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Some fine Daily Kos diarists, as well as the estimable Charles P. Pierce at Esquire, have highlighted the new article in Politico today called "Taxpayers fund creationism in the classroom"...

In addition to the now-customary attacks on biological evolution, the article refers to other domains of modern knowledge that are considered suspect by the religious, including areas of modern mathematics, and in particular, "set theory".  Now I've seen this article linked on a few other lefty blogs I read in addition to Mr. Pierce, as well as by several friends on Facebook, and the usual response to the sentences about modern math are generally met with a comment like "Creationists are so dumb they probably still think Pi=3!".  But the fact that the article specifically singles out set theory immediately jumped out at me, and for reasons which I would like to explore below the fleur-de-Kos, I think signifies something rather more important than many of us are giving it credit for.  I would like to share directly with the DKos community something I wrote earlier on the Esquire piece...

I would like to offer an somewhat-informed opinion on the whole math & set theory thing, on the basis that if we are to be forced to debate these kinds of people, we should understand what it is they're really on about. I'm sure the creationist doesn't deny "carry the five" type arithmetic. Since the article specifically mentions set theory, allow me to posit something related to that which I have noticed cropping up a great deal in the modern Christian "apologetics" movement, particularly propounded by such folks as William Lane Craig (and I apologize but this comment is of necessity going to be kinda long)

Craig is the leading proponent of the so-called "Kalam Cosmological Argument" which uses some propositions based on formal logic and a lot of sophistry to deduce "therefore God did it." One of the supporting legs of this argument is the essential impossibility of a real infinity, and to explain this Craig uses a "thought experiment" known as Hilbert's Hotel (named for English mathematician David Hilbert)

The essence of Hilbert's Hotel is really an argument drawing from set theory, much like the following: take the "set" (or grouping) of normal positive integers (1, 2, 3, 4, 5, ... 10... 27 ... a billion...). It goes on forever, there are an infinite number of positive integers. Now consider the set of positive EVEN integers (integers evenly divisible by 2: 2, 4, 6, 8, 10, 12, ... a billion...). Are there fewer even integers than positive integers (even & odd)? There are an infinite number of members of both sets, but in some sense there's only "half as many" even integers... or alternately you could think of it as you get the set of evens by taking away the set of odds (1, 3, 5, 7, ... also an infinite set) away from the set of all integers. But in fact there are EXACTLY the same number of even integers as odd integers as all integers. How to prove it? You can map every single integer to every single even integer simply by doubling it: 1:2, 2:4, 3:6, 4:8, etc. In this way you can see that there must be one, and ONLY one, corresponding even integer to every member of the integer set we started with. Therefore the two sets must be exactly the same size.

But does this work for all real numbers? Real numbers include both integers and fractions and irrational numbers (numbers that can't be represented exactly as a fraction, such as Pi and the square root of 2). Is the infinity of real numbers the same size as the infinity of integers? As it turns out, no, it's not, it's a BIGGER infinity. Georg Cantor proved this again using this concept of mapping from one set to the other, and you CANNOT find a mapping which will include all possible real numbers. (check the links for the details or read the classic "Gödel, Escher, Bach" by Douglas Hofstader (where I was first introduced to these ideas back in college). Because of this difference the infinity of real numbers is not denoted by the familiar "sideways 8" character, but by the hebrew letter "Aleph".

Now how does this relate to Craig and creationism? Because the knowledge that there are higher order infinities which cannot be mapped in the simple way I described undermines the "Hilbert Hotel" argument, which is part of the underpinning of the Kalam Cosmological model which serves as a smart-sounding philosophical "proof" of the necessity of God.

I first came to understand this when I came upon a "William Lane Craig and Hilbert's Hotel" video on Youtube a while back, and tried to explain this in the comments: whereupon I was met with responses along the lines of "Hurr Durr Infinity is the biggest there is you are stupid and Craig has a PhD" before I was blocked from the channel.

I am far, far more annoyed and frightened by guys like Craig than I am by armies of Ken Hams (owner of the "Creation Museum" and recent debate opponent of "Bill Nye the Science Guy") and Roy Comforts (aka "Banana Man"). Those latter guys are never going to persuade anyone to creationism who wasn't already firmly in their camp to begin with. But Craig is different: he is a very smart sounding and soft spoken guy whose arguments sound very rational, even to the fairly educated. You won't understand the holes in them without knowing these kinds of fairly obscure branches of modern mathematics, philosophy, and so on. And that's why it's so easy for Craig to pull the wool over the eyes of otherwise smart people.

I wanted to share this because I feel it's important for "Team Science" to understand the arguments of the enemy to successfully refute them in public. The reason they don't want kids to learn about set theory (among other things) is NOT because "It ain't in the Bible, that's good enough fer me" type yokel-ism (which is how I see this being characterized in a lot of the comments on this news on various websites), but because the vanguard of the assault on modern science (including guys like Craig) understand that knowledge of set theory undermines part of their pseudo-intellectual rationales for creationism; and if enough people learn about it, they won't be as easily able to get away with their intellectual chicanery.

Originally posted to Rheinhard on Mon Mar 24, 2014 at 12:56 PM PDT.

Also republished by Community Spotlight.

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  •  Tip Jar (168+ / 0-)
    Recommended by:
    annieli, Garrett, Ojibwa, ferg, rduran, serendipityisabitch, Cassandra Waites, defluxion10, kevinpdx, dougymi, oortdust, semiot, GeorgeXVIII, FG, ends with space, ontheleftcoast, Calvino Partigiani, Vatexia, Rogneid, Constantly Amazed, Catte Nappe, David54, political mutt, gfv6800, Matt Z, MarkW53, LeftCoastTom, rbird, tarkangi, OHdog, tgrshark13, pat bunny, profundo, Shippo1776, dewtx, xaxnar, Nowhere Man, pvasileff, citizen dan, mommyof3, cama2008, barleystraw, Aunt Pat, No one gets out alive, celdd, Rhysling, Luma, hubcap, Daulphin, i saw an old tree today, greenotron, Debby, Yosef 52, DavidMS, El Bloguero, franziskaner, whl, scurvy turbo, Ahianne, elfling, Kingsmeg, eOz, Dbug, sny, Urban Space Cowboy, liberaldregs, Black Max, radarlady, The Geogre, JVolvo, itsjim, arlene, on the cusp, david78209, JBL55, funningforrest, J M F, HiKa, tobendaro, MartyM, I am a Patriot, p gorden lippy, ChemBob, Jeffersonian Democrat, Pescadero Bill, pixxer, Cedwyn, StrayCat, leeleedee, DeminNewJ, maryabein, Kobuk Sands, Trevin, JosephK74, tle, cordgrass, kumaneko, eru, zozie, muddy boots, GreenMother, bbctooman, tj iowa, Aureas2, jomi, blackjackal, Glacial Erratic, P Carey, dksbook, papercut, Shockwave, smokeymonkey, kbman, wasatch, gnbhull, JerryNA, Gowrie Gal, GreenPA, TracieLynn, Sharon Wraight, jrooth, Mimikatz, Jackson L Haveck, NYFM, happymisanthropy, JayC, Yellow Canary, 1BQ, turn blue, millwood, tacet, wader, milkbone, Nicci August, trumpeter, Yoshimi, Bluesee, Brown Thrasher, bakeneko, adamsrb, VTCC73, TheMeansAreTheEnd, mndan, madgranny, unclejohn, MKinTN, slowbutsure, RJDixon74135, cybersaur, Dumbo, Emerson, ArthurPoet, zerelda, Temmoku, petulans, enhydra lutris, BlueMississippi, splashy, zooecium, snazzzybird, hwy70scientist, 207wickedgood, Sandino, Mostserene1, Alfred E Newman, lehman scott, sciguy, lastamendment

    ----------------------- "Zu jeder Zeit, an jedem Ort, bleibt das Tun der Menschen das gleiche..."

    by Rheinhard on Mon Mar 24, 2014 at 12:56:16 PM PDT

  •  numerology ain't ontology or "Math is hard" (38+ / 0-)

    the onward rush to low-information fascism....

    Thanks for this diary

    Set theory is the branch of mathematical logic that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.
    The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.
    Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals

    Warning - some snark may be above‽ (-9.50; -7.03)‽ eState4Column5©2013 "I’m not the strapping young Muslim socialist that I used to be" - Barack Obama 04/27/2013 (@eState4Column5).

    by annieli on Mon Mar 24, 2014 at 01:01:24 PM PDT

    •  I majored in Philosophy and tutored Logic. Take: (5+ / 0-)
      Recommended by:
      NYFM, wader, bakeneko, ArthurPoet, Temmoku


      {Symbolic logic with identity}

      {Set theory}

      Three symbol-systems for the same concepts.  Take any one of the three and make the appropriate "symbol definitions" and you can derive both of the other two.

      We did it in one of my logic classes.

      “Sin lies only in hurting other people unnecessarily. All other "sins" are invented nonsense. (Hurting yourself is not sinful -- just stupid.)” ― Robert A. Heinlein, Time Enough for Love

      by midgebaker on Tue Mar 25, 2014 at 09:00:59 AM PDT

      [ Parent ]

    •  The Kalam argument contains.. (1+ / 0-)
      Recommended by:

      something like 6 logical fallacies in his three sentences.  These guys mirror themselves after Aristotle in believing they can think "proofs" into existence.  For example, they claim that if you can imagine a god, then one must exist.

      But Aristotle, though brilliant, was wrong on many of his ideas.  Along came the scientific method, where hypotheses were to be tested and replicated and falsified.  

      The Bible makes many testable claims, and everyone fails.  A supernatural all-powerful being has been utterly disproven by science, and there has never been the slightest bit of reliable evidence that a supernatural being actually exists.

      But of course faith is being in something without, or contrary to, evidence.  

      People believe in gods because that is how they were raised.  And very few can break out of that indoctrinated mindset.

  •  But Nobody Thinks Pi = 6. nt (7+ / 0-)

    We are called to speak for the weak, for the voiceless, for victims of our nation and for those it calls enemy.... --ML King "Beyond Vietnam"

    by Gooserock on Mon Mar 24, 2014 at 01:10:34 PM PDT

  •  Thanks for this explanation. (19+ / 0-)

    I'm a 60+ engineer type who loves math stuff, and you taught me something.

    You can't spell CRAZY without R-AZ.

    by rb608 on Mon Mar 24, 2014 at 01:12:44 PM PDT

  •  Interesting position, there (38+ / 0-)

    that those guys have: there is no actual (as opposed to mathematical--they dispute about this) infinite, and therefore the universe must have a beginning, and a Beginner, and therefore God, by definition, exists.

    Or, in their words, there must be a cause, and the "cause of the universe must be a personal, uncaused, beginningless, changeless, immaterial, timeless, spaceless, enormously powerful, and enormously intelligent being."  Which you would have to call God.

    Okay, nice armchair theoretical subject for philosophical debate (and it is interesting that many of the disputants are Muslim).

    But where it goes horribly, stupidly wrong, is in its distillation to "and thus the [insert holy book here] is infallibly correct, and we know exactly what God thinks and what he intends and how we should behave.  Stop kissing that man."

    •  I think this is more of a case of (1+ / 0-)
      Recommended by:
      old possum

      conservative cantankerousness over New Math than a theological conniption.

    •  question here: where does a beginning (7+ / 0-)

      imply a creator or primary force?  This concept seems to be based on the idea that takes the idea that all effects must have a cause to the conclusion that all causes must have a First Mover.

      Of course, this whole argument is based on the whole concept of "cause", which means one thing in ordinary discourse vs the scientific meaning of the term, the same as with "theory"

    •  The more we learn, the further away God seems (2+ / 0-)
      Recommended by:
      gnbhull, JerryNA

      to reside.

      In the 30th century, anyone pointing to an argument like the Kalam Cosmological Argument will seem so totally stuck in the 20th century.

      "We must make our choice. We may have democracy, or we may have wealth concentrated in the hands of a few, but we can't have both." - Louis Brandies

      by Pescadero Bill on Tue Mar 25, 2014 at 07:08:15 AM PDT

      [ Parent ]

      •  Really? They had better arguments (1+ / 0-)
        Recommended by:

        for the existence of God back in the middle ages, like Bishop Berkeley's flawed argument, which at least raised interesting questions about the nature of reality.  By comparison, this seems rather primitive and stupid.

        So we're moving backwards, folks.

    •  I'm actually surprised (1+ / 0-)
      Recommended by:

      to see this argument coming from them as theology has historically defined God as the only actual infinity.  Suggesting that God is a potential infinity would imply that God isn't omnipotent and perfect.  Nice to see some discussion of set theory here (which isn't an obscure branch of mathematics, but one of the leading contenders for the foundation of all mathematics; along with category theory).  I suspect that the reason they're hostile to set theory is that it defends the existence of multiple infinities that can't be totalized or unified into a single infinity.  That spells the ruin of their conception of God.

      •  plus (1+ / 0-)
        Recommended by:

        the infinite number of curves is greater than the infinity of rational numbers. It doesn't stop with the first aleph.

      •  Actually, I don't see that as a problem. (0+ / 0-)

        In advanced algebra, which is where we really got into set theory, this talk about different infinities is described as talking about cardinalities.  E.g., real numbers and integers have different cardinalities, different sized infinities.

        However, even though you can't map the set of integers onto real numbers, you can map the set of real numbers onto the set of integers.  For instance, if X is some real number, then

        For every real number X, there exists a non-unique integer number N, which encompasses all numbers of the integer set.

        N = round down to the nearest integer(X).

        Don't want to make this too complicated, but whatever.

        The implication of this is that there some cardinalities are large enough to encompass other smaller cardinalities.  

        I don't know if there's a max cardinality out there.  That's a good question, one I can't recall learning, and I'm not sure how you would prove it.

    •  It is all based on the concept of time... (3+ / 0-)
      Recommended by:
      side pocket, Bluesee, Vericima

      As Socrates believed, time is simply the measurement of motion.  Time as a measurement is a completely different animal than time as a philosophical concept of history.  To go back in time is like measuring the distance a car travels in reverse.  There is nothing mystic about it; it is distance either way.  

      Zeno's Paradox used to keep me up at night.  I would lay in bed for hours trying to solve the problem.  I do not believe that set theory adaquately explains it.  The fundamental issue is that our measurement of motion is entirely arbitrary comparisson of another moving object (the earth in relative to the sun, the hand of a watch to the numbers on the dial, etc...) and therefor easy to indescriminately assign values of infinity to them.  Understanding the relativistic limitations to an arbitrary measurement system helps to see why Zeno's paradox is really an argument about the limitations of our made up system.  I believe that motion is actually an additional dimension that works similar to a one plank second per frame shutter speed (but that is really neither here nor there).

      Set theory has it's limitations.  I think that it kind of throws its hands up and says that since the concept of infinity is really hard, if a number gets infinitely close to another number then it may as well be considered that number.  I have always been troubled by the concept that 0.9 ad infinitum is supposedly equal to 1.  I have seen the proofs and they make sense (and throughout my life, I have actually gone back and forth on the concept in somewhat of an internal argument) but have never completely accepted it.  I have always liked the idea that the first positive real number is equal to one minus zero point nine ad infinitum.

      "Perhaps the sentiments contained in the following pages, are not YET sufficiently fashionable to procure them general favour..."

      by Buckeye Nut Schell on Tue Mar 25, 2014 at 07:57:36 AM PDT

      [ Parent ]

      •  For Zeno's paradox, use calculus. (2+ / 0-)

        Splitting something into infinitely tiny pieces seems to me to be the bread and butter of calculus.

        (Did you mean Planck?)

        I am become Man, the destroyer of worlds

        by tle on Tue Mar 25, 2014 at 09:14:46 AM PDT

        [ Parent ]

        •  You don’t even really need calculus: (1+ / 0-)
          Recommended by:
          Buckeye Nut Schell

          all you need is a rigorous notion of the sum of a convergent infinite series.

          •  Implicit in that is that an infinite series (1+ / 0-)
            Recommended by:
            Buckeye Nut Schell

            CAN have a sum.  There are implicit assumptions all over the place, you know.

            I remember in a math class, a teacher was explaining the induction step part of induction proofs.  An induction proof goes like this:

            1. Prove that it's true for F(1).

            2. (The induction step) Prove that if something is true for F(X), then it must be true for F(X+1).

            3. If these two things are true, then it's true for all F(n).

            You can use that to prove all kinds of dumb things, like that any two even numbers added together gives you an even result.

            HOWEVER, one student was having trouble understanding it, and the teacher pointed to the board, and said, you get step 1, right?  Yes.  Do you accept step 2?  And she said no.

            NO????  The teacher paused for a moment and said, you know, that's brilliant in a way.  He wasn't mocking her.  The induction step is very useful, but it's one of the rules of the game that we invented, and a different type of math (probably less useful or interesting) would arise if you chucked it out.  It's not inescapably true on its merits.  

            The induction step is necessary to proving the result of many infinite series problems.  Chuck it out the window, and any result is debatable.

            •  It’s not an assumption at all: (0+ / 0-)

              we define what we mean by the sum of a convergent series, and then prove that this definition gives the sum the properties that we expect it to have.

              I suspect that you’ve misremembered what occurred in the discussion of mathematical induction, since in the abstract there is nothing to accept or not to accept in clauses (1) and (2): the only question is whether clause 3 actually follows from clauses (1) and (2), and this is is easily proved in any of the formalizations of the relevant mathematics.  There is nothing there that can be thrown out the window, unless of course you want to abandon logic altogether.

              In the case of a specific proof by induction one might accept or fail to accept that (1) or (2) had actually been demonstrated, but that’s a different matter altogether.  If the incident occurred as you’ve reported it, dealing with the nature of inductive proofs in general, the teacher was (a) wrong, because the student’s response makes no sense, and (b) not very good, since his questions also make no sense.

              •  It can't be proven, though. (1+ / 0-)
                Recommended by:
                Buckeye Nut Schell

                You are correct.  It's not the induction step but the conclusion that is in question here.  Ultimately, we can prove it if we fall back on definitions that force the conclusion, such as the definition of the natural numbers as "naturally ordered" set.  But what if we have different rules about the natural number set?  We assume that n+1 is always similar to n going out to infinity because that's the way things have been defined.  A different definition would give different results.  

                For instance, what if their is a maximum number in the N series that is finite but undefinable?  

                In a way, that is the way most people deal with the difficulties of the concept of infinity in their head.  They translate it into some fuzzy maximum number way out there in the woowoo.  No wonder somebody comes along then, like that woman in that class, and says she's not sure about all this induction business.

                The natural number set is defined using infinity.  You can't PROVE that it's infinite -- it's just a pointless tautology.  Another math system, messier but plausible, could be based on the idea of an undefinable maximum.  This would prohibit the induction step from even being attempted.  Just like how the rules of division require you to always add the caveat, divisor n <> 0.

                •  The nature of the set of natural numbers (1+ / 0-)
                  Recommended by:
                  Buckeye Nut Schell

                  really isn’t contentious: it’s no accident that the various formalizations all reflect the same underlying intuition.  If you define something else, you’re no longer talking about the natural numbers.

                  You also write:

                  In a way, that is the way most people deal with the difficulties of the concept of infinity in their head.  They translate it into some fuzzy maximum number way out there in the woowoo.  
                  I have no idea whether this is true, even after some 40 years of teaching the subject.  It doesn’t really matter, save insofar as it might affect how one goes about trying to teach these ideas.
                  No wonder somebody comes along then, like that woman in that class, and says she's not sure about all this induction business.
                  While many people certainly do have trouble understanding mathematical induction, it does not in general seem to result from the picture that you suggest.  The underlying problem is usually failure to understand the logic, relatively simple though it is, not failure to understand the underlying model of the natural numbers.  And even when it does involve the infinite, it’s not usually a problem with the picture of the natural numbers: it’s an unintended consequence of the all-too-common string of dominos metaphor, which encourages students to think of a proof by induction as an infinite process rather than the once-and-for-all argument that it actually is.
                  Another math system, messier but plausible, could be based on the idea of an undefinable maximum.
                  This is by no means clear even without but plausible, which in my view is simply false.  It would be difficult, if not impossible, to formalize such a notion of the natural numbers at all, let alone in a way that would agree with the usual notion in all areas of practical interest – and if you don’t have that agreement, you can’t claim to be talking about the natural numbers.
                  •  Formalizing it... (1+ / 0-)
                    Recommended by:
                    Buckeye Nut Schell

                    A look at the Wikipedia entry on infinity lead to this:


                    Main idea

                    The main idea of finististic mathematics is not accepting the existence of infinite objects like infinite sets. While all natural numbers are accepted as existing, the set of all natural numbers is not considered to exist as a mathematical object. Therefore quantification over infinite domains is not considered meaningful. The mathematical theory often associated with finitism is Thoralf Skolem's primitive recursive arithmetic.

                    In the entry on ultrafinitism, it says this, which goes to the point I was making about a tautology...
                    Edward Nelson criticizes the classical conception of natural numbers because of the circularity of its definition. In classical mathematics the natural numbers are defined as 0 and numbers obtained by the iterative applications of the successor function to 0. But the concept of natural number is already assumed for the iteration. In other words, to obtain a number like 2^^^6 one needs to perform the successor function iteratively, in fact exactly 2^^^6 times to 0.
                    I'm not arguing against set theory or the natural number set as it's used in abstract algebra.  I'm just pointing out that there are assumptions about infinity built into it.  

                    A good question, one I wanted to ask, and you're an expert (I'm just an old man with a BS in computer engineering) is this: How would you explain the concept of infinity in a useful way without referring in some way to the natural number set?  If you try to explain it through analogies involving counting, then you have to define counting in a way that doesn't itself implicitly depend on the definition of infinity.  Again, I'm not trying to deny the existence of infinity -- just pointing out that I have always been struck by the circularity of this.  Can you define counting without first defining infinity, and can you define infinity without first defining counting?

        •  Yes, yes I did mean Planck... (1+ / 0-)
          Recommended by:

          I do mathmatics better than I do spelling...

          However, calculus only solves Zeno's stadium paradox (or any of his others) if you accept the premise that the sum of an infinite geometric series is equal to a finite, rational number.  For example that 0.9 ad infinitum is equal to one or that 1+1/2+1/4+1/8...=2.  Although for all practical mathmatics that may be true, for non-commutative algebraic geometry, I believe that it does not necessarily equal a rational number.  Strictly my opinion.

          Zeno's stadium Paradox is one of those things I have thought about for 35 years and sometimes I think I understand it and then a new thought comes to mind or I read something new and apply it to the manner in which I thought I understood it and then I change my opinion again and relook at the puzzle with a new respect.  It is not an easy thing to dismiss.  The more you think about it, the more complicated it becomes.  I always liked Diagonese's solution... He just got up and walked out and by doing so, proved (as far as he was concerned) that motion was indeed possible and therefore Zeno was full of crap.

          "Perhaps the sentiments contained in the following pages, are not YET sufficiently fashionable to procure them general favour..."

          by Buckeye Nut Schell on Tue Mar 25, 2014 at 12:20:47 PM PDT

          [ Parent ]

      •  Plato gave Socrates seven standards (0+ / 0-)

        Chaos, Mythos, Eros, Holos, Logos, Chronos, Cosmos or
        in English, Similarity, Difference, Motion, Rest, Number, Sequence and Consequence,

        All are paired opposites which in their totality are Becoming a Consequence, which is Being their essence.

        If you go back and look at the Egyptian concept of Ma'at which is often described as the doing of what is right and proper, the important thing about it is thinking in the process terms of Becoming (Eros or Motion).

        All the numerical analysis, the proofs the Greeks loved, measuring, weighing and judging things in an orderly manner were ideally sequential according to standards proceeding through the proof with all the steps in the right order rather than simply quantum jumping to what is and was and always will be the Platonic ideal, the ideal Consequence.

        Live Free or Die --- Investigate, Incarcerate

        by rktect on Tue Mar 25, 2014 at 05:33:48 PM PDT

        [ Parent ]

    •  Maybe God got bored (1+ / 0-)
      Recommended by:

      so he picked up his guitar and started playing - AKA string theory.

      "When I see I am nothing, that is wisdom. When I see I am everything, that is love. My life is a movement between these two." - Nisargadatta Maharaj.

      by mkor7 on Tue Mar 25, 2014 at 08:09:09 AM PDT

      [ Parent ]

      •  I dreamed once that it was a harp (2+ / 0-)
        Recommended by:
        Vericima, mkor7

        so vast its strings stretched from one end of eternity to the other, and encompassed the length and breadth of the universe. And the strings vibrated to produce a grand symphony of sound, but I couldn't see a player. It was awesome.

        Then I saw a flying 'ship' crossing a set of the strings, dragging a silver thread behind it. I tried to move out of its way, but a great huge batten caught me in the string as it swept across at a right angle.

        Turned out it wasn't a harp at all, or even a guitar. It was a loom, and the fabric was history - time. Into which tapestry all our lives are woven.

        A matter, it seems, of perspective. Relatively speaking... §;o)

        There are three kinds of men. The one that learns by reading. The few who learn by observation. The rest of them have to pee on the electric fence for themselves. - Will Rogers

        by Joieau on Tue Mar 25, 2014 at 12:46:55 PM PDT

        [ Parent ]

        •  Very cool (1+ / 0-)
          Recommended by:
          Into which tapestry all our lives are woven.
          The Great Mandala.  Peter Paul & Mary I believe.

          "When I see I am nothing, that is wisdom. When I see I am everything, that is love. My life is a movement between these two." - Nisargadatta Maharaj.

          by mkor7 on Wed Mar 26, 2014 at 06:40:20 AM PDT

          [ Parent ]

    •  To mangle a phrase, (0+ / 0-)

      There is no thus there.

      Not that the antirational folks would get it, any more than they would Gertrude Stein.

      I am become Man, the destroyer of worlds

      by tle on Tue Mar 25, 2014 at 09:01:54 AM PDT

      [ Parent ]

    •  I think that the kalam cosmological argument (0+ / 0-)

      is a somewhat good one, and "reasonable."  The problem is that it has no bearing at all on the authority of the Bible, which can just be seen as a cultural product like any other.

  •  You've just highlighted a larger problem (13+ / 0-)

    with the conservative publishing niche.  That is its unhealthy, sporadic dismissal of anything the editors and writers didn't pick up in their formative years.  

    We don't know specifically why A Beka Book, the PCC tax vehicle playing the heavy in this piece, takes a silly swipe at "modern theories" in mathematics.  Haven't even read the materials, it's quite possible the text doesn't live up to the brochure.  We don't even know if the editor who wrote this blurb knows what "set theory" is. And wingnuts aren't of one mind on the matter; BJU Press at least thinks set theory has a place in K-12 education.

    Bottom line, this is likely unconnected to creationism but instead a symptom of broader conservative anti-intellectualism.

    •  I was exposed to the rudiments in grade school (12+ / 0-)

      That tiny introduction to set theory eased the shock of Serge Lang's leaden exposition when I dove into topology in grad school.

      You never know when your education is going to pay off, which is why the fundies are so paranoid in choking off things that might be dangerous one day.

      o caminho d'ouro, uma pinga de mel: Parati

      by tarkangi on Mon Mar 24, 2014 at 03:35:38 PM PDT

      [ Parent ]

      •  KLEIN BOTTLES, BITCHEZZ!!!!! (1+ / 0-)
        Recommended by:
      •  OMG Lang (3+ / 0-)
        Recommended by:
        tarkangi, JerryNA, ferg


        Abstract algebra can have a real beauty and elegance. I'm sure Lang loves it for that, but his writing, lord, its like trying to read a brick.

        "What could BPossibly go wrong??" -RLMiller "God is just pretend." - eru

        by nosleep4u on Tue Mar 25, 2014 at 06:43:30 AM PDT

        [ Parent ]

        •  Clear as mud (2+ / 0-)
          Recommended by:
          JerryNA, Nowhere Man

          That's how we put it.

          What I still don't understand is why the professor thought we might learn something from these books.

          And if I may be allowed a digression, this is why I am skeptical of MOOCs.  Learning is not about gathering the pearls that drip from the mouth of some lecturer.  Lectures are important, reading is important, homework is important, but discussions are really really important - with the instructor as well as with your fellow students, and I find that internet discussions are a pale imitation of the real thing.

          Elsewhere in this thread someone makes a brilliant comment about Goedel.  1) it's the kind of insightful off the cuff remark that I love and 2) I can love it only because I came adequately prepared.

          o caminho d'ouro, uma pinga de mel: Parati

          by tarkangi on Tue Mar 25, 2014 at 07:54:25 AM PDT

          [ Parent ]

      •  See my post below. I learned sets early (1+ / 0-)
        Recommended by:

        and now at 60 am doing something unique with it.

        "You can die for Freedom, you just can't exercise it"

        by shmuelman on Tue Mar 25, 2014 at 09:55:30 AM PDT

        [ Parent ]

      •  Set Theory - Grad School Concepts For Grade School (0+ / 0-)
        That tiny introduction to set theory eased the shock of Serge Lang's leaden exposition when I dove into topology in grad school.

        You never know when your education is going to pay off, which is why the fundies are so paranoid in choking off things that might be dangerous one day.

        That's my beef with the idea of teaching set theory - it's something that most people will never hear about again unless they go to grad school.  For kids, this time would be better spend on the fundamentals.

        Men are so necessarily mad, that not to be mad would amount to another form of madness. -Pascal

        by bernardpliers on Wed Mar 26, 2014 at 08:20:19 AM PDT

        [ Parent ]

  •  I would have assumed "new math" was their (17+ / 0-)

    problem with set theory. I'm not doubting this diary, but it seems an awfully specific reason to not like set theory.

    As a note: this concept of sizes of infinity is the basis of Godel's proof that there are more "true" statements than proofs (where "true" and "proof" are specific mathematical concepts.)

  •  I think it's simpler than that (20+ / 0-)

    Set theory is emblematic of "New Math."  Both New Math and the BSCS biology curriculum, which explicitly adopted the teaching of evolution, have been right-wing targets since they were put forth in the late 1950s to jumpstart American science and math education after the shock of the USSR's Sputnik launch.

    "Well, I'm sure I'd feel much worse if I weren't under such heavy sedation..."--David St. Hubbins

    by Old Left Good Left on Mon Mar 24, 2014 at 01:19:36 PM PDT

    •  I'd expect those programs (3+ / 0-)
      Recommended by:
      Ojibwa, Aunt Pat, rduran

      to have rubbed off on them, then - or else they never actually got any of those in their classes.
      (I wish someone would reprint those early-60s 'new math' texts, the ones that were softcover and looked like they were typed. I'd find a way to buy a set.)

      (Is it time for the pitchforks and torches yet?)

      by PJEvans on Mon Mar 24, 2014 at 02:24:22 PM PDT

      [ Parent ]

    •  I think it's even simpler (14+ / 0-)

      I think it's because they, and the parents their schools are aimed at, don't understand it. Any of it. At all.

      And since they don't understand it, and don't even know where to start to understand it, they would have to say 'I don't know' if their kids asked for help with something.

      This is a bunch that absolutely LOATHES having to admit they don't know something, no matter what it is. So if the school doesn't teach it, and actively seeks to undermine teaching anything about it, that means the parents feel 'safe' when the kid asks questions, because what's being taught is pretty much only things they know or understand about.

      It's not about the kids. It's about the PARENTS.

      •  You could probably count the number of people who (2+ / 0-)
        Recommended by:
        radarlady, J M F

        understand it--any of it--within a day.  That's pretty unfortunate, since it's so damned useful.  That said, New Math was a bad idea, but I don't think math educators have taken any empirical lessons from the experience.  Which is why the standards are all over the map where it concerns topics like sets--ranging from never touching it to simple things like unions and intersections to set builder notation.

        •  The so-called New Math wasn’t in itself (13+ / 0-)

          a bad idea: the point of it, after all, was to emphasize understanding rather than rote manipulation.  The SMSG (School Mathematics Study Group) texts, which covered K-12 and then some, were quite good, but they were never published commercially.  (The Illinois Project texts, which covered only grades 9-12, were in my opinion less good: they were a bit on the fussy side.)

          There were two real problems.  One is clearly described by Ralph A. Raimi:

          Despite all the furor, the “new math” never ran very deep in American schools, and most textbooks either ignored it or included its ideas superficially or in garbled form.   For a few years most commercial textbooks touted themselves as "new math", which they seldom were, and then they backed off.
          The other is that the vast majority of teachers were not themselves prepared to teach the concepts.  Summer institutes helped, but they just scratched the surface.  (I’m actually even more familiar with this phenomenon in chemistry: my father spent 1960-1962 working with Larry Strong on the Chemical Bond Approach Project, partly contributing to the text proper, but mostly teaching high school teachers in summer institutes and travelling round the country helping them during the school year.)

          A secondary problem was that too many people went too far and forgot that one needs a certain amount of mechanical proficiency both for practical reasons and to provide a solid basis for the concepts.

          •  And there was no orientation for the parents (eom) (2+ / 0-)
            Recommended by:
            itsjim, BMScott
          •  ah SMSG (1+ / 0-)
            Recommended by:

            I remember being taught SMSG in grade school in Delaware.  ( In the 1960s)   Originally we were issued loseleaf pamphlets with 3 ring binder holes.  the second and third years we received soft cover books.  The had video cameras inthe room ( High tech!) to watch us learn.

            As my father used to say,"We have the best government money can buy."

            by BPARTR on Tue Mar 25, 2014 at 06:42:49 AM PDT

            [ Parent ]

            •  Years ago I had the complete set (1+ / 0-)
              Recommended by:
              Brown Thrasher

              of SMSG books with the monochromatic soft covers, but at some point they disappeared.  If I remember correctly, I even had the teacher’s manuals.  As I recall, my father was able to get them thanks to his connection with the CBA Project.  There really were some good ideas in those books.

    •  Not simply conservative targets (3+ / 0-)
      Recommended by:
      radarlady, J M F, tarkangi

      New Math had a wide range of opposition (Tom Lehrer's no right wing nut), which is why it never made it out of the 1950s.  Conservatives, being conservatives, love holding a grudge and are especially animated by government programs that actually manage to die.

  •  Minor correction (45+ / 0-)

    Not that it's important, but as a mathematician specializing in formal logic I can't let this go by:

    Because of [the difference between the infinity of the integers and that of the real numbers] the infinity of real numbers is not denoted by the familiar "sideways 8" character, but by the hebrew letter "Aleph".
    This is not true. The transfinite cardinals (informally, the "infinite numbers") start with Aleph-null, written as an aleph with subscript zero. Aleph-null is exactly the infinity of the integers. The infinity of the real numbers is some larger transfinite cardinal: aleph-one, or aleph-two, or some other aleph.

    (Well, which one is it? This is a complex question. The "continuum hypothesis" says that the infinity of the real numbers is exactly aleph-one, that is, there is no infinity bigger than the integers and smaller than the reals. Cantor tried to prove this for years and it literally drove him insane. In fact, it can't be proved from the standard axioms of mathematics.)

    The eight-on-the-side symbol for infinity is not an infinite number like the alephs are, and it's generally not used in transfinite mathematics. It's used only in expressions like "N -> infinity" meaning that N grows without bound.

    Sorry for the pedantry, but that's what I'm all about. Other than this particular sentence, I'm fully in agreement with the diarist. It frightens me when people use the Heisenberg principle to prove that telepathy is possible.

    •  I love pedantry ... (19+ / 0-)

      ... when posited by people who actually know their shit.  So thanks for the comment!

      Another W. L. Craig Youtube lecture I saw that bugged me (thanks to principles first encountered in "Gödel, Escher, Bach") included a comment to the general effect that "Materialist Science demands that everything should be observable and demonstrable, but they do not include this proposition in that list!  They don't demonstrate a proof that everything should be observable!"  

      To which I got really angry, thinking to myself: This guy is basically demanding that science should be an formal system that contains its own axioms!  I may not be an expert in formal logic, but I remember Kurt Gödel proved that no axiomatic system can be both complete AND self-consistent.  It was in all the major papers (and by that I mean science publications in the early 20th century)!  But here is Craig essentially demanding something that is proved to be impossible!  And what's more, as a formally trained philosopher, he has to know this already!  So here he is putting one over on the audience, making himself sound really smart and getting over on those uppity scientists, only because most of his audience doesn't really understand what he's asking!

      I'm really interested in what people who really know more about this stuff than I do have to say about Craig.

      ----------------------- "Zu jeder Zeit, an jedem Ort, bleibt das Tun der Menschen das gleiche..."

      by Rheinhard on Mon Mar 24, 2014 at 01:57:38 PM PDT

      [ Parent ]

      •  I'm not sure I'm following you (0+ / 0-)

        or agree with you.  The statement "to the general effect that "Materialist Science demands that everything should be observable and demonstrable," is pretty much true, although phrased poorly there.  

        Science, like set theory, is a game with its set of rules and assumptions, and one of those assumptions is the existence of an objective material universe.  It's monistic in this sense -- it can't operate or discuss things that are outside this domain.  For instance, a question like "Is Sarah Palin an obnoxious, arrogant twit?" would seem to most of us to have an obvious true answer, but science can't assess it.  The question of whether Darth Vader is Luke's father is one we all know the answer to, but science has no answer for it, because Darth Vader and Luke are fictional characters.  These things are just outside the domain of things science can discuss.

        It doesn't mean science is wrong.  It just means that science itself offers a limited perspective on reality.  Backward engineering from the limitations of science's intentionally limited metaphysics to a suggestion that that's all there is... that IS like a cardinality dispute.

    •  Godel & Cohen proved that the Continuum Hypothesis (9+ / 0-)

      is consistent with, but not provable from, the standard axioms of set theory.  In other words, it is impossible from the standard axioms to construct or describe a set larger than Aleph-null, but smaller than C, the cardinality of the continuum.  (Godel)

      However, if you merely assume there is such an intermediate-size infinite set, Cohen proved you can never demonstrate a contradiction with the standard axioms.  

      So either the Continuum Hypothesis (CH) or its negation (~CH) is consistent with the other axioms.  

      If you don't believe that axioms come from God, or the FSM, it is a matter of taste and utility whether or not to add CH to the standard axioms.

      There's no such thing as a free market!

      by Albanius on Mon Mar 24, 2014 at 08:35:17 PM PDT

      [ Parent ]

      •  I think *that* is their problem with Set Theory (1+ / 0-)
        Recommended by:
        So either the Continuum Hypothesis (CH) or its negation (~CH) is consistent with the other axioms.  

        If you don't believe that axioms come from God, or the FSM, it is a matter of taste and utility whether or not to add CH to the standard axioms.

        You can choose different sets of axioms, each producing different results; yet all results are self-consistent, and none are obviously superior to the others.

        In other words: In Set Theory, truth is relative.

        (Let's not tell them about non-Euclidean geometries, shall we?)

        Let us all have the strength to see the humanity in our enemies, and the courage to let them see the humanity in ourselves.

        by Nowhere Man on Tue Mar 25, 2014 at 11:47:29 AM PDT

        [ Parent ]

    •  a correction to your correction (1+ / 0-)
      Recommended by:
      J M F

      [ain't pedantry grand?]

      The omegas and alephs are used to denote two very different concepts.

      The omegas (sidewise eights) refer to ordinals, which are based on the notion of successor.   In this realm, you can prove that there is no X such that X + 1 = omega, but you can also prove that omega < omega + 1.    You can't find the integer just before omega, but there is one just after it, and it is larger.  In fact, you can go on to omega + omega, omega * omega, etc.

      The alephs refer to cardinals, as you note, which are based on the notion of size.  In this realm, it's true that that size of the integers with one of them removed is the same as the size of all integers.  It's also true that aleph0 + 1 = aleph0, since the added number can be matched to 0, and all other matches can be shifted to accommodate.

      There are an infinite number of both omegas and alephs. In general each aleph is the size of an infinite number of different omegas, which is sensible since the alephs are effectively a quotient set over the omegas, where the equivalence relation looks for one-to-one matching and ignores ordering.

      On a meta level, this distinction between ordinals and cardinals is emblematic of many advances in math and science where two conflated concepts had to be painstakingly teased apart:  energy and momentum, force and power, heat and temperature, etc.

      •  This is not entirely correct insofar as (0+ / 0-)

        it concerns actual usage.  It’s true that the alephs almost always denote cardinals, but many of us use the omegas for both ordinals and cardinals and rarely use the alephs at all – especially those of us who automatically think in terms of von Neumann cardinals.

        Of course the distinction between ordinals and cardinals itself is indeed crucial.

      •  omegas aren't sideways eights (0+ / 0-)

        they are just greek letters.

      •  about this... (0+ / 0-)

        "The omegas (sidewise eights) refer to ordinals, which are based on the notion of successor.   In this realm, you can prove that there is no X such that X + 1 = omega, but you can also prove that omega < omega + 1.    You can't find the integer just before omega, but there is one just after it, and it is larger.  In fact, you can go on to omega + omega, omega * omega, etc."

        I should start by thanking you for adding to the discussion.  You are correct regarding the cardinals.  Cardinals are a way to put an order on sets based on whether there is a mapping from one set into another set.  Cardinal numbers only care about the size of a set.

        But ordinals don't concern themselves solely with the successor operator, but with the general concept of well-ordering.  The successor operator alone will not get you to omega + 1.  Omega is the ordinality of the natural numbers.  One gets omega + 1 by considering the ordinality of the set of N (union) {infinity}, where the usual order on N (the natural numbers) is augmented by declaring n < (infinity) for all natural numbers N.  

        Ordinals aren't distinct from cardinals so much as they are a refinement of them.  

        •  I didn't quite follow (0+ / 0-)

          but 'omega + 1' is exactly 'successor(omega)'

          You can't get to omega with the successor operation (you need to take a limit to do that).

          But once you have omega, it is entirely straightforward to pick up again with more successors.

          And ordinals are quite distinct from cardinals, in much the same way that 1/1 and 1.0 are distinct from 1 -- they exist at different levels.

    •  thanks for the contribution (0+ / 0-)

      As you say, "Aleph" is not a single cardinal number.
      Aleph_0 is the smallest of the infinite cardinal numbers, the cardinality of the integers.  

      Usually logicians use c to denote the cardinality of the real numbers, and the continuum hypothesis  is Aleph_1 = c.

      This is true "In fact, it [CH]can't be proved from the standard axioms of mathematics."

      But there's more!  CH isn't merely unprovable, it's independent of the basic axioms of set theory.  By which I mean that one can add an extra axiom to the basic set of axioms to prove CH is true, but one can also add an extra axiom that proves that CH is false.  CH is literally beyond the scope of the usual axioms of set theory.

  •  Some corrections: (17+ / 0-)

    David Hilbert was German, not English; he was born either in Königsberg or in Wehlau (now Znamensk), and he taught first at the University of Königsberg (1886-95) and then at the University of Göttingen (1895-1943).

    The usual notations for the cardinality of the set of real numbers are 𝔠 (lower-case fraktur c), 2אo, and 2ω.  The Hebrew letter א by itself isn’t generally used for any specific cardinality.

    Constructions like that of Hilbert’s hotel are possible with every infinite cardinality, not just with אo = ωo, the cardinality of the non-negative integers; thus, the existence of higher cardinalities has nothing to do with Craig’s ‘argument’.  I use scare quotes because it actually just boils down to a statement of personal incredulity.  (The brief discussion here may be helpful in this connection.)

    (By the way, I’m a set-theoretic topologist, so this is very familiar ground.)

  •  Actually (14+ / 0-)

    Pie really equals six if you cut it up right.

    -- GW Bush

    Best Scientist Ever Predicts Bacon Will Be Element 119 On The Periodic Table

    by dov12348 on Mon Mar 24, 2014 at 02:07:54 PM PDT

  •  Shazam! (2+ / 0-)
    Recommended by:
    Ojibwa, i saw an old tree today

    You can't make this stuff up.

    by David54 on Mon Mar 24, 2014 at 02:12:44 PM PDT

  •  their argument fails so many ways (4+ / 0-)

    That the Hilbert Hotel cannot exist in actuality (spacial infinity) does not disprove the possibility of a temporal infinity.

    If we reject the possibility of a temporal infinity, that still does not say anything about the nature of the "first cause".  

  •  Here's what A Beka Book covers (3+ / 0-)

    Grade 7 and up:

  •  Thanks Rheinhard for a very interesting diary. (2+ / 0-)

    But if the cause be not good, the king himself hath a heavy reckoning to make, ... there are few die well that die in a battle; ... Now, if these men do not die well, it will be a black matter for the king that led them to it; — Shakespeare, ‘Henry V’

    by dewtx on Mon Mar 24, 2014 at 05:16:06 PM PDT

  •  Math and religion - dangerous stuff (12+ / 0-)

    The third episode of COSMOS mentioned that Isaac Newton put a tremendous effort into comparing different versions of the Bible in order to elucidate hidden wisdom. He also dabbled in alchemy. It's ironic that the man who gave us the "Clockwork Universe" was into what can only be called mysticism.

    In the case of Craig, it sounds like he's constructed an elaborate mathematical scaffolding to hold up his world views, albeit one with structural flaws. He's reversed things a bit - conclusion first, evidence 'hammered to fit'. Not all that different I suppose from the old medieval arguments over how many angels could dance on the head of a pin. It's the kind of intellectual rigor that demands heretics be burned - it leaves no room for differences of opinion because Craig's opinions are never in doubt. He's proved them mathematically.

    I am reminded the late Robert A. Heinlein, who had a high opinion of mathematics, nonetheless made a point in at least one of his juvenile fiction novels about the content of mathematics.

    IIRC, it was this - mathematics has none. It's possible to devise a system of internally consistent mathematics that works very well to model the world - but that doesn't mean the world is bound by it. Newtonian mathematics seemed to be perfectly adequate to describe the universe - until we got to the point where we could start detecting relativistic effects.

    Craig's determination of the infinite nature of God is self defeating - because he's devised an understanding of God that purports He can be bound within the limits of the human mind.

    "No special skill, no standard attitude, no technology, and no organization - no matter how valuable - can safely replace thought itself."

    by xaxnar on Mon Mar 24, 2014 at 05:33:44 PM PDT

    •  Right you are (and so was Heinlein). (4+ / 0-)
      Recommended by:
      itsjim, JerryNA, xaxnar, Nowhere Man

      "Mathematics is ontologically neutral."

      "...logic and mathematics are formal sciences [which is to say] that they have no ontological commitment, i.e., that they do not assume the existence of any real entities.  In other words, logic and mathematics, and a fortiori metalogic and metamathematics, are not about concrete things but about constructs:  predicates, propositions, and theories."

      --Mario Bunge, Moderate Mathematical Fictionism, Scientific Realism, Prometheus Books, New York, 2001, p. 190

      This is a good example of why religionists like Craig should be very careful when treading onto the grounds of logic and mathematics. Craig assumes the real (as opposed to strictly conceptual) existence of an entity he calls "God", and then finds some esoteric mathematics to back up his religious claim.  Had Craig claimed to believe in the existence of God only after studying set theory, well then, maybe we would listen to whether he had something to say.  But he didn't.  He assumed God to be real (based solely on religious faith, of course) to begin with, and then went looking for facts that supported his hypothesis.  BAD SCIENCE!!  Good science works the other way around:  you construct a hypothesis that supports the facts, and when facts falsify your hypothesis, you modify or abandon the hypothesis.  Good science does not permit you to cling to your sacred beliefs and then go fact-shopping to shore them up.

  •  if your a female Mathematician and you encounter (10+ / 0-)

    a crowd of these guys on your chariot ride home from the Library, don't try to broaden their perspective by engaging them in a Socratic dialog, don't try to help them shed their irrational delusions don't try to point out where their arguments fall apart at the seems.

    Especially if they're carrying oyster shells.

    They won't care that you invented the plane astrolabe, the graduated brass hydrometer and the hydroscope.

    They'll just want to rip the flesh off your bones with those oyster shells to please their God.

  •  It's pretty much a schtick (3+ / 0-)
    Recommended by:
    Back In Blue, JerryNA, itsjim

    It's more like he's giving separate speeches.  You can go back and find him repeating nearly word for word the same arguments twenty years ago.  

    It still boils down to the idea that he knows because he knows.  


    by otto on Mon Mar 24, 2014 at 06:50:54 PM PDT

  •  Just another your're wrong so I'm right argument? (2+ / 0-)
    Recommended by:
    funningforrest, JerryNA

    Is this not just that?  I've been hearing these all my life from various people trying to assert their beliefs on others.  Basically, your science can't prove what you say so it is unknowable and therefore proves that God must exist.

    It proves nothing. And there's far more proof that what is now "unknowable" someday will won't be.

    Pure hucksterism.

    America, where a rising tide lifts all boats! Unless you don't have a boat...uh...then it lifts all who can swim! Er, if you can't swim? SHAME ON YOU!

    by Back In Blue on Mon Mar 24, 2014 at 07:39:23 PM PDT

  •  Can God make a number so big, he cannot count it? (7+ / 0-)

    I found this, from someone brought up with a fundamentalist education, trying to explain it as more of a culture and values objection.

    Set theory, particularly the stuff about infinity, has a bit of that wibbly-wobbly, timey-wimey flavor to it. It doesn't make sense on the level of "common sense". It's dealing with things that aren't standard, simple numbers. It makes links between nice, factual math and floppy, subjective philosophy. If you're raised in Christian fundamentalist culture, all of that—every last bit—absolutely reeks of modernism. It's easy to see how somebody at A Beka would look at set theory and conclude that it's really just modernist propaganda. To them, set theory is just a step on the road to godless atheism.

    What do Christian fundamentalists have against set theory?

    •  that 's omega, in some sense (1+ / 0-)
      Recommended by:

      The successor function is the basis of counting:  the successor of 0 is 1, the successor of 1 is 2, etc.

      Using 0 and successor, you can define all the natural numbers (non-negative integers) simply by counting through them.

      Omega is the set of all such natural numbers, and can also be viewed as the (unreachable) limit of repeated applications of successor -- it's the first number you can't reach by just adding one to some other number.   Once you add omega, though, you are free to add one to that, and one to that, etc..  2 * omega (i.e., omega + omega) is then the first number you can't reach in that new series.  And so on, forever.

      Just as omega is a limit beyond the natural numbers reachable via successor, you can define larger and larger limits of of numbers that are beyond the series of numbers reached though other operations such as powersets.

      If you carry this far enough, it leads you to inaccessible cardinals, which are infinities so large that you can't reach them through any sequence of operations on smaller infinities.

      As with the axiom of choice and the continuum hypothesis, you can have consistent theories with or without the existence of these insanely large numbers.  

    •  This is right up there in my book with --who would (2+ / 0-)
      Recommended by:
      JerryNA, itsjim

      win--Batman or Superman.

      I don't really want to have to care.

      "It were a thousand times better for the land if all Witches, but especially the blessing Witch, might suffer death." qtd by Ehrenreich & English. For Her Own Good, Two Centuries of Expert's Advice to Women pp 40

      by GreenMother on Tue Mar 25, 2014 at 08:22:37 AM PDT

      [ Parent ]

    •  George Carlin as set theorist (0+ / 0-)

      "Hey Fadda!  If God is all powerful can he make a rock so heavy that he cannot lift it himself?"

  •  OH my. (3+ / 0-)
    Recommended by:
    Ahianne, Cassandra Waites, mkor7

    In the most basic lay language, if I may (hat tip to David Foster Wallace's Infinity and More (which is a must read)).

    Two basic kinds of infinity (forgive in infelicity of that statement please, and I'll leave out the footnotes). Stick with me here, I'm not Borges.

    There's the kind of infinity where there is no highest number.  Whatever the purported highest number might be, there is always another +1. That kind of infinity you imagine as infinitely growing.

    The other common infinity is the one that is between the integers. A person wants to cross the street by going half way across. Then half way across the remaining half. And half way across what remains. The crosser never gets to the other side because there's an infinity between the curbs, there's always a smaller slice of the street.

    I suspect that pervasiveness of infinity actually frightens some folks. That they'd like their god to be more finite.


  •  How I learned set theory (2+ / 0-)
    Recommended by:
    Ahianne, itsjim

    As an undergraduate math major I had little exposure to set theory and little interest. In 1964 I taught my first class, college algebra. I was not enthusiastic about set theory but it was in the course and I quickly picked it up. I had students who felt put upon by 'new math.' I thought of ways to make concrete examples of sets and their unions and intersections. I began to appreciate set theory as a tool.
    Set theory was not really new in 1964; it simply was new to public education. Its foundations had been made by Georg Cantor in the 19th century.
    When I talked with other teacher buddies I heard there was a lot of resistance to set theory by many teachers, who did not want to learn something new. I gather that 'new math' has atrophied in most schools, but I only taught on the university level.
    In one class I tried presenting infinite set theory. That did not work out well, even with the best students.

    Censorship is rogue government.

    by scott5js on Mon Mar 24, 2014 at 08:13:53 PM PDT

    •  Interesting. (1+ / 0-)
      Recommended by:

      I started first grade in 1964. For 12 years, all the math I learned was grounded in set theory. Then when my kids were in elementary and high school from about 1995 till 2005, I did not see much, if any reference to set theory in any of their math books. It was really apparent when I tried to help them understand the domain of functions. They had trouble grasping the concept because they had so little understanding of sets. I've always wondered why set theory had apparently fallen out of favor.

      So endith the trick.

      by itsjim on Tue Mar 25, 2014 at 07:00:17 AM PDT

      [ Parent ]

    •  Lack of co-operation, I think (1+ / 0-)
      Recommended by:

      It seems many old-time teachers did not care to learn set theory. They may have done so, but not well enough to make it interesting. So, after a few years, it came to be considered a failure.
      My response was different. I was reluctant at first, but then I started assimilating it, and then I learned about infinite sets. Not long ago I investigated various proofs that the Liouville numbers form an uncountable set.

      Censorship is rogue government.

      by scott5js on Tue Mar 25, 2014 at 07:10:10 AM PDT

      [ Parent ]

  •  The Kalam is rubbish. (3+ / 0-)

    It fails on every point.  If they think attacking set theory is going to help, they are mistaken.

  •  Wow, first time in the Community Spotlight! (5+ / 0-)
    Recommended by:
    eOz, Ahianne, otto, funningforrest, GreenMother

    Thanks for the comments, especially the folks who know more set theory than me!

    ----------------------- "Zu jeder Zeit, an jedem Ort, bleibt das Tun der Menschen das gleiche..."

    by Rheinhard on Mon Mar 24, 2014 at 08:25:41 PM PDT

    •  It's a good feeling, ain't it? (0+ / 0-)

      Some years back I hosted WYFP here on Daily Kos (under a different user name; I've been absent for awhile), and for a brief time I was on the Rec List.

      Made my day.  Made my year!

      Congrats, Rheinhard, and on such a cerebral subject no less.

  •  I don't think this explanation makes sense (1+ / 0-)
    Recommended by:

    As sophomoric as this cosmological argument is, the existence of higher-order infinities doesn't seem to contradict or challenge it at all.

    As far as I can tell, the gist of Craig's argument is, "Hilbert's hotel is absurd, and therefore infinity is not metaphysically possible."  If you point to higher-order infinities, what leg of that argument is broken?

    The existence of higher-order infinities does not overturn or contradict the Hilbert hotel paradox; whatever you can say about real numbers, it doesn't change the fact that the natural numbers can be placed in 1-to-1 correspondence with any infinite subset of the natural numbers.  Nor does any higher-order infinity alter Craig's (again, sophomoric) premise that the Hilbert hotel is "absurd."

    Indeed, there are similar and more complicated absurdities that are possible with the reals, so if anything it would just serve as more fuel for the "infinity is absurd" argument.  It doesn't seem like this would provide any reason for a creationist to target set theory.  I think it's more likely that they are targeting it as part of the "new math."

    Taking jokes seriously is the exact mirror activity of laughing if someone says they have cancer. --jbou

    by Caj on Mon Mar 24, 2014 at 09:04:02 PM PDT

  •  Douglas Hofstader's book would (1+ / 0-)
    Recommended by:

    really blow their minds.  Maybe we should put copies in hotel rooms, replacing Gideon's Bibles!

    Great book.

  •  One problem with this theory (0+ / 0-)

    The people who went after set theory would have no idea what you're talking about.  Although IIRC set theory shows up in new math, a well known communist plot from the 1960s.

    Now, I don't doubt someone in the fundie-space figured out the connection.  But you'll find that, at this point, none of the people who write the fundie curriculums have any idea about what any of this about.  I mean, these are the people who think that God pronounced the value of pi as 3.

  •  Good explanation (0+ / 0-)

    I, by the way, also have a serious problem with the "greater and lesser infinities," so it's not a sore spot only there. The truth is that this concept is a paradox. It serves, therefore, to either confirm that mathematics is an idealized language not tied to reality (but irrational numbers themselves should point at the same thing -- there is either a flaw with our math or with our circles, and it's much easier to say that we have no perfect circles than that the practicality of the math should be discarded) or that we humans run about with illegal categories in our heads (as Immanuel Kant said about both zero and infinity).

    In other words, finding problems in greater/lesser infinity isn't new, and it doesn't lead only where they want it to lead. Mathematics proposes things that cannot translate to physical reality. It's not "about" physical reality, and paradox is simply paradox.

    (Yes, I know. It's not a paradox mathematically. It is a paradox when you translate it to "peas" or "ducks.")

    "man, proud man,/ Drest in a little brief authority,. . . Plays such fantastic tricks before high heaven/ As make the angels weep; who, with our spleens,/ Would all themselves laugh mortal." -- Shakespeare, Measure for Measure II ii, 117-23

    by The Geogre on Tue Mar 25, 2014 at 03:46:41 AM PDT

  •  I don't get it. If you use normal limit theory (1+ / 0-)
    Recommended by:

    from calculus, you get the ratio of the size of the set of even integers to the size of the all integers = 1/2.  The 'infinity' of all integers is twice the size of the 'infinity' of all even integers.
         Set theory seems to say they are the same size because a 1:1 mapping can be generated.  But the 1:1 mapping can only be maintained if you extend the bounds of the even integer set along the number line at twice the rate of the bounds of all integers, thus the different answers from set theory and limit theory.
         So, I guess in set theory you can demonstrate infinities where 1:1 mappings cannot be created no matter how the bounds get extended, and that is the basis for 'higher' infinities...???

    Ok, maybe i get it.

    And we love to wear a badge, a uniform / And we love to fly a flag But I won't...let others live in hell / As we divide against each other And we fight amongst ourselves

    by ban48 on Tue Mar 25, 2014 at 03:47:11 AM PDT

    •  You’re looking at two different notions (0+ / 0-)

      of size.  One is cardinality: two sets A and B have the same cardinality if there is a one-to-one matching of their elements.  Thus, the set of positive integers has the same cardinality as the set of even positive integers, via the matching 1:2, 2:4, 3:6, ... .  This notion applies to all sets.

      The other, which is applicable specifically to sets of positive integers, is asymptotic (or natural) density.  The asymptotic density of the set of even positive integers is 1/2: if en is the number of even integers in the range [1, n], the ratio en/n approaches 1/2 as n increases indefinitely.  In other words, if n is a reasonably large positive integer, approximately half of the positive integers ≦ n are even.

      Both notions are useful, but they’re very distinct concepts.

  •  They see themselves at war . . . (2+ / 0-)
    Recommended by:
    JerryNA, happymisanthropy

    . . .with Satan. And anything you say to dispute them is just Satan rocking you like a sock puppet.

    Far more scary is that I have two different friends, both highly educated and formerly very nice people who have succumbed to this worldview. Your explanations were marvelous but it will never, ever reach these people. They have been infected with a malignant Jesus virus (not to be confused with people of faith that actually retain some measure of rationality).

  •  To quote/paraphrase (2+ / 0-)
    Recommended by:
    funningforrest, mkor7

    The Right Reverend Howard Johnson, "Lord, will we fool these evil men and save our town of Rock Ridge; or, are we just jackin' off?"
    (H/T Blazing Saddles)

  •  Thanks - that was very informative. I had (1+ / 0-)
    Recommended by:

    forgotten all those infinities, in the decades since my last math class :) IIRC [part of?] the argument that you can't map the integers onto the reals is that between every two integers, there are an infinity of real numbers.

    •  Reals (4+ / 0-)

      Actually, thats not the argument. Between every two integers, there are also infinitely many rationals, but the set of rationals is also denumerable (ie, the same size as the set of integers). The argument for proving that the set of reals is bigger than the set of integers involves a questionable (but generally accepted in math today) technique called Cantor's [second] diagonalization argument.

      Diagonalization was controversial. Many [secular] academics questioned its legality since the proof requires an infinite process and its not clear whether legal math proofs have to be finite (there are foundations of math that explicitly reject such techniques). Theologicians of Cantor's time also considered it as  questioning a supreme being though I doubt today's theologicians criticizing it are smart enough to realize this connection.

      Unfortunately, most (but far from all) college math courses these days concentrate on applied math over continuous domains (calc, real analysis, etc) rather than the discrete domains that are needed for stuff like this. But this crosses over into philosophy and computer science where they want to model reasoning.

      •  There is nothing at all questionable (0+ / 0-)

        about the diagonal argument.  It’s a constructive algorithm that accepts as input any infinite sequence of reals and produces as output a real number not in the range of the sequence, thereby showing that the set of reals must be uncountable.

  •  IMO these guys are cultural terrorists... (0+ / 0-)

    ...bent on destroying the edifice of science that has taken thousands of years to build, at least in America because as you know America has been chosen by their god.

    Daily Kos an oasis of truth. Truth that leads to action.

    by Shockwave on Tue Mar 25, 2014 at 09:07:31 AM PDT

  •  I spend my days doing matrix problems (1+ / 0-)
    Recommended by:

    I have written an "expert system" using  matrices to address the possible solution sets. So like a gps that looks at city, state, house number and street to pinpoint a location, this looks at a large number of factors and finds the solution through addressing rather than through a large number of if-then-else statements. It is quite deterministic and straightforward to expand the factors. The code on the other hand is quite abstract because it is basically a solution calculus for merge/purge of the matrix sets  (which are sometimes made of sets, sets of sets, sets of sets of sets).
    My point being that I learned sets and number systems quite early through grade school and junior high "new math" curriculum. Even at that time, there were quite a bit of "conservatives" who objected to that curriculum and felt that "carry the five" is what teh maths were really about. No child left behind, learn to do the arithmetic you need to run a cash register. You know - one step up from pi = 3 crowd - all common-sensey. It is the early education I received that allowed me to imagine a system where I can present a coherent visualization of solutions to the experts (this is of critical importance because of the large level of factors, how do you know you are getting the correct solution?) through set theory, as well as the actual calculation of solutions sets that made the use of these abstract techniques possible to me.

    "You can die for Freedom, you just can't exercise it"

    by shmuelman on Tue Mar 25, 2014 at 09:51:55 AM PDT

    •  Now that is interesting (1+ / 0-)
      Recommended by:

      Merge/purge: I like that phrasing.

      Is your method an evolutionary search, or a clever way of organizing an intelligent exhaustive search?

      (My apologies if our dialects are mutually incomprehensible.)

      Once upon a time I was putting together a program that needed groups broken down into their irreproducible representations, so I handed the module over to a bright young student.

      He came back with what was essentially a one line formula, where the data went in and the representations came out.  The damned thing worked, perfectly, but I never understood why it works or how he came up with it.

      And that is a systematic issue with being young and ambitious: all the incentives are for plugging it in and getting it out the door, with no time to develop a deep understanding of what you are actually doing.  The next time someone tells me to just pick the low hanging fruit, I'm going to punch him in the throat.

      o caminho d'ouro, uma pinga de mel: Parati

      by tarkangi on Tue Mar 25, 2014 at 10:56:23 AM PDT

      [ Parent ]

      •  It is a clever way of organizing an intelligent... (1+ / 0-)
        Recommended by:

        I can't talk too much about it, but it is for dosing antibiotics. So first you build sets of available treatments based on a number of factors (merge). Then you do your "take-aways" (purge). You remove drugs based on allergies, resistances in the community, etc.. Then you purge the resulting sets that no longer contain drugs you want to use. So there are many treatment fallback positions. If you are left with no viable treatments because of the "take-aways" the patient is in medical terms "f*cked."
        It is all based on very rigorous modeling of the many factors that decide optimal treatment sets. It is all done through a modeling / authoring system that pyramids the treatments from drugs and other relationships. It reminds me of a "Babbage engine." You set up all the gears where each gear is a factor, you pull the lever, all the gears turn and a set of treatments come out. This is what I mean by deterministic results rather than executing huge amounts of procedural if-then-else rules where you can never really know where you have gone astray.
        The art of a great software architect is to know what corners you can cut (low-hanging fruit). There were none. Fortunately I saw this from the beginning, so as the treatments extend through the different body sites ( am working on peritonitis and necrotizing pancreatitis now), there are hadly any additional programming changes. Sometime I have to turn four gears(treatment factors)  into five, but I suppose this is a little cryptic. I spent many months just on the modeling and the authoring system.
        Hopefully it will start savings lives in the next year.

        "You can die for Freedom, you just can't exercise it"

        by shmuelman on Tue Mar 25, 2014 at 12:05:21 PM PDT

        [ Parent ]

        •  That is excellent (1+ / 0-)
          Recommended by:

          Saving lives makes you feel real good.

          Saving a billion dollars for the company makes you feel real good.

          And you illustrate the funny thing about modeling: you start with a simple model, build it up to make it more accurate, then it becomes impossibly complicated, until you wander into a regime where an alternative model makes things simple again.

          Best wishes for your project.

          o caminho d'ouro, uma pinga de mel: Parati

          by tarkangi on Tue Mar 25, 2014 at 01:53:49 PM PDT

          [ Parent ]

  •  this has been going on a long while (1+ / 0-)
    Recommended by:

    when my husband had an interview at Whittier College, 10 years ago now, he encountered a student who brought up the question of infinity and how we can never understand it because duh, infinity. he proceeded to show that yes, we can, using set theory and the integers example. he thought it an odd question, but looking back, he realized it was a religious student trying to justify their beliefs by attempting to put him on the spot, thinking there was no suitable answer, and he, as student extraordinaire, would prove how brilliant he was (math professors get this a lot from the "i'm brilliant let me trip you up" students and laypeople). i'm sure this question gets asked over and over in a college setting, with students thinking they can get the same reaction from their professors as the characters do in a jack chick comic.

  •  As 1 who has the equivilant of a Phd in Set Theory (1+ / 0-)
    Recommended by:

    I found this most hilarious. I guess these Christians also have a problem with programming in SQL (Structured Query Language) and think it's a sin, because SQL is based entirely upon Set Theory. Basically, all one is doing at the highest levels is defining and joining SETS. In fact, all Database design and architecture is, is SET THEORY.

    * * * DONATE/VOLUNTEER: Marianne Williamson for CA-33 * * * #CampaignFinanceReform is the lynchpin of our democracy. #AIKIDOPROVERBMoveSoonerNotFaster ~

    by ArthurPoet on Tue Mar 25, 2014 at 01:25:22 PM PDT

  •  OK (0+ / 0-)

    I read the diary and skimmed the comments and I still can't figure out how set theory or any other element of math has anything to do with the Bible one way or the other.  Even the Book of Numbers does not posit any alternatives to how we understand math to work:)  At least we can read Genesis and understand where Creationism comes from.

  •  the distrust of set theory (1+ / 0-)
    Recommended by:

    I think some of it stems from the New Math curriculum of the 1960s which tried to teach set theory before algebra and other types of advanced math.  It was very unpopular at the time.  My personal opinion is that we're better off doing the more concrete math first and then later go more abstract.  

    •  I totally agree (0+ / 0-)

      As I said up thread, why teach grad school conselts to grade schoolers?  And why should we be haggling over this stuff 50 years later, when nobody ever generates empirical data about what works? To me it has always seemed like  folie à plusieurs ("madness of many") where a group shares delusion beliefs.

      That tiny introduction to set theory eased the shock of Serge Lang's leaden exposition when I dove into topology in grad school.

      You never know when your education is going to pay off, which is why the fundies are so paranoid in choking off things that might be dangerous one day.

      Men are so necessarily mad, that not to be mad would amount to another form of madness. -Pascal

      by bernardpliers on Wed Mar 26, 2014 at 08:36:02 AM PDT

      [ Parent ]

  •  anything confusing is a proof of God (1+ / 0-)
    Recommended by:
    Calamity Jean

    You can start with anything complicated or confusing, hide a fallacy in it, and use it to prove whatever you want. Lead people into the dark woods and get them lost, and then "save" them.

    Read the Weekly Sift every Monday afternoon.

    by Pericles on Wed Mar 26, 2014 at 04:20:54 AM PDT

  •  This problem is easily solved (0+ / 0-)

    God can just send William Lane Craig a vision with a proof against Cantor. We'll wait.

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