Set Theory, Now?

During my time in New York, I had lunch with some friends from England. We were discussing evolution and creationism, and religious fundamentalism more generally. Somewhere along the line I mentioned that creationists routinely use mathematical arguments in their writing, and one of my friends replied that he had heard that some fundamentalists even have a problem with set theory.

I stared at him. I thought I was up on the latest in pseudomathematics, but this one was entirely new to me. But with a few taps on his phone he showed me what he was talking about. He was referring to this paragraph, from the website for A Beka Book. They are a fundamentalist Christian publisher whose materials will now be supported by taxpayer dollars in Louisiana.

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. Man's task is to search out and make use of the laws of the universe, both scientific and mathematical.

A Beka Book provides attractive, legible, and workable traditional mathematics texts that are not burdened with modern theories such as set theory. These books have been field-tested, revised, and used successfully for many years, making them classics with up-to-date appeal. Besides training students in the basic skills needed for life, A Beka Book traditional mathematics books teach students to believe in absolutes, to work diligently for right answers, and to see mathematical facts as part of the truth and order built into the real universe.

Of all the crazy things! What could possibly be wrong with set theory?

Maggie Koerth-Baker tries to help us out.

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.

I think this is right. Indeed, Georg Cantor, a pioneer in set theory, faced precisely this objection to his original work. People get weird when you start talking about infinity. They think you're talking about God and religion and whatnot.

This, of course, is the sheerest madness. Mathematicians rarely talk about infinity in the abstract. We mostly talk abut infinite sets, and when we do we have in mind a rigorously defined abstract construction no different from anything else we study. To say that there are more real numbers than positive integers, for example, is just to say that the set of real numbers cannot be placed into one-to-one correspondence with the set of natural numbers, but the positive integers can be placed into one-to-one correspondence with a proper subset of the reals. That's just true as a matter of logic. Nothing about God, or even the real world, for that matter.

The A Beka paragraph quoted above also refers to the laws of mathematics being absolute and God-given. That's one I have heard before, as I describe in the first chapter of Among the Creationists. At a home-schooler's convention in Richmond, VA, I heard a speaker going off on how she hated math in high school, because her teachers never told her that math was about learning God's laws for the universe. It is because of God's faithfulness and constancy, she told us, that we can be sure that 1+1 will always equal 2. Seriously.

After the talk I met a math professor from Liberty University. That's Jerry Falwell's outfit, you might recall. He rolled his eyes at the speaker. He also told me that at Liberty they have a college-wide faculty meeting at the start of every year to discuss how the curricula of the different academic units will contribute to the religious mission of the school. He then told me that usually they just skip right over the math department. So I guess this sort of thing is just too crazy even for some fundamentalists.

Interestingly, I heard something similar when I met a mathematician from Brigham Young University. This particular professor was Jewish. A group of us had dinner with him, and someone asked him what it was like to teach at a Mormon university. He replied that almost no one in the math department was a Mormon. As far as the school was concerned, the math department existed because a school that wants to be taken seriously must have a math department. But the department was entirely separate from the religious mission of the school. So they actually have a pretty good deal. The school leaves them alone, and they don't challenge the religious mission of the school.

I hope the A Beka folks don't start giving them ideas.

Anyway, Koerth-Baker's article has a lot of other good stuff, so go read the whole thing.

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I wouldn't be surprised if (theological rejection of) Godel also had something to do with their antipathy. His work is also set theory. If one cannot prove that certain statements about mathematical sets are true, then one cannot know the proof...and that puts a bit of a crimp in the omniscience of God.

I saw this mentioned on another blog. I thought it hilarious.

I checked conservapedia, they don't seem to have any problems with set theory.

Of course starting with nothing (the content of the empty set), foundations of mathematics does build up the whole of arithmetic. Perhaps that challenges the common apologetics argument that you can't get something from nothing.

You wrote "Mathematicians rarely talk about infinity in the abstract." That's going to confuse people who think of all of mathematics as being in the abstract.

By Neil Rickert (not verified) on 13 Aug 2012 #permalink

Actually, the problem started when a mathematician started a talk with "we define God as a sphere of radius R"

Neil --

You wrote “Mathematicians rarely talk about infinity in the abstract.” That’s going to confuse people who think of all of mathematics as being in the abstract.

Good point! I could probably have phrased that more artfully. Actually, though, the way mathematicians talk about infinity is often confusing to those on the outside. For example, in calculus we often say things like, “The limit as x goes to infinity...” which makes it sound like infinity is some sort of final destination for wandering variables.

How Satan counts: {}, {{}}, {{}, {{}}}, ...

By Tom English (not verified) on 13 Aug 2012 #permalink

More than a few people went mad during the development of set theory as told in "Naming Infinity: A True Story of Religious Mysticism", by Loren Graham and Jean-Michel Kantor. Contemplating varying levels of infinity is difficult. I think that the Fundie objection is that you are contemplating the transcendent, and maybe you are only suppose to know what has been revealed.

I think there might be a simpler explanation. I was never taught set theory in school, and I think the same is true of most of my generation. I learned some later, but it's not intuitive with me as the rest of school math now is. Therefore, I'm less confident about helping children with their homework when the topic is set theory than with other subjects. Non-academic parents might welcome the assurance that there is one less unfamiliar topic for them to worry about.

By David Evans (not verified) on 14 Aug 2012 #permalink


My first thought when reading the objection was "Um, it's not the same sort of 'infinity'". Mathematical infinity is not the theological/philosophical infinity, so if set theory implies multiple infinities that doesn't mean that there isn't one real infinity, God. The argument about the basis of the certainty of mathematics is an interesting one, but that's a problem for the philosophy of mathematics, not mathematics itself, which I think if I'm remembering abstract algebra right wouldn't necessarily think that 1+1=2 IS a certainty; it depends on what "1", "2" and "+" are defined to be, and that's specific to a number system, not global.


Do you have any evidence that theologically Godel's theory is considered problematic? It's certainly not part of mainstream theology or folk theology, which is what you seem to want to constrain yourself to. I have, in fact, never heard anyone actually raise the claim you've raised, which seems to me to be for good reason because it's shaky epistemologically and theologically.

By Verbose Stoic (not verified) on 14 Aug 2012 #permalink

They may be using "set theory" as a synonym for "new math," which to them is a synonym for "crazy radical educational experiments that teach our children useless gibberish that they can't use in ordinary life."

Besides, it has "theory" in it, and everyone knows that anything that is "just a theory" is dangerous. ;)

I first learned about set theory in first or second grade, at the elementary school run by the Madison College education department. Unfortunately, I believe JMU has not kept the school in operation.

By David Winfrey (not verified) on 14 Aug 2012 #permalink

Goedel's famous paper is called (in German) "On Formally Undecidable Propositions of Principia Mathematica and Related Systems," and that is a fair description. It does not put limitations on the god of your choice, but rather shows that consistent axiom systems in which one can do arithmetic (i.e., elementary number theory) are intrinsically limited as to what can be proved using that axiom system. I cannot understand how a result that demonstrates human limitations (unless one thinks that there is a divinely-revealed axiomatization of something that includes the counting numbers) could offend fundamentalists--but there are other thngs about them I don't understand, including their objections to set theory.

By Corvus illustris (not verified) on 14 Aug 2012 #permalink


Do you have any evidence that theologically Godel’s theory is considered problematic?

Nope, just a guess. That's why I started out my post with "I wouldn't be surprised if..." As in: some theists evidently have a problem with multiple infinities. I wouldn't be surprised if those same folks had a problem with incompleteness.

Because geologists use math to determine the age of fossils (obvious work of Satan), math is to be avoided because it promotes evolution and homosexuality.

Note that many famous men who used math were homosexuals.

(Note, this post is only a little tongue-in-cheek.)

Don't Christians believe in the empty set?

@Tom etc: I thought set theory was in Genesis, "In the beginning was the void" and all that? Clearly referring to the empty set. :-)

@Xuuths: Whose little tongue in whose cheek?

By Jon Wharf (not verified) on 14 Aug 2012 #permalink

Dr. Rosenhouse, for our edification, would you use "math" to display the contents of John 1:1... please?

I think that creations, and more generally fundamentalists, fear things such as set theory because it threatens their most basic view of the world because it threatens the most basic ways in which they think.

I'm an engineer who visualizes things in order to understand them. My vision of set theory is that of a scaffold hanging in air. New theorems add new beams cantilevering some prior work and some times linking back to form trusses with other cantilevers. The important part is that all this floats on air above the ground; there may have been some coins on Georg Cantor's desk, but they are gone now.

Bible reasoning must float on air, too, There is no written record of Creation, or of the Flood, or (first hand) of Elijah being caught up into heaven. These are the beams of Biblical reasoning, ready to be cantilevered into new configurations leading to new ideas.

But Biblical reasoners cannot do that. Their structures must lead to pre-ordained conclusions. Their beams must be bent, their joints must be forced until they reach just the right result. Imagine taking the steel skeleton of New York City's Chrysler Building and using it to bridge the Mississippi; it might work but would you want it to.

Creative thought, especially with the tools of abstract thought provided by disciplines such as set theory.

By Peter Olsen (not verified) on 14 Aug 2012 #permalink

I'm with David W. above -- it's essentially a culture war thing. Set theory = new math = radical socialist hippy commie atheist liberal Alinskyite plot. The rest of the arguments are backfill.

A Beka Book teaches that the laws of mathematics are a creation of God and thus absolute.

That's about the most sensible thing anyone who believes in a Creator God has ever said. It's even grounded in observable reality! Too bad they had to blow their roll by deliberately misunderstanding their God's creation. (Poor God must feel a lot like a woman who spends all day cooking a perfect roast turkey for her husband, only to hear him take a few bites and say "Chicken again?!")

By Raging Bee (not verified) on 14 Aug 2012 #permalink

Don’t Christians believe in the empty set?

That reminds me of a movie I saw in a math class, way back in grade-school, in which math is portrayed as magic, and some guy is accused of heresy for saying there's a "zero spirit," and he has to prove the existence of zero to avoid being burned at the stake.

By Raging Bee (not verified) on 14 Aug 2012 #permalink

Über einen ehrlichen und deswegen anerkennungswürdigen Evolutionsforscher
Als Herr Svante Pääbo einmal von Herrn Stefan Klein im ZEITmagazin beim Thema Genetik (Titel: "Alle Menschen sind miteinander verwandt") gefragt wurde, warum die Frage, woher wir kommen, die Fantasie der Menschen dermaßen anrege, antwortete dieser, weil wir wissen wollen, wer wir eigentlich sind und weil wir hoffen, die Geschichte werde es uns verraten. Zum Schluss wurde er dann gefragt, ob er verstünde, warum sich die Menschen nach einfacheren religiösen Erklärungen sehnen und antwortete:
(Svante Pääbo ZEITmagazin): Ich verstehe, dass Menschen, die vor existenziellen Herausforderungen stehen, religiöse Bedürfnisse haben. Die habe ich auch. Manchmal stelle ich mir vor, dass ich mit mir nahestehenden Verstorbenen geistig Kontakt aufnehmen kann. Das hilft mir dann, die Trennung zu verarbeiten. Trotzdem finde ich es naiv, an einen persönlichen Schöpfer zu glauben.
Daraufhin wurde er gefragt, ob das nicht inkonsequent sei und er antwortete:
(Svante Pääbo ZEITmagazin): Wer ist schon immer konsequent? Ich hatte einmal einen Doktoranden, der war fundamentalistischer Muslim. Er litt, denn natürlich glaubte er an die Schöpfung. Doch wir konnten uns einigen. Denn können wir kategorisch ausschließen, dass es einen allmächtigen und unergründlichen Gott gibt? Vielleicht ist die molekulare Evolution sein Plan, den wir nur nicht durchschauen.
Ein zweifelsohne brillanter diplomatischer Forscher – der den Mut zur Offenheit hat. Denn diese Inkonsequenz, auf der einen Seite wie jeder Mensch, schwere Schicksalsschläge verarbeiten und am Ende den größten Verlust einsehen müsste, wohl wissend aber auf der anderen Seite die kritische Rationalität als Wissenschaftler nicht aus den Augen zu verlieren zu dürfen, führt in der gesamten Forschung zu einem Bruch der Objektivität und dies wird in ganz natürlicher Weise übersehen.
Was ist ein persönlicher Gott? Ist es ein deistischer Gott, der die Dominoreihe anschubst, und wer schubst dann seine Existenz auf die gleiche Weise an und in diesem Moment meinen Gedanken, ganz gleich wie abstrakt und allgemein sich dieser über das Ganze vorentwickelt, gefangen in der Detektion einer Kernspinresonanz? Genau hier beginnt die Naivität und Inkonsequenz der naturwissenschaftlichen Forschung letztendlich durch uns alle, da wir glauben, unsere Gedanken seinen prinzipiell materiell nicht existent, nicht detektierbar, unterschlagen diese, wie von Geisterhand und eben „Totengeistern“. Ist diese Kette aller „fallenden Dominosteine“, seiner ganzen Verzweigungen, dann eine Menge, genauer die Menge aller möglichen Mengen und existiert diese? Und mit welcher bildet diese eine Identität mit der Russelschen Antinomie oder Ihrem exakten Gegenteil? Wir können also behaupten, dass wir selbst dort drinnen nicht enthalten seien, dann müssten wir selbst die Obermenge der Russelschen Antinomie verkörpern, und könnten damit unsere Klasse nicht mehr ad hoc verbieten, um ein Widerspruch zu bilden, also eben wie dann zu erwarten ist nicht existieren, durch solche unmögliche Gedanken. Daraus folgt dann das Gegenteil, und diese Menge, all der Mengen, die sich selbst enthalten, bildet mit der Menge aller Mengen eine Identität (Johannes 6 NT), die Russelsche Menge schließt sich also auch dadurch ganz aus. Enthält sich die Menge aller Mengen, könnten wir uns fragen und schon würde in unsere Endlichkeit die Allmenge aufs Neue hineinlegt, unendlich oft. Wir liegen aber eben in dieser und der Grund, warum diese über alle Zeiten sich selbst enthält, liegt daran, dass der bereits dargelegte Widerspruchsbeweis, über alle Zeiten, schon längst über alle Zeiten gesetzt ist, auch das Nachvollziehen in der Gegenwart, man denke auch dabei etwa an die vollständige Induktion. Dass auch Allgemeingültigkeit, man denke an die physikalischen Gesetze, immer gilt, liegt daran, weil das determiniert ist, also so beschlossen ist. Der lebensunmögliche ewigliche Zustand des Big Chill müsste also schon längst aus statistischen Gründen eingetroffen sein und das ist nicht der Fall, weil der Zufall unendlich, also in einem unendlichen Willen, in dem wir gesetzt sind, ausgeschlossen ist. Sonst wäre also dieser Wahn, diese Staubwolke des Big Chill, gleich einer allumfassenden Nuklearexplosion aus den besagten statistischen Gründen längst eingetreten. Aktuell und in einem endlichen Rahmen kann angenommen werden, dass die Entropie zunimmt, aber auf sehr lange Sicht, endet diese nicht in der Divergenz, im Kältetod, sondern die unendliche Real-Existenz gegen den Zufall, wie ich es ausdrücken würde, endet, weil partikuläre Systeme, eben nie abgeschlossen sein können, in ewiger Existenz einer Persönlichkeit, die nun verstanden werden kann, in der wir gesetzt wandeln. Sie werden also, wenn auch in für Sie nicht vorstellbaren Zeit- oder Entwicklungsabschnitten zurückgerufen, weil alles andere ausgeschlossen ist. Weitere Informationen stelle ich unter dort dann unter Konsequenzen einer bestimmten Welt, Nathanprogramm, Jesus ein wachsendes Vermächtnis, bereit. Bitte die Adresse genau in Ihre Adressleiste kopieren. Gott kann also nur in Bestimmung bewiesen werden, weil er sonst ganz durch das Erklärungswerk unserer Unvollkommenheit, der Immanenz relativiert würde. Ich schicke das noch nicht einmal als Leserbrief, sondern nur als Kommentar, als Anregung, still und leise, den Wert dieser Nachricht einzuschätzen, stelle ich dem Leser allerdings anheim.

By Friedrich Heß … (not verified) on 14 Aug 2012 #permalink

@Friedrich Heß: Difficile est, saturam non scribere (use a translator)

By Corvus illustris (not verified) on 15 Aug 2012 #permalink

I'dont need a translater to understand, but some understand everything und nothing!

By Friedrich Heß … (not verified) on 15 Aug 2012 #permalink

Yup, still content free there, fred.

My suggestion was that this is due to the fact that set theory allows for the argument that mathematics is constructed rather than having some ghostly magical ethereal Platonic existence in its own right, and that disrupts all kinds of sophistamacated arguments for the existence of God a la our friend VS.

And since most mathematicians are apparently Platonists (anecdotally anyway) it allows creationists to enlist them on their side, willingly or not.

I think creationists actually like Godel because he was very smart and believed in God. I've seen Godel invoked in all kinds of arguments from authority before.

Friedrich, it's incredibly rude to vomit all over someone else's place and then refuse to clean it up. It's especially rude to do it a second time. Would you please stop?

Well, as a Dutchman I happen to read German pretty well. Don't bother. It has nothing to do with set theory and otherwise is hardly interesting.