there was only one virgin birth, the odds of Mary being the one is that number. Then he inflates it more, because, of course, there was only one virgin birth in the history of humanity. So he can use the entire number of women ever, and use that as the denominator of the probability. (Big numbers, perspective errors, fake numbers, and misshapen space - all colliding in one great, hideous pileup.)

This second example of how he argues about probabilities makes his tactics and his errors extra clear.
First of all, there's the fact that the numbers are, pretty much, pulled out of his ass. But more
importantly, they measure the wrong things. They combine perspective errors and bad space errors, by claiming to be computing the probability of one thing, when in fact, they're using fake numbers to compute the probability of some entirely different thing. When someone wants to know what's the probability of something like the supposed virgin birth, they're not asking: "if you assume that there was a virgin birth and that Mary was the mother, what are the a priori odds
that Mary would have been the mother?". They're asking: "what is the probability of something as
out of the ordinary or inexplicable as a virgin birth?". But Bittinger, through this section, assumes
the truth of his religious beliefs, and calculates probabilities of irrelevant things, as if they somehow
mean something.

It's really, frankly, just bizarre. Bittinger is a smart guy, not a moron. How can he make arguments this nonsensical, and think that they're going to be convincing? How can he put his reputation as
a mathematician on the line for garbage like this? What in hell was he thinking?

No responses yet

  • Anonymous says:

    Gosh, I have a simpler explanation for the 30 pieces. Could the New Testament writer possibly have actually read the earlier text and then incorporated the imagery to bolster his credibility (or perhaps simply to have the story flow better)?
    I think Hume had something to say about supposed 'miracles'. Perhaps Bittinger should re-read Hume.

  • spudbeach says:

    I'm amused, but not really amazed. When people with a set of preconceived ideas try to justify the ideas, there isn't a whole lot of stupid things they won't do.
    All of the big numbers he comes up with do make me think, though. The odds of a royal flush are 1 in 649,350. If I get three royal flushes in three hands, I know that something isn't right. My sample space enlarges to admit the possibility that a magician or a cheat is dealing. When seeing long odds realized, one must enlarge one's sample space to allow for the possibility that the situation is not as one supposed.
    So, why doesn't Bittinger enlarge his. When seeing odds like he calculated, why doesn't he enlarge his sample space to admit the possibility that this is all just a made up story?

  • John Marley says:

    why doesn't he enlarge his sample space to admit the possibility that this is all just a made up story?

    Maybe he realizes that "the possibility that this is all just a made up story" approaches 1:1

  • Then one of the Twelve--the one called Judas Iscariot-- went to the chief priests and asked, "What are you willing to give me if I hand him over to you?"
    So they counted out for him thirty silver coins. From then on Judas watched for an opportunity to hand him over.
    MATTHEW 26:14-16
    Wikipedia has an article on this, in which the citations are hotlinks, and why it seems to be (as Mark CC suggests) a "made up story:
    {start wikipedia excerpt}... The traditional Christian explanation is that Judas hanged himself in the field, and afterwards the rope snapped, and his body burst open on the ground. These views do not explain why the accounts differ in their details both of the death and what happened to the money afterward: the Matthew account says Judas returned it and the priests used it to buy a field called the Potter's Field as a burial ground for strangers, whereas the Acts account says Judas used it to buy the field he died in. Concerning the money and what was bought with it, there is also another view put forward by E.W. Bullinger. According to this view the accounts of Acts and Matthew speak about two different transactions and two different fields.[The Journal of Biblical Accuracy: "The two fields of blood". Retrieved on 2007-10-13]
    The consensus among scholars is that the Matthew account is a midrashic exposition that allows the author to present the event as a fulfilment of prophetic passages from the Old Testament books of Zechariah and Jeremiah. They argue that the author adds imaginative details such as the thirty pieces of silver, and the fact that Judas hangs himself, to an earlier tradition about Judas's death.[ Reed, David A. (2005). "Saving Judas"--A social Scientific Approach to Judas's Suicide in Matthew 27:3-10. Biblical Theology Bulletin. Retrieved on 2007-06-26.]{end wikipedia excerpt}
    Then other sources show how shady is the background of "30 pieces of silver", what Old Testmanet passage was cited by Matthew (Jeremiah or Zechariah?) and what happened to the money. For example:
    The Skeptical Review: 1996: September/October: A Response to Alleged Difficulties in Matthew 27:9-10 [pp. 1-2]
    A Response to Alleged Difficulties in Matthew 27:9-10
    by Wilhelm E. Schmitt
    The Septuagint, and also its revision by Symmachus, reads in verse 13, "cast them, i.e., the 30 pieces of silver) into the furnace" (Greek: eis to choneuterion). This shows that before the Gospel of Matthew was written, yotzer was interpreted as referring not to a "potter" but to a fashioner of metals.
    It seems to me a particularly weak text upon which to build a probabilistic model. My Math background makes me wonder "why 30?" Because of a numerological attribution to the Hebrew letters for 30?
    In feminine form: shloshim sfarim : 30 books.
    Because it is a primorial (1st prime * 2nd prime * 3rd prime = 2*3*5)? Because it is the sum of the squares of the integers 1, 2, 3 and 4? That makes it a square pyramidal number, and pyramids were in Egypt, where the Jews were slaves?
    But I don't bother to calculate the odds that Judas' alleged pay is the same number as {again wikipedia}:
    * The atomic number of zinc
    * The number of days in the months April, June, September and November (and in unusual circumstances February - see February 30)
    * The total number of major and minor keys in Western tonal music, including enharmonic equivalents
    * In years of marriage, the pearl wedding anniversary
    * The duration in years of the Thirty Years' War
    * Used to indicate the end of a wire service story. (Possibly a corruption of German fertig - "finished, ready" - or in journalistic context "end of story"); in several Superman stories from various titles, failure by a Daily Planet employee to use this signature proved to be a plot point revealing a character's impersonation, mind control, etc.
    * The code for international direct dial phone calls to Greece
    * The house number of 30 St Mary Axe (The Gherkin)
    * The designation of Interstate 30, a freeway that runs from Texas to Arkansas
    * The designation of U.S. Route 30, a highway that runs from Oregon to New Jersey
    * Various other routes have been numbered "30"; for example, New York State Route 30 which runs from the Pennsylvania border to the Canadian border
    * The designation of E30, the European route from Cork to Samara
    * The number of tracks on The Beatles' eponymous album, usually known as The White Album
    * The number of cars in the Nintendo 64 game F-Zero X
    * -30-, a motion picture about work in a Los Angeles newspaper, starring Jack Webb and William Conrad
    * A stage in young adulthood
    * Part of the name of:
    o Thirty odd foot of grunts, the band fronted by actor Russell Crowe
    o The movie title 13 Going on 30, starring Jennifer Garner
    o The title of the Food Network show 30 Minute Meals
    * The uniform number of Maury Wills when he played for the Los Angeles Dodgers.
    * Judas Iscariot betrayed Jesus for 30 pieces of silver. Matthew 26:15.
    What are the odds that Judas forshadows both the Beatles and the Dodgers? I think this reductio ad absurdum makes its point.
    But what happens to all the many biblkical prophecies that did NOT come true? That's a traditional source of data for Bible skeptics. My point here being that Bittinger has cherry-picked a prophecy that another verse in the Bible says was the fulfilment of a prophecy, rather than picking a prophecy at random, and factoring the ratio of fulfilled to unfulfilled to "don't know yet."
    And what are the odds that the recently translated archaeological find of "The Book of Judas" shows (as it does) extremely good agreement with the plot of "Jesus Christ Superstar?" At least the film has a well explored motivation for Judas, and some good songs.

  • Blake Stacey says:

    Instead of just being more of Bittinger's self-obsessed babbling, it looks like he's using his skills as a mathematician to support his arguments, even though he isn't. It's pure nonsense, wrapped up in mathematical notation, just to make it look as if there was something serious there, rather than the total bullshit.

    I keep waiting for him to equate the square root of -1 with the male generative organ. . . .
    (Come on, I know you Alan Sokal fans are out there!)

    Gosh, I have a simpler explanation for the 30 pieces. Could the New Testament writer possibly have actually read the earlier text and then incorporated the imagery to bolster his credibility (or perhaps simply to have the story flow better)?

    Most of the "prophecies" in the book of Matthew have exactly this character; he seems to have been trying to impress people steeped in the Biblical lore of the day. And, as it happens, most of Matthew's allusions are horribly off-target. For example, he says (Matthew 27) that the thirty-silver-pieces story was foretold by Jeremiah, but there's nothing about thirty silver pieces in Jeremiah. The closest we get is a story about somebody buying a field from his cousin for seventeen silver shekels, in Jeremiah 32.
    Zechariah 11:12-13 has a closer approximation, but Matthew didn't say Zechariah, did he? Besides, the story in Zechariah is about a shepherd who gets insulted because he's paid so little, and gives his thirty silver pieces to "the potter in the house of the LORD." This has nothing to do with betrayal, blood money or anything else of significance in the story Matthew is telling. In fact, I have to wonder what the heck a potter is doing in "the house of the LORD"; maybe he's making jars for the money-changers? "Potter" might not even be the right word. The Revised Standard Version says "treasury" instead, on the grounds that yatsar ("potter") might be a corruption of atsar (treasurer, paymaster), and a shepherd who is piqued by his low wages is more likely to throw the coins at the man responsible rather than a poor schlemazel who's just making pots nearby.
    All the "prophecies" trotted out in these debates break down into a few categories.
    1. Stuff anybody could have done, if they wanted to be called the Messiah. Somebody says the Messiah should arrive in Jerusalem riding a young donkey? Well, go grab yourself a donkey.
    2. Stuff made up by people who knew about the old prophecies. Everything Matthew says about the childhood years of Jesus seems intended to make Jesus' life parallel that of Moses. This might explain why the stories Matthew tells don't appear in Mark, Luke or John.
    3. Stuff based on mistranslation of old prophecies. The "virgin birth" thing appears only in Matthew and, possibly, in a parenthetical remark of Luke (which, for all we know, was inserted later by some monk who "just knew" what Luke "had to say"). It's supposed to be a fulfillment of Isaiah 7:14, right? But in Isaiah 7, the prophet is trying to provide to the king a sign that the prophet knows what he's talking about and that the king should listen to the prophet in order to preserve the kingdom. Saying that a child will be born seven hundred years later is not exactly what a smart prophet would do under the circumstances. And, behold: in Isaiah 8, a young woman — Isaiah's wife — gives birth to a child who fulfills exactly the criteria Isaiah set forth in chapter 7. Notice how people in the New Testament aren't actually running around calling Jesus "Emmanuel"? Well, that's because the prophecy of "Emmanuel" was actually fulfilled by Isaiah's son, Maher-shalal-hash-baz, one chapter later. (The two names have the same symbolism: Emmanuel, "God is with us", implies good fortune for Judah, and Maher-shalal-hash-baz, "Hurrying to the spoil he has made haste to the plunder", implies bad fortune for the people attacking Judah at the time.)
    All of these "prophecies" carry exactly the weight of Harry Potter fulfilling in Deathly Hallows something foretold in Goblet of Fire.

    Again, fake numbers - why specifically choose seven for this? Is including "labor" appropriate? Are all seven equally probably? It's just a way of adding a factor of seven for no particularly good reason.

    Clearly, it's because seven is a particularly holy number, as evidenced by the heptadic structure of Matthew. I mean, it's the sum of the Elements (4) and the Trinity (3), so it has to be important! Notice that the product of the Elements and the Trinity is 12 (the Tribes, the Zodiac, etc.). Truly, the arithmetical ways of the LORD are marvelous and subtle!

  • Pierce R. Butler says:

    Mark C. Chu-Carroll: ... what's the probability that my birthday is in July? 31/365. Then, separately, what's the probability that my birthday is in summer? 1/4. So the probability of my being born during the summer in July is 1/4×31/365.

    Er, what are the odds of you having been born in the northern hemisphere?

  • MartinM says:

    Well, go grab yourself a donkey.

    Or two, if you misread the original prophecy.

  • Thomas says:

    How insightful. Stupid Christians. I for one am so stupid I can't even recall the last time you bashed Jews like this. Which I'm sure you do all the time, being a free-minded thinker and all.

  • g says:

    1. Do you have some specific examples of laughably bad mathematics written by Jews that Mark ought to be commenting on, or do you just have a general wish to see more Jew-bashing?
    2. He isn't bashing "Christians", he's bashing one particular book written by one particular Christian. (With a few more general swipes at *fundamentalist Christianity*, which is not at all the same thing as Christianity-in-general.)
    3. There are approximately 1500 times as many Christians as Jews in the world. It's therefore reasonable to expect on the order of 1500 times more stupidity coming from Christians than from Jews.
    (Though on that basis you'd also expect 1500 times more strokes of genius coming from Christians than from Jews, and that seems pretty high -- a surprising number of Jews win Nobel prizes, for instance. Which suggests that maybe there's something in Jewish culture or genetics that biases Jews as a population towards cleverness rather than stupidity, which would further reduce the amount of insanity Mark would find to comment on from them.)
    Since Mark is in the US, perhaps a more relevant figure is the ratio in the US. That would be about 40:1.

  • I've heard it said from one of my Jewish freinds (I'm Jewish by descent myself) that Jewish inteilligence is bimodal.
    There is one peak above the mean of the general population, from which comes those 1/3 of Nobel Laureates (or whatever the precise ratio may be). There is another peak just below the mean. Allegedly, these two segments of the Jewish population originate from, on the one hand, care for the differently abled with coherent communities in a faith-based safety net, and, on the other hand, millennia of allowing the most intelligent and literate member of each village to become the rabbi, with his choice of which young woman in the village for a wife. Evolution by natural selection. So the allegation goes, anyway.
    Today I am half of the pair debating "Does God Exist" at the Golden Eagle Ballroom, 2-4 p.m., California State University, Los Angeles. It's basically a debate between atheism and agnosticism, with mutual attack on Intelligent Design. My distinguished opponent is a minister turned militant Atheist, who still draws a royalty streams for his stage and TV Christian Musicals. I'll trot out those "axioms" on the existence of God, Physical Reality, and the efficacy of Mathematics that I first blogged here, in a related thread.

  • Mark C. Chu-Carroll says:

    Thomas:
    I'd happily attack a fellow jew for bad math. I have in the past - back before my blogging days, I was one of the guys trying to debunk the hebrew bible skip code crap which was produced by the Aish HaTorah yeshivah. I've also done some stuff on gematria. And if anyone sends me a good fisk-worthy website or book containing bad math from a Jewish perspective, I promise that I'll write about it, and that I'll be no easier on it than I've been on Bittinger.
    But let's face it - as an American, there's a hell of a lot more threat from the kind of rubbish spewed by Christian fundamentalist than from Jewish loonies. Just compare ow much ability to influence government and education in America the Lubavitcher jews have in comparison to the Southern Baptists?
    Bittinger is also important because he's a relatively famous guy. A lot of people know the name from college calculus; that's why *I* knew his name - my freshman calc book at Rutgers was one of his texts. His book has been intensely promoted by the publisher, on the net, on the radio, all over the place, and the publicity always talks about him as a mathematician and author of math textbooks who's applying his mathematical skills to Christian apologetics. The attention that Bittinger is getting as a result of his record as an author of textbooks makes pointing out how bad the math in *this* book all the more important.
    Where is there a book about the mathematical proof of Judaism that's been promoted the way that Bittinger's book has? Show it to me, and I'll cover it in as much detail as Bittingers.

  • "Where is there a book about the mathematical proof of Judaism..."
    *-algebras and the Star of David, by John von Neumann, Vito Volterra, Norbert Wiener, Paul Erdös, Izrail Gelfand, André Weil, Alexander Grothendieck. Just kidding. But see:
    Jews in Mathematics
    http://www.jinfo.org/Mathematics.html
    which has a breakdown of Math awards by percentage of jewish recipients:
    # Jewish Fields Medalists (25% of recipients)
    # Jewish Recipients of the Wolf Prize in Mathematics (40% of recipients)
    # Jewish Recipients of the Leroy P. Steele Prize for Lifetime Achievement (59% of recipients)
    # Jewish Recipients of the Bôcher Memorial Prize (42% of recipients)
    # Jewish Recipients of the Frank Nelson Cole Prizes in Algebra and Number Theory (44% of recipients)

  • I knew that Wildberger stuff looked familiar. It seems that I blogged about him at page 4 of 225 comments on mathematics and God (the first 3 pages have quite a number of comments about God and Math, including several versions of the alleged Euler proof (spoof) to Diderot in audience with Russia's Empress):
    http://magicdragon2.livejournal.com/3457.html?nc=225&page=4#comments
    Modern mathematics as religion: N. J. Wildberger
    magicdragon2
    2006-04-12 04:31 pm UTC (link)
    Modern mathematics as religion
    Set Theory: Should You Believe?
    N. J. Wildberger
    School of Maths UNSW Sydney NSW 2052 Australia
    "I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction." (Gauss)
    "I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there." (Kronecker)
    "...classical logic was abstracted from the mathematics of finite sets and their subsets...Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory ..." (Weyl)
    Modern mathematics as religion
    Modern mathematics doesn't make complete sense. The unfortunate consequences include difficulty in deciding what to teach and how to teach it, many papers that are logically flawed, the challenge of recruiting young people to the subject, and an unfortunate teetering on the brink of irrelevance.
    If mathematics made complete sense it would be a lot easier to teach, and a lot easier to learn. Using flawed and ambiguous concepts, hiding confusions and circular reasoning, pulling theorems out of thin air to be justified `later' (i.e. never) and relying on appeals to authority don't help young people, they make things more difficult for them.
    If mathematics made complete sense there would be higher standards of rigour, with fewer but better books and papers published. That might make it easier for ordinary researchers to be confident of a small but meaningful contribution. If mathematics made complete sense then the physicists wouldn't have to thrash around quite so wildly for the right mathematical theories for quantum field theory and string theory. Mathematics that makes complete sense tends to parallel the real world and be highly relevant to it, while mathematics that doesn't make complete sense rarely ever hits the nail right on the head, although it can still be very useful.
    So where exactly are the logical problems? The troubles stem from the consistent refusal by the Academy to get serious about the foundational aspects of the subject, and are augmented by the twentieth centuries' whole hearted and largely uncritical embrace of Set Theory.
    Most of the problems with the foundational aspects arise from mathematicians' erroneous belief that they properly understand the content of public school and high school mathematics, and that further clarification and codification is largely unnecessary. ...Difficulties of Set Theory arise from the insistence that there exist "infinite sets"...
    In perpetuating these notions, modern mathematics takes on many of the aspects of a religion. It has its essential creed---namely Set Theory, and its unquestioned assumptions, namely that mathematics is based on `Axioms', in particular the Zermelo-Fraenkel `Axioms of Set Theory'. It has its anointed priesthood, the logicians...
    Training of the young is like that in secret societies---immersion in the cult involves intensive undergraduate memorization of the standard thoughts before they are properly understood, so that comprehension often follows belief instead of the other (more healthy) way around....
    The large international conferences let the fellowship gather together and congratulate themselves on the uniformity and sanity of their world view, though to the rare outsider that sneaks into such events the proceedings no doubt seem characterized by jargon, mutual incomprehensibility and irrelevance to the outside world. The official doctrine is that all views and opinions are valued if they contain truth, and that ultimately only elegance and utility decide what gets studied. The reality is less ennobling---the usual hierarchical structures reward allegiance, conformity and technical mastery of the doctrines, elevate the interests of the powerful, and discourage dissent.

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