May 01 2014

Set theory and god

Science and religion share a long history of controversy and even hostility. Mathematics and religion, not so much. There could be many reasons for this, the primary one being that there is some similarity in the way that both mathematics and theology operate. Both seek to create self-contained systems based on axioms that are assumed to be true. In the case of mathematics, the axioms depend upon the field of mathematics being studied while in the case of theology, the fundamental axiom is that ‘god exists’.

Some sophisticated theologians will likely disagree with me and argue that their field is more like empirical science and that far from being an axiom, the existence of god is a deductive inference based on knowledge of the empirical world while in the case of mathematics, agreement with empirical data is not necessary since those usually involve extra assumptions outside of the mathematical framework.

For natural scientists, the existence of a god who can interfere with the workings of the laws of nature poses a direct conflict whereas it does not pose any problems for mathematics. What is contained in religious books do not pose any problems for mathematics. Mathematicians may choose, if they wish, to believe in a god and not be immediately confronted with difficulties in their field. This lack of direct contradiction may explain why in a survey of members of the National Academy of Scientists, while there was overwhelming levels of disbelief in a god in all categories, belief was highest among mathematicians (14.3%) and lowest among biologists (5.5%).

But it turns out that religious people do have problems with mathematics. The website of a religious fundamentalist publishing house that produces textbooks designed to be taught at religious institutions lays it out.



What’s the problem with set theory? I had not heard of this issue until Maggie Koerth-Baker pointed it out.

It turns out that there are problems if you dig deeply enough into set theory and that is because of the concept of ‘nesting’, the idea that one set can be a subset of another, even if both sets are infinite. So for example, the idea that the set of positive integers is a subset of the set of real numbers seems to be uncontroversial. Theologically speaking, as long as we don’t look too closely at what being a subset involves, things are fine.

But it turns out that the idea of a hierarchy of infinities leads to some theological consternation since the idea of god as the single ultimate infinite being conflicts with it. Roughly put, since there is only one true god and he is infinite, there can only be one infinity (god) and Georg Cantor’s idea of a hierarchy of infinities is an affront to it. Religious people trace the source of this heresy of multiple infinities back to set theory. I can see how this might matter to sophisticated theologians who worry about such things.

Mathematician Jason Rosenhouse says that he too had not heard of this problem that set theory posed for religion and recounts a discussion that he had with a mathematics faculty member from Liberty University, the school founded by Jerry Falwell, that is fundamentalist to the core.

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.

Mathematicians at religious fundamentalist universities tend keep a low profile and benefit from from the idea that mathematics has nothing to do with religion. While this particular math-religion conflict has not reached the high profile level of the evolution-religion one, I think it is only a matter of time. Some religious fundamentalist at some point is going to object to teaching set theory in the math curriculum, or that if they do teach it, schools must also teach alternatives to it that are consistent with there being just one infinity, and we would have ourselves another glorious fight.


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  1. 1

    Uh oh. Now you’ve done it.

    This seems to be the favorite topic of some of the kookiest kooks on the internet. Someone will probably stop by presently to explain to us how Cantor’s diagonalization argument is a tool of satan, and that there “obviously” must be just as many rationals as there are reals.

  2. 2

    Oh, see also a whole bunch of posts on Good Math, Bad Math. It’s humorous for about the first 50,000 words. :)

  3. 3

    I am skeptical of your predictions in the final paragraph. This stuff about set theory and a hierarchy of infinities is subtle and requires some rarified knowledge. If you manage to put the theological objection to set theory as pithily as “my grandpappy ain’t no monkey!”, then get back to me ;)

    The set theory thing, I think, will remain the domain of people who spend a good fraction of their time thinking about this sort of thing… the folks who have a whole complex story to tell involving flood geology, etc. They are in the minority, though they help drive a larger opposition based on simple appeal to incredulity. There is no simple appeal to the set theory thing, so I just can’t see it catching on.

  4. 4

    Shh, don’t tell them that some infinities are bigger than others.

  5. 5

    Holy crap!

    Set theory and the hierarchy of infinities are the logical deductions made from reasonable premises that have been shown to work in all the rest of mathematics. Suddenly, they don’t like logical conclusions based on reasonable premises. Which is funny considering that is exactly what they are claiming to do when they “prove” the existence of God.

    And remember that the premises of set theory were developed to take care of logical problems like Russell’s paradox. Without these rigorous premises, then it appears that God has created a logical mess. Which is strange since usually the existence of rigorous, logical mathematics is supposed to be more evidence of God’s existence.

    Man, these people really like cutting off the branch they’re sitting on, don’t they?

  6. 6

    Axioms are not things that mathematicians arbitrarily assume to be true. They come about in a number of ways: 1) They are an irreducible postulate arrived at by trying to formally prove things that we basically already believe. Set theory’s axioms for example allow one to prove that 1 + 1 = 2. Those axioms would not have been chosen that way if they proved 1 + 1 = 3. 2) Axioms can be chosen just to find out what one can prove if they are so chosen. Complex numbers were just mathematicians’ plaything until someone found a paradigm where they were useful (describing quantum phenomena for example). Neither of the ways is the way the theists assume a god.

  7. 7

    They have a lot of targets to hate beside evolution. It’s just about anything at one time or another, other sciences, history, polysci, social sciences, climatology.

    You can sum it up easily. They hate reality!!! For good reason, reality is the true opponent of fundie xianity.

    These days secondary targets include history which they just lie about i.e. David Barton et al.. In the sciences it is neurobiology because they never found the soul. Embryology for some reason. Astronomy because of the Big Bang. Geology because they keep insisting that the universe is older than 6,000 years.

    It would be far harder to find some subject the fundies don’t hate.

  8. 8

    Wait, what? But the real numbers are provably more cardinality than the rational numbers. There’s plenty of proofs you can find online. Or just take the powerset of the rational numbers.

    I suppose God can’t be the “Set of All Sets” because that yields to Russell’s Paradox. But that’s taking mathematics way too literally. It just doesn’t fit the definition of a set. You can still have the “Category of All Sets” or something like that. I guess you could always take the powerset of God, or something, but this is practical nonsense. I can talk about the powerset of my cells. It’s not like that logically disproves me because it’s larger than the number of my cells. How bizarre.

    Model Theory, in my opinion, actually more directly goes against religious belief, because it makes you consider Occam’s Razor from a different perspective (the “can you tell the difference between this and that” perspective). Basically, I was a deist before I learned model theory.

  9. 9

    The A Becka curriculum is designed to produce brainwashed Zombies with minimal education. This is the ideal of the death cult xians. It emphasizes obedience to authority meaning the cult leaders.

    I’ve never met anyone educated like that, even online that I know of. Either they escape the programming or are so mentally imprisoned they can’t turn on a computer.

    In our modern Hi Tech society, they are just setting their kids up to fail. And then they fail.

  10. 10

    Raven @ #7: “It would be far harder to find some subject the fundies don’t hate.”

    There may not be one. A fundie running for the Lt Governorship in South Carolina is on a crusade to abolish public education entirely.

  11. 11
    Rik van

    Interested to know what Gödels theorem might mean in this context.

  12. 12

    I waver between thinking these sorts of people have it childishly easy or they’re undertaking some amazingly herculean task, albeit of their own invention. On the easy side the argument is pretty simple. You can tilt at windmills anytime you like if you imagine there’s one nearby anytime it would be convenient. For the hard interpretation it’s more like whack-a-made-up-mole or a children’s game where you try to avoid stepping on cracks, which might pop up anywhere because you’re hunting anything anyone anywhere might possibly consider to be picking a fight with your imaginary friend. “Wait, wait, you can’t sit down! My imaginary friend might be sitting there. No don’t walk either! He may be laying on the floor.” And it only gets sillier from there.

    I suppose everyone needs a hobby and I guess it could be comforting to pick one where you can readily imagine yourself being as good at it as you feel like.

  13. 13

    You might enjoy Underwood Dudley’s book Mathematical Cranks.

    One Roman Catholic Priest couldn’t live with non-Euclidean geometry and thought that he had a proof that Euclidean geometry was the only geometry (e. g., that one can prove the parallel postulate from the previous ones).

    You might also enjoy reading about the trouble the religious had with pi and with (-1)^(1/2) .

  14. 14

    You might also enjoy this:


  15. 15

    While it is true on its face that there is no direct, day-to-day conflict between mathematics and theism (as opposed to, say, biology and theism), mathematics is ultimately incompatible with almost every kind of theistic belief structure. For example, it is a brute logical fact that Noetherian rings are finitely-generated. They cannot be otherwise; if you give me a minimal generating set for a ring that has infinitely many elements, I will construct for you a non-stabilising ascending chain of ideals in that ring, which precludes it from being Noetherian by definition.

    No god is powerful enough to square that circle; if you insist that your god *can* produce an infinitely-generated Noetherian ring (because your god is omnipotent and can do everything and anything), then you’ve abandoned any notion of logical consistency to the Universe. You’ve destroyed the fundamental precept of all of mathematics. No mathematics can exist in such an arbitrary Universe. Therefore the conflict between theism and mathematics is no less deep than between theism and every other intellectual endeavour. It just takes more work to understand where the conflict lies.

    (Yes, I know this argument reduces to ‘god cannot create a stone too heavy for god to lift’ and similar, but it let me show of some flashy math. ;)

  16. 16


    I have to agree with jamesweet; I doubt this issue will become all that important as long as the deep ideas of infinite sets don’t become a regular part of the public school curriculum.

  17. 17

    Re corwyn @ #6

    Analysis using complex numbers are required to understand why Taylor Series expansions of functions like 1/(1 + x^2) diverge for x >= 1.

  18. 18
    Pierce R. Butler

    Does it help anybody to understand the, ahem, logic of this by pointing out that A Beka Corp comes from Florida?

  19. 19
    Mano Singham


    I don’t get the connection.

  20. 20
    Ollie Nanyes

    17: colnago80

    You mean, of course, Taylor series expansions which are centered at 0. :-)

    But yes, these show why even the expansion centered at, say, x = 1 will have a radius of convergence of 2^(1/2)

  21. 21
    Lassi Hippeläinen

    “God made the integers, all the rest is the work of man.” -Leopold Kronecker
    I wonder what Kronecker would say about his own delta function…

    @Mano #19: Flodira is the only state that has a tag in Fark.

  22. 22
    Mano Singham

    @Lassi #21,

    It looks like I am more clueless than I thought! What does having a tag in Fark say about Florida?

  23. 23
    Pierce R. Butler

    Florida has rightfully claimed the crown of National Crazy from California, and we work hard to maintain our championship status in every arena!

  24. 24
    Lassi Hippeläinen

    Fark isn’t your regular news aggregator. Appearing so often in Fark that you get your own tag is… priceless.

  1. 25
    Mathematics and God…. « blueollie

    […] Mano Singham talks about how there might be religious objections to…set theory? […]

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