What do Christian fundamentalists have against set theory?

I've mentioned here before that I went to fundamentalist Christian schools from grade 8 through grade 11. I learned high school biology from a Bob Jones University textbook, watched videos of Ken Ham talking about cryptozoology as extra credit assignments, and my mental database of American history probably includes way more information about great revival movements than yours does. In my experience, when the schools I went to followed actual facts, they did a good job in education. Small class sizes, lots of hands-on, lots of writing, and lots of time spent teaching to learn rather than teaching to a standardized test. But when they decided that the facts were ungodly, things went to crazytown pretty damn quick.

All of this is to say that I usually take a fairly blasé attitude towards the "OMG LOOK WHAT THE FUNDIES TEACH KIDS" sort of expose that pops up occasionally on the Internet. It's hard to be shocked by stuff that you long ago forgot isn't general public knowledge. You say A Beka and Bob Jones University Press are still freaked about Communism, take big detours into slavery/KKK apologetics, and claim the Depression was mostly just propaganda? Yeah, they'll do that. Oh, the Life Science textbook says humans and dinosaurs totally hung out and remains weirdly obsessed with bombardier beetles? What else is new?

Well, for me, this is new:

"Unlike the "modern math" theorists, who believe that mathematics is a creation of man and thus arbitrary and relative, A Beka Book teaches that the laws of mathematics are a creation of God and thus absolute....A Beka Book provides attractive, legible, and workable traditional mathematics texts that are not burdened with modern theories such as set theory." — ABeka.com

Wait? What?

First off, let's establish what set theory actually involves.

Sets are exactly what you think they are—groups of things. Prime numbers, unicorns, cats, whatever ... you can make a set of it. Set theory is just a way of talking about what sets do and what they are like.

On the surface, this sounds pretty simple. For instance, most of what I learned about set theory when I was in college came through classes in anthropological linguistics. That's because sets, being made of anything you damn well please, have applications outside of pure math. Ted Sider, a professor of philosophy at Cornell University has some good examples of this in a set theory primer he's written:

In linguistics, for example, one can think of the meaning of a predicate, ‘is red’ for instance, as a set — the set of all red things. Here’s one hint of why sets are so useful: we can apply this idea to predicates that apply to other predicates. For instance, think of ‘red is a color’. ‘Is a color’ is a predicate, and it looks like it holds of things like the meaning of ‘is red’. So we can think of ‘is a color’ as meaning a set containing all the colors. And these colors in turn are sets: the set of all red things, the set of all green things, etc. This works because i) a set collects many things into one, and ii) many sets themselves can be put together to form a new set.

So far, so good. In this basic form, sets are involved in lots of things. They come up in musical notation, they help define the way we communicate with computers, and they are the things that make Venn Diagram jokes possible.

But sets and set theory can also be a lot more complicated. For instance, you can make up sets that contradict themselves. The classic example is a set made up of barbers who shave everyone in town (including themselves) and who only shave the people who don't shave themselves. Oops. Another problem: Sets that are too broadly defined, so you don't know if you're actually putting the right stuff in there. A set made up of the favorite things of a tall person, say. Paradoxes like this are what really drive set theory, much of which centers on defining rules for sets and how they work so that we don't just go around assuming certain sets exist when they clearly can't—andso that we can still use the valuable logic and math of sets even when we can't prove that the stuff we're sticking into a set actually exists in the real world. Basically, set theory has a lot to do with creating rules and helping us apply a rule-based system in weird, hypothetical situations.

All of which turns out to be really important when you want to talk about the idea of infinity. Set theory actually has its origins in attempts to define infinity and deal with it in a concrete way in mathematics. Checking Wikipedia, you'll learn that this "modern" theory was actually established in 1874. Why 1874? Because that was when a guy named Georg Cantor proved that there are different infinities and that not all infinities are created equal.

Again, what?

This is really where set theory starts to sound like something you thought up while high and later forgot about.

You can have an infinite set of numbers, right? That makes sense. But, Cantor figured out that an infinite set of, say, whole numbers, is smaller than an infinite set of decimal numbers. They're both infinite. But they're not the same. This TEDEd video explains it really, really well:

So what does all of this have to do with Christian fundamentalists? I have to admit, when I first read that Mother Jones piece, I was stumped. I don't remember anybody disparaging set theory at the schools I went to. And, I'll be honest, I didn't remember enough about what set theory was to be able put the pieces together. (I was also somewhat disappointed to find that the Conservapedia entry didn't offer much help.)

But after re-acquainting myself with this stuff, I think I see a couple of things happening that would make set theory problematic for some Christian fundamentalists.

First: Some of these folks get very touchy about the idea of infinity. Mark Chu-Carroll is a software engineer at foursquare and a math blogger. Unlike me, he was already aware of the fundamentalist objection to set theory, because he's actually had people show up in his comment section railing about how the theory is an affront to God. Particularly the part about multiple infinities. Chu-Carroll told me that one commenter explained the problem this way: "There is only one infinity, and that is God." Basically, this perspective looks at set theory and Georg Cantor and sees humankind trying to replace the divine with numbers and philosophy.

The second problem is a little more complex. Remember how the modern idea of set theory really isn't all that modern? That's because I'm pretty sure A Beka doesn't mean "modern" as in "recent", but "modern" as in "modernist".

I can tell you from experience that A Beka (and Bob Jones University Press) are stridently against modernism in all its forms. (I'm assuming they're against post-modernism, too, but you have to understand that the opinions and perspectives this sort of Christian fundamentalism has about society and culture were formed between the late 1920s and early 1970s and, because of this, the culture wars that they are fighting often come across as confusingly antiquated. Thus, the ongoing obsession with the imminent threat of Communism. See also: Why I sat through multiple sermons on the evils of rock n' roll in the late 1990s.)

If you associate modernism primarily with abstract art, Scandinavian furniture, and houses made out of glass, then all of this is probably just as confusing as set theory, itself. But art isn't really what the fundamentalists are thinking about when they think about modernism.

Instead, they see modernism as the opposing worldview to their own. They are all about tradition (or, at least, what they have decided is traditional). Modernism is a knee-jerk rejection of tradition in favor of the new. Obviously, they think a very specific sort of Christian God should be the center of everything and all parts of society, public and private. Modernists prefer ideas like secular humanism and think God is something you should be doing in private, on your own time. They believe strongly in the importance of power hierarchies and rules. Modernism smashes all of that and says, "Hey, just do your own thing. Nobody's ideas are any better or worse than anybody else's. There's no right and wrong. Go crazy, man!" [Insert obligatory bongo drumming session]

I am hamming this up a bit, but you get the picture. Modernism, to the publishers of A Beka math books, is sick and wrong. The idea is that if you reject their specific idea of God and their specific idea of The Rules, then you must be living in a crazy, dangerous world. You could kill people, and you would think it was okay, because you're a modernist and you know there's really no such thing as right and wrong. Basically, they've bumped into a need to separate themselves from the almost inhuman Other on a massive scale, and latched on to modernism as a shorthand for how to do that. It doesn't matter what you or I actually believe, or even what we actually do. They know what we MUST believe and what we MUST be like because of the tenets of modernism.

More importantly, they know that we are subtle, and use sneaky means to indoctrinate children and lure adults into accepting modernist values. So the art, the literature, the jazz—probably the Scandinavian furniture, too, though I never heard anyone mention that specifically—are all just traps. They're ways of getting us to reject to One True Path a little bit at a time. (I should note that, up to this point, I am basing my analysis on what I was taught in Baptist school. After this, I'm speculating, and attempting to connect the ideas I know are present in this subculture with set theory.)

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.

Add in the historical fact that Georg Cantor's ideas weren't terribly popular at first, and they can easily create a narrative where true math is being suppressed so that false, modernist math can corrupt the minds of children.

If this sounds crazy ... you're right. It's pretty crazy. In fact, it's this kind of thinking, and my realization that it was based fundamentally on lying about everybody who wasn't a member of your religious tribe, that led me away from religion to begin with. Ironically. But there is a coherent thought process going on here, and I want you to understand that. If all you do is point and laugh at the fundies for calling set theory evil, then you are missing the point. This isn't about them being stupid. It's about who they think you are.

Mother Jones on the wacky things you can find in Christian school textbooks
Wikipedia on set theory; some interesting history, but not great for helping you understand this stuff.
Ted Sider, a philosophy professor at Cornell, has a pdf document that is a must-read if you are starting from scratch and Wikipedia's set theory explanation makes your brain hurt.
• A link to the TEDEd video embedded in this story, which explains some of the weirdness with infinite sets quickly and simply.
• Vanderbilt University math professor Eric Schechter has a page about the Axiom of Choice, one of the rules in set theory that allows you to play with hypothetical sets and overlaps with some of the problems of infinity. Includes links to other great resources.
• Mark Chu-Carroll's math blog Good Math, Bad Math

Image: Venn Diagram, a Creative Commons Attribution (2.0) image from scottjacksonx's photostream


    1. If only they actually drank strychnine. But no, they somehow cherry pick even single sentences and only selectively follow Jesus’s command about drinking poison because he will protect them.

    2. I still own a Creationist dinosaur book that does in fact actually go on & on about bombardier beetles & how that way dinosaurs could breathe fire & that is totally how medieval knights fought dragons.  

      It is not a joke.  Well, it IS a joke, but they aren’t joking.

  1. These ambiguities, redundancies and deficiencies remind us of those which doctor Franz Kuhn attributes to a certain Chinese encyclopaedia entitled ‘Celestial Empire of benevolent Knowledge’. In its remote pages it is written that the animals are divided into: (a) belonging to the emperor, (b) embalmed, (c) tame, (d) sucking pigs, (e) sirens, (f) fabulous, (g) stray dogs, (h) included in the present classification, (i) frenzied, (j) innumerable, (k) drawn with a very fine camelhair brush, (l) et cetera, (m) having just broken the water pitcher, (n) that from a long way off look like flies.

        1.  Dude, I wish had a retort for that, but you’ve liked summed it up so summarily, there is nothing left to be unsaid, or nothing, or something or something.

      1. Ha – I just assumed it was a quote from Calvino’s Invisible Cities.

        There is a set of writers who like lists. Calvino is in that set. Borges is in a smaller set within the set of writers who like lists: Borges is a writer who likes lists of sets.

        (I apologize if my math-grammar is wonky there)

        1. You can tell it’s Borges because he not only made up (“interpolated”) the passage in question, but also attributed it to a book that (probably) doesn’t exist, written by someone who (probably) never lived.

          It’s like William Goldman and his “S. Morganstern” rabbit hole, only deeper!

          1. Actually, I did some looking into it, and the book definitely doesn’t exist.

            (Although I can’t know this for sure since he’s not around to ask. In contrast, Neil Gaiman referenced a book of Japanese folkstories at the end of _Dreamhunters_, and after failing to find any trace of it, I asked him and he eventually did admit he was pulling a Borges.)

      1.  From your comment I see you have never been to the library of Babel. BTW, I pity you for that terrible lack in your education.

        1. I went to the library of Babel book sale and came away with a volume of holy writ printed in Bombay. Frankly, I couldn’t finish it.

    1. Having been studying US export regulations lately, that JLB passage sounds bone-chillingly familiar.

          1. Careful! Just because Jesus never married and hung around with a bunch of guys, don’t go thinking he was – you know.

    1. More like child neglect.  Sad and regrettable, but not so bad that it is actionable.

      These kids might be getting a less than ideal education, maybe even a warped mind, but real abuse is so much worse.  Having known someone who worked in child protective services, I learned that child abuse can be truly horrific.  Sick sick stuff.  Abused kids would be lucky to trade places with a fundy kid.  Calling a fundy education child abuse does a disservice to truly abused kids.

      We can press to fix the problem without mixing two issues.

      1. Unless they’re lesbian, gay, trans, autistic, or otherwise considered born-abominations.

        Less than 10% of everyone in America are lesbian, gay, bi, or trans.

        More than 40% of homeless children in America are lesbian, gay, bi, or trans.

        A lot of fundamentalist parents are abusing these children, and some are kicking these children into the streets.

    2. I feel you. I’m baffled when I think about the stuff I used to believe back then. Bombardier beetles, holy crap. They thought it was literally the dumbest thing in the world that they could have evolved. People also thought (or still think) that there used to be a race of giant evil humans that got busy with angels and God got mad and flooded the earth. But like the author said, people raised with this ideology still do get a decent education in other areas. Believe it or not, they can be a pretty academically rigorous group, if you take out the insane stuff. I grew up in it, and seeing A Beka come up here was incredibly surreal. I left all of their crazy behind, but my love of learning came from the people who taught from and believed this curriculum.

    3.  Well because these people are effectively the neo-Amish. We don’t prevent Amish people from teaching their kids that electricity is evil. If the neo-Amish want to teach their kids that certain kinds of math is evil, then I suppose it is their right.

      1. There are issues with the Amish, but believing that electricity is “evil” isn’t one of them.

  2. Might as well blame the Catholics, they were the first Christian sect to challenge modern mathematics with their “three equals one” theory.

          1. And equality is also an equivalence relation. But as far as I know, not all equivalence relations are equal (har har).

    1. Actually, they were among the first to challenge science with the whole “Galileo is wrong” thing.  The “three equals one” thing is not a theory, it is just a bit of mystical metaphor. 

      1. I feel obliged to point out that not all Catholics opposed Galileo (Although most Catholics, and most academics at the time did).  The Jesuits actually defended him very strongly.

        It’s also notable that there was much more furor over his descriptions of the Moon than over his solar system model.

        But really the whole heliocentrism thing didn’t really catch on until a bit after Kepler anyways.

        PS: If you want a real challenge to “science,” check out Al-Ghazali’s “Rescuer from Error.” That book basically ruined the entire empirical school of Islamic philosophers.

  3. If all you do is point and laugh at the fundies for calling set theory evil, then you are missing the point.

    I guess I am, because even understanding the coherent thought process leading up to this, it only looks like a textbook case of taking things to risible lengths. As an honest question, what besides a cautionary tale were you hoping I could see here?

    1.  I think the ‘taking things to ridiculous lengths’ bit is almost exactly the point. This type of culture has no breaking mechanism. In order for them to stand back and say “Hey, I think we have gone about as far down this road as needs to happen” requires that they can stand back from themselves in the first place.

      The ability to moderate the thought process before it spirals into the ridiculous explicitly requires self awareness and rationality. Without those, ANY thought process will spiral off into the insane.

    2. Formerly there were those who said: You believe things that are incomprehensible, inconsistent, impossible because we have commanded you to believe them; go then and do what is unjust because we command it. Such people show admirable reasoning. Truly, whoever is able to make you absurd is able to make you unjust. If the God-given understanding of your mind does not resist a demand to believe what is impossible, then you will not resist a demand to do wrong to that God-given sense of justice in your heart. As soon as one faculty of your soul has been dominated, other faculties will follow as well. And from this derives all those crimes of religion which have overrun the world.
      Voltaire, Questions sur les miracles (1765)

      These absurd attacks on modern thought are a real threat to the fundamental achievements of the Enlightenment: the ideals of reason and democracy. We have to understand the nature of these attacks in order to defeat them.

        1. Which just proves that the best way to get kids to read something is to tell them they’re not allowed to.

  4. I spent four years in a Fundamentalist Baptist church. I’m still going to point and laugh.

    1. I spent nine years at a fundamentalist Christian school (Pensacola Christian, home of A Beka Books) and 19 years at a fundamentalist Baptist church. I just sigh and shake my head.

      1. I went to a fundamentalist Christian grade school, and lived at home (and thus had to go to church) until my early 20’s. And my father was a physics teacher, of all things, as well as a fundamentalist Christian.

        I can confirm the anti-Communism; we got shown the same film every year about the perils of Communism. Set theory was too high a level of mathematics for grade school, but the one set fundamentalist Christians really want you, yes, you to belong to is the set of fundamentalist Christians.

        I’m still not able to point and laugh. It just wasn’t funny.

  5. Thanks for this, Maggie.  I’d actually just read the Mother Jones article, and the fundamentalist objection to math really stuck out at me.  As a liberal arts guy, the Wikipedia entry on set theory was a little dense, and this explained things perfectly.

  6. David Foster Wallace’s ‘Everything and More’ breaks down Cantor’s insights into set theory and different infinities better than anything else I’ve ever read. I had my mind blown 6 times in the first 10 pages and even when it gets into the really complicated stuff it’s still treading on solid ground because DFW did such a good job laying a foundation of knowledge.

    1.  I concur. I would recommend that book to anybody. Sure, parts of it will be slow going, but still other part are quite graspable even though they kinda make your head hurt at first.

        1.  It’s true :) If you read Infinite Jest, you will know that he totally had the chops to do this. Of course, you have to read all the footnotes, endnotes…..etc. Infinite Jest may be a bit of a slog, but it is WELL worth it. Took me something like 5 years to finish, but it is just one of those odd ducks that you can put down for quite a long time and then pick up again and know precisely where you are at.

  7. Interesting piece.   I do want to say though, that I attend a Christian college where we actually don’t find any issue with teaching set theory, modernism, or (parts) of post-modernism.  I was taught to accept the idea that multiple infinites did actually exist (Though we were a little wary when it came to whether actual infinities could exist in reality.  But that was a philosophical issue similar to that of the Hilbert Hotel instead of a theological issue).  

    I also know of quite a few apologists who find no issue with set theory, most notably William Lane Craig and John Byl.
    Now, it’s worth noting that A Beka and Bob Jones are each their own entity, with authors who have different opinions on different topics.  So, I do ask that this judgment isn’t made too widely.
    Thanks guys, and I hope this helps clarify any potential misconceptions.

    1. Christopher,  I think I tried to be pretty fair here, and clarify that I’m talking about a specific subculture of Christian fundamentalism, of which Bob Jones and A Beka tend to be a part. Apologies if I made you feel attacked. That was not my intention. You should know better than others that Christianity isn’t a single thing. 

      1. It’s an unfortunate fact of modern life that certain subcultures of Christian fundamentalism put many decent, kind Christians in the position of having to say “We’re not all like that.”

        1. It’s an unfortunate fact of modern life that certain n put many decent, kind n in the position of having to say “We’re not all like that.”

          Where n=a whole lot of groups, including, but not limited to, Muslims.

        2. Re: “We’re not like that.” Who is this “we”? When some foible or enormity of a particular Christian sect or text is revealed, there’s always at least one Christian who is suddenly like the pope, speaking with a “we.” The people putting you on the spot are your risible or shocking coreligionists, not the person who has gone to the trouble to point it out.

          As to the “modern life” bit, well, it’s objectionable, a sort of Stalinist history erasing. Christianity has a long history of “we’re not like that,” from its very inception to the schism that just happened thirty seconds ago somewhere. And if by “we’re not like that” you mean that Christianity is not about the commission of various enormities, you must at least confess that innumerable enormities have been committed in its name.

          If you’re going to wear the jersey and wave the big foam finger, you’re also going to have to take the lumps when your team loses.

          1. “If you’re going to wear the jersey and wave the big foam finger, you’re also going to have to take the lumps when your team loses.”

            That’s like calling all the participants in a golf tournament a team.

          2. You forgot the “all” when you quoted Christopher. “We’re not all like that.” That, apart from making his statement self-evidently true instead of arguably false, puts most of the rest of your comment into the set of all comments that make no sense. Particularly the pope bit.

          3. Yeah, right, Saltine. As if Christianity entails a commitment on set theory. Foam finger metaphors, for all their merit, are not quite robust enough to make that case. You see, being a Christian is hardly a sufficient condition for being committed to other Christians’ views on just anything and everything. This would be like saying that because the Unabomber believed that cabin living makes for a darn fine way of life, and you believe that cabin living makes for a darn fine way of life, then you’re committed to all his craziness.

          4. Just because I chose to apply a label to myself doesn’t mean that I have to accept the behaviour and beliefs of everyone else who uses the same label.

  8. So the art, the literature, the jazz—probably the Scandinavian furniture, too, though I never heard anyone mention that specifically—are all just traps.

    The product labels in the IKEA catalog are actually the names of DEMONS. If you read them off in order, you will summon an unholy legion of bloodthirsty myrmidons. They will obey you . . . at the cost of your soul!

      1. They come flat packed, in boxes made of the compressed souls of farm boys who diddled goats back in the 1st century C.E.

        And every orifice you stick the Allen key in is as sensitive as your urethra, so by the time your done they’ll hate you as well.

        (The above isn’t true, but if it keeps you from summoning Ikea demons or diddling goats, A Beka books will print it. For the good of the children.)

      2. The demons arrived flat-packed, much to Stabenaw’s chagrin. Thirteen long black boxes chased in silver crosses. The UPS guy did his best not to grunt and swear but boxes like these are what grunting and swearing were invented for. He set the boxes down upside down.

        This wasn’t why Stabenaw did not tip the man. In a few minutes . . . a few hours . . . unfolding the instructions, pictures of tabs and slots and curling horns and blister-packs of virgin blood, he thought maybe *days* might be more like it, but, fuck it, at *some* point this millenium tips were going to become a thing of high irrelevance. 

        “Oh, honey,” Janice said when she saw the wreckage spread across the living room floor. “You know the Hendershotts will be here in fifteen minutes, right? I told you not to start this until–”  

        Janice was a sweet woman, beloved wife and mother to Stabenaw’s seven amazing children and he wouldn’t trade her for a buschel basket of Bünschens, but the allen wrench had recently skinned his fingertip and the *last* fucking thing he wanted to hear about right now was Clarence Hendershott’s latest hemarrhoid lasering and the most recent car their juvie spawn had stolen.

        Consequently, he invited her to leap from the nearest window. The Blasphemous Heart of Hrrjen (M) glistened with fresh dripped blood (not shown), and Janice did not bother putting on her shoes before she leapt. So that was settled. 

        Part of Stabenaw was appalled, but as he’d already gotten so far as to insert Talons (V) into Toebones (Q) and wrench them into place, the greater part thought that the Hendershotts would make a pretty fine meal regardless of whether Janice was there to cook.

      3. I agree. When I get to hell I’m going to be in senior management; most likely in charge of the fun ‘n debauchery department.

    1. There is an IKEA catalog on my desk right next to me. I never realized that it was a veritable grimoire of Scandinavian demons. I’m going to start reading product names out loud right no–

  9. “They are all about tradition (or, at least, what they have decided is traditional). Modernism is a knee-jerk rejection of tradition in favor of the new.”

    Religious fundamentalism is a modernist phenomenon. Without the the context of modernism it makes absolutely no sense, because religious fundamentalism is not a true representation of tradition so much as it is a reaction to modernism. You had to have the Enlightenment first before you could have the Great Awakening.

    1. It’s a phenomenon of modernity, but not Modernist, which is a really precise historical-cultural-artistic-etc. term.

      1. Could you please be more specific about how it’s precise and what is the precise form?  Looking at the wikipedia article and considering my own experience with usage of the term, “Modernism” is all over the map.  I also think of religious fundamentalism as a reaction to Modernism in the sense jetfx is talking about.

        1. To me the Wikipedia entry backs up what I’m saying: Modernism is Virginia Woolf, Marx, cubism, Futurism, Stravinsky, etc. The list of artists between 1900-1930 would be a good rubric for Modernism.

          Modernity is modern times, which go back at least as far as the Renaissance, depending who’s doing the reckoning.

          I too agree fundamentalism is a reaction to Modernism, just not that it’s itself modernist. It’s anti-modern, anti-change, anti-progress . . . all while being a modern-day phenomenon impossible without modernity.

          1. Edit: Removed stuff to conserve vertical space.

            I haven’t experienced “mordernity” used in the way you mention either; in most cases, I’ve heard “modernism” used instead.

            I too agree fundamentalism is a reaction to Modernism, just not that it’s itself modernist. It’s anti-modern, anti-change, nti-progress . . . all while being a modern-day phenomenon impossible without modernity.

            That does not logically follow, actually.  The fact that mongooses kill predatory animals does not imply that mongooses are not predators.

            Edit: It’s a semantic issue, not an issue of logic. “The content of this ideology is anti-modernist, therefore the ideology itself is anti-modernist” does not logically follow unless you define the ideology in terms of its content. Which is what you’re doing, and that’s cool too. There are some good reasons not to do that, though (the contents of ideologies are not necessarily consistent with the actions or goals of adherents).

          2. Behold, I tell you a mystery:

            – You don’t need to put a line break before a blockquote; our style creates more than enough space.
            – Don’t leave even a single character’s worth of space after a blockquote; back the following text right up to the >.
            – Disqus, for no good reason and apparently irreparably, inserts a single character blank space into the comment box. If you start your comment with a blockquote and you don’t nuke that space, it creates extra space at the top of your comment. If you don’t get rid of that space, it also has a nasty habit of inserting itself into masked URLs, making them not work and making me spend a significant amount of time fixing non-working links.

          3.  Modernism is more than just art. It also includes the philosophies and ideologies of modernity (like Marxism for example). The term is most specifically and popularly applied to art, but more broadly modernism is about the individual and social ways of being that modernity engenders (industrialization, urbanization, secularism, capitalism, mass media, consumerism, etc, etc). But we’re arguing semantics here, and we mostly agree on my main point.

            Fundamentalism is anti-modern, but it’s not attacking modernity from any traditional pre-modern perspective. That’s what makes it modernist. Religious fundamentalism is reacting to the rapid social change and instability of the modern age, by proposing a solution that amounts to a religious version of modern utopianism.

          4. @wysinwyg:disqus Can’t reply to you directly, so here goes:

            Yes, as a PhD professor-type with a background in this stuff, I would guess that I’m using it in a precise way. The Wikipedia entry itself speaks of the “broadest definition,” which I’m specifically moving away from. I prefer, from habit and training, the hard particularity of a Rilke poem or Joyce’s Ulysses to a baggier concept. I don’t think anything I said negates those broader definitions, but I don’t think they’re incredibly useful when so many particular, distinct, unique examples of Modernism are so readily to hand.

            Yes, please use the term as you wish! :D I assure you that modernity and Modernism are often used the way I’m using them–you have Wikipedia as a guide, if you wish–but please use the terms as you’re comfortable.

            Actually, what I said at the end perfectly logically follows, pace mongooses. Not all historical events obtaining at a given time are part of the generalized Weltanschauung or paradigm of that period: thus, we have some stone-age tribes coexisting whilst we’re talking away on the Internets. Those tribesfolks are not modern. Similarly (though with critical differences), anti-modern, anti-Modernist movements have obtained during the twentieth century, coexisting and living during modernity, yet acting in continual opposition to it. jetfx said as much in the original post.

          5. @boingboing-8430445454669181ee6a757a6cd3c53a:disqus O, yeah, it’s just that the art movements are most easily defined by example for most folks, in my experience. It’s a bit harder to shoehorn Marx in there, but cubism, Joyce, etc. tend to be viscerally appreciable as Modernist. Use Henry Ford as an example if you wish.

            I strongly disagree that there’s nothing premodern to fundamentalism, though. The Bible is stunningly premodern, and it’s a cornerstone of Christian fundamentalism and its anti-modern agendas. It’s resoundingly not modernist. The only way you could make this argument would be to look at fundamentalism’s way of organizing itself, its use of mass media, etc.: but this would be “modern,” not “Modernist,” in the strict sense. And then we’re into fascinating McCluhanite arguments, like Is fundamentalism anti-modern because of its content (the Bible, say), or modern because of its means of expressing that content (mass media)? So perhaps calling contemporary fundamentalism a hybrid–modern in means, anti-modern in outlook and values–would work best.

            Fun discussion. Good times!

          6. I think that Modernism as you define it is not quite appropriate to the OP’s point. The Modernism you define was in many ways an intensification of the challenges that traditional authoritarian belief structures already faced starting from the Enlightenment and earlier. (To use a term-of-art, “The Early Modern.”)

          7. Timothy Krause, I should have been more clear. I don’t disagree that much of the content of religious fundamentalism is pre-modern, but I’m arguing that fundamentalism as a form of religion is modern. While the Bible is also pre-modern, fundamentalist readings tend to be more modern, because they highlight things that would not have been of concern to earlier Christians. For example, the Book of Genesis was always considered to be an allegory, and not literal until at least Martin Luther. It wasn’t until science began to give an accurate age for Earth until you started to see this creationist push back. Another example is the Temperance Movement. The consumption of alcohol was never really an issue for Christianity (Jesus’ first public miracle is to turn water into wine), but the industrialized production of alcohol wrought enormous destruction on factory slums to the point where temperance became a major focus of fundamentalist Christianity. Or the focus on homosexuality. And so on. I don’t think it’s water tight, but I think there is a good case to be made for religious fundamentalism as a modernist version of pre-modern belief systems, but with a great deal of elements that are wholly modern.

            I don’t think it’s much of a stretch to put Marx in there, as his thinking is one of the cornerstones of modernist philosophy.

          8. Can I just comment bomb to say that I love everything about this argument? I mean, seriously. You guys are awesome. I love that this is the kind of thing that creates a giant thread on this site. 

          9. It’s anti-modern, anti-change, anti-progress . . . all while being a modern-day phenomenon impossible without modernity.

            Not quite. I’m gonna use Islamic fundamentalism, because I’m more familiar with that, but I should imagine that this applies to different forms of fundamentalism in different ways.

            Many forms of Islamic fundamentalism purport to be a return to a “golden era” of Islam — the most common one I’ve heard is the call to a return to the Madinah Charter — basically a time when Muslims basically numbered in the hundreds, were restricted to one city, and were primarily agrarian, and led by the Prophet.

            The problem is, this sort of attitude didn’t exist prior to Modernism. As is, I mean the ideas that led to Islamic fundamentalism — the paring down of ideas to their essentials, the removal of all “useless fripperies” like saint-veneration, intellectual discourse beyond a form of Neo-Platonism, mystical movements — they came from Modernism’s push to remove all the “useless fripperies” of tradition and superstition for progress.

            Basically, fundamentalism — at least in its Islamic forms, although I am pretty sure it can be applied to other variants, but I leave that as an exercise to you — while outwardly claiming to be a return to a “purer” form of Islam, uses techniques that initially find root in modernist thinking and modernity.

            The thing is, I feel like this is often unacknowledged aspect of fundamentalism — witness groups like Boko Haram using automatic weapons and techniques of guerilla warfare despite saying that they’re against all Western thought, to the Taliban adopting techniques that the CIA thought them, to Al-Qaeda adopting propaganda methods and memes (like, suspiciously, rhetoric that resembles some forms of Marxism) that simply weren’t available not only during the Prophet’s time, but even 200 years ago.

            And this kind of explains how come so many Muslim engineers tend to find themselves becoming fundamentalists. When you are a colonized, and you were brought up traditionally, and then suddenly thrust into a system created by modernism — the modern colonial-formed education system — the only thing you can do to maintain your self and identity is to turn to fundamentalism.

            And since while you were brought up initially as traditional Muslim, and suddenly you were given mental tools that expanded your thinking — by allowing you to use reductionism and reducing things to their essential forms mentally — and yet alienated you from your tradition — because let’s face it, when you’re a Muslim you are constantly reminded of how inferior, dangerous and barbaric you are — all you can do is build a political identity with the tools you have.

            Hence, fundamentalism is a modernist movement, although you’d have a hell of a time getting them to agree with you on that.

        2.  The reason you should be able to make a distinction between modern and “Modernism” is simple, modern will always be whatever is new, while “modernist” will apply to specific types of expressions, the fact that postmodernism is a reaction to “modernism” itself suggests that while postmodernism is modern, it is very clearly not “modernist”. I blame whoever thought of coining the term “modern art” as if there was no way to go from there.

          Having said that, the point is clearly understood, no need to defend any use of modern, people who don’t know the difference won’t care and people who do have had this conversation before.

          1. This is why current design at any given moment is usually referred to as contemporary rather than modern.

    2. They should be rejecting all the trappings of modernism, cell phones, cars, airplanes, modern medicine…

      1. Oh, no. They are all over human dominion of the world, and technology is necessary for that.

    3.  Karen Armstrong’s book “The Battle for God” is excellent on the subject of why modern fundamentalism is the way it is (in all the major Western religions). It is somewhat helpful to read her book, “A History of God” first, tho.

      1. George Marsden’s “Fundamentalism and American Culture” is likely the go-to book for understanding American Christian fundamentalism.  Marsden’s book largely supersedes Sandeen’s earlier “Roots of Fundamentalism”, though Sandeen is still worth reading. If you just want one book on fundamentalism, though, go with Marsden.

    4. This is not essentially different from the sort of rules the Platonists or the Pythagoreans enforced in ancient times. You could not do or teach any geometry that could not be done with a ruler and compass. You could not use the quadratrix (which can made with a compass that travels down a ruler as it turns) so you could not trisect an angle. Or a conchoid. Or a cycloid. Only conical sections were pure. Which effectively limits you to 2nd power curves.

      Oh yes: being a total jerk has a long and noble history.

    5. Religious fundamentalism is a modernist phenomenon? 

      What about 1st century BC Israel? Savonarola?

      You mean “Protestant Christian Fundamentalism as we know it today,” right?

      1.  Religious fanatics pop up in particular times and places throughout history, but fundamentalism differs from them, because it is not the product of local circumstances, but rather a reaction to modern society as a whole. Savonarola kind of prefigures this, as he is on the leading edge of Modernity (Early Modern Period), and is reacting against, as well as borrowing from, Renaissance humanism.

    6.  What (I think) is weird is that someone like Yeats, who from what I can tell bemoaned much of what the Enlightenment gave us, is “modernist.”

      I never could get my head around Postmodern, for that matter.  Mainly, I can’t tell if it’s bemoaning or celebrating what it describes. (Though I suppose the answer is “both at the same time, that’s what makes it Postmodern!”)

  10. There may be another aspect to this.  What I think of as the fundamental debate in philosophy of mathematics is whether mathematics actually exists (Platonism) or whether it’s constructed by mathematicians as they go along (Constructivism).  The latter isn’t necessarily as crazy as it sounds; consider the fact that the assumption for about 2300 years was that Euclidean geometry was the only geometry.  Now it’s been pretty conclusively demonstrated that the real universe isn’t Euclidean in the first place.  Essentially, Euclid constructed Euclidean geometry by making the assumptions he did as part of putting together the postulates; the downstream effects of these assumptions aren’t necessarily intuitively obvious, but the effects are constructed nonetheless because they depend entirely on the truth of the initial assumptions.

    On the other hand, anecdotally it seems a pretty strict majority of mathematicians are Platonists.  Mathematicians tend to have very strong insights that they are “exploring” a Platonic realm accessible to their minds rather than “exploring” the consequences of an initial set of assumptions.  Goedel in particular was very sure of this intuition (his arguments for it were weak but amusing), and his Goddy-quotes are often cited by Evangelicals in desperate attempts at arguments from authority. 

    There’s also the fact that some forms of the cosmological rely on the non-existence of actual infinities (IIRC) which is an important open question in philosophy of mathematics.  Also, Platonism, if true, would demonstrate a variety of truth that does not seem to depend at all on any actions of human beings; it would also demonstrate directly that most forms of naturalism are false.

    So I think there are deeper philosophical motives behind this sort of argument.  It goes something like:
    -Platonism is antithetical to most forms of naturalism and friendly to theism
    -Platonism provides justification for believing in a form of truth that is independent of human mind
    -Most mathematicians are Platonists, allowing Evangelicals to enlist them (probably unwillingly, as usual) to their cause

    1. This is an interesting thought, but I wonder how it might fit in. I thought historically it was mainly constructivists who were hostile to Cantor’s set theory; this at least was true for Kronecker. If you are tying this opposition to Platonism, which is what it sounds like, that would be a strange reversal.

      1. Set theory in general is the constructivist’s wet dream.  It allows all mathematicians to do to analysis what Gauss did to geometry.  I’d be interested in learning more about constructivist objections to Cantor if you have any cites.

        1. I’m afraid I’m not an expert, and for Cantor’s time I can’t point to much beyond Kronecker’s strong objections.

          However, the way set theory is usually used, it is definitely not constructive. On the contrary, things like the axiom of choice and law of the excluded middle let you say all kinds of sets exist with no idea how to determine membership. Sometimes you can even prove there is no way, like in the Banach-Tarski paradox.

          During the 60s and 70s there was a push to provide an alternative, and constructive set theories were developed. Before this, though, I’m not sure if there was any good way for constructivists to approach the subject.

          1. To explain what I mean in terms of your example, the axiom of choice is an assumption.  One can take the negation of the axiom of choice as an axiom instead of the positive form and you get a different form of mathematics.  Which is the “true” mathematics?  It’s not clear that either of them is.  The downstream effects of these assumptions isn’t always obvious — it wasn’t obvious that the axiom of choice implies the continuum hypothesis.  Nevertheless, assuming the truth of the axiom of choice ensures the truth of the continuum hypothesis.  Or one can take the truth of the continuum hypothesis as an axiom and derive the axiom of choice as a theorem.  Which is the “correct” form of the true Platonic mathematics; axiom of choice or continuum axiom?

            So my view is that the truth or falsehood of pretty much any theorem is dependent on the mathematical theory in which it is contextualized.  Truth is constructed from an initial set of assumptions.  This is probably a non-standard interpretation given your links.

            Edit: No one has ever been able to explain to me what they mean by the word “exist” in the sentence: “Mathematical objects exist.” So I disagree with your constructivists too.

          2. @wysinwyg:disqus My impression is that though many mathematicians think mathematical objects have some sort of Platonic existence, few expect there are unique “true” axioms. Mathematical constructivism comes closer to that in placing limits on what you should assume; so that explains the reversal in terms! Maybe they use them differently in philosophy.

          3. I think the motive can be tied to a paraphrase of Kronecker’s objection. “God created the natural numbers. Everything else was created by the devil.”

            We also need to remember that Cantor was a mystic. He claimed that his understanding of transfinite numbers were communicated to him by God. So from the beginning the controversy over Cantor’s theories had a religious nature to it.

            My guess is that antipathy to set theory among Fundamentalists began early, and is just carried on to this day out of tradition and tribalism. 

            It would take someone doing actual work (so not me) to look at the historical record to see if this is the origins or whether (as also seems possible) that it was just a reaction to “new math” in the 1970s.

    2. I’m an atheist and a Platonist. People usually think of physics as fundamentally real and math as a way to model it, but given that it’s possible to discover the equivalence of matter and energy using only math, I suspect it may be the other way around: That is, math is fundamental and physics models math.

      Suppose you have an infinite set of possible states of existence and an infinite set of mathematical rules to chain them together over time and space (or what have you) and although all possible universes exist, the anthropic principle dictates that we live in a universe that chains those states together logically in a way that supports life and consciousness.

      Just a thought.

      1. That’s why I said “most forms of naturalism.” =D

        People usually think of physics as fundamentally real and math as a way to model it, but given that it’s possible to discover the equivalence of matter and energy using only math, I suspect it may be the other way around: That is, math is fundamental and physics models math.

        I disagree pretty strenuously with this line of argument.  It wasn’t enough for Einstein to demonstrate mathematically that mass and energy were equivalent under special relativity.  It also had to be demonstrated that the theory of special relativity actually described reality and that cannot be derived from pure mathematics, only from empirical comparisons of predictions of the theory to experimental results.

        In other words, it’s precession of the perihelion of Mercury and gravitational lensing that demonstrate the equivalence of mass and energy (from my perspective) rather than a completely mathematical demonstration.  I find the anthropic principle to be a monumentally unsatisfying solution to any philosophical problem and I still don’t buy the “reality is just math” line (though that may have more to do with stoned idiots in college than any really rigorous philosophical objection).

        1. I haven’t decided how I feel about the anthropic principle. It’s an unsatisfying explanation because it presumes the existence of multiple possibilities, (multiple universes or sets of mathematical givens in this case,) and I’m not sure that probability really works in a way that allows you to talk about which of those universes you “end up living in.” However, if there are many universes and probability really does work that way, it’s as elegant an explanation as survival of the fittest.

          1. I can see why you’d compare the multiverse argument with evolution by natural selection, but I don’t think “elegance” is intrinsic to an argument; wrt to evolution, I think its elegance has more to do with its explanatory power than its form.  Natural selection explains particular features of the real world and would not if those features were other than what they were.  Evolution could have been wrong. 

            Compare with the infinite universe/strong anthropic principle argument.  There is no set of facts it cannot explain.  A theory that can explain anything does not explain what actually is.  It simply has no explanatory power — it explains experimental outcome A as well as it does experimental outcome B.  It’s very similar to “God did it” in that respect; there is no universe that God could not create, so no matter what our universe was like, “God did it” would be consistent with the evidence.  That’s pretty much why I’m an atheist in the first place.

          2. A theory that can explain anything does not explain what actually is.

            Good point. I think you’re saying a theory should be not just explanatory, but predictive.

      2. As to “…math is fundamental and physics models math”, well, that just reminded me of a book (which really shouldn’t be as obscure as it seems to be) called “Quantum Philosophy” by the French physicist Roland Omnès. Don’t let the title fool you into thinking it is one of those touchy-feely types of physics books (such as “Tao of Physics”). Omnès main thing is to show how common sense (and thus reality as we normally perceive it) can be directly derived from quantum mechanics. It is very impressive and along the way he demonstrates why the Schrödinger cat ‘paradox’ really is not an actual problem. Awesome :)

        1. Sure. Platonism may have been popular with Christian theologians, but believing in concepts that exist somewhat independently of the material world is a long way from believing in a deity who created the universe and sent his son to die for our sins.

          1. Unless you’re a materialist like me, in which case you’re in a similar camp what with the Platonism and all, at least as long as you can’t explain what mathematical objects ARE, metaphysically speaking.

    3. I’m reminded that you can use set theory and Cantor’s proofs to construct an alternate mathematics that is consistent.

      Lets say you construct a whole-number-like set of all of the subsets of the set of the whole-number powers of 2, each subset could represent the sum of its members.  Having done that you have a set that has all the whole numbers as a subset but also has the order of the continuum and has an infinite number of infinite elements.


      If you allow negative powers of 2 then you have the (non-negative) real numbers too.

      Mathematical thinking gets you to non-standard models easily and quickly.

    4. “God made the integers: all else is the work of man”?  (all right, you have no need of that hypothesis).  

      There is a theory that (representational) sculpture is much easier than drawing.  The human figure is in there all the time – all the sculptor has to do is to  chip off the superfluous material.  Is mathematics like that?

      How could you have a ‘physical infinity?  How would you check it, physically, in a finite amount of time ? Wouldn’t you need to check it?

  11. Maggie, I love your writing but this is the best. Thanks so much for a great article. Mind opening as usual.


  12. “This isn’t about them being stupid.  It’s about who they think you are.”

    Fantastic line.

  13. Still can’t keep myself from pointing and laughing, but I try to keep it on the inside, since I don’t like to be rude if I can help it.  No, seriously.

    But I wonder why a fundamentalist should have much of a problem with differing values of “infinite.”  At a glance, sure, I could imagine it’s due to a poverty of imagination and an unwillingness to consider infinity as anything outside the Deity Himself.  But if faced with a person who claimed that God is Himself infinite, then it wouldn’t trouble me too much to ask that person to consider fractions of His being.  If God’s power is infinite, then how do we characterize the number of miracles he could perform on Mondays, Wednesdays, and Fridays alone?  Would it not make sense that that number would approximate 3/7 of the total infinitude of miracles He could perform?  Still infinite, sure, but somehow lacking that numberless yet not insignificant quantity of Sunday, Tuesday, and Thursday miracles, right?  (There are no Saturday miracles of course, but that’s made up for by the double dose on Sundays.  Our Lord takes His rest on Saturdays, does He not?)

    I was a Christian who considered himself “born again” until the latter years of high school, and initially (I tell myself that this is because I was a kid) I held a fairly literal view of Scripture.  I used to read Jack Chick’s Crusaders comic books unironically.  Even then, I would have fit what I understand about set theory comfortably within my own theology without any perceived conflict, so I find it hard to imagine that anyone who distrusts even that particular subset of mathematics as a threat to Christian thought does not deserve hoots of derision from even halfway-educated people of faith, let alone godless high-school graduates such as myself.

    If you’re right in your analysis of their distrust of “modern” math concepts such as set theory, I’ll have to re-evaluate my approach to conversation with this type of believer.  They’ve dug themselves into such a counterfactual hole, it’s no wonder they bare their fangs so readily these days, when they seemed so friendly and harmless in my youth… they feel cornered and persecuted by everyone who keeps innocently (or not so innocently) trying to present them with factual data that doesn’t sit well with their comforting mythology.

    Sorry… just musing out loud here.  I gotta think about this some more.

    1. Regarding 3/7 of infinity: Cantor was the one who put this on solid ground.  3/7 of an infinite set is exactly the same infinite size as the original infinite set.  Even better, there are just as many points in a line as there are in the plane.  This is great fun to explain to students, who sometimes want to denounce it as well.

      1. Foolishness, sez I!  Any infinite list of weekday miracles that doesn’t count Sunday’s is necessarily a smaller infinity than the total week’s infinity!

        Burn, witch!

    2. from what little i’ve read of classical ideas of infinity, it showed up as a kind of shorthand for the absolute. it’s like perfect morality was considered as what humanity would eventually reach “in the limit,” and that this coincided with paradise. i can see how it would be disorienting to learn that infinity is just another thing subject to counterintuitive results once you start thinking about it. i guess infinity was to them sort of what quantum mechanics is to deepak chopra today.

      as far as i’m concerned, dynamical systems theory is the more imminent threat to them since it shows that certain complex systems will enter a loop or even dissipate, rather than reliably converge to something absolute. but i don’t think they could understand dynamical systems theory, so griping about infinity it is.

      and at the risk of godwin’s wrath, it’s worth noting soberly that the undesirable mathematics here does bear some similarities to the “jewish mathematics” of nazi germany; e.g. being “overly” axiomatic; reductionist; and unintuitive. (and it’s not technically a godwin, because the campaign against jewish mathematics started a few decades before hitler.)

      1.  I’ll grant you your godwins’ pass, but it is worth noting that ‘dynamical systems theory’ is just another way of saying ‘chaos theory’, which I’ll admit  I am more used to. But then that statement of  ‘enter a loop or even dissipate’ becomes problematical. If a system did this, it would be not only NOT be chaotic, it would not be dynamic; it would be static. I AM, just now, starting to like the word ‘dynamic’ over ‘chaotic’ simply because the latter word seems to carry too much baggage for the, er, great unwashed. Then again, I embraced the term ‘chaos’ because the ‘catastrophe’ models sounded so damn alarming :)

  14. It might also be a reference to New Math – which was a horrible experimental idea of a thing like Whole Word Learning where you have the child derive mathematics from the ground up so they understand what they’re doing when they add two numbers. Every number is a set – you have the empty set, a set with one set (1), etc and the arithmetic operations are related to set theory. All theoretically sound and interesting, but way too confusing for kids who haven’t even learned to add yet.

    Or they could just think sets are the unholy creation of man.

    1. Ironically, the purpose of the New Math was to fight communists (by solving the perceived pipeline problem that was believed to be giving the Soviet Union an advantage in the Space Race).

    2. Exactly. I was a grade schooler at the tail end of the New Math era. While the idea was good — to get kids into math by teaching interesting things like set theory rather than boring arithmetic, in practice it ended up alienating more students than it inspired (in part because many of the teachers didn’t really understand what they were teaching — grade school teachers were generally not math and science majors).  I can well see how set theory would be interpreted as “that worthless crap they taught us instead of the multiplication tables from the Good Old Days” in the conservative world view.

      1. Also it seemed to tend to freak out the kids’ parents, because the *parents* didn’t understand it. Perhaps because their thinking had solidified and they just didn’t want to relearn math again so they could still feel smarter than their kids.

    3.  Actually, I was a student of New Math back in my childhood. Ended up going to MIT, getting a Masters degree in engineering, and now design the computer chip you used to make the comment.  Deriving math from the ground up has its problems but is far superior to the rote memorization and mechanical crank-turning that most schools “teach.”

      1. It worked for some people. I survived it, kept going in math. But like Jonathan said, it doesn’t help when even the teachers obviously didn’t understand it and I saw a lot of kids turned off. It would have worked much better as a supplementary lesson than the mainline for kids who actually liked math already.

        And surely there is a better middle ground between rote memorization and derive from the ground up. You at least want to assure that almost everyone can do simple arithmetic, because it’ll really cost them (and everyone else) later. That was New Math’s big failure.

        1. I think the problem is in that your teachers didn’t understand it. Nor did mine. But we should be cautious for faulting a pedagogy as presented by those who don’t understand it.

          1.  Given the leap in conceptual complexity from ‘memorize your addition tables’ to New Math, the question ‘Can we teach this effectively?’ seems like an issue you can’t ignore when considering a methodology.

            It sounds like it did exactly what it was supposed to do with smart kids and well educated teachers. I certainly expect they did classroom trials first – with real math teachers who were well trained in it. Then with those positive results, rolled it out nationwide, and conditions weren’t nearly so optimum.

            Theoretically good, in practice not. And given the math teachers I remember, it couldn’t have been.

        2. The math my daughter is learning seems to have the same idea of teaching the kids to think in terms of math and more or less have them figure themselves out the rules. But not in the… um… dogmatic?… way that set theory was taught, or at least how it was taught in the school I went to. I can appreciate now what they then were _trying_ to do, but in my opinion it was just doomed to fail. It was doing the same ol’ same ol’… just with set theory. Not teaching children to think for themselves. I’m waaaaaaay more impressed with how my daughter is taught math now. (I’m honestly impressed… I have absolutely no complaints!)

          1.  That’s great, sounds like they found that middle ground I was hoping for earlier. It certainly seemed like you could have swept a lot of the set theory under the rug (till later)  and still gotten the message across.

        3. I thought I had posted a comment last night that I suspected that hostility to “the New Math” and Bourbaki lie behind the hostility to set theory, but it ended up in /dev/null. Anyway, many fundamentalists reject *any* innovation in pedagogy. Most homeschoolers learn basic arithmetic and Euclidean geometry, and learn phonics; some of this is caused by the fact that parents think they should know more about what their kids are learning than the kids do, and are confused when they see that more modern pedagogies are so unfamiliar to them.

          New Math was not the problem. The way that New Math was introduced was the problem, and the rejection of the New Math with “well, how is he going to learn how to make change at the store” was the problem.

    4. I was going to make this comment, but you got to it first. I think it’s really a lumping of New Math with other dangerous and subversive Modernist ideas, like Scandinavian furniture and key exchange parties.

      1. There’s similar “it’s different than the way I was taught, so it must be an EEEEVIL work of the devil” thinking about teaching kids to read.  That’s the phonic vs whole language debate. 

        1. I had a boyfriend at university who was taught phonics at his fundie schools.  He couldn’t spell for shit.

    5. Whole language learning is one way children learn reading. I’m not sure where you’re coming from, but it is very common for Christian, right-wing home-schoolers to attack whole language, and that’s one source of the popularity of “phonics,” which is a pedagogy based on another way that children acquire language. Each mode of learning is appropriate, and one or the other will work better at different developmental stages.

    6. In the early 60s I was enrolled in a second grade New Math pilot program and consider myself lucky to have been exposed to it. It was the opposite of horrible — I learned to use mathematical operations in a very fluid and intuitive way — so much so that I pretty much coasted through the rest of elementary school arithmetic. It seemed to me that it was the kids who weren’t exposed to it who were confused — they were learning to add without fully understanding what adding meant.

      1.  Pilot programs often have much better results than actual implementations. This is because everyone involved knows what they are doing. No one runs a pilot program by picking a teacher at random, giving them a guidebook, and then checking back in twelve months. Instead everyone involved tends to be an expert with the new methodology.

        Such is not the case when it comes time to roll the same thing out to ten thousand classrooms.

    7. Oh, yeah… I did that. I remember how on the first page of my math workbook I drew an empty set. I didn’t quite see the point of it. When we had sets again in… um… I don’t remember what grade… high school?… it was soooo easy as I had had it already in bloody first grade!!!  (Strangely, it didn’t seem as easy to everybody.) Anyway, I think the biggest fail was that the teachers (or at least the one I had) just had absolutely no idea about how to teach it. I don’t think she understood it completely herself.

    8. Yes, I’ll chime in here and say I was wondering the same thing (whether this isn’t a response to the New Math).  It certainly jibes with Maggie’s general point about the “antiquated” quality of a lot of the fundamentalists’ battles.  Forty years ago, the New Math was an innovative and controversial approach to math; this current wacky episode certainly has the appearance of the fundies wandering onto a forty-years-deserted pedagogical battlefield and sounding their trumpets to charge.

    9. I share your guess that it’s about the evils of New Math and Anti-Phonics Whole Word Learning systems, by people who tend to be concrete thinkers rather than abstract thinkers.  And since your handle here is “oldtaku”, I’m guessing you and I are both old fogies.  The school district I was in taught us Old Math for a couple of years, so I learned arithmetic up through long and short division, and then switched over to New Math, so I learned how all the stuff we already knew was Really Cool when you derived it from scratch, which made it easy to use hex and octal for computers, and the sixth grade math teacher used ancient beat-up math books so we not only learned fractions but learned how to think about word problems so we could apply math to the real world, and after that we did all the set theory stuff.

      As somebody who’s good at abstract thinking, this worked really well for me, but I suspect that for a non-abstract thinker it would have really sucked.  On the other hand, as somebody who’s better at intuitive reasoning than at learning from close observation, I really can’t evaluate Whole Word reading, but I suspect it would have really sucked for me; we didn’t call what we did “phonics”, we just called it “learning how to read”, which we did by learning what sounds the letters made, learning how they fit together, so by the end of first grade we could read any piece of paper that had ideas we could understand, and then doing lots of practice so we could learn all the horrible things English spelling does, and start learning more about how sentences worked.  (Gym class mostly used the “Do it like this” “No, not like that, like this!” “Choke up on the bat so you can hit it right!” approach, which got me labelled as even more uncoordinated than I naturally was; I didn’t really get the “Put your back into the swing” bit until I was doing tai chi as an adult.)

    10. Legend has it that New Math was born of a post-war request to the Institute for Advanced Study in Princeton from the US Navy for something to help Navy staff to understand and manage complex weapons procurement contracts. This request was passed down the IAS hierarchy until it landed on the desk of a young faculty member called Harlan D Mills (later of IBM and one of the fathers of Structured Programming). Mills advised that they needed at least a sound foundation in logic and ran a short, but successful, course in predicate calculus and set theory. The Department of Education got interested when they discovered that there were no prerequisites for this course which could therefore be taught to young children. There was nothing wrong in principle with this idea. The problem was that, as nothing in the rest of the junior school curriculum built on, or even referred to, those formal foundations, they were dismissed as irrelevant. More of a missed opportunity than failed experiment.

    1. From the context – “what we MUST believe and what we MUST be like” –  obviously this was intended as Tennants of modernism, suffused with wibbly-wobbly, timey-wimey flavor.

    2. Living in Palm Springs, I have many friends who are tenants of modernism and even more who are homeowners of modernism.

  15. “because of the tenants of modernism.”

    well, I liked this post.  but I still don’t get the ven diagram, although I’ve seen Clerks many times.  maybe if i got I.M. Pei to design my house, I could be a tenant of modernism, and all would become clear?

    edit: when i wrote mine, there were only 8 comments posted. not only did dexitrobopper have the same idea, his post was directly before me. i swear I’m not a biter ( ._.)

  16. Most of the mathematicians I know–and I know quite a few of them–are neo-Platonists, in the sense that they believe that mathematics exists somewhere out there in the land of Platonic ideals and it is the job of mathematicians to discover mathematical truth, not invent it. This is quite independent of religious belief. Paul Erdos, an atheist, subscribed to the semi-serious notion of a Supreme Fascist who possessed The Book of perfect proofs and goad of mathematicians was to recreate these proofs as best they could.

    Perhaps the textbook authors were thinking of Leopold Kronecker’sf amous statement that “God made the integers. Everything else is the work of man.” Kronecker was indeed an opponent of Cantor, but his objection to trans-finite set theory was at a much deeper level. 

    In any event, Cantorian set theory is as deeply embedded in in mathematics as evolution is in biology. 

    1. My father-in-law never really believed in negative numbers.   It wasn’t a religious thing; he was a raised-Catholic atheist, but to him, numbers count things or measure things, and negative numbers were just a bad representation of counting the wrong thing.  For instance, you don’t have a negative balance in your checking account, you either have money or you have a debt you owe the bank, sometimes even both at once.  (He was fine with fractions and even square roots and sines/cosines.)

      My wife tried discussing this with him when she was a kid.  It was sufficiently unsuccessful that she decided that telling him about imaginary numbers was just not going to work. 

      1. You father-in-law would have been in the mainstream of mathematics as late as the 17th century, when negative numbers were not accepted as quite valid. Descartes realized that polynomials had to have negative roots, but he called them “false” roots. A similar problem arose a century or so later with the term “imaginary” numbers.

      2. Hmm… well… when put like that… I don’t disagree with your father-in-law. I would say it’s just a matter of how you look at things. When you have -100 in your bank account, you really don’t have 100 minus moneys that you can go and withdraw, you have a debt. He was looking at it from a real world representation of the numbers, the -100 is an abstract representation.

      3.  I spent an entertaining half-hour explaining to a drunken Scotsman what prime numbers were, and why really big ones were valuable last night. He’s probably still counting if his wife hasn’t killed him yet…

  17. Just want to say: Great post, Maggie.

    Lots of great info and insights to things I wouldn’t have known otherwise.Thank you!

  18. Great article! You squarely hit a rarely touched crossing of two of my interests; math and religious wackjobery 

  19. There was a hilarious satire post a few years ago titled “Mathematics is an Atheist Lie”, but sadly the site is no more. Here’s a discussion that gets into the basis of the screed, that a bible verse in Kings states pi=3, thus if you are a godly person you take that as the fact, not that squishy 3.14 etc. etc. number: http://home.eyesonff.com/archive/t-106596.html

    I wondered at the time if Abeka and BJU maths texts taught that way.  It’s simultaneously disconcerting and deeply amusing when a joke turns out to be the way someone really thinks.

    1. I’ve heard a rationalization of the “pi=3” claim from a Christian perspective. The passage is referring to the large bowl that was used for ceremonial washing in the Temple of Solomon. I think the idea is that the diameter given could be the diameter of the bowl including the rim, while the circumference is the inner circumference of the bowl. When you calculate the circumference based on this, it is at least closer to 3.14.

      It’s difficult to be accurate in looking at these accounts, as cubits and hand-breadths are not standard measurements, although the ratio between them should be fairly consistent. In addition, it’s quite clear that the numbers weren’t meant to be used in a very precise way from the start; the parallel passage in Chronicles gives the volume of the bowl to be 50% larger than in this account. The spiritual significance of the numbers used would also have been more important in the specifications for the temple than an accurate version of pi, so it’s not safe to assume that the Jews of the time didn’t know pi to a few decimal places (as others in the area that traded with the Jews did).

      I never did see the rest of the A Beka series, although I grew up reading the graded readers.

  20. I’m sure I’m doing many things that make them think much worse things about me than my acceptance of set theory. 

  21. I wonder if it might have to do with Godel’s proof.  Because Cantor gets some crazy infinity going on, but Godel pretty much blew mathematics out of the water when he demonstrated that there are theorems which are true, but unprovable, because if proven they would prove themselves to be false.

    Whitehead and Russell pretty much hated set theory, because of its internal contradictions, and worked themselves into pretzels trying to  work around it.  So maybe some fundies read Principia Mathematica, and didn’t realize the authors were godless Communists.

    Also, tenets are not tenants.

    1. You cannot imagine how confused I was, after entering public schools, to learn that Josef Stalin was totally against modernism. http://en.wikipedia.org/wiki/Modernism#Criticism_and_hostility

      1. Too much popular writing about evolution is essentially Lysenkoist.  “The Bird Flu hasn’t figured out how to pass from human to human yet.”  “This mutation was successful, that one wasn’t.”  Or else it’s representing some belief that evolution is “progress”, e.g. “This animal is really primitive.” “Those fundies are really Neanderthal.” which goes along with the Marxist mystical beliefs about how history develops just as well as it does with the right-wing Social-Darwinist belief that stronger people (especially the ruling class) are better. 

    2.  Whitehead and Russell’s relation to set theory was even more complex than that; the Principia Mathematica is a huge (failed) attempt to derive all of math from set theory, basically to replace the Formalist (really the modernist position) arbitrary grounding with one in the very nature of logic.  Their failure demonstrated there wasn’t an arbitrarily absolutely valid position in math, only arbitrary choices based on what you wanted to describe.  Which, of course, is another thing to make the funjadelicals soil their shorts.

    3. Goedel’s proof really says that humans aren’t smart enough to know everything.  Anti-intellectuals are just fine with a mathematician saying that, since God knows what things are really true and what aren’t.  (Shhh, don’t push it!)

      I used to live near the town of Tennent NJ.  Lots of apartment buildings in that whole part of the state had egregiously misspelled signs about who could park where.

      1. Not quite. It would only imply this if-and-only-if the nature of human consciousness is algorithmic in nature. If our consciousness arises from a non-deterministic mechanism then none of Godel’s incompleteness theory would apply.

  22. Set theory is also one of the (many) ways to prove evolution. All those overlapping venn diagrams between dinosaurs and birds, why, it gives me the vapors!

    Which reminds me, I’m curious if any fundies have railed against the recent discoveries that T-Rex was plumed,  on the grounds that it feminizes the manly thunder lizards they grew up with.

  23. The problem of multiple infinities, and the divine implications thereof, isn’t new at all. Cantor himself dealt with it while working on modern set theory. He got around it by calling the new infinities that he discovered “transfinite numbers”, in order to make clear that these were not supposed to be equivalent to the Absolute Infinity of God. In retrospect, this was stupid, but it seems to have worked, in the respect that he didn’t get burned at the stake or anything.

  24. “Oh, the Life Science textbook says humans and dinosaurs totally hung out and remains weirdly obsessed with bombardier beetles?”

    If we categorize birds as dinosaurs, an idea which is gaining traction within the scientific community as more feathered dinosaur fossils come to light, then the question of whether dinosaurs and humans coexisted becomes an obsolete litmus test, as roughly ten thousand species of dinosaurs coexist with humans today.

    As for the textbook authors having an inordinate fondness for beetles, perhaps they’ve misunderstood J.B.S. Haldane.

      1.  Leetle tiiiiny dinosaurs…. it’s easier if you swap out “seed” for “watermelon” and “shower rod” for “Niagra falls cliff edge”. That said, I get the giggles in natural history museums, looking at T Rex and imagining him green and blue and chirping delicately at a terrified Eohippus-snack.

  25. Just to sort of paint the other side here: I am currently a PhD student in Mathematical Logic, of which Set Theory is a branch.  In my experience, logic is a much more religious (particularly Christian and Jewish) branch of math than most, in terms of who’s actually doing the research.  There are some famous examples which Wikipedia would gladly supply, but I’m more thinking about the individuals I’ve actually met at conferences, etc.  Which includes me.

    All of which makes this article super weird to read.  I guess not everybody’s getting the same memos?

    1. I believe there’s a sentiment among fundamentalist Christians that liberal Christianity is another tool of the dreaded Modernism to tear their children away from the One True Faith.

      1. just like protestantism tears (would-be) true catholics further away from christ. 

          1. Just don’t invite the Arians, or someboy is going to go all saint Nicolas on their ass.  (One the the realtively concrete stories about Bishop Nicolas is that he struck Arias for not believing in the trinity)

    2. This doesn’t surprise me entirely; some of the esoteric-sounding debates in theoretical mathematics and logic have a wiff of the theological about their abstractions.  I’m not saying that they are equally futile or meaningless, but that they might attract the same sorts of people.

      Years ago, at an informal gathering of some philosophers and mathematicians talking about the foundations of logic, I overheard one person ask another, known to be Catholic, how he squared his views on the identity relation with his views on the trinity.  He replied, ” the stuff I believe about theology isn’t *half* as strange as the stuff I believe about logic.”

  26. This clarification is very helpful, and your general point – that we should understand how and why people think these crazy things – is important, too.

    But I’m not sure what you mean at the very end. I don’t think that “ALL” that most people do is “point and laugh” at the fundies. Mostly, critics make the cogent point that fundamentalist ideas and beliefs are false, even stupidly and crazily so, and that imposing them on the public is both abusive and dangerous. The fact that those ideas are “coherent” – simultaneously believable, given a false starting point and a complete lack of critical thinking – doesn’t mean they’re not also stupid.

    As for the fact that fundies believe crazy things because they hold fearful and crazy perspectives on how they think I am, that hardly matters. Nothing about what they think of others justifies the false things they also think; the fact that those falsehoods are created in order to protect their reactionary and weird beliefs about the modern world makes them worse, not more justifiable.

    Your explanation clarifies why they think as they do about set theory. It doesn’t make it any less ludicrous.

  27. Really nice post! I am a huge fan of trying to suss out the internal rationale behind the “whacko” evangelical/fundamentalist stuff.  You can’t just call millions of people making the same, consistent life choices “crazy” and really get anywhere.  Super enjoyed it, thanks

    1. I think it’s interesting to consider which direction people are trying to go in their thinking and practices. For many fundamentalists, they aren’t looking to the future for advances in philosophy, culture, morality etc. There seems to be a lot of wishing for a return to whatever ideal situation was present in one of the great revivals, the reformation, whenever Americans all believed in Jesus, the early church or whenever. The idea seems to be that once Christian foundations are set in a culture, all of these insidious lies from the Atheists and other anti-Christians will be shown to be false (in some people’s thinking, these people are not just doing their own thing, they are actively opposing a truth that they cannot fully avoid).

      It was really odd to go to university and talk with a number of conservative Muslims who were students and teachers there. Some of their arguments and ways of thinking were uncanny in their similarity to how I was brought up (and how I still thought at the time).

  28. I’m under the impression that the objection to teaching set theory in primary school instruction is more general than fundamentalism; I believe it’s disparaged because it’s part of the New Math, introduced in the 1960s and deviating from the TRADITIONAL!!1! arithmetic instruction that dates back to the turn of the century.

    Not everybody thinks this.  There is a note from 2005 here, An informal analysis of the A Beka curriculum, that suggests the objection is to Cantor sets and such.  However, it doesn’t provide any statements from actual fundamentalists or other folks that use the curriculum that that is, indeed, the problem.  It’s speculative.

    Poking around further reveals actual statements from folks using the curriculum that are echoing all the old objections to New Math.  Example, from Our Mathematics Adventure: “It seems, and, of course, this is just my opinion, that Miquon is what was termed “new math” in the late 60s.  It reminded me a little of some of the math that I did in first grade which was called new math.  I looked up `new math’ in our new (at the time) CD encyclopedia and read it.  (See Grolier’s Academic American Encyclopedia from 1998 through 1993, the article entitled `Mathematics, New.’) After reading that, I decided that for sure we would head toward a more balanced arithmetic program.  But which one? . . . I had read somewhere of test results (which I can’t find, so if you know about these, please let me know!) that indicated that homeschoolers were above average in *understanding how to get the answer* (concepts), but were only average in being able to *get the right answer* (computation).  And after reading the Grolier’s encyclopedia article, I was looking for something that didn’t bring up `set theory.’  I wanted a program that supported my view concerning memorization of addition and multiplication tables.”

    Obviously, the right people to ask about the motivations behind the design decisions made in the development of the A Beka curriculum would be the folks at A Beka Book… assuming it’s possible to get a straight answer out of them on this point, of course.

    1. Hmmmm, yeah, you could be right about that. But I also suspect that their objection to New Math would be an objection built out of modernism, too. With the fact that New Math didn’t work very well just being an example of how awful and flawed modernism really is. 

  29. ” Chu-Carroll told me that one commenter explained the problem [about set theory] this way: ‘There is only one infinity, and that is God.'”

    A friend of mine who edits a technical publication for electrical engineering professionals told me they once ran an article on evolutionary computing (programs that compete to find optimal solutions to a problem). They got letters of complaint demanding equal time for creationist computing.

    1. “Creationist computing” is all that top-down programming stuff they tried to teach us in CS100.  Nobody really believed it, especially the bits about Goto being Harmful, or checking for input errors or testing assertions about array overruns  :-)

  30. Great post, one question though: is set theory taught in the mainstream curriculum? In Canada is isn’t. It’s only mentioned as asides to the “more important” branches of math (ie. the “math that you’ll need when you’re an engineer, kid”.) I learned about it in university, or doing my own research. BTW “Logicomics” (biography of Bertrand Russel) gives a brilliant explanation to the evil that lurks in set theory.

    1. I remember getting a teeny bit of this (in an extremely confusing manner) in high school calculus my senior year. Other than that, I don’t remember being taught it anywhere else. 

      1. I remember being taught set theory in secondary (high) school in the UK in the 80’s, along with alternative base systems and matrices. We were utterly baffled as to what use it all was and I think the teachers were too. (Except for alternative bases, which were just inexplicably cool).
        I finally found it useful when learning a new language. If you think of words as sets, then the sets of different meanings/definitions of words in different languages overlap but don’t necessarily include all of the same definitions  —my ‘red’ is not necessarily quite the same as your ‘红色’ or his ‘rouge’.

        1.  I, too, was taught set theory & number bases in the ’80s in Northern England, by an awesome (and awesomely bat-shit crazy) maths teacher by the name of Miss Kinch. I liked it. ‘course, the pinnacle of computing at the start of my secondary school years was the ZX81, then the Spectrum, so base 16 seemed to be a perfectly reasonable idea (cheat codes were complex things back then). I was  surprised to find that, on enrolling on a maths course a couple of years ago in order to do a degree, that topology, sets & number bases were considered ‘too advanced’.

      2. The New math curriculum was big on set theory, but even in its heyday, the secondary curriculum never got into transfinite sets. Infinity doesn’t enter in any serious way until calculus and even in calculus there’s a tendency to avoid thinking about it very much.

      3. We got a whole section of it in 7th grade in the late 60s.  (I’d had a couple years of traditional arithmetic in elementary school, and then my district bought into the New Math so we relearned it all from scratch.)  My mom ended up borrowing my 7th grade math books to prep for her GREs when she went back to grad school, since they hadn’t done set theory in school when she was a kid.

    1. Oh, for pete’s sake, I fixed that and it should have shown up by now. 

      Edit: The version I can see shows “tenets”. If you’re still seeing “tenants”, please try refresh. If it’s still not fixing, let me know.

  31. Sorry to quibble, but the statement

    “That’s because sets, being made of anything you damn well please, have applications outside of pure math.”

    is not quite true. You cannot make sets out of anything you damn well please. For example, the collection of interest in Russell’s paradox fails to be a set. Some collections are called “classes” because they are not “sets”.

        1.  Then making a set of all sets that don’t contain themselves, might not be for you. Also, don’t think about the “smallest counting number not describable in less than eleven words.” Basically, don’t try to do any math about math.

  32.  one commenter explained the problem this way: “There is only one infinity, and that is God.”

    I realize that this is nutpicking one commenter, but this would be painfully dimwitted even from a point of view sympathetic to the commenter’s: why not just say God is the greatest infinity of all, inconceivably more important than any mere mathematical notion?
    And the claim is bad strategy, to boot: with simple arithmetic, you can understand that the universe of integers is infinite and yet must be in some way smaller or less densely packed than the universe of real numbers. Once you have some ninny saying this obvious fact must be false for your God to be true, it will become awfully easy to start pulling at that loose string and wind up completely disbelieving in your God.

    1.  Because if God is an infinity, then the existence of ANY other infinities is going to make a certain sub-set (heh) of Christians nervous.

      And it doesn’t matter if you can prove it with logic. Just by relying on logic for your argument you have already lost.

  33. Maggie, 

    I think the criticism is one of relativism, primarily. The thought is this: If mathematics was made up by humans, then humans could have made it different than they did. Thus, the “truths” of mathematics could have been different than they are. Therefore, mathematics is relativistic, i.e., in principle, it could have been false that 2+2=4, just as long as humans had decided other than they actually did. William Lane Craig, Christian apologist and philosopher of religion, has used this argument for the existence of God: Either humans made up the mathematical truths, or God did. If humans made up the mathematical truths, those truths are relativistic. Mathematical truths are not relativistic. Therefore, God made up the mathematical truths. 

    So this is really an argument over mathematical realism vs. anti-realism, with some philosophy of religion thrown in. 

    However, the inference from ‘mathematical realism is true’ to ‘God exists’ is invalid. The traditions of Platonism and Idealism, most prominently, can countenance an atheistic worldview and mathematical realism. The Platonist says that that mathematical truths are abstract objects, not present in space and time, but that they are nonetheless objective. The Idealist says that while humans have a hand in creating the mathematical truths, that creation could not have gone any different than it did; humans were bound to create a single set of objective, universal mathematical truths. 

    This is all very complicated, of course. This interpretation of the criticism makes sense of the charge of relativism in the quotation, however.-Andrew

    1.  ” If mathematics was made up by humans, then humans could have made it different than they did.”

      Not necessarily so – in my worldview, mathematics was made up by humans, but unlike literature or art, there are constraints as to how we could make it up. These constraints don’t come from Platonic ideals or objectivism, but from the fact that those doing the thinking up are biological beings, with particular neural and physical and social capabilities, the result of eons of evolution where problem-solving and pattern-noticing were adaptive. The mathematics we are capable of creating is the mathematics we are capable of thinking about, with the brains and bodies we have now. If we had starfish shaped bodies, or 8 appendages like octopi, or existed as a distributed colony of organisms like slime mold, our mathematics might have developed in quite a different way than it has…

      1. If we had starfish shaped bodies, or 8 appendages like octopi, or existed as a distributed colony of organisms like slime mold, our mathematics might have developed in quite a different way than it has…

        Or if we didn’t have such an overwhelming psychological preference for certainty.

    2. Isn’t that merely semantics?  Math is a language that describes itself.  If humans had decided that 2+2=5, all that would mean would be that we were defining the word “five” to mean what we now know as “four.” To a Japanese person, 2+2=四. It doesn’t change any mathematical truth.

  34. I adore the fact that you included a Dr. Who quote in this article. As a former homeschooler who got fed A Beka books, I to this day find dismaying gaps in my education. Thanks for speaking up!

  35. I used the A Beka correspondence school courses from 8th grade through high school. However, with neither parent home to enforce the “school” in “home school” I just read or watched old movies and cable news all day. I rejected any notion of dong anything related to the course work when my younger brother told me that his courses (5th grade) had told him that “Rock and Roll is the devil’s music”.  Other than the crippling lack of social skills after graduating college, I actually did quite nicely despite my lack of “education”; scoring a 28 on my ACT. Which would have been nearly perfect if not for math (should have watched the math videos).

  36. I have often had an argument put to me that mathematics is proof of Intelligent design; that geometry and numbers prove that our universe was assembled by a vastly superior being, “God”. I think that perhaps the reason why Set Theory is the “Devil’s Math” is that it exposes that there are flaws or unanswerable questions in mathematics. For scientifically minded individuals, finding flaws isn’t an issue because it opens up opportunities for more exploration. However, for individuals with a fundamentalist or evangelical mindset finding flaws in mathematics could be a faith destroying event.

  37. IIRC the barber shaves everyone who don’t shave themselves (and no one else). The paradox arises when deciding whether or not he should shave himself (he should shave himself iff he doesn’t shave himself).
    It seems your barbers just shave everybody, and the cute furry set seems fine, just empty.

      1. Noted. Thanks guys. I’m a bit wiped intellectually, and may not get to this until the morning, after I can do some re-reading and make sure you’re right (I think you are). Just want to get my brain in a place where the correction is likely to be accurate.

        I hate you, set theory. Almost as much as I love you. 

        1. I think you should have either barbers or sets, but not both:

          The barber shaves everyone who doesn’t shave themselves (does he shave himself?), or
          The set of all sets that don’t contain themselves (does it contain itself?).

          The wording for Russell’s Paradox was simpler back in the days when it didn’t occur to anyone that women could be barbers.

  38. To be honest Set Theory is really, really confusing and I don’t blame the Christians for wanting no part of it. One of the major issues with the subject is that you have to be careful as to how you define what a set is in order not to fall foul of Russell’s paradox. There are several ways around that one of which is to introduce the notion of a class, and then define a set to be a class that is a member of another class, and a proper class to be a class that’s not a set. There’s also ZF set theory which gives explicit instructions on how to construct a set. None of the various candidates that have been proposed as a rigorous foundation for the notion of a set seems to be more “correct” than any of the others. Indeed it’s all rather a muddle and takes us quite far from the intuitive notion of a set as being nothing more than a simple collection of objects.

  39. They must have had a change of heart about set theory since the late  70’s. I attended Pensacola Christian School (home of  ABeka Books) through 8th grade, and we were taught some basic set theory in 3rd or 4th grade. I think it was part of the “New Math” curriculum that was popular at the time. 

    They were still developing their curriculum at the time I was there, so we used some carefully selected “secular” school textbooks. We also got xerox galleys of textbooks that were in production. Yep, they used us as beta testers. 

  40. I also have to wonder whether the Christians oppose set theory because it provides tools that people could use, if they understood set logic, to understand how holding two opposing views on related topics makes no logical sense.  This is to say that when leaders suggest that red is blue one day, and then suggest that yellow is blue on the next day, a logical person might think that everything is then blue.  In contrast, the leader only needs the followers to believe that red is blue on one day and that yellow is blue on another day in order to retain control of the flock.

  41. i was home schooled 6th-12th grade, we started out with abeka and it was too easy and cartoonish, tried bob jones and it was insanely labor intensive, we ended up using saxon books (for math).   The saxon books were pretty straight forward math books, i dont remember there being anything other than math in them, lots and lots of math

    im sure theres some kind of info about them out there that would make all fundamentalist atheists here scoff, but they seemed to get the job done

    1. I don’t have any complaints of the math textbooks I used in fundamentalist Christian school. Like I say, the school did good work right up until the point that their ideology conflicted with reality. 

      Case in point: Grammar. I diagramed more sentences than I can count, and hated every minute of it, and it’s done me good. 

      1. The only textbook of any kind that I remember from school is my first year Latin book. Not sure if that’s because I’m twice your age and the melon’s going soft or because we had more teacher time back then and relied less on reading. As I remember the teachers and the classes quite vividly, I’m guessing that it’s the latter.

      2. Hating grammar did me a lot of good, too; that’s how I ended up teaching it in such a way that kids didn’t hate it.

      3.  >Case in point: Grammar. I diagramed more sentences than I can count, and hated every minute of it, and it’s done me good.

        that’s irony, right?

    2. WTF is a fundamentalist atheist?  As a person who doesn’t believe in deities, I’m curious as to what you’re on about.

  42. FWIW, this business of not buying Cantor’s set theory doesn’t necessarily just stem from religious considerations, but from mathematical philosophical ones. Cantor himself was at least nominally Lutheran, and his choice of א to represent his infinities wasn’t just because all the letters of the Greek alphabet were taken. :-)

    I’m Lutheran, too, and the set theory I buy is http://en.wikipedia.org/wiki/Constructive_set_theory#Aczel.27s_constructive_Zermelo.E2.80.93Fraenkel because, as a matter of physics, I find no evidence for anything resembling a “completed infinity.” If you can’t demonstrate that you can compute something in a finite number of discrete steps in finite time, then I conclude you’re talking nonsense. Yes, this means that goodly chunks of stuff like measure theory, in particular the “Hausdorff sphere paradox,” aren’t paradoxes at all; they’re simply meaningless, on par with “If God is omnipotent, can he make a rock so heavy even He can’t lift it?” Grammatically impeccable. Semantically null and void.

    As a matter of mathematical pedagogy, I really wish, too, that people would quit saying “Cantor invented set theory and proved there are multiple infinities. The end.” as if the entirety of the rest of the development of mathematical logic and, in particular, information theory, from the late 19th century to the present, hasn’t happened. There are good reasons to be skeptical of Cantor. Bourbaki was a disaster. “Pure mathematics,” i.e. mathematics that doesn’t follow the lead of what’s actually physically possible, is as unhinged from reality as you would expect.

    It’ll be pretty funny if Kurt Gödel’s ontological argument for the existence of God turns out to be correct, and the Christian fundamentalists are right, and we all eventually meet God, who then says “My child Cantor got oh so close, but… *sigh*”

    1. I don’t think there’s supposed to be evidence for mathematics, nor is it supposed to follow the lead of what’s physically possible, nor is it supposed to be hinged to reality.

      Mathematics is entirely mental construction, strictly following rules one has happened to come up with, providing structures that may or may not happen to be applicable to problems one wishes to solve.

      And sometimes, different mathematicians like different rules. Brouwer rejected the law of the excluded middle, in opposition to David Hilbert. Other mathematicians reject various notions of infinity. Esenin-Volpin goes further and even rejects large finite numbers (see Harvey Friedman’s story about this).

      Who is correct? They all are, they’re just drawing out the consequences of different rules.

  43. Wow.  This explains a lot.  I was having dinner with my very nice, fundamentalist aunt-in-law.  We get along very well despite our religious differences.  And I jokingly told her how I was teaching my son about multiple levels of infinity (I’m not a mathematician, but I did take discrete mathematics in undergrad and the multiple infinities is the one thing I still remember from it…awesome stuff).

    When I told her this, she suddenly flipped into religious talk, asking me questions about my belief and what I believe in and why.  Not in a rude way, but it was unlike her since she generally just doesn’t go in that direction when she’s around me.

    Now I know why she got so excited  She must have internalized this notion that there can only be one infinity and that is god.  Thanks for the explanation.

  44. TIL that fundies will not produce a cadre of good computer programmers, or anything good with computers. As a consequence it’s one more highly lucrative job market their children will not get into, yay.

  45. I’m surprised Christians don’t use the multiple infinities thing as support for the Trinity doctrine – the idea that The Father, The Son, and The Holy Spirit are all distinct entities but also one God. I remember that one puzzling lots of my fellow children in Sunday School.

    When I was in high school a fundamentalist Christian teacher was teaching the class about the theory of spontaneous generation. She told the class that since spontaneous generation had been disproven by Louis Pasteur in the 19th century anyone who believed in evolution was a fool. Maggots don’t spontaneously generate from rotten meat, after all.

    1. Yeah, difficult to explain metaphors without admitting that they exist …
      I just heard a nice thing the other day: All important religious works were written by people with a great intellect. And people with great intellects use language that is not literal but uses metaphors to explain abstract concepts.
      There’s no problem with religion, there’s a problem with denying the existence of figures of speech in the bible.

      … I think it was Michael Palin on “Life of Brian” …

      1. Most religions are, at their core, metaphorical philosophies that 90-99% of their adherents don’t comprehend.

        1. Really? It’s my impression that most religions are, at their core, a set of weird rules to enforce social cohesion and allow for robbing or killing your neighbors.
          All the sensible stuff – halfway sensible hygiene rules, rule of law, metaphysics – seems to be tacked on by people who had to stay in the framework.  Unless they wanted to get robbed or killed.

  46. Based on “A Beka Book teaches that the laws of mathematics are a creation of God and thus absolute” I wonder if the actual problem with set theory is that it leads to Goedel’s Incompleteness Theorem.

    1. Bingo!  God, by his very properties, knows everything.  Goedel tells us that there are necessarily true things no one (god included) can prove to be so.  This would be very challenging to the core fundamentalist concept god has it all together.  It would be much easier to dismiss Goedel’s ideas on the basis that it is a consequence of linguistic absurdity rather than deep mathematical truth.

      1. to be honest I don’t think it follows. I’ve known quite a few religious people use Godel’s incompleteness theorem(s) to argue that they reveal a limitation on the capacity of human beings to discover truths using the tools at our disposal, one of the most important ones being of course the axiomatic method in mathematics; whereas God has no need of these methods, since He has direct access to all truth, and indeed knows everything. I had a discussion once with someone who argued that since God couldn’t enumerate all the items in an uncountable set (which would take longer than infinity) and that there were likely at least as many truths as there were real numbers that therefore God couldn’t know everything. To which I replied, yes, but if God were all knowing why wouldn’t God know everything simultaneously, why would He have to enumerate it all?

  47. +1 for working “wibbly-wobbly, timey-wimey” in without forcing it.

    In fact, +infinity. So long as it’s not in your own timestream.

  48. One thing I’ve noticed from internet discussions is that Cantor’s proofs get a lot of interest from amateurs.

     They’re actually very easy to understand and talking about infinity makes them sound exotic and interesting.

    I read a comment on a blog suggesting that fundamentalists don’t like set theory because it’s hard, I suggest the problem is the opposite.  It IS easy enough for them to understand – and having students think deeply about anything other than the bible is what they’re afraid of.

    You’ll note that fundies are also known for opposing relativity but not quantum mechanics.  I suggest the reason is the same, it’s not hard to understand at least special relativity, so once again it can be an easy conduit into non-religious thinking (in this case, physics).

    But quantum physics is deeply confusing so they don’t mind that.

    1. Good point.  It’s not like they’re objecting to partial differential equations or Fourier analysis or some other relatively advanced or involved topic.  Set theory is easy to get your head around.  (Hence the New Math, which taught set theory to elementary school students.)

  49. God, please, nobody tell them set theory was created by evolution. (Reference: Where Mathematics Comes From, George Lakoff and Rafael Nunez).

  50. I guess there are two reasons for fundamentalist Christians (or any religious fanatics really) to have a problem with set theory.
    One: When it was introduced to schools (at least in Germany, but presumably also elsewhere) it was “the hip new thing”, and it came together with liberal education ideas that are of course a problem with some people. 
    And to be honest: The first incarnation of both concepts had problems. I’ve had a teacher who was so “liberal” (mind the quotation marks!) she wasn’t able to get the class to sit still for a minute, and a few years later I saw what kind of abuse she was willing to take from her own son (who interestingly turned out quite well, another few years later). Likewise, the first curricula of set theory (which I had in second grade) left me with an impression of … “well, yeah, wasn’t that ovious? If we have a set og squares and one of blue shapes, then a blue square is both a sqare and a blue shape, duh!”. That’s because set theory was new in schools and noone knew how to handle it. It was praised by mathematicians as big important thing, taught by teachers who didn’t quite see it that way in a curriculum developed by school authorities who didn’t know what it even meant. Much like CS a decade later.
    After that, I’m not angry at anyone who thinks it doesn’t need to be taught. Of course, it is still the basis of logic, and that’s a big thing. Especially when dealing with lots and lots of false binary decisions in public discussions, I wish people knew more about set theory.

    The second reason goes  deeper than the article makes it sound: Set theory can be used to conclusively prove that not only mathematics is infinite (i.e. cannot be described by a finite set of rules and definitions), but that this extends to every other logical system. That means: Physics is infinite! Good bye to grand unified theory (at least as some imagine it), and more important: This proves that _not_ every aspect of life, the universe’s creation and the world in general is covered by the bible. Gosh! This means there are moments where we have to go and think for ourselves! So this is where math becomes philosophical, and where not just Cantor but also Gödel and others faced stiff opposition for a long time from mathematicians, physicists and philosophers (like, the whole Vienna Circle), because it was such a revolutionary idea about not just some obscure math bit but the universe and logic itself.
    As I said, there’s a mathematical proof for all of this, but I kind of get that some people are not willing to accept it. Even though I do wonder how many people even know about this sort of thing, given that usually you don’t learn about it unless you study number theory or  epistemology … it’s also actually a bit unsettling sometimes, regardless of religious affiliation: No logical construct can ever be complete … think about that the next time you have an argument!

  51. Reminds me of the claim that Alice in Wonderland was written as a response to the introduction of non-Euclidean math, and the horrors that would bring.

  52. Modernism smashes all of that and says, “Hey, just do your own thing. Nobody’s ideas are any better or worse than anybody else’s. There’s no right and wrong. Go crazy, man!” [Insert obligatory bongo drumming session]

    I’m gonna mostly agree with what you say here, Maggie, but I have one quibble, and it’s with the above quoted bit.

    Mostly because I need to remind you that Taylorism, Objectivism and collectivist movements like Marxism are also modernist movements, and their philosophy was less, “do your own thing” and more, “we must destroy the Old Ways of superstition and unscientificness, and replace it with a better, more logical system!”

    Which, I suspect, are what many fundamentalists think modernists are doing anyway.

    1. Well, I’m not so much describing modernism here, as I am describing modernism as it was taught to me by Baptist school and Jack Chick tracts. Both of which tend to focus on the “do your own thing!” mantra at the expense of any other modernist idea. 

  53. Hard to believe there’s a discussion of transfinite numbers and Cantor on boingboing without mention of Rudy Rucker’s novel White Light, which was my first introduction to the infinity of infinities. http://en.wikipedia.org/wiki/White_Light_(Rudy_Rucker_novel)

  54. Great article. I have a question though: Cantor and others have offered proofs of the various levels of infinity, and some of those proofs are even quite simple. Do fundamentalist churches have to deny the legitimacy of mathematical proofs in order to justify their dismissal of the theorems of set theory? How can they simply ignore an entire field of mathematics, given that the field contains well-established proofs of theorems about infinity?

    1. I thought about making that same joke. Only with the burrito so big even He cannot eat it all in one sitting. Then I wasn’t sure whether that was accurately related to set theory or not. 

  55. Both the article itself and the comments here have been amazing… lots of stuff to look into further I was only vaguely familiar with and we somehow managed to avoid the standard anti-religion arguments that happen in every other comment thread about religion (I’m atheist and agree with most of the arguments but it’s just the same thing every time, except this time).

    BTW Maggie, William Gibson praised this article on Twitter, check it out if you missed it. I agree with him and while I don’t have anything intelligent to add to the conversation, I didn’t want to skip out on telling you how much I enjoyed all this.

  56. How much of this do you think is actually a rejection of the New Math and Bourbaki (rightly, in my opinion) emphasizing the teaching of set theory in the early grades over arithmetic?

  57. Watching these folks trying to teach modern math and science their way is kind of cute, like a cargo cult trying to steal the white man’s mojo with crude imitation and invocations.

  58. There is another aspect of the aversion to set-theory based infinity which you didn’t mention – and if it was mentioned elsewhere in the comments I apologize but I stopped trying to read them all after about aleph null.

    It’s this: “Infinity” sounds like something that’s hard to understand (it’s not), universal, all encompassing. Ineffable. The “number to rule them all.” So it was a convenient metaphor or even an equivalence for GOD in nineteenth century philosophy. So if you mess with that you mess with a lot of the ways Christianity tried to smuggle God into science and mathematics and philosophy. What’s greater than all things, contains all things…? God. What’s greater than that? What, a bigger infinity? You blaspheme!

  59. This thread represents BB at its best: civil, informative, and entertaining. Thanks to all of you, especially MK-B for inspiring it.

    Of course we anthropologist types think maths are human languages, thus replete with paradoxes and obfuscations — and no small amount of beauty.

  60. I wonder if their problem with set theory is even deeper – that it clashes with the dysfunctional compartmentalized thinking that seems required for a lot of modern conservatism.

    From my personal experience, many conservatives are only able to avoid dealing with the contradictions inherent in their belief system by roughly compartmentalizing them off from each other.

    For example, its ok for boys to want sex but it’s not ok for girls to want sex, but boys must only want sex with girls. The only way to avoid the cognitive dissonance of mores like that, is to never think about boys, girls and gays as intersecting sets.

    This same siloing of different sets occurs with the distinction of ‘good’ science (electronics) from ‘bad’ science (climate science, evolution, modern economics), of Republican-initiated military action vs. Democratic military action, even (or especially) when the Democratic action is less costly and more effective, medicine (viagra should be covered but contraception should not be), and so on.

    So, there may be a very instinctive antipathy to the integration of information that is essential to set theory. They may *want* to keep different sets of information separate, as a key to avoiding their own cognitive dissonance as well as a way to ‘teach kids right’ and not unravel the mess their parents’ heads are in.

  61. Great video, but the version of Cantor’s diagonal proof starting around 3:40 is flawed. Any number that can be written as a decimal numeral that ends with an infinite sequence of 9s can also be written as a different decimal numeral ending with an infinite sequence of 0s (e.g., 0.9999… is the same number as 1.0000…), so proving that a particular decimal representation of a number doesn’t appear in the list doesn’t prove that the number isn’t represented in the list.

    The versions of this proof that I’ve seen deal with this by ensuring that each numeral used in the proof represents a different number (e.g., by only using numerals that don’t contain 9s). That way, you can prove that a subset of the real numbers is uncountable, so the real numbers themselves must also be.

    1. “0.9999… is the same number as 1.0000”

      Isn’t that approaching infinitely close to an asymptote rather than being the same number?

      1. Intuitively, it would seem that way, but it’s really the same number. You can think of it as the limit of the series 9/10 + 9/100 + … + 9/1000 + … as the number of terms approaches infinity.

        It turns out there’s an entire Wikipedia page about it.

  62. Can I add one more thing into the mix here? When I was a primary-age kid (we started at 5 in the UK in those days) we had plastic rings that could be used to make Venn diagrams on the carpet, and our teachers talked to us about the basics of set theory in a simple way. When I was 18 and studying mathematics for A-level exams, set theory reappeared (much more comprehensively, of course!). Our teacher at the time said that the reason it had appeared on the British primary school curriculum in the late 70s and early 80s was part of a movement called “logicism”, which was hoping to prove that all mathematics could be reduced to logic. Philosophers like Bertrand Russell were very keen on this, and it got into the consciousness of educationalists. “If set theory is the basis of all of arithmetic, and possibly all of mathematics – then we must teach it to our children!” was what the educationalists thought. Of course, years later the interest in logicism faded – too much mathematics wasn’t reduceable to logic – but I wonder if this may also have been the root of the fundamentalists dislike of set theory? Anything that connects maths to humanist philosophy is going to be bad in their book…

    For me, learning all that set theory did pay off – I got interested in computers, and later in electronics. My biggest learning epiphany was being shown Claude Shannon’s idea that logic circuits (made from switches or logic gates) can be represented as Boolean algebra – and that simplifying the expression and translating back to a circuit results in a simpler circuit. Pure genius, and the underpinnings of the whole digital revolution!

  63. Another book that satisfactorily explains the Platonist and non-Platonist (including Brouwer and the Intuitionists) points of view about the foundations of mathematics is Morris Kline’s ‘Mathematics: The Loss of Certainty’ (1980).

  64. I used to have a book on formal logic (1950s vintage at a guess) which insisted on all the traditional moods and figures of the syllogism, with occasional denunciations of modern symbolic logic as a godless Commie plot to sap the moral fibre of the West. Or something very like that.

    (Hey, my BoingBoing login — admittedly not used for a while — no longer works. Did I miss a “we deleted all the old passwords” post? Over to OpenID …)

    1. Over a year ago when we switched from Movable Type to WordPress, you had to reset your password. If you still have access to the e-mail address that you used then, you should still be able to reset it. Probably.

  65. I’m a professional mathematician and happened to read Beka Books when I was a kid. I did notice this comment about set theory a few years ago and wondered what kind of set theory they were talking about.

    Beka do respond to emails (I have emailed them). Unfortunately, many of their old authors no longer work for them. They try to track them down if you ask, but they sometimes can’t get answers.

    I strongly suspect this has nothing to do with Venn diagrams or simple set theory as it is learned at school, but to do with modern set theory, which, as a professional mathematician, I would not burden kids with (nor anyone else in this thread).

    Concrete concepts are required until a certain age. In fact, most members of the public, atheist or not, get through their entire life without ever understanding any modern mathematics.

  66.  And here I was thinking Andrew Schlafly’s insanity (General Relativity leads to moral relativism! Black holes are a liberal plot! Complex numbers are inherently suspect!) was unique.

  67. I encounter something similar when discussing the foundations of ethics with fundies.  I argue that ethics rests on a consistent and thorough use of the golden rule, which is really just a simplified and only slightly misleading version of a more precise principle philosophers call “moral supervenience.”  Anyway, the fundies don’t like this approach, despite the obvious support it gets from the Bible, both explicitly in the new testament and implicitly in some stories from the older bits.  Why?  Well, first off a lot of the conclusions that would follow about morality contradict other moral views in the Bible, especially those they want to foist on other people (would you want others to impose rules from their culture’s ancient scriptures on you?  well then…)  But more basically, it’s too rational, too empowering, too non-theological.  The GR is found in every culture and religion, not just Christianity; and it’s really just an expression of the requirement to be consistent in our moral judgments.  It’s something rational people can and have figured out for themselves with no help from religion, and so allows us to work out morality without divine intervention–or, in the absence of that, the interpretations of ministers and the traditions of human churches.  It would mean that morality doesn’t need religion.  And the possibility that they don’t own morality frightens the S*** out of them.

    And so when I talk about how a little complexity enters in when higher-order thinking is needed to correct certain naive mistakes in using the GR (“no, the masochist who pokes other people with pins is *not* really following the GR, because he wouldn’t want other people whose desires he didn’t share to impose those desires on him, like the guy who likes being tickled by feathers, or something else which the masochist can’t stand…”) they try hard not to listen or understand the point.  They try to somehow hear this as just another arbitrary idea about ethics, to act like I’m trying to confuse them with obfuscations and subtleties, and the devil must be behind all that, somehow.  

    1. It’s funny (in a depressing way) how little complexity needs to enter in to trigger such stonewalling.  If a believer puts forward the idea that “God is the only manifestation of infinity,” when confronted with a simple rational number like 1/3 presented in decimal form, with its infinitely repeating threes after the decimal point, do they throw their hands up in terror and decry repeating decimals as the work of the Adversary?  Do they pick an arbitrary number of significant figures and just assume that 0.333 (for example) is close enough for ecclesiastical work?  Can a devout Christian find work as a precision machinist?

      That is pretty inarguably silly.  I don’t personally know any Christians with an allergy to the secular infinite… nor ones who ascribe any divinity to such mundane things as repeating decimals.  Do they actually exist and hold down jobs and move in respectable circles?  After all, a smart one might realize that 1/3 written in base 3 would simply be 0.1 with no infinite repeating necessary, right?  Wouldn’t it be far simpler to just reject the premise that “Infinity=God” and realize that if one insists on worshipping an omnipowerful, omnipresent, and omniscient Deity, there’s no good reason why one’s faith in Him should feel threatened by all those harmless little 3s following the decimal point.

      Like George Carlin used to ask the priests back when he was in Catholic school, “Hey, Faddah, if God is all-powerful, can He make a rock so big that He Himself can’t lift it?”  Even schoolkids can grasp the idea of that paradox: if God can’t make a rock too heavy for Himself to lift, then He’s not quite all-powerful, is He?  But that just illustrates how stupidly useless such terms as “all-powerful” are; the limitation isn’t so much placed on the Deity as it is on the simplistic terminology, and that’s what leads to these pitfalls in the paths of such weak imaginations who are quick to proclaim that only God can be infinite.  Recent Popes have the imagination and resilience of faith to acknowledge evolution as “an effectively proven fact.”  Why should mathematical concepts threaten anyone’s faith, if said faith is worth having in the first place?

      I lose track of all the simplistic, childish concepts to which some fundamentalist Christians must cling in order to preserve their worldview.  They’d get a lot farther if they’d just try to make their language and thought processes a bit more sophisticated.

  68. Obviously, the author doesn’t know a lot about set theory. But this is an interesting article nevertheless. The main reason that Christian Fundamentalists would object to Cantorian set theory is NOT that it mixes the subjective with mathematics, but that it gives an account of the infinite. Yet according to many Christians only God is infinite and the infinite is not comprehensible nor theorizable. The infinite is not to be found in the natural world. Indeed, this debate between anti-infinity Christians and pro-infinity mathematicians goes back to the 1700s (at least) when Bishop Berkeley argued that the introduction of the calculus would lead to atheism. The calculus presupposes actual infinities in order to be given a rigorous foundation. Berkeley knew this. And he knew that by theorizing the infinite and placing it IN Nature, God would be left out. In a sense, he was right. 

    1. Looks like you’re the one who doesn’t know much about mathematics.  The calculus does not suppose presuppose actual infinities; that’s part of what the method of limits helps us with.  It certainly doesn’t ever suppose that there are “infinities” in the real world, except in the sense that one can measure real objects in an infinite number of potential ways, and sum up those potential ways using an algorithm that spares us from the impossible task of actually doing each one, to tell us something interesting about the real objects.  But you don’t start or end with an actual “infinity,” any more than an ordinary circle contains “infinity” somehow simply because the decimal expansion of pi is non-repeating.

      1. surely if you’re working with the real numbers then not only are you presupposing that you have an infinite set of values to work with you’re also assuming that between any two points on the real line there are an infinity of other points…also I believe that calculus was originally founded on the notion of infinitesimals and the whole notion of limits came much later, which would at least explain ecosophy’s comment re:Berkely.

        1. also come to think of it, how would limits actually work unless you had an infinite set? if a limit of a function as it travels toward some point is infinite then there is no bound to how large that function can get as you get closer to that point and so essentially you’re dealing with infinity.

        2. But the “infinite values” are not necessarily sitting out there in the world for you to observe.  Even if they were, we would never have had time to observe them, and never will, so any such claim is entirely irrelevant to the math itself.  Hence whatever we might be doing in any branch of mathematics or life, observing infinite sets of values is no part of it.  Again:  we’re just noticing that it is possible, in principle, to make observations/measurements without limit, from an infinite number of points, or infinitely recursively, etc.  We never actually *do* this.  But we’ve found clever ways to figure out what some of those observations would approach, as a limit, without even actually going through with such a measurement, or even making the absurd claim that, “well, this is the result we’d get *if* we made an infinite number of observations,” since that obviously could never be done and so there is no such result (of course we can speak this way metaphorically, but it’s only a substitute for the more subtle and accurate language of limits).

  69. Another point to make here that was missed by the author: The Cosmological argument (for the existence of God) rests on the Aristotelian/Thomistic assumption that an actual infinity (in Nature) is impossible. If physics describes the world and it uses a mathematical theory (i.e., calculus) that presupposes the existence of an actual infinity (as is does) then it would seem that there are actual infinities in the natural world after all. So, the assumption of the Cosmological argument is incorrect. William Lane Craig (one of the few fundamentalist Christian philosophers) continues to employ the argument nevertheless and has to disparage Georg Cantor in order to do so. BTW, it is interesting that Cantor did believe in God and incorporated the latter into his overall philosophical approach (but not in his mathematical theory).

  70. This is an excellent post.  I don’t know fundamentalists or fundamentalism so my guess that you are exactly right is worth nothing.  

    On the other hand, your effort to describe impossible sets is unsuccessful.  The sets you describe exist.  They are the null set.  You have just discovered that the same set can be defined in different ways.  It is the empty set.  It is the set of all barbers who shave everyone but not themselves.  It is the set of all cute furry animals who shave everyone but not themselves.  The fact that the same set can be defined using different words is not problematic.
    The problem is that you want a simple example and the examples of impossible sets are not simple.  The one I know of is Russell’s paradox.  With the definition that a set is just a bunch of things, one can imagine a set whcih contains itself (the set S contains a cute furry thing, all whole numbers, a barber and the set S).  These are weird sets.  They seem impossible (just because infinity seems impossible) but nothing has ruled them out.  So we can consider the set of weird sets (each of which contains itself).  So we can consider the set of non weird sets (the set of all sets which don’t have themselves as an element).  Is the set of non weird sets weird ?  If so it is not a member of the set of all non weird sets so it is not weird.  If it is not weird then it is a member of the set of all non weird sets so it is weird.  

    The set of all non weird sets can’t exist.  It isn’t empty.  It contradicts itself.

    Now this example is confusing.  It has to be.  The impending logical contradiction disorders our brains (it’s what they do).  But there is no way to give a simple clear description of a set so strange that it can’t exist.

    1. It is not correct to say that that contradictory sets are the “null set.”  (This is a common mistake in informal discussions of this topic.)  The null set is an empty set, a set with no elements.  It’s not a complicated or tricky set; it’s a set that basically says, “Nothing in here.”  So saying that the kinds of sets described by Russell’s paradox are examples of, or described by, the null set, is saying *definitively* that the contradictory sets “don’t exist.”  But the whole point of contradictory sets is that they’re undecidable — they can’t be said to “exist” or “not exist” because they can’t be coherently described.  They don’t “belong” to the null set, they derail the whole system of set-theoretical description.  The null set itself doesn’t derail anything; it’s a basic building-block of that system.

  71. It’s funny. Through all of the math courses I’ve had, they never really covered the different sizes of infinity. But in these cases I more imagined that the multiple infinities were the same size, but that they had different densities. It amounted to how quickly you “traveled to infinity”. Still, the concept appears the same. You’re just taking the denser infinities and spreading it out to match the less-dense infinities before comparing the size (so to speak).

    Now for religion and math/science, I figured God created the universe bound by certain rules based upon structure and modeling. Math is an attempt to abstractly model the conceptual basics, and (like physics) every so often issues in what we are thinking the modeling is proves to be off and needs correction.

    So the issue with the fundamentalists is that they already think they know the mind of God and all of the rules everything is based on, whereas the mathematicians and scientists recognize that sometimes mistakes are made or gaps were missed. (Yes, anyone who challenges an established view is harshly criticized by others, but that is rightly so. It’s the survival through extremely harsh scrutiny that helps prove the new idea is correct… or more correct than the previous model. It’s better than being accepting of ideas which turn out to be complete crap and act as steps backwards for everyone.)

    But getting back to these fundies v. mathematicians/scientists, if you look at the two groups and how they perceive flexibility, then these fundamentalists are showing not only arrogance, but also extreme hubris. “We know what God did, and that’s that. If you speak against this, you deny God’s word.” *sigh* New ideas in math and science have never tried to disprove God. All it does is state, “The universe operates a little differently than how we thought.” For those who believe (like myself), that simply means “God made things a bit more wondrous than how we previously thought.”

    And I like that.

  72. Hi Maggie – nice article but unfortunately both your examples of self-contradictory sets are perfectly consistent as written! The first one, the set of cute furry animals, is simply an empty set if furry animals are not cute. And Russell’s barber paradox is phrased here in a way that makes it straight forward (if trivial): you start by saying the barber shaves everyone AND […]. Since the union of everyone and any other set of people is everyone, he simply shaves everyone (including himself).
    Also, it’s not quite right to say that an infinite set of whole numbers has smaller cardinality than an infinite set of decimal numbers. Here is an infinite set of decimal numbers: {0.5, 1.5, 2.5, 3.5, 4.5, …}. This set has the same cardinality as the set of whole numbers (aleph nought). What you mean to distinguish is the difference between the cardinality of the set of all whole numbers, and the cardinality of the set of ALL decimal numbers (strictly speaking it should be “real numbers”, as the quantities do not have to be represented as decimal numbers, which is a notational feature of a quantity, not an intrinsic one). Cantor’s diagonal proof that these are not the same is simple enough to explain to a middle-schooler, and an awesome example of the power of simple rational thought to arrive at a completely unexpected conclusion.

  73. Must say that the A Beka Books grammar workbooks that I used daily in the 80s are much better than anything my kids are given these days. 

    Maybe proper punctuation sits outside the set corrupting modernism?

  74. Maggie,  thanks for the BEST post I’ve read on BoingBoing in a long, long time. You presented the scientific view in a respectful way and free of the knee-jerk anti-Christian intolerance displayed by all too many.  

  75. Seems like the main Christian opposition to set theory would have to do with the incompleteness theorem, which is at odds with the idea of God’s omniscience and omnipotence.

  76. As a Christian, I have no problem with set theory.  I do believe there is only one God who made all that ever was, is and will be – and I also believe you can perceive it all as divided as you choose to.  See there may be an infinite number of whole numbers and an entirely different infinite amount of decimal number, each with a varying degree of value…but the BOTH share the one trait: infinite.  Where did that trait come from? Where did the theory of infinity itself come from? How can we even fathom that concept? Maybe because we are part and parcel of creation as a whole, which includes the aspect of infinity – which is a part of God’s nature and hence inherently expressed in His creation.

    I would challenge those fundamentalists in this way: what is going to happen to your soul when your body dies? Will it not, according to your understanding, continue on infinitely with God? As with all the other saved souls? Okay, well then there is the SET of souls who will live out infinitely in heaven – and then there is God, also living out infinitely in heaven. You cannot say that God is us and we are God, they are separate, of differing value, yet both infinite.

    Once again, religion compliments science and vice versa – they cannot truly conflict, for science is the discovery of God’s work.

  77. This is strange.  Numbers aren’t real, they only represent reality.  Sure, the set {n, n+1, 1, empty set} is real in mathematics, but is it true in life?  I mean, are there really an infinite set of basketballs – or anything represented by whole numbers – to know and see?

    It’s strange to think this way.  This is exactly the issue with fundamentalism across the board – thinking up new ways to exclude others and the “world”. 

  78. You’re right, it’s not about the fundies being stupid, it’s about them cynically — albeit correctly —  assuming their target audience is stupid.

    1. I’m going to say there’s some wilful doublethink going on here, also. They want to fool the rubes – but it’s also because they want to think the rubes need to be fooled for their own good, because Ultimate Truth is above Icky Tricky Liberal Secular Logical Truth.

      It’s that special category of chosen delusion…

  79. “If all you do is point and laugh at the fundies for calling set theory evil, then you are missing the point. This isn’t about them being stupid. It’s about who they think you are.”
    I fail to see what you can possibly do besides point and laugh at them. Reasoning with them is utterly impossible, so mockery is the only remaining recourse.

  80. Richard Feynman had some real problems with teaching set theory in American schools .. just so you know.

  81. I do believe this is the best article I’ve ever read on BoingBoing. So good that BoingBoing’s won me back.

    As one who also suffered a fundie/evvie/ Jesus-Camp schooling (until I escaped), I’d also suggest that it’s possible to put more thinking into fundie logic than they themselves justify. I’ve heard “It’s not in the Bible so it’s false” thrown around – the same reasoning that tries to strain itself into believing that Pi=3 because that’s the measurements given in Solomon’s temple.

  82. Just another note of thanks for your article.  I’m always happy to hear about the thought processes that go into what seem to be incomprehensible decisions.  The set theory issue had me stumped.

    And it’s always nice to hear explanations about cultures vastly different from our own that aren’t filled with rancor.

    1. I agree that this is the correct statement. This “paradox”, by the way, is easy – there is no such barber. So it isn’t really a paradox.

      What you can have is actual paradoxes phrased a bit like that. Russell’s paradox: Let S be the set of all sets that do not contain themselves. Does S contain itself?

      If you try to say “there is no such set”, you have just stated that we are *not* allowed to define sets however we want, as made out of completely arbitrary entities; a valid view, but one that then one has to keep in mind whenever one is doing set theory. Moreover, which sets can’t one build? Do you forbid yourself from talking about the “set of all sets”? Or from the notion of “containing oneself”? The latter could happen in indirect ways, so it is then necessary to rank sets, assigning to them different “types”, with the definition of a set being allowed to refer only sets at a lower level.

  83. They must have real problems with  Quantum Mechanics.  I mean all that weirdness about things not being in one set place, but instead being in a probability cloud (and probability has to be something they also have problems with).  God would not allow such a thing, would he.  Newton would have to be the last thing in physics.

  84. @boingboing-7160c7db52df96e5fe196a6c9ce73f83:disqus  “It’s hard to be shocked by stuff that you long ago forgot isn’t general public knowledge.”

    Thank you from the bottom of my heart for posting this. I can’t tell you how much this particular line resonated with me. 

    I went to a Fundie school in Minneapolis from 1-9th grade. In history we learned all about revivals and nothing about America’s involvement in Vietnam – all from Bob Jones, who routinely came to our weekly chapel sessions to convince us to avoid “secular” universities, etc.

    There weren’t too many schools like that; it would be an incredible coincidence if we were alumni.

  85. Do you really think the wealthy 1% will be sending their offspring to these schools teaching superstition and nonsense? It is a plan so elegantly simple and densely diabolical … teach the proles mythology, teach my kids science. Imagination is not required to envision the victorious in this scheme. http://inkandpaperguy.wordpress.com/2012/08/09/usaandreligion/

  86. Just remember fundamentalist christians are a set. Not all Christians are in that weird set. We may be weird though, which puts us in the weird set with other weirdos. I for one have no problem with evolution, especially because it means that God didn’t make me this awkward on purpose.

  87. So, A Beka is more likely pre-modernist, not post-modernist. The reason that a pre-modernist might be against modernism is because it puts a possibly unwarrented focus on the ability of man.

    Like a modernist, a pre-modernist believes in absolute truth. It has nothing to do with  wanting to “just do your own thing”.  I’m not sure how the author argued this, considering that he just accused fundamentalists of believing mathematical truth to be absolute in reality.

    The biggest flaw with this article I’d say is that the author doesn’t appear to know enough Mathematics or philosophy.  The idea alluded to with his first quote

    “Unlike the “modern math” theorists, who believe that mathematics is a creation of man and thus arbitrary and relative, A Beka Book teaches that the laws of mathematics are a creation of God and thus absolute…”

    is not limited to theists. The debate about whether logic/truth is arbitrary or necessary extends far beyond theism/atheism. There are strong transcendental arguments as to why it would be necessary, and people do believe this- theist or not.  Beck only teaches Platonism as an implication of Theism, which is not wrong in itself.

  88. It’s because when people draw these they inadvertently end up drawing vaginas, and there’s nothing that makes a fundamentalist more angry than a vagina.

  89. I dunno — sounds like the author of the post may be confused.

    For a start, looking at set theory as the same as cantor’s work is a bit confusing. You need to look at the concept of new math, http://en.wikipedia.org/wiki/New_Math which was a trend in the US in the 1960s to teach primary school kids set theory and working in different bases, rather than focussing on elementary arithmetic. Which is why so many US people educated then can’t add up. It’s a pedagogically questionable decision, and selling your textbooks as working with traditional maths instead of “modern set theory” does actually make sense and is potentially a good idea. Even if it is somewhat usual these days, and is using “modern” to mean “the cutting-edge ideas of the ’60s”. This seems a lot more likely than fundamentalists being worried about concerns of the Generalised Continuum Hypothesis taking youngsters’ focus away from the Almighty.

  90. modernism DOES NOT SAY ALL IDEAS ARE EQUAL, nor that there is no right and wrong

    You are thinking of – and also misstating, though not quite as badly – postmodernism.

    Modernism posits right and wrong and true and false, it just says those are arrived at through reason and science, not through obeying authority or reading the bible.

    man, you are SO FAR OFF the mark with that.


  91. The discussion of the fundamentalist attitude to the infinite took me back to 1990. Sitting in the Deadwood in Iowa City, I was introduced as a professor of mathematics to a very personable young lady, who immediately said that she’d heard that mathematicians had proved that there were different kinds of infinity. I allowed as how that was so, and that while it wasn’t exactly suited to bar conversation, it wasn’t very difficult to understand, and I’d be happy to explain in a quieter setting of her choosing. I was assailed with a long rant, not on sexism or bad pick-up techniques, but on how I was misusing the word “infinite”, which could apply only to God. Attempts to suggest that she could use “infinite” her way and I would continue to use the technical sense were met with scorn. I was eventually allowed to return to my beer, and resolved never to use the word again except amongst like-minded perverts.

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